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Sample records for zeroth order approximation

  1. Zeroth order regular approximation approach to electric dipole moment interactions of the electron.

    PubMed

    Gaul, Konstantin; Berger, Robert

    2017-07-07

    A quasi-relativistic two-component approach for an efficient calculation of P,T-odd interactions caused by a permanent electric dipole moment of the electron (eEDM) is presented. The approach uses a (two-component) complex generalized Hartree-Fock and a complex generalized Kohn-Sham scheme within the zeroth order regular approximation. In applications to select heavy-elemental polar diatomic molecular radicals, which are promising candidates for an eEDM experiment, the method is compared to relativistic four-component electron-correlation calculations and confirms values for the effective electric field acting on the unpaired electron for RaF, BaF, YbF, and HgF. The calculations show that purely relativistic effects, involving only the lower component of the Dirac bi-spinor, are well described by treating only the upper component explicitly.

  2. Zeroth order regular approximation approach to electric dipole moment interactions of the electron

    NASA Astrophysics Data System (ADS)

    Gaul, Konstantin; Berger, Robert

    2017-07-01

    A quasi-relativistic two-component approach for an efficient calculation of P ,T -odd interactions caused by a permanent electric dipole moment of the electron (eEDM) is presented. The approach uses a (two-component) complex generalized Hartree-Fock and a complex generalized Kohn-Sham scheme within the zeroth order regular approximation. In applications to select heavy-elemental polar diatomic molecular radicals, which are promising candidates for an eEDM experiment, the method is compared to relativistic four-component electron-correlation calculations and confirms values for the effective electric field acting on the unpaired electron for RaF, BaF, YbF, and HgF. The calculations show that purely relativistic effects, involving only the lower component of the Dirac bi-spinor, are well described by treating only the upper component explicitly.

  3. A gauge-independent zeroth-order regular approximation to the exact relativistic Hamiltonian—Formulation and applications

    NASA Astrophysics Data System (ADS)

    Filatov, Michael; Cremer, Dieter

    2005-01-01

    A simple modification of the zeroth-order regular approximation (ZORA) in relativistic theory is suggested to suppress its erroneous gauge dependence to a high level of approximation. The method, coined gauge-independent ZORA (ZORA-GI), can be easily installed in any existing nonrelativistic quantum chemical package by programming simple one-electron matrix elements for the quasirelativistic Hamiltonian. Results of benchmark calculations obtained with ZORA-GI at the Hartree-Fock (HF) and second-order Møller-Plesset perturbation theory (MP2) level for dihalogens X2 (X=F,Cl,Br,I,At) are in good agreement with the results of four-component relativistic calculations (HF level) and experimental data (MP2 level). ZORA-GI calculations based on MP2 or coupled-cluster theory with single and double perturbations and a perturbative inclusion of triple excitations [CCSD(T)] lead to accurate atomization energies and molecular geometries for the tetroxides of group VIII elements. With ZORA-GI/CCSD(T), an improved estimate for the atomization energy of hassium (Z=108) tetroxide is obtained.

  4. Relativistic nuclear magnetic resonance J-coupling with ultrasoft pseudopotentials and the zeroth-order regular approximation

    SciTech Connect

    Green, Timothy F. G., E-mail: tim.green@materials.ox.ac.uk; Yates, Jonathan R., E-mail: jonathan.yates@materials.ox.ac.uk

    2014-06-21

    We present a method for the first-principles calculation of nuclear magnetic resonance (NMR) J-coupling in extended systems using state-of-the-art ultrasoft pseudopotentials and including scalar-relativistic effects. The use of ultrasoft pseudopotentials is allowed by extending the projector augmented wave (PAW) method of Joyce et al. [J. Chem. Phys. 127, 204107 (2007)]. We benchmark it against existing local-orbital quantum chemical calculations and experiments for small molecules containing light elements, with good agreement. Scalar-relativistic effects are included at the zeroth-order regular approximation level of theory and benchmarked against existing local-orbital quantum chemical calculations and experiments for a number of small molecules containing themore » heavy row six elements W, Pt, Hg, Tl, and Pb, with good agreement. Finally, {sup 1}J(P-Ag) and {sup 2}J(P-Ag-P) couplings are calculated in some larger molecular crystals and compared against solid-state NMR experiments. Some remarks are also made as to improving the numerical stability of dipole perturbations using PAW.« less

  5. Scalar relativistic computations of nuclear magnetic shielding and g-shifts with the zeroth-order regular approximation and range-separated hybrid density functionals

    SciTech Connect

    Aquino, Fredy W.; Govind, Niranjan; Autschbach, Jochen

    2011-10-01

    Density functional theory (DFT) calculations of NMR chemical shifts and molecular g-tensors with Gaussian-type orbitals are implemented via second-order energy derivatives within the scalar relativistic zeroth order regular approximation (ZORA) framework. Nonhybrid functionals, standard (global) hybrids, and range-separated (Coulomb-attenuated, long-range corrected) hybrid functionals are tested. Origin invariance of the results is ensured by use of gauge-including atomic orbital (GIAO) basis functions. The new implementation in the NWChem quantum chemistry package is verified by calculations of nuclear shielding constants for the heavy atoms in HX (X=F, Cl, Br, I, At) and H2X (X = O, S, Se, Te, Po), and Temore » chemical shifts in a number of tellurium compounds. The basis set and functional dependence of g-shifts is investigated for 14 radicals with light and heavy atoms. The problem of accurately predicting F NMR shielding in UF6-nCln, n = 1 to 6, is revisited. The results are sensitive to approximations in the density functionals, indicating a delicate balance of DFT self-interaction vs. correlation. For the uranium halides, the results with the range-separated functionals are mixed.« less

  6. Relativistic Zeroth-Order Regular Approximation Combined with Nonhybrid and Hybrid Density Functional Theory: Performance for NMR Indirect Nuclear Spin-Spin Coupling in Heavy Metal Compounds.

    PubMed

    Moncho, Salvador; Autschbach, Jochen

    2010-01-12

    A benchmark study for relativistic density functional calculations of NMR spin-spin coupling constants has been performed. The test set contained 47 complexes with heavy metal atoms (W, Pt, Hg, Tl, Pb) with a total of 88 coupling constants involving one or two heavy metal atoms. One-, two-, three-, and four-bond spin-spin couplings have been computed at different levels of theory (nonhybrid vs hybrid DFT, scalar vs two-component relativistic). The computational model was based on geometries fully optimized at the BP/TZP scalar relativistic zeroth-order regular approximation (ZORA) and the conductor-like screening model (COSMO) to include solvent effects. The NMR computations also employed the continuum solvent model. Computations in the gas phase were performed in order to assess the importance of the solvation model. The relative median deviations between various computational models and experiment were found to range between 13% and 21%, with the highest-level computational model (hybrid density functional computations including scalar plus spin-orbit relativistic effects, the COSMO solvent model, and a Gaussian finite-nucleus model) performing best.

  7. Calibration Method to Eliminate Zeroth Order Effect in Lateral Shearing Interferometry

    NASA Astrophysics Data System (ADS)

    Fang, Chao; Xiang, Yang; Qi, Keqi; Chen, Dawei

    2018-04-01

    In this paper, a calibration method is proposed which eliminates the zeroth order effect in lateral shearing interferometry. An analytical expression of the calibration error function is deduced, and the relationship between the phase-restoration error and calibration error is established. The analytical results show that the phase-restoration error introduced by the calibration error is proportional to the phase shifting error and zeroth order effect. The calibration method is verified using simulations and experiments. The simulation results show that the phase-restoration error is approximately proportional to the phase shift error and zeroth order effect, when the phase shifting error is less than 2° and the zeroth order effect is less than 0.2. The experimental result shows that compared with the conventional method with 9-frame interferograms, the calibration method with 5-frame interferograms achieves nearly the same restoration accuracy.

  8. Noise is the new signal: Moving beyond zeroth-order geomorphology (Invited)

    NASA Astrophysics Data System (ADS)

    Jerolmack, D. J.

    2010-12-01

    The last several decades have witnessed a rapid growth in our understanding of landscape evolution, led by the development of geomorphic transport laws - time- and space-averaged equations relating mass flux to some physical process(es). In statistical mechanics this approach is called mean field theory (MFT), in which complex many-body interactions are replaced with an external field that represents the average effect of those interactions. Because MFT neglects all fluctuations around the mean, it has been described as a zeroth-order fluctuation model. The mean field approach to geomorphology has enabled the development of landscape evolution models, and led to a fundamental understanding of many landform patterns. Recent research, however, has highlighted two limitations of MFT: (1) The integral (averaging) time and space scales in geomorphic systems are sometimes poorly defined and often quite large, placing the mean field approximation on uncertain footing, and; (2) In systems exhibiting fractal behavior, an integral scale does not exist - e.g., properties like mass flux are scale-dependent. In both cases, fluctuations in sediment transport are non-negligible over the scales of interest. In this talk I will synthesize recent experimental and theoretical work that confronts these limitations. Discrete element models of fluid and grain interactions show promise for elucidating transport mechanics and pattern-forming instabilities, but require detailed knowledge of micro-scale processes and are computationally expensive. An alternative approach is to begin with a reasonable MFT, and then add higher-order terms that capture the statistical dynamics of fluctuations. In either case, moving beyond zeroth-order geomorphology requires a careful examination of the origins and structure of transport “noise”. I will attempt to show how studying the signal in noise can both reveal interesting new physics, and also help to formalize the applicability of geomorphic

  9. Design of experiments for zeroth and first-order reaction rates.

    PubMed

    Amo-Salas, Mariano; Martín-Martín, Raúl; Rodríguez-Aragón, Licesio J

    2014-09-01

    This work presents optimum designs for reaction rates experiments. In these experiments, time at which observations are to be made and temperatures at which reactions are to be run need to be designed. Observations are performed along time under isothermal conditions. Each experiment needs a fixed temperature and so the reaction can be measured at the designed times. For these observations under isothermal conditions over the same reaction a correlation structure has been considered. D-optimum designs are the aim of our work for zeroth and first-order reaction rates. Temperatures for the isothermal experiments and observation times, to obtain the most accurate estimates of the unknown parameters, are provided in these designs. D-optimum designs for a single observation in each isothermal experiment or for several correlated observations have been obtained. Robustness of the optimum designs for ranges of the correlation parameter and comparisons of the information gathered by different designs are also shown. © 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

  10. Zeroth-order design report for the next linear collider. Volume 1

    SciTech Connect

    Raubenheimer, T.O.

    1996-05-01

    This Zeroth Order Design Report (ZDR) for the Next Linear Collider (NLC) has been completed as a feasibility study for a TeV-scale linear collider that incorporates a room-temperature accelerator powered by rf microwaves at 11.424 GHz--similar to that presently used in the SLC, but at four times the rf frequency. The purpose of this study is to examine the complete systems of such a collider, to understand how the parts fit together, and to make certain that every required piece has been included. The design presented here is not fully engineered in any sense, but to be assured that themore » NLC can be built, attention has been given to a number of critical components and issues that present special challenges. More engineering and development of a number of mechanical and electrical systems remain to be done, but the conclusion of this study is that indeed the NLC is technically feasible and can be expected to reach the performance levels required to perform research at the TeV energy scale. Volume one covers the following: the introduction; electron source; positron source; NLC damping rings; bunch compressors and prelinac; low-frequency linacs and compressors; main linacs; design and dynamics; and RF systems for main linacs.« less

  11. Zeroth-order design report for the next linear collider. Volume 2

    SciTech Connect

    Raubenheimer, T.O.

    This Zeroth-Order Design Report (ZDR) for the Next Linear Collider (NLC) has been completed as a feasibility study for a TeV-scale linear collider that incorporates a room-temperature accelerator powered by rf microwaves at 11.424 GHz--similar to that presently used in the SLC, but at four times the rf frequency. The purpose of this study is to examine the complete systems of such a collider, to understand how the parts fit together, and to make certain that every required piece has been included. The ``design`` presented here is not fully engineered in any sense, but to be assured that the NLCmore » can be built, attention has been given to a number of critical components and issues that present special challenges. More engineering and development of a number of mechanical and electrical systems remain to be done, but the conclusion of this study is that indeed the NLC is technically feasible and can be expected to reach the performance levels required to perform research at the TeV energy scale. Volume II covers the following: collimation systems; IP switch and big bend; final focus; the interaction region; multiple bunch issues; control systems; instrumentation; machine protection systems; NLC reliability considerations; NLC conventional facilities. Also included are four appendices on the following topics: An RF power source upgrade to the NLC; a second interaction region for gamma-gamma, gamma-electron; ground motion: theory and measurement; and beam-based feedback: theory and implementation.« less

  12. Exploration of zeroth-order wavefunctions and energies as a first step toward intramolecular symmetry-adapted perturbation theory

    NASA Astrophysics Data System (ADS)

    Gonthier, Jérôme F.; Corminboeuf, Clémence

    2014-04-01

    Non-covalent interactions occur between and within all molecules and have a profound impact on structural and electronic phenomena in chemistry, biology, and material science. Understanding the nature of inter- and intramolecular interactions is essential not only for establishing the relation between structure and properties, but also for facilitating the rational design of molecules with targeted properties. These objectives have motivated the development of theoretical schemes decomposing intermolecular interactions into physically meaningful terms. Among the various existing energy decomposition schemes, Symmetry-Adapted Perturbation Theory (SAPT) is one of the most successful as it naturally decomposes the interaction energy into physical and intuitive terms. Unfortunately, analogous approaches for intramolecular energies are theoretically highly challenging and virtually nonexistent. Here, we introduce a zeroth-order wavefunction and energy, which represent the first step toward the development of an intramolecular variant of the SAPT formalism. The proposed energy expression is based on the Chemical Hamiltonian Approach (CHA), which relies upon an asymmetric interpretation of the electronic integrals. The orbitals are optimized with a non-hermitian Fock matrix based on two variants: one using orbitals strictly localized on individual fragments and the other using canonical (delocalized) orbitals. The zeroth-order wavefunction and energy expression are validated on a series of prototypical systems. The computed intramolecular interaction energies demonstrate that our approach combining the CHA with strictly localized orbitals achieves reasonable interaction energies and basis set dependence in addition to producing intuitive energy trends. Our zeroth-order wavefunction is the primary step fundamental to the derivation of any perturbation theory correction, which has the potential to truly transform our understanding and quantification of non

  13. Exploration of zeroth-order wavefunctions and energies as a first step toward intramolecular symmetry-adapted perturbation theory

    SciTech Connect

    Gonthier, Jérôme F.; Corminboeuf, Clémence, E-mail: clemence.corminboeuf@epfl.ch

    2014-04-21

    Non-covalent interactions occur between and within all molecules and have a profound impact on structural and electronic phenomena in chemistry, biology, and material science. Understanding the nature of inter- and intramolecular interactions is essential not only for establishing the relation between structure and properties, but also for facilitating the rational design of molecules with targeted properties. These objectives have motivated the development of theoretical schemes decomposing intermolecular interactions into physically meaningful terms. Among the various existing energy decomposition schemes, Symmetry-Adapted Perturbation Theory (SAPT) is one of the most successful as it naturally decomposes the interaction energy into physical and intuitivemore » terms. Unfortunately, analogous approaches for intramolecular energies are theoretically highly challenging and virtually nonexistent. Here, we introduce a zeroth-order wavefunction and energy, which represent the first step toward the development of an intramolecular variant of the SAPT formalism. The proposed energy expression is based on the Chemical Hamiltonian Approach (CHA), which relies upon an asymmetric interpretation of the electronic integrals. The orbitals are optimized with a non-hermitian Fock matrix based on two variants: one using orbitals strictly localized on individual fragments and the other using canonical (delocalized) orbitals. The zeroth-order wavefunction and energy expression are validated on a series of prototypical systems. The computed intramolecular interaction energies demonstrate that our approach combining the CHA with strictly localized orbitals achieves reasonable interaction energies and basis set dependence in addition to producing intuitive energy trends. Our zeroth-order wavefunction is the primary step fundamental to the derivation of any perturbation theory correction, which has the potential to truly transform our understanding and quantification of non

  14. On the zeroth-order hamiltonian for CASPT2 calculations of spin crossover compounds.

    PubMed

    Vela, Sergi; Fumanal, Maria; Ribas-Ariño, Jordi; Robert, Vincent

    2016-04-15

    Complete active space self-consistent field theory (CASSCF) calculations and subsequent second-order perturbation theory treatment (CASPT2) are discussed in the evaluation of the spin-states energy difference (ΔH(elec)) of a series of seven spin crossover (SCO) compounds. The reference values have been extracted from a combination of experimental measurements and DFT + U calculations, as discussed in a recent article (Vela et al., Phys Chem Chem Phys 2015, 17, 16306). It is definitely proven that the critical IPEA parameter used in CASPT2 calculations of ΔH(elec), a key parameter in the design of SCO compounds, should be modified with respect to its default value of 0.25 a.u. and increased up to 0.50 a.u. The satisfactory agreement observed previously in the literature might result from an error cancellation originated in the default IPEA, which overestimates the stability of the HS state, and the erroneous atomic orbital basis set contraction of carbon atoms, which stabilizes the LS states. © 2015 Wiley Periodicals, Inc.

  15. Analysis of forward scattering of an acoustical zeroth-order Bessel beam from rigid complicated (aspherical) structures

    NASA Astrophysics Data System (ADS)

    Li, Wei; Chai, Yingbin; Gong, Zhixiong; Marston, Philip L.

    2017-10-01

    The forward scattering from rigid spheroids and endcapped cylinders with finite length (even with a large aspect ratio) immersed in a non-viscous fluid under the illumination of an idealized zeroth-order acoustical Bessel beam (ABB) with arbitrary angles of incidence is calculated and analyzed in the implementation of the T-matrix method (TTM). Based on the present method, the incident coefficients of expansion for the incident ABB are derived and simplifying methods are proposed for the numerical accuracy and computational efficiency according to the geometrical symmetries. A home-made MATLAB software package is constructed accordingly, and then verified and validated for the ABB scattering from rigid aspherical obstacles. Several numerical examples are computed for the forward scattering from both rigid spheroids and finite cylinder, with particular emphasis on the aspect ratios, the half-cone angles of ABBs, the incident angles and the dimensionless frequencies. The rectangular patterns of target strength in the (β, θs) domain (where β is the half-cone angle of the ABB and θs is the scattered polar angle) and local/total forward scattering versus dimensionless frequency are exhibited, which could provide new insights into the physical mechanisms of Bessel beam scattering by rigid spheroids and finite cylinders. The ray diagrams in geometrical models for the scattering in the forward half-space and the optical cross-section theorem help to interpret the scattering mechanisms of ABBs. This research work may provide an alternative for the partial wave series solution under certain circumstances interacting with ABBs for complicated obstacles and benefit some related works in optics and electromagnetics.

  16. Is the choice of a standard zeroth-order hamiltonian in CASPT2 ansatz optimal in calculations of excitation energies in protonated and unprotonated schiff bases of retinal?

    PubMed

    Wolański, Łukasz; Grabarek, Dawid; Andruniów, Tadeusz

    2018-04-10

    To account for systematic error of CASPT2 method empirical modification of the zeroth-order Hamiltonian with Ionization Potential-Electron Affinity (IPEA) shift was introduced. The optimized IPEA value (0.25 a.u.), called standard IPEA (S-IPEA), was recommended but due to its unsatisfactory performance in multiple metallic and organic compounds it has been questioned lately as a general parameter working properly for all molecules under CASPT2 study. As we are interested in Schiff bases of retinal, an important question emerging from this conflict of choice, to use or not to use S-IPEA, is whether the introduction of the modified zeroth-order Hamiltonian into CASPT2 ansatz does really improve their energetics. To achieve this goal, we assessed an impact of the IPEA shift value, in a range of 0-0.35 a.u., on vertical excitation energies to low-lying singlet states of two protonated (RPSBs) and two unprotonated (RSBs) Schiff bases of retinal for which experimental data in gas phase are available. In addition, an effect of geometry, basis set, and active space on computed VEEs is also reported. We find, that for these systems, the choice of S-IPEA significantly overestimates both S 0 →S 1 and S 0 →S 2 energies and the best theoretical estimate, in reference to the experimental data, is provided with either unmodified zeroth-order Hamiltonian or small value of the IPEA shift in a range of 0.05-0.15 a.u., depending on active space and basis set size, equilibrium geometry, and character of the excited state. © 2018 Wiley Periodicals, Inc. © 2018 Wiley Periodicals, Inc.

  17. Zeroth order Fabry-Perot resonance enabled ultra-thin perfect light absorber using percolation aluminum and silicon nanofilms

    DOE PAGES

    Mirshafieyan, Seyed Sadreddin; Luk, Ting S.; Guo, Junpeng

    2016-03-04

    Here, we demonstrated perfect light absorption in optical nanocavities made of ultra-thin percolation aluminum and silicon films deposited on an aluminum surface. The total layer thickness of the aluminum and silicon films is one order of magnitude less than perfect absorption wavelength in the visible spectral range. The ratio of silicon cavity layer thickness to perfect absorption wavelength decreases as wavelength decreases due to the increased phase delays at silicon-aluminum boundaries at shorter wavelengths. It is explained that perfect light absorption is due to critical coupling of incident wave to the fundamental Fabry-Perot resonance mode of the structure where themore » round trip phase delay is zero. Simulations were performed and the results agree well with the measurement results.« less

  18. Zeroth Law, Entropy, Equilibrium, and All That

    ERIC Educational Resources Information Center

    Canagaratna, Sebastian G.

    2008-01-01

    The place of the zeroth law in the teaching of thermodynamics is examined in the context of the recent discussion by Gislason and Craig of some problems involving the establishment of thermal equilibrium. The concept of thermal equilibrium is introduced through the zeroth law. The relation between the zeroth law and the second law in the…

  19. Morphing Continuum Theory: A First Order Approximation to the Balance Laws

    NASA Astrophysics Data System (ADS)

    Wonnell, Louis; Cheikh, Mohamad Ibrahim; Chen, James

    2017-11-01

    Morphing Continuum Theory is constructed under the framework of Rational Continuum Mechanics (RCM) for fluid flows with inner structure. This multiscale theory has been successfully emplyed to model turbulent flows. The framework of RCM ensures the mathematical rigor of MCT, but contains new material constants related to the inner structure. The physical meanings of these material constants have yet to be determined. Here, a linear deviation from the zeroth-order Boltzmann-Curtiss distribution function is derived. When applied to the Boltzmann-Curtiss equation, a first-order approximation of the MCT governing equations is obtained. The integral equations are then related to the appropriate material constants found in the heat flux, Cauchy stress, and moment stress terms in the governing equations. These new material properties associated with the inner structure of the fluid are compared with the corresponding integrals, and a clearer physical interpretation of these coefficients emerges. The physical meanings of these material properties is determined by analyzing previous results obtained from numerical simulations of MCT for compressible and incompressible flows. The implications for the physics underlying the MCT governing equations will also be discussed. This material is based upon work supported by the Air Force Office of Scientific Research under Award Number FA9550-17-1-0154.

  20. Zeroth-order phase-contrast technique.

    PubMed

    Pizolato, José Carlos; Cirino, Giuseppe Antonio; Gonçalves, Cristhiane; Neto, Luiz Gonçalves

    2007-11-01

    What we believe to be a new phase-contrast technique is proposed to recover intensity distributions from phase distributions modulated by spatial light modulators (SLMs) and binary diffractive optical elements (DOEs). The phase distribution is directly transformed into intensity distributions using a 4f optical correlator and an iris centered in the frequency plane as a spatial filter. No phase-changing plates or phase dielectric dots are used as a filter. This method allows the use of twisted nematic liquid-crystal televisions (LCTVs) operating in the real-time phase-mostly regime mode between 0 and p to generate high-intensity multiple beams for optical trap applications. It is also possible to use these LCTVs as input SLMs for optical correlators to obtain high-intensity Fourier transform distributions of input amplitude objects.

  1. APPROXIMATING SYMMETRIC POSITIVE SEMIDEFINITE TENSORS OF EVEN ORDER*

    PubMed Central

    BARMPOUTIS, ANGELOS; JEFFREY, HO; VEMURI, BABA C.

    2012-01-01

    Tensors of various orders can be used for modeling physical quantities such as strain and diffusion as well as curvature and other quantities of geometric origin. Depending on the physical properties of the modeled quantity, the estimated tensors are often required to satisfy the positivity constraint, which can be satisfied only with tensors of even order. Although the space P02m of 2mth-order symmetric positive semi-definite tensors is known to be a convex cone, enforcing positivity constraint directly on P02m is usually not straightforward computationally because there is no known analytic description of P02m for m > 1. In this paper, we propose a novel approach for enforcing the positivity constraint on even-order tensors by approximating the cone P02m for the cases 0 < m < 3, and presenting an explicit characterization of the approximation Σ2m ⊂ Ω2m for m ≥ 1, using the subset Ω2m⊂P02m of semi-definite tensors that can be written as a sum of squares of tensors of order m. Furthermore, we show that this approximation leads to a non-negative linear least-squares (NNLS) optimization problem with the complexity that equals the number of generators in Σ2m. Finally, we experimentally validate the proposed approach and we present an application for computing 2mth-order diffusion tensors from Diffusion Weighted Magnetic Resonance Images. PMID:23285313

  2. Zeroth Law, Entropy, Equilibrium, and All That

    NASA Astrophysics Data System (ADS)

    Canagaratna, Sebastian G.

    2008-05-01

    The place of the zeroth law in the teaching of thermodynamics is examined in the context of the recent discussion by Gislason and Craig of some problems involving the establishment of thermal equilibrium. The concept of thermal equilibrium is introduced through the zeroth law. The relation between the zeroth law and the second law in the traditional approach to thermodynamics is discussed. It is shown that the traditional approach does not need to appeal to the second law to solve with rigor the type of problems discussed by Gislason and Craig: in problems not involving chemical reaction, the zeroth law and the condition for mechanical equilibrium, complemented by the first law and any necessary equations of state, are sufficient to determine the final state. We have to invoke the second law only if we wish to calculate the change of entropy. Since most students are exposed to a traditional approach to thermodynamics, the examples of Gislason and Craig are re-examined in terms of the traditional formulation. The maximization of the entropy in the final state can be verified in the traditional approach quite directly by the use of the fundamental equations of thermodynamics. This approach uses relatively simple mathematics in as general a setting as possible.

  3. Strictly stable high order difference approximations for computational aeroacoustics

    NASA Astrophysics Data System (ADS)

    Müller, Bernhard; Johansson, Stefan

    2005-09-01

    High order finite difference approximations with improved accuracy and stability properties have been developed for computational aeroacoustics (CAA). One of our new difference operators corresponds to Tam and Webb's DRP scheme in the interior, but is modified near the boundaries to be strictly stable. A unified formulation of the nonlinear and linearized Euler equations is used, which can be extended to the Navier-Stokes equations. The approach has been verified for 1D, 2D and axisymmetric test problems. We have simulated the sound propagation from a rocket launch before lift-off. To cite this article: B. Müller, S. Johansson, C. R. Mecanique 333 (2005).

  4. Errors from approximation of ODE systems with reduced order models

    SciTech Connect

    Vassilevska, Tanya

    2016-12-30

    This is a code to calculate the error from approximation of systems of ordinary differential equations (ODEs) by using Proper Orthogonal Decomposition (POD) Reduced Order Models (ROM) methods and to compare and analyze the errors for two POD ROM variants. The first variant is the standard POD ROM, the second variant is a modification of the method using the values of the time derivatives (a.k.a. time-derivative snapshots). The code compares the errors from the two variants under different conditions.

  5. Wave vector modification of the infinite order sudden approximation

    SciTech Connect

    Sachs, J.G.; Bowman, J.M.

    1980-10-15

    A simple method is proposed to modify the infinite order sudden approximation (IOS) in order to extend its region of quantitative validity. The method involves modifying the phase of the IOS scattering matrix to include a part calculated at the outgoing relative kinetic energy as well as a part calculated at the incoming kinetic energy. An immediate advantage of this modification is that the resulting S matrix is symmetric. We also present a closely related method in which the relative kinetic energies used in the calculation of the phase are determined from quasiclassical trajectory calculations. A set of trajectories ismore » run with the initial state being the incoming state, and another set is run with the initial state being the outgoing state, and the average final relative kinetic energy of each set is obtained. One part of the S-operator phase is then calculated at each of these kinetic energies. We apply these methods to vibrationally inelastic collinear collisions of an atom and a harmonic oscillator, and calculate transition probabilities P/sub n/1..-->..nf for three model systems. For systems which are sudden, or nearly so, the agreement with exact quantum close-coupling calculations is substantially improved over standard IOS ones when ..delta..n=such thatub f/-n/sub i/ is large, and the corresponding transition probability is small, i.e., less than 0.1. However, the modifications we propose will not improve the accuracy of the IOS transition probabilities for any collisional system unless the standard form of IOS already gives at least qualitative agreement with exact quantal calculations. We also suggest comparisons between some classical quantities and sudden predictions which should help in determining the validity of the sudden approximation. This is useful when exact quantal data is not available for comparison.« less

  6. Wave vector modification of the infinite order sudden approximation

    NASA Astrophysics Data System (ADS)

    Sachs, Judith Grobe; Bowman, Joel M.

    1980-10-01

    A simple method is proposed to modify the infinite order sudden approximation (IOS) in order to extend its region of quantitative validity. The method involves modifying the phase of the IOS scattering matrix to include a part calculated at the outgoing relative kinetic energy as well as a part calculated at the incoming kinetic energy. An immediate advantage of this modification is that the resulting S matrix is symmetric. We also present a closely related method in which the relative kinetic energies used in the calculation of the phase are determined from quasiclassical trajectory calculations. A set of trajectories is run with the initial state being the incoming state, and another set is run with the initial state being the outgoing state, and the average final relative kinetic energy of each set is obtained. One part of the S-operator phase is then calculated at each of these kinetic energies. We apply these methods to vibrationally inelastic collinear collisions of an atom and a harmonic oscillator, and calculate transition probabilities Pn1→nf for three model systems. For systems which are sudden, or nearly so, the agreement with exact quantum close-coupling calculations is substantially improved over standard IOS ones when Δn=‖nf-ni‖ is large, and the corresponding transition probability is small, i.e., less than 0.1. However, the modifications we propose will not improve the accuracy of the IOS transition probabilities for any collisional system unless the standard form of IOS already gives at least qualitative agreement with exact quantal calculations. We also suggest comparisons between some classical quantities and sudden predictions which should help in determining the validity of the sudden approximation. This is useful when exact quantal data is not available for comparison.

  7. An Extension of the Krieger-Li-Iafrate Approximation to the Optimized-Effective-Potential Method

    SciTech Connect

    Wilson, B.G.

    1999-11-11

    The Krieger-Li-Iafrate approximation can be expressed as the zeroth order result of an unstable iterative method for solving the integral equation form of the optimized-effective-potential method. By pre-conditioning the iterate a first order correction can be obtained which recovers the bulk of quantal oscillations missing in the zeroth order approximation. A comparison of calculated total energies are given with Krieger-Li-Iafrate, Local Density Functional, and Hyper-Hartree-Fock results for non-relativistic atoms and ions.

  8. Antiferromagnetic order and the structural order-disorder transition in the Cd6Ho quasicrystal approximant

    NASA Astrophysics Data System (ADS)

    Kreyssig, Andreas; Beutier, Guillaume; Hiroto, Takanobu; Kim, Min Gyu; Tucker, Gregory S.; de Boissieu, Marc; Tamura, Ryuji; Goldman, Alan I.

    2013-09-01

    It has generally been accepted that the orientational ordering of the Cd4 tetrahedron within the Cd6 R quasicrystal approximants is kinetically inhibited for R = Ho, Er, Tm and Lu by steric constraints. Our high-resolution X-ray scattering measurements of the Cd6Ho quasicrystal approximant, however, reveal an abrupt (first-order) transition to a monoclinic structure below T S = 178 K for samples that have 'aged' at room temperature for approximately one year, reopening this question. Using X-ray resonant magnetic scattering at the Ho L 3-edge we have elucidated the nature of the antiferromagnetic ordering below T N = 8.5 K in Cd6Ho. The magnetic Bragg peaks are found at the charge forbidden H + K + L = 2n + 1 positions, referenced to the high-temperature body-centred cubic structure. In general terms, this corresponds to antiferromagnetic arrangements of the Ho moments on adjacent clusters in the unit cell as previously found for Cd6Tb.

  9. An approximation technique for predicting the transient response of a second order nonlinear equation

    NASA Technical Reports Server (NTRS)

    Laurenson, R. M.; Baumgarten, J. R.

    1975-01-01

    An approximation technique has been developed for determining the transient response of a nonlinear dynamic system. The nonlinearities in the system which has been considered appear in the system's dissipation function. This function was expressed as a second order polynomial in the system's velocity. The developed approximation is an extension of the classic Kryloff-Bogoliuboff technique. Two examples of the developed approximation are presented for comparative purposes with other approximation methods.

  10. Finite-dimensional approximation for optimal fixed-order compensation of distributed parameter systems

    NASA Technical Reports Server (NTRS)

    Bernstein, Dennis S.; Rosen, I. G.

    1988-01-01

    In controlling distributed parameter systems it is often desirable to obtain low-order, finite-dimensional controllers in order to minimize real-time computational requirements. Standard approaches to this problem employ model/controller reduction techniques in conjunction with LQG theory. In this paper we consider the finite-dimensional approximation of the infinite-dimensional Bernstein/Hyland optimal projection theory. This approach yields fixed-finite-order controllers which are optimal with respect to high-order, approximating, finite-dimensional plant models. The technique is illustrated by computing a sequence of first-order controllers for one-dimensional, single-input/single-output, parabolic (heat/diffusion) and hereditary systems using spline-based, Ritz-Galerkin, finite element approximation. Numerical studies indicate convergence of the feedback gains with less than 2 percent performance degradation over full-order LQG controllers for the parabolic system and 10 percent degradation for the hereditary system.

  11. First and second order approximations to stage numbers in multicomponent enrichment cascades

    SciTech Connect

    Scopatz, A.

    2013-07-01

    This paper describes closed form, Taylor series approximations to the number product stages in a multicomponent enrichment cascade. Such closed form approximations are required when a symbolic, rather than a numeric, algorithm is used to compute the optimal cascade state. Both first and second order approximations were implemented. The first order solution was found to be grossly incorrect, having the wrong functional form over the entire domain. On the other hand, the second order solution shows excellent agreement with the 'true' solution over the domain of interest. An implementation of the symbolic, second order solver is available in the freemore » and open source PyNE library. (authors)« less

  12. Zeroth Poisson Homology, Foliated Cohomology and Perfect Poisson Manifolds

    NASA Astrophysics Data System (ADS)

    Martínez-Torres, David; Miranda, Eva

    2018-01-01

    We prove that, for compact regular Poisson manifolds, the zeroth homology group is isomorphic to the top foliated cohomology group, and we give some applications. In particular, we show that, for regular unimodular Poisson manifolds, top Poisson and foliated cohomology groups are isomorphic. Inspired by the symplectic setting, we define what a perfect Poisson manifold is. We use these Poisson homology computations to provide families of perfect Poisson manifolds.

  13. Physical Applications of a Simple Approximation of Bessel Functions of Integer Order

    ERIC Educational Resources Information Center

    Barsan, V.; Cojocaru, S.

    2007-01-01

    Applications of a simple approximation of Bessel functions of integer order, in terms of trigonometric functions, are discussed for several examples from electromagnetism and optics. The method may be applied in the intermediate regime, bridging the "small values regime" and the "asymptotic" one, and covering, in this way, an area of great…

  14. The Approximation of Two-Mode Proximity Matrices by Sums of Order-Constrained Matrices.

    ERIC Educational Resources Information Center

    Hubert, Lawrence; Arabie, Phipps

    1995-01-01

    A least-squares strategy is proposed for representing a two-mode proximity matrix as an approximate sum of a small number of matrices that satisfy certain simple order constraints on their entries. The primary class of constraints considered defines Q-forms for particular conditions in a two-mode matrix. (SLD)

  15. Breakdown of the single-exchange approximation in third-order symmetry-adapted perturbation theory.

    PubMed

    Lao, Ka Un; Herbert, John M

    2012-03-22

    We report third-order symmetry-adapted perturbation theory (SAPT) calculations for several dimers whose intermolecular interactions are dominated by induction. We demonstrate that the single-exchange approximation (SEA) employed to derive the third-order exchange-induction correction (E(exch-ind)((30))) fails to quench the attractive nature of the third-order induction (E(ind)((30))), leading to one-dimensional potential curves that become attractive rather than repulsive at short intermolecular separations. A scaling equation for (E(exch-ind)((30))), based on an exact formula for the first-order exchange correction, is introduced to approximate exchange effects beyond the SEA, and qualitatively correct potential energy curves that include third-order induction are thereby obtained. For induction-dominated systems, our results indicate that a "hybrid" SAPT approach, in which a dimer Hartree-Fock calculation is performed in order to obtain a correction for higher-order induction, is necessary not only to obtain quantitative binding energies but also to obtain qualitatively correct potential energy surfaces. These results underscore the need to develop higher-order exchange-induction formulas that go beyond the SEA. © 2012 American Chemical Society

  16. First-order approximation error analysis of Risley-prism-based beam directing system.

    PubMed

    Zhao, Yanyan; Yuan, Yan

    2014-12-01

    To improve the performance of a Risley-prism system for optical detection and measuring applications, it is necessary to be able to determine the direction of the outgoing beam with high accuracy. In previous works, error sources and their impact on the performance of the Risley-prism system have been analyzed, but their numerical approximation accuracy was not high. Besides, pointing error analysis of the Risley-prism system has provided results for the case when the component errors, prism orientation errors, and assembly errors are certain. In this work, the prototype of a Risley-prism system was designed. The first-order approximations of the error analysis were derived and compared with the exact results. The directing errors of a Risley-prism system associated with wedge-angle errors, prism mounting errors, and bearing assembly errors were analyzed based on the exact formula and the first-order approximation. The comparisons indicated that our first-order approximation is accurate. In addition, the combined errors produced by the wedge-angle errors and mounting errors of the two prisms together were derived and in both cases were proved to be the sum of errors caused by the first and the second prism separately. Based on these results, the system error of our prototype was estimated. The derived formulas can be implemented to evaluate beam directing errors of any Risley-prism beam directing system with a similar configuration.

  17. New second order Mumford-Shah model based on Γ-convergence approximation for image processing

    NASA Astrophysics Data System (ADS)

    Duan, Jinming; Lu, Wenqi; Pan, Zhenkuan; Bai, Li

    2016-05-01

    In this paper, a second order variational model named the Mumford-Shah total generalized variation (MSTGV) is proposed for simultaneously image denoising and segmentation, which combines the original Γ-convergence approximated Mumford-Shah model with the second order total generalized variation (TGV). For image denoising, the proposed MSTGV can eliminate both the staircase artefact associated with the first order total variation and the edge blurring effect associated with the quadratic H1 regularization or the second order bounded Hessian regularization. For image segmentation, the MSTGV can obtain clear and continuous boundaries of objects in the image. To improve computational efficiency, the implementation of the MSTGV does not directly solve its high order nonlinear partial differential equations and instead exploits the efficient split Bregman algorithm. The algorithm benefits from the fast Fourier transform, analytical generalized soft thresholding equation, and Gauss-Seidel iteration. Extensive experiments are conducted to demonstrate the effectiveness and efficiency of the proposed model.

  18. Examples of the Zeroth Theorem of the History of Science

    SciTech Connect

    Jackson, J.D.

    2007-08-24

    The zeroth theorem of the history of science, enunciated byE. P. Fischer, states that a discovery (rule,regularity, insight) namedafter someone (often) did not originate with that person. I present fiveexamples from physics: the Lorentz condition partial muAmu = 0 definingthe Lorentz gauge of the electromagnetic potentials; the Dirac deltafunction, delta(x); the Schumann resonances of the earth-ionospherecavity; the Weizsacker-Williams method of virtual quanta; the BMTequation of spin dynamics. I give illustrated thumbnail sketches of boththe true and reputed discoverers and quote from their "discovery"publications.

  19. Testing approximate theories of first-order phase transitions on the two-dimensional Potts model

    SciTech Connect

    Dasgupta, C.; Pandit, R.

    The two-dimensional, q-state (q > 4) Potts model is used as a testing ground for approximate theories of first-order phase transitions. In particular, the predictions of a theory analogous to the Ramakrishnan-Yussouff theory of freezing are compared with those of ordinary mean-field (Curie-Wiess) theory. It is found that the Curie-Weiss theory is a better approximation than the Ramakrishnan-Yussouff theory, even though the former neglects all fluctuations. It is shown that the Ramakrishnan-Yussouff theory overestimates the effects of fluctuations in this system. The reasons behind the failure of the Ramakrishnan-Yussouff approximation and the suitability of using the two-dimensional Potts model asmore » a testing ground for these theories are discussed.« less

  20. Relaxation approximations to second-order traffic flow models by high-resolution schemes

    SciTech Connect

    Nikolos, I.K.; Delis, A.I.; Papageorgiou, M.

    2015-03-10

    A relaxation-type approximation of second-order non-equilibrium traffic models, written in conservation or balance law form, is considered. Using the relaxation approximation, the nonlinear equations are transformed to a semi-linear diagonilizable problem with linear characteristic variables and stiff source terms with the attractive feature that neither Riemann solvers nor characteristic decompositions are in need. In particular, it is only necessary to provide the flux and source term functions and an estimate of the characteristic speeds. To discretize the resulting relaxation system, high-resolution reconstructions in space are considered. Emphasis is given on a fifth-order WENO scheme and its performance. The computations reportedmore » demonstrate the simplicity and versatility of relaxation schemes as numerical solvers.« less

  1. Semiclassical approximation of the Wheeler-DeWitt equation: arbitrary orders and the question of unitarity

    NASA Astrophysics Data System (ADS)

    Kiefer, Claus; Wichmann, David

    2018-06-01

    We extend the Born-Oppenheimer type of approximation scheme for the Wheeler-DeWitt equation of canonical quantum gravity to arbitrary orders in the inverse Planck mass squared. We discuss in detail the origin of unitarity violation in this scheme and show that unitarity can be restored by an appropriate modification which requires back reaction from matter onto the gravitational sector. In our analysis, we heavily rely on the gauge aspects of the standard Born-Oppenheimer scheme in molecular physics.

  2. High-order harmonic generation in solid slabs beyond the single-active-electron approximation

    NASA Astrophysics Data System (ADS)

    Hansen, Kenneth K.; Deffge, Tobias; Bauer, Dieter

    2017-11-01

    High-harmonic generation by a laser-driven solid slab is simulated using time-dependent density functional theory. Multiple harmonic plateaus up to very high harmonic orders are observed already at surprisingly low field strengths. The full all-electron harmonic spectra are, in general, very different from those of any individual Kohn-Sham orbital. Freezing the Kohn-Sham potential instead is found to be a good approximation for the laser intensities and harmonic orders considered. The origins of the plateau cutoffs are explained in terms of band gaps that can be reached by Kohn-Sham electrons and holes moving through the band structure.

  3. Low-frequency approximation for high-order harmonic generation by a bicircular laser field

    NASA Astrophysics Data System (ADS)

    Milošević, D. B.

    2018-01-01

    We present low-frequency approximation (LFA) for high-order harmonic generation (HHG) process. LFA represents the lowest-order term of an expansion of the final-state interaction matrix element in powers of the laser-field frequency ω . In this approximation the plane-wave recombination matrix element which appears in the strong-field approximation is replaced by the exact laser-free recombination matrix element calculated for the laser-field dressed electron momenta. First, we have shown that the HHG spectra obtained using the LFA agree with those obtained solving the time-dependent Schrödinger equation. Next, we have applied this LFA to calculate the HHG rate for inert gases exposed to a bicircular field. The bicircular field, which consists of two coplanar counter-rotating fields having different frequencies (usually ω and 2 ω ), is presently an important subject of scientific research since it enables efficient generation of circularly polarized high-order harmonics (coherent soft x rays). Analyzing the photorecombination matrix element we have found that the HHG rate can efficiently be calculated using the angular momentum basis with the states oriented in the direction of the bicircular field components. Our numerical results show that the HHG rate for atoms having p ground state, for higher high-order harmonic energies, is larger for circularly polarized harmonics having the helicity -1 . For lower energies the harmonics having helicity +1 prevails. The transition between these two harmonic energy regions can appear near the Cooper minimum, which, in the case of Ar atoms, makes the selection of high-order harmonics having the same helicity much easier. This is important for applications (for example, for generation of attosecond pulse trains of circularly polarized harmonics).

  4. Fast and Analytical EAP Approximation from a 4th-Order Tensor.

    PubMed

    Ghosh, Aurobrata; Deriche, Rachid

    2012-01-01

    Generalized diffusion tensor imaging (GDTI) was developed to model complex apparent diffusivity coefficient (ADC) using higher-order tensors (HOTs) and to overcome the inherent single-peak shortcoming of DTI. However, the geometry of a complex ADC profile does not correspond to the underlying structure of fibers. This tissue geometry can be inferred from the shape of the ensemble average propagator (EAP). Though interesting methods for estimating a positive ADC using 4th-order diffusion tensors were developed, GDTI in general was overtaken by other approaches, for example, the orientation distribution function (ODF), since it is considerably difficult to recuperate the EAP from a HOT model of the ADC in GDTI. In this paper, we present a novel closed-form approximation of the EAP using Hermite polynomials from a modified HOT model of the original GDTI-ADC. Since the solution is analytical, it is fast, differentiable, and the approximation converges well to the true EAP. This method also makes the effort of computing a positive ADC worthwhile, since now both the ADC and the EAP can be used and have closed forms. We demonstrate our approach with 4th-order tensors on synthetic data and in vivo human data.

  5. Fast and Analytical EAP Approximation from a 4th-Order Tensor

    PubMed Central

    Ghosh, Aurobrata; Deriche, Rachid

    2012-01-01

    Generalized diffusion tensor imaging (GDTI) was developed to model complex apparent diffusivity coefficient (ADC) using higher-order tensors (HOTs) and to overcome the inherent single-peak shortcoming of DTI. However, the geometry of a complex ADC profile does not correspond to the underlying structure of fibers. This tissue geometry can be inferred from the shape of the ensemble average propagator (EAP). Though interesting methods for estimating a positive ADC using 4th-order diffusion tensors were developed, GDTI in general was overtaken by other approaches, for example, the orientation distribution function (ODF), since it is considerably difficult to recuperate the EAP from a HOT model of the ADC in GDTI. In this paper, we present a novel closed-form approximation of the EAP using Hermite polynomials from a modified HOT model of the original GDTI-ADC. Since the solution is analytical, it is fast, differentiable, and the approximation converges well to the true EAP. This method also makes the effort of computing a positive ADC worthwhile, since now both the ADC and the EAP can be used and have closed forms. We demonstrate our approach with 4th-order tensors on synthetic data and in vivo human data. PMID:23365552

  6. Some spectral approximation of one-dimensional fourth-order problems

    NASA Technical Reports Server (NTRS)

    Bernardi, Christine; Maday, Yvon

    1989-01-01

    Some spectral type collocation method well suited for the approximation of fourth-order systems are proposed. The model problem is the biharmonic equation, in one and two dimensions when the boundary conditions are periodic in one direction. It is proved that the standard Gauss-Lobatto nodes are not the best choice for the collocation points. Then, a new set of nodes related to some generalized Gauss type quadrature formulas is proposed. Also provided is a complete analysis of these formulas including some new issues about the asymptotic behavior of the weights and we apply these results to the analysis of the collocation method.

  7. High Order Approximations for Compressible Fluid Dynamics on Unstructured and Cartesian Meshes

    NASA Technical Reports Server (NTRS)

    Barth, Timothy (Editor); Deconinck, Herman (Editor)

    1999-01-01

    The development of high-order accurate numerical discretization techniques for irregular domains and meshes is often cited as one of the remaining challenges facing the field of computational fluid dynamics. In structural mechanics, the advantages of high-order finite element approximation are widely recognized. This is especially true when high-order element approximation is combined with element refinement (h-p refinement). In computational fluid dynamics, high-order discretization methods are infrequently used in the computation of compressible fluid flow. The hyperbolic nature of the governing equations and the presence of solution discontinuities makes high-order accuracy difficult to achieve. Consequently, second-order accurate methods are still predominately used in industrial applications even though evidence suggests that high-order methods may offer a way to significantly improve the resolution and accuracy for these calculations. To address this important topic, a special course was jointly organized by the Applied Vehicle Technology Panel of NATO's Research and Technology Organization (RTO), the von Karman Institute for Fluid Dynamics, and the Numerical Aerospace Simulation Division at the NASA Ames Research Center. The NATO RTO sponsored course entitled "Higher Order Discretization Methods in Computational Fluid Dynamics" was held September 14-18, 1998 at the von Karman Institute for Fluid Dynamics in Belgium and September 21-25, 1998 at the NASA Ames Research Center in the United States. During this special course, lecturers from Europe and the United States gave a series of comprehensive lectures on advanced topics related to the high-order numerical discretization of partial differential equations with primary emphasis given to computational fluid dynamics (CFD). Additional consideration was given to topics in computational physics such as the high-order discretization of the Hamilton-Jacobi, Helmholtz, and elasticity equations. This volume consists

  8. Approximate solution of space and time fractional higher order phase field equation

    NASA Astrophysics Data System (ADS)

    Shamseldeen, S.

    2018-03-01

    This paper is concerned with a class of space and time fractional partial differential equation (STFDE) with Riesz derivative in space and Caputo in time. The proposed STFDE is considered as a generalization of a sixth-order partial phase field equation. We describe the application of the optimal homotopy analysis method (OHAM) to obtain an approximate solution for the suggested fractional initial value problem. An averaged-squared residual error function is defined and used to determine the optimal convergence control parameter. Two numerical examples are studied, considering periodic and non-periodic initial conditions, to justify the efficiency and the accuracy of the adopted iterative approach. The dependence of the solution on the order of the fractional derivative in space and time and model parameters is investigated.

  9. Approximate optimal guidance for the advanced launch system

    NASA Technical Reports Server (NTRS)

    Feeley, T. S.; Speyer, J. L.

    1993-01-01

    A real-time guidance scheme for the problem of maximizing the payload into orbit subject to the equations of motion for a rocket over a spherical, non-rotating earth is presented. An approximate optimal launch guidance law is developed based upon an asymptotic expansion of the Hamilton - Jacobi - Bellman or dynamic programming equation. The expansion is performed in terms of a small parameter, which is used to separate the dynamics of the problem into primary and perturbation dynamics. For the zeroth-order problem the small parameter is set to zero and a closed-form solution to the zeroth-order expansion term of Hamilton - Jacobi - Bellman equation is obtained. Higher-order terms of the expansion include the effects of the neglected perturbation dynamics. These higher-order terms are determined from the solution of first-order linear partial differential equations requiring only the evaluation of quadratures. This technique is preferred as a real-time, on-line guidance scheme to alternative numerical iterative optimization schemes because of the unreliable convergence properties of these iterative guidance schemes and because the quadratures needed for the approximate optimal guidance law can be performed rapidly and by parallel processing. Even if the approximate solution is not nearly optimal, when using this technique the zeroth-order solution always provides a path which satisfies the terminal constraints. Results for two-degree-of-freedom simulations are presented for the simplified problem of flight in the equatorial plane and compared to the guidance scheme generated by the shooting method which is an iterative second-order technique.

  10. Infinite order sudden approximation for rotational energy transfer in gaseous mixtures

    SciTech Connect

    Goldflam, R.; Green, S.; Kouri, D.J.

    1977-11-01

    Rotational energy transfer in gaseous mixtures has been considered within the framework of the infinite order sudden (IOS) approximation. A new derivation of the IOS from the coupled states Lippmann--Schwinger equation is given. This approach shows the relation between the IOS and CS T matrices and also shows in a rather transparent fashion Sencrest's result that the IOS method does not truncate closed channels but rather employs a closure relation to sum over all rotor states. The general CS effective cross section formula for relaxation processes is used, along with the IOS approximation to the CS T matrix, to derivemore » the general IOS effctive cross section.Factorization permits one to calculate other types of cross sections if any one type of cross section has been obtained by some procedure. The functional form can also be used to compact data. This formalism has been applied to calculate pressure broadening for the systems HD--He, HCl--He, CO--He, HCN--He, HCl--Ar, and CO/sub 2/--Ar. To test the IOS approximation, comparisons have been made to the CS results, which are known to be accurate for all these systems. The IOS approximation is found to be very accurate whenever the rotor spacings are small compared to the kinetic energy, provided closed channels do not play too great a role. For the systems CO--He, HCN--He, and CO/sub 2/--Ar, these conditions are well satisfied and the IOS is found to yield results accurate to within 10%--15%.« less

  11. Asymptotic orderings and approximations of the Master kinetic equation for large hard spheres systems

    NASA Astrophysics Data System (ADS)

    Tessarotto, Massimo; Asci, Claudio

    2017-05-01

    In this paper the problem is posed of determining the physically-meaningful asymptotic orderings holding for the statistical description of a large N-body system of hard spheres, i.e., formed by N ≡1/ε ≫ 1 particles, which are allowed to undergo instantaneous and purely elastic unary, binary or multiple collisions. Starting point is the axiomatic treatment recently developed [Tessarotto et al., 2013-2016] and the related discovery of an exact kinetic equation realized by Master equation which advances in time the 1-body probability density function (PDF) for such a system. As shown in the paper the task involves introducing appropriate asymptotic orderings in terms of ε for all the physically-relevant parameters. The goal is that of identifying the relevant physically-meaningful asymptotic approximations applicable for the Master kinetic equation, together with their possible relationships with the Boltzmann and Enskog kinetic equations, and holding in appropriate asymptotic regimes. These correspond either to dilute or dense systems and are formed either by small-size or finite-size identical hard spheres, the distinction between the various cases depending on suitable asymptotic orderings in terms of ε.

  12. Improved phase shift approach to the energy correction of the infinite order sudden approximation

    SciTech Connect

    Chang, B.; Eno, L.; Rabitz, H.

    1980-07-15

    A new method is presented for obtaining energy corrections to the infinite order sudden (IOS) approximation by incorporating the effect of the internal molecular Hamiltonian into the IOS wave function. This is done by utilizing the JWKB approximation to transform the Schroedinger equation into a differential equation for the phase. It is found that the internal Hamiltonian generates an effective potential from which a new improved phase shift is obtained. This phase shift is then used in place of the IOS phase shift to generate new transition probabilities. As an illustration the resulting improved phase shift (IPS) method is appliedmore » to the Secrest--Johnson model for the collinear collision of an atom and diatom. In the vicinity of the sudden limit, the IPS method gives results for transition probabilities, P/sub n/..-->..n+..delta..n, in significantly better agreement with the 'exact' close coupling calculations than the IOS method, particularly for large ..delta..n. However, when the IOS results are not even qualitatively correct, the IPS method is unable to satisfactorily provide improvements.« less

  13. A third-order approximation method for three-dimensional wheel-rail contact

    NASA Astrophysics Data System (ADS)

    Negretti, Daniele

    2012-03-01

    Multibody train analysis is used increasingly by railway operators whenever a reliable and time-efficient method to evaluate the contact between wheel and rail is needed; particularly, the wheel-rail contact is one of the most important aspects that affects a reliable and time-efficient vehicle dynamics computation. The focus of the approach proposed here is to carry out such tasks by means of online wheel-rail elastic contact detection. In order to improve efficiency and save time, a main analytical approach is used for the definition of wheel and rail surfaces as well as for contact detection, then a final numerical evaluation is used to locate contact. The final numerical procedure consists in finding the zeros of a nonlinear function in a single variable. The overall method is based on the approximation of the wheel surface, which does not influence the contact location significantly, as shown in the paper.

  14. A Note on Substructuring Preconditioning for Nonconforming Finite Element Approximations of Second Order Elliptic Problems

    NASA Technical Reports Server (NTRS)

    Maliassov, Serguei

    1996-01-01

    In this paper an algebraic substructuring preconditioner is considered for nonconforming finite element approximations of second order elliptic problems in 3D domains with a piecewise constant diffusion coefficient. Using a substructuring idea and a block Gauss elimination, part of the unknowns is eliminated and the Schur complement obtained is preconditioned by a spectrally equivalent very sparse matrix. In the case of quasiuniform tetrahedral mesh an appropriate algebraic multigrid solver can be used to solve the problem with this matrix. Explicit estimates of condition numbers and implementation algorithms are established for the constructed preconditioner. It is shown that the condition number of the preconditioned matrix does not depend on either the mesh step size or the jump of the coefficient. Finally, numerical experiments are presented to illustrate the theory being developed.

  15. High-order above-threshold ionization beyond the electric dipole approximation

    NASA Astrophysics Data System (ADS)

    Brennecke, Simon; Lein, Manfred

    2018-05-01

    Photoelectron momentum distributions from strong-field ionization are calculated by numerical solution of the one-electron time-dependent Schrödinger equation for a model atom including effects beyond the electric dipole approximation. We focus on the high-energy electrons from rescattering and analyze their momentum component along the field propagation direction. We show that the boundary of the calculated momentum distribution is deformed in accordance with the classical three-step model including the beyond-dipole Lorentz force. In addition, the momentum distribution exhibits an asymmetry in the signal strengths of electrons emitted in the forward/backward directions. Taken together, the two non-dipole effects give rise to a considerable average forward momentum component of the order of 0.1 a.u. for realistic laser parameters.

  16. Corrections to the thin wall approximation in general relativity

    NASA Technical Reports Server (NTRS)

    Garfinkle, David; Gregory, Ruth

    1989-01-01

    The question is considered whether the thin wall formalism of Israel applies to the gravitating domain walls of a lambda phi(exp 4) theory. The coupled Einstein-scalar equations that describe the thick gravitating wall are expanded in powers of the thickness of the wall. The solutions of the zeroth order equations reproduce the results of the usual Israel thin wall approximation for domain walls. The solutions of the first order equations provide corrections to the expressions for the stress-energy of the wall and to the Israel thin wall equations. The modified thin wall equations are then used to treat the motion of spherical and planar domain walls.

  17. Examining the accuracy of the infinite order sudden approximation using sensitivity analysis

    SciTech Connect

    Eno, L.; Rabitz, H.

    1981-08-15

    A method is developed for assessing the accuracy of scattering observables calculated within the framework of the infinite order sudden (IOS) approximation. In particular, we focus on the energy sudden assumption of the IOS method and our approach involves the determination of the sensitivity of the IOS scattering matrix S/sup IOS/ with respect to a parameter which reintroduces the internal energy operator h/sub 0/ into the IOS Hamiltonian. This procedure is an example of sensitivity analysis of missing model components (h/sub 0/ in this case) in the reference Hamiltonian. In contrast to simple first-order perturbation theory a finite result ismore » obtained for the effect of h/sub 0/ on S/sup IOS/. As an illustration, our method of analysis is applied to integral state-to-state cross sections for the scattering of an atom and rigid rotor. Results are generated within the He+H/sub 2/ system and a comparison is made between IOS and coupled states cross sections and the corresponding IOS sensitivities. It is found that the sensitivity coefficients are very useful indicators of the accuracy of the IOS results. Finally, further developments and applications are discussed.« less

  18. Homogenization of one-dimensional draining through heterogeneous porous media including higher-order approximations

    NASA Astrophysics Data System (ADS)

    Anderson, Daniel M.; McLaughlin, Richard M.; Miller, Cass T.

    2018-02-01

    We examine a mathematical model of one-dimensional draining of a fluid through a periodically-layered porous medium. A porous medium, initially saturated with a fluid of a high density is assumed to drain out the bottom of the porous medium with a second lighter fluid replacing the draining fluid. We assume that the draining layer is sufficiently dense that the dynamics of the lighter fluid can be neglected with respect to the dynamics of the heavier draining fluid and that the height of the draining fluid, represented as a free boundary in the model, evolves in time. In this context, we neglect interfacial tension effects at the boundary between the two fluids. We show that this problem admits an exact solution. Our primary objective is to develop a homogenization theory in which we find not only leading-order, or effective, trends but also capture higher-order corrections to these effective draining rates. The approximate solution obtained by this homogenization theory is compared to the exact solution for two cases: (1) the permeability of the porous medium varies smoothly but rapidly and (2) the permeability varies as a piecewise constant function representing discrete layers of alternating high/low permeability. In both cases we are able to show that the corrections in the homogenization theory accurately predict the position of the free boundary moving through the porous medium.

  19. Examining the accuracy of the infinite order sudden approximation using sensitivity analysis

    NASA Astrophysics Data System (ADS)

    Eno, Larry; Rabitz, Herschel

    1981-08-01

    A method is developed for assessing the accuracy of scattering observables calculated within the framework of the infinite order sudden (IOS) approximation. In particular, we focus on the energy sudden assumption of the IOS method and our approach involves the determination of the sensitivity of the IOS scattering matrix SIOS with respect to a parameter which reintroduces the internal energy operator ?0 into the IOS Hamiltonian. This procedure is an example of sensitivity analysis of missing model components (?0 in this case) in the reference Hamiltonian. In contrast to simple first-order perturbation theory a finite result is obtained for the effect of ?0 on SIOS. As an illustration, our method of analysis is applied to integral state-to-state cross sections for the scattering of an atom and rigid rotor. Results are generated within the He+H2 system and a comparison is made between IOS and coupled states cross sections and the corresponding IOS sensitivities. It is found that the sensitivity coefficients are very useful indicators of the accuracy of the IOS results. Finally, further developments and applications are discussed.

  20. First-order approximation for the pressure-flow relationship of spontaneously contracting lymphangions.

    PubMed

    Quick, Christopher M; Venugopal, Arun M; Dongaonkar, Ranjeet M; Laine, Glen A; Stewart, Randolph H

    2008-05-01

    To return lymph to the great veins of the neck, it must be actively pumped against a pressure gradient. Mean lymph flow in a portion of a lymphatic network has been characterized by an empirical relationship (P(in) - P(out) = -P(p) + R(L)Q(L)), where P(in) - P(out) is the axial pressure gradient and Q(L) is mean lymph flow. R(L) and P(p) are empirical parameters characterizing the effective lymphatic resistance and pump pressure, respectively. The relation of these global empirical parameters to the properties of lymphangions, the segments of a lymphatic vessel bounded by valves, has been problematic. Lymphangions have a structure like blood vessels but cyclically contract like cardiac ventricles; they are characterized by a contraction frequency (f) and the slopes of the end-diastolic pressure-volume relationship [minimum value of resulting elastance (E(min))] and end-systolic pressure-volume relationship [maximum value of resulting elastance (E(max))]. Poiseuille's law provides a first-order approximation relating the pressure-flow relationship to the fundamental properties of a blood vessel. No analogous formula exists for a pumping lymphangion. We therefore derived an algebraic formula predicting lymphangion flow from fundamental physical principles and known lymphangion properties. Quantitative analysis revealed that lymph inertia and resistance to lymph flow are negligible and that lymphangions act like a series of interconnected ventricles. For a single lymphangion, P(p) = P(in) (E(max) - E(min))/E(min) and R(L) = E(max)/f. The formula was tested against a validated, realistic mathematical model of a lymphangion and found to be accurate. Predicted flows were within the range of flows measured in vitro. The present work therefore provides a general solution that makes it possible to relate fundamental lymphangion properties to lymphatic system function.

  1. Electromagnetic enhancement of ordered silver nanorod arrays evaluated by discrete dipole approximation.

    PubMed

    Wei, Guoke; Wang, Jinliang; Chen, Yu

    2015-01-01

    The enhancement factor (EF) of surface-enhanced Raman scattering (SERS) from two-dimensional (2D) hexagonal silver nanorod (AgNR) arrays were investigated in terms of electromagnetic (EM) mechanism by using the discrete dipole approximation (DDA) method. The dependence of EF on several parameters, i.e., structure, length, excitation wavelength, incident angle and polarization, and gap size has been investigated. "Hotspots" were found distributed in the gaps between adjacent nanorods. Simulations of AgNR arrays of different lengths revealed that increasing the rod length from 374 to 937 nm (aspect ratio from 2.0 to 5.0) generated more "hotspots" but not necessarily increased EF under both 514 and 532 nm excitation. A narrow lateral gap (in the incident plane) was found to result in strong EF, while the dependence of EF on the diagonal gap (out of the incident plane) showed an oscillating behavior. The EF of the array was highly dependent on the angle and polarization of the incident light. The structure of AgNR and the excitation wavelength were also found to affect the EF. The EF of random arrays was stronger than that of an ordered one with the same average gap of 21 nm, which could be explained by the exponential dependence of EF on the lateral gap size. Our results also suggested that absorption rather than extinction or scattering could be a good indicator of EM enhancement. It is expected that the understanding of the dependence of local field enhancement on the structure of the nanoarrays and incident excitations will shine light on the optimal design of efficient SERS substrates and improved performance.

  2. Solute transport in aquifers: The comeback of the advection dispersion equation and the First Order Approximation

    NASA Astrophysics Data System (ADS)

    Fiori, A.; Zarlenga, A.; Jankovic, I.; Dagan, G.

    2017-12-01

    Natural gradient steady flow of mean velocity U takes place in heterogeneous aquifers of random logconductivity Y = lnK , characterized by the normal univariate PDF f(Y) and autocorrelation ρY, of variance σY2 and horizontal integral scale I. Solute transport is quantified by the Breakthrough Curve (BTC) M at planes at distance x from the injection plane. The study builds on the extensive 3D numerical simulations of flow and transport of Jankovic et al. (2017) for different conductivity structures. The present study further explores the predictive capabilities of the Advection Dispersion Equation (ADE), with macrodispersivity αL given by the First Order Approximation (FOA), by checking in a quantitative manner its applicability. After a discussion on the suitable boundary conditions for ADE, we find that the ADE-FOA solution is a sufficiently accurate predictor for applications, the many other sources of uncertainty prevailing in practice notwithstanding. We checked by least squares and by comparison of travel time of quantiles of M that indeed the analytical Inverse Gaussian M with αL =σY2 I , is able to fit well the bulk of the simulated BTCs. It tends to underestimate the late arrival time of the thin and persistent tail. The tail is better reproduced by the semi-analytical MIMSCA model, which also allows for a physical explanation of the success of the Inverse Gaussian solution. Examination of the pertinent longitudinal mass distribution shows that it is different from the commonly used Gaussian one in the analysis of field experiments, and it captures the main features of the plume measurements of the MADE experiment. The results strengthen the confidence in the applicability of the ADE and the FOA to predicting longitudinal spreading in solute transport through heterogeneous aquifers of stationary random structure.

  3. Orbitally invariant internally contracted multireference unitary coupled cluster theory and its perturbative approximation: theory and test calculations of second order approximation.

    PubMed

    Chen, Zhenhua; Hoffmann, Mark R

    2012-07-07

    A unitary wave operator, exp (G), G(+) = -G, is considered to transform a multiconfigurational reference wave function Φ to the potentially exact, within basis set limit, wave function Ψ = exp (G)Φ. To obtain a useful approximation, the Hausdorff expansion of the similarity transformed effective Hamiltonian, exp (-G)Hexp (G), is truncated at second order and the excitation manifold is limited; an additional separate perturbation approximation can also be made. In the perturbation approximation, which we refer to as multireference unitary second-order perturbation theory (MRUPT2), the Hamiltonian operator in the highest order commutator is approximated by a Mo̸ller-Plesset-type one-body zero-order Hamiltonian. If a complete active space self-consistent field wave function is used as reference, then the energy is invariant under orbital rotations within the inactive, active, and virtual orbital subspaces for both the second-order unitary coupled cluster method and its perturbative approximation. Furthermore, the redundancies of the excitation operators are addressed in a novel way, which is potentially more efficient compared to the usual full diagonalization of the metric of the excited configurations. Despite the loss of rigorous size-extensivity possibly due to the use of a variational approach rather than a projective one in the solution of the amplitudes, test calculations show that the size-extensivity errors are very small. Compared to other internally contracted multireference perturbation theories, MRUPT2 only needs reduced density matrices up to three-body even with a non-complete active space reference wave function when two-body excitations within the active orbital subspace are involved in the wave operator, exp (G). Both the coupled cluster and perturbation theory variants are amenable to large, incomplete model spaces. Applications to some widely studied model systems that can be problematic because of geometry dependent quasidegeneracy, H4, P4

  4. Many particle approximation of the Aw-Rascle-Zhang second order model for vehicular traffic.

    PubMed

    Francesco, Marco Di; Fagioli, Simone; Rosini, Massimiliano D

    2017-02-01

    We consider the follow-the-leader approximation of the Aw-Rascle-Zhang (ARZ) model for traffic flow in a multi population formulation. We prove rigorous convergence to weak solutions of the ARZ system in the many particle limit in presence of vacuum. The result is based on uniform BV estimates on the discrete particle velocity. We complement our result with numerical simulations of the particle method compared with some exact solutions to the Riemann problem of the ARZ system.

  5. Second order accurate finite difference approximations for the transonic small disturbance equation and the full potential equation

    NASA Technical Reports Server (NTRS)

    Mostrel, M. M.

    1988-01-01

    New shock-capturing finite difference approximations for solving two scalar conservation law nonlinear partial differential equations describing inviscid, isentropic, compressible flows of aerodynamics at transonic speeds are presented. A global linear stability theorem is applied to these schemes in order to derive a necessary and sufficient condition for the finite element method. A technique is proposed to render the described approximations total variation-stable by applying the flux limiters to the nonlinear terms of the difference equation dimension by dimension. An entropy theorem applying to the approximations is proved, and an implicit, forward Euler-type time discretization of the approximation is presented. Results of some numerical experiments using the approximations are reported.

  6. Thomas-Fermi approximation for a condensate with higher-order interactions

    SciTech Connect

    Thoegersen, M.; Jensen, A. S.; Zinner, N. T.

    We consider the ground state of a harmonically trapped Bose-Einstein condensate within the Gross-Pitaevskii theory including the effective-range corrections for a two-body zero-range potential. The resulting nonlinear Schroedinger equation is solved analytically in the Thomas-Fermi approximation neglecting the kinetic-energy term. We present results for the chemical potential and the condensate profiles, discuss boundary conditions, and compare to the usual Thomas-Fermi approach. We discuss several ways to increase the influence of effective-range corrections in experiment with magnetically tunable interactions. The level of tuning required could be inside experimental reach in the near future.

  7. Approximate series solution of multi-dimensional, time fractional-order (heat-like) diffusion equations using FRDTM

    PubMed Central

    Singh, Brajesh K.; Srivastava, Vineet K.

    2015-01-01

    The main goal of this paper is to present a new approximate series solution of the multi-dimensional (heat-like) diffusion equation with time-fractional derivative in Caputo form using a semi-analytical approach: fractional-order reduced differential transform method (FRDTM). The efficiency of FRDTM is confirmed by considering four test problems of the multi-dimensional time fractional-order diffusion equation. FRDTM is a very efficient, effective and powerful mathematical tool which provides exact or very close approximate solutions for a wide range of real-world problems arising in engineering and natural sciences, modelled in terms of differential equations. PMID:26064639

  8. Approximate series solution of multi-dimensional, time fractional-order (heat-like) diffusion equations using FRDTM.

    PubMed

    Singh, Brajesh K; Srivastava, Vineet K

    2015-04-01

    The main goal of this paper is to present a new approximate series solution of the multi-dimensional (heat-like) diffusion equation with time-fractional derivative in Caputo form using a semi-analytical approach: fractional-order reduced differential transform method (FRDTM). The efficiency of FRDTM is confirmed by considering four test problems of the multi-dimensional time fractional-order diffusion equation. FRDTM is a very efficient, effective and powerful mathematical tool which provides exact or very close approximate solutions for a wide range of real-world problems arising in engineering and natural sciences, modelled in terms of differential equations.

  9. High order filtering methods for approximating hyberbolic systems of conservation laws

    NASA Technical Reports Server (NTRS)

    Lafon, F.; Osher, S.

    1990-01-01

    In the computation of discontinuous solutions of hyperbolic systems of conservation laws, the recently developed essentially non-oscillatory (ENO) schemes appear to be very useful. However, they are computationally costly compared to simple central difference methods. A filtering method which is developed uses simple central differencing of arbitrarily high order accuracy, except when a novel local test indicates the development of spurious oscillations. At these points, the full ENO apparatus is used, maintaining the high order of accuracy, but removing spurious oscillations. Numerical results indicate the success of the method. High order of accuracy was obtained in regions of smooth flow without spurious oscillations for a wide range of problems and a significant speed up of generally a factor of almost three over the full ENO method.

  10. Fourth order exponential time differencing method with local discontinuous Galerkin approximation for coupled nonlinear Schrodinger equations

    DOE PAGES

    Liang, Xiao; Khaliq, Abdul Q. M.; Xing, Yulong

    2015-01-23

    In this paper, we study a local discontinuous Galerkin method combined with fourth order exponential time differencing Runge-Kutta time discretization and a fourth order conservative method for solving the nonlinear Schrödinger equations. Based on different choices of numerical fluxes, we propose both energy-conserving and energy-dissipative local discontinuous Galerkin methods, and have proven the error estimates for the semi-discrete methods applied to linear Schrödinger equation. The numerical methods are proven to be highly efficient and stable for long-range soliton computations. Finally, extensive numerical examples are provided to illustrate the accuracy, efficiency and reliability of the proposed methods.

  11. High order filtering methods for approximating hyperbolic systems of conservation laws

    NASA Technical Reports Server (NTRS)

    Lafon, F.; Osher, S.

    1991-01-01

    The essentially nonoscillatory (ENO) schemes, while potentially useful in the computation of discontinuous solutions of hyperbolic conservation-law systems, are computationally costly relative to simple central-difference methods. A filtering technique is presented which employs central differencing of arbitrarily high-order accuracy except where a local test detects the presence of spurious oscillations and calls upon the full ENO apparatus to remove them. A factor-of-three speedup is thus obtained over the full-ENO method for a wide range of problems, with high-order accuracy in regions of smooth flow.

  12. Higher and lowest order mixed finite element approximation of subsurface flow problems with solutions of low regularity

    NASA Astrophysics Data System (ADS)

    Bause, Markus

    2008-02-01

    In this work we study mixed finite element approximations of Richards' equation for simulating variably saturated subsurface flow and simultaneous reactive solute transport. Whereas higher order schemes have proved their ability to approximate reliably reactive solute transport (cf., e.g. [Bause M, Knabner P. Numerical simulation of contaminant biodegradation by higher order methods and adaptive time stepping. Comput Visual Sci 7;2004:61-78]), the Raviart- Thomas mixed finite element method ( RT0) with a first order accurate flux approximation is popular for computing the underlying water flow field (cf. [Bause M, Knabner P. Computation of variably saturated subsurface flow by adaptive mixed hybrid finite element methods. Adv Water Resour 27;2004:565-581, Farthing MW, Kees CE, Miller CT. Mixed finite element methods and higher order temporal approximations for variably saturated groundwater flow. Adv Water Resour 26;2003:373-394, Starke G. Least-squares mixed finite element solution of variably saturated subsurface flow problems. SIAM J Sci Comput 21;2000:1869-1885, Younes A, Mosé R, Ackerer P, Chavent G. A new formulation of the mixed finite element method for solving elliptic and parabolic PDE with triangular elements. J Comp Phys 149;1999:148-167, Woodward CS, Dawson CN. Analysis of expanded mixed finite element methods for a nonlinear parabolic equation modeling flow into variably saturated porous media. SIAM J Numer Anal 37;2000:701-724]). This combination might be non-optimal. Higher order techniques could increase the accuracy of the flow field calculation and thereby improve the prediction of the solute transport. Here, we analyse the application of the Brezzi- Douglas- Marini element ( BDM1) with a second order accurate flux approximation to elliptic, parabolic and degenerate problems whose solutions lack the regularity that is assumed in optimal order error analyses. For the flow field calculation a superiority of the BDM1 approach to the RT0 one is

  13. Approximating recreation site choice: the predictive capability of a lexicographic semi-order model

    Treesearch

    Alan E. Watson; Joseph W. Roggenbuck

    1985-01-01

    The relevancy of a lexicographic semi-order model, as a basis for development of a microcomputer-based decision aid for backcountry hikers, was investigated. In an interactive microcomputer exercise, it was found that a decision aid based upon this model may assist recreationists in reduction of an alternative set to a cognitively manageable number.

  14. Higher order approximation to the Hill problem dynamics about the libration points

    NASA Astrophysics Data System (ADS)

    Lara, Martin; Pérez, Iván L.; López, Rosario

    2018-06-01

    An analytical solution to the Hill problem Hamiltonian expanded about the libration points has been obtained by means of perturbation techniques. In order to compute the higher orders of the perturbation solution that are needed to capture all the relevant periodic orbits originated from the libration points within a reasonable accuracy, the normalization is approached in complex variables. The validity of the solution extends to energy values considerably far away from that of the libration points and, therefore, can be used in the computation of Halo orbits as an alternative to the classical Lindstedt-Poincaré approach. Furthermore, the theory correctly predicts the existence of the two-lane bridge of periodic orbits linking the families of planar and vertical Lyapunov orbits.

  15. Approximate tensor-product preconditioners for very high order discontinuous Galerkin methods

    NASA Astrophysics Data System (ADS)

    Pazner, Will; Persson, Per-Olof

    2018-02-01

    In this paper, we develop a new tensor-product based preconditioner for discontinuous Galerkin methods with polynomial degrees higher than those typically employed. This preconditioner uses an automatic, purely algebraic method to approximate the exact block Jacobi preconditioner by Kronecker products of several small, one-dimensional matrices. Traditional matrix-based preconditioners require O (p2d) storage and O (p3d) computational work, where p is the degree of basis polynomials used, and d is the spatial dimension. Our SVD-based tensor-product preconditioner requires O (p d + 1) storage, O (p d + 1) work in two spatial dimensions, and O (p d + 2) work in three spatial dimensions. Combined with a matrix-free Newton-Krylov solver, these preconditioners allow for the solution of DG systems in linear time in p per degree of freedom in 2D, and reduce the computational complexity from O (p9) to O (p5) in 3D. Numerical results are shown in 2D and 3D for the advection, Euler, and Navier-Stokes equations, using polynomials of degree up to p = 30. For many test cases, the preconditioner results in similar iteration counts when compared with the exact block Jacobi preconditioner, and performance is significantly improved for high polynomial degrees p.

  16. Hyperspherical lowest-order constrained-variational approximation to resonant Bose-Einstein condensates

    NASA Astrophysics Data System (ADS)

    Sze, M. W. C.; Sykes, A. G.; Blume, D.; Bohn, J. L.

    2018-03-01

    We study the ground-state properties of a system of N harmonically trapped bosons of mass m interacting with two-body contact interactions, from small to large scattering lengths. This is accomplished in a hyperspherical coordinate system that is flexible enough to describe both the overall scale of the gas and two-body correlations. By adapting the lowest-order constrained-variational method, we are able to semiquantitatively attain Bose-Einstein condensate ground-state energies even for gases with infinite scattering length. In the large-particle-number limit, our method provides analytical estimates for the energy per particle E0/N ≈2.5 N1 /3ℏ ω and two-body contact C2/N ≈16 N1 /6√{m ω /ℏ } for a Bose gas on resonance, where ω is the trap frequency.

  17. Microwave vector radiative transfer equation of a sea foam layer by the second-order Rayleigh approximation

    NASA Astrophysics Data System (ADS)

    Wei, En-Bo

    2011-10-01

    The microwave vector radiative transfer (VRT) equation of a coated spherical bubble layer is derived by means of the second-order Rayleigh approximation field when the microwave wavelength is larger than the coated spherical particle diameter. Meanwhile, the perturbation method is developed to solve the second-order Rayleigh VRT equation for the small ratio of the volume scattering coefficient to the extinction coefficient. As an example, the emissive properties of a sea surface foam layer, which consists of seawater coated bubbles, are investigated. The extinction, absorption, and scattering coefficients of sea foam are obtained by the second-order Rayleigh approximation fields and discussed for the different microwave frequencies and the ratio of inner radius to outer radius of a coated bubble. Our results show that in the dilute limit, the volume scattering coefficient decreases with increasing the ratio of inner radius to outer radius and decreasing the frequencies. It is also found that the microwave emissivity and the extinction coefficient have a peak at very thin seawater coating and its peak value decreases with frequency decrease. Furthermore, with the VRT equation and effective medium approximation of densely coated bubbles, the mechanism of sea foam enhancing the emissivity of a sea surface is disclosed. In addition, excellent agreement is obtained by comparing our VRT results with the experimental data of microwave emissivities of sea surface covered by a sea foam layer at L-band (1.4 GHz) and the Camps' model.

  18. A new approximation of Fermi-Dirac integrals of order 1/2 for degenerate semiconductor devices

    NASA Astrophysics Data System (ADS)

    AlQurashi, Ahmed; Selvakumar, C. R.

    2018-06-01

    There had been tremendous growth in the field of Integrated circuits (ICs) in the past fifty years. Scaling laws mandated both lateral and vertical dimensions to be reduced and a steady increase in doping densities. Most of the modern semiconductor devices have invariably heavily doped regions where Fermi-Dirac Integrals are required. Several attempts have been devoted to developing analytical approximations for Fermi-Dirac Integrals since numerical computations of Fermi-Dirac Integrals are difficult to use in semiconductor devices, although there are several highly accurate tabulated functions available. Most of these analytical expressions are not sufficiently suitable to be employed in semiconductor device applications due to their poor accuracy, the requirement of complicated calculations, and difficulties in differentiating and integrating. A new approximation has been developed for the Fermi-Dirac integrals of the order 1/2 by using Prony's method and discussed in this paper. The approximation is accurate enough (Mean Absolute Error (MAE) = 0.38%) and easy enough to be used in semiconductor device equations. The new approximation of Fermi-Dirac Integrals is applied to a more generalized Einstein Relation which is an important relation in semiconductor devices.

  19. A novel condition for stable nonlinear sampled-data models using higher-order discretized approximations with zero dynamics.

    PubMed

    Zeng, Cheng; Liang, Shan; Xiang, Shuwen

    2017-05-01

    Continuous-time systems are usually modelled by the form of ordinary differential equations arising from physical laws. However, the use of these models in practice and utilizing, analyzing or transmitting these data from such systems must first invariably be discretized. More importantly, for digital control of a continuous-time nonlinear system, a good sampled-data model is required. This paper investigates the new consistency condition which is weaker than the previous similar results presented. Moreover, given the stability of the high-order approximate model with stable zero dynamics, the novel condition presented stabilizes the exact sampled-data model of the nonlinear system for sufficiently small sampling periods. An insightful interpretation of the obtained results can be made in terms of the stable sampling zero dynamics, and the new consistency condition is surprisingly associated with the relative degree of the nonlinear continuous-time system. Our controller design, based on the higher-order approximate discretized model, extends the existing methods which mainly deal with the Euler approximation. Copyright © 2017 ISA. Published by Elsevier Ltd. All rights reserved.

  20. Representation of the exact relativistic electronic Hamiltonian within the regular approximation

    NASA Astrophysics Data System (ADS)

    Filatov, Michael; Cremer, Dieter

    2003-12-01

    The exact relativistic Hamiltonian for electronic states is expanded in terms of energy-independent linear operators within the regular approximation. An effective relativistic Hamiltonian has been obtained, which yields in lowest order directly the infinite-order regular approximation (IORA) rather than the zeroth-order regular approximation method. Further perturbational expansion of the exact relativistic electronic energy utilizing the effective Hamiltonian leads to new methods based on ordinary (IORAn) or double [IORAn(2)] perturbation theory (n: order of expansion), which provide improved energies in atomic calculations. Energies calculated with IORA4 and IORA3(2) are accurate up to c-20. Furthermore, IORA is improved by using the IORA wave function to calculate the Rayleigh quotient, which, if minimized, leads to the exact relativistic energy. The outstanding performance of this new IORA method coined scaled IORA is documented in atomic and molecular calculations.

  1. Newton's Zeroth Law: Learning from Listening to Our Students

    NASA Astrophysics Data System (ADS)

    Scherr, Rachel E.; Redish, Edward F.

    2005-01-01

    Modern instructional advice encourages us to not just tell our students what we want them to know, but to listen to them carefully. This helps us to find out "where they are" in order to better understand what tasks to offer them that might help them learn the physics most effectively. Sometimes, listening to students and trying to understand their intuitions not only helps them, it helps us—giving us new insights into the physics we are teaching. We had such an experience in the fall of 2003 in our algebra-based physics class at the University of Maryland.

  2. Anharmonic phonons and second-order phase-transitions by the stochastic self-consistent harmonic approximation

    NASA Astrophysics Data System (ADS)

    Mauri, Francesco

    Anharmonic effects can generally be treated within perturbation theory. Such an approach breaks down when the harmonic solution is dynamically unstable or when the anharmonic corrections of the phonon energies are larger than the harmonic frequencies themselves. This situation occurs near lattice-related second-order phase-transitions such as charge-density-wave (CDW) or ferroelectric instabilities or in H-containing materials, where the large zero-point motion of the protons results in a violation of the harmonic approximation. Interestingly, even in these cases, phonons can be observed, measured, and used to model transport properties. In order to treat such cases, we developed a stochastic implementation of the self-consistent harmonic approximation valid to treat anharmonicity in the nonperturbative regime and to obtain, from first-principles, the structural, thermodynamic and vibrational properties of strongly anharmonic systems. I will present applications to the ferroelectric transitions in SnTe, to the CWD transitions in NbS2 and NbSe2 (in bulk and monolayer) and to the hydrogen-bond symmetrization transition in the superconducting hydrogen sulfide system, that exhibits the highest Tc reported for any superconductor so far. In all cases we are able to predict the transition temperature (pressure) and the evolution of phonons with temperature (pressure). This project has received funding from the European Union's Horizon 2020 research and innovation programme under Grant agreement No. 696656 GrapheneCore1.

  3. An infinite-order two-component relativistic Hamiltonian by a simple one-step transformation.

    PubMed

    Ilias, Miroslav; Saue, Trond

    2007-02-14

    The authors report the implementation of a simple one-step method for obtaining an infinite-order two-component (IOTC) relativistic Hamiltonian using matrix algebra. They apply the IOTC Hamiltonian to calculations of excitation and ionization energies as well as electric and magnetic properties of the radon atom. The results are compared to corresponding calculations using identical basis sets and based on the four-component Dirac-Coulomb Hamiltonian as well as Douglas-Kroll-Hess and zeroth-order regular approximation Hamiltonians, all implemented in the DIRAC program package, thus allowing a comprehensive comparison of relativistic Hamiltonians within the finite basis approximation.

  4. Well-Balanced Second-Order Approximation of the Shallow Water Equations With Friction via Continuous Galerkin Finite Elements

    NASA Astrophysics Data System (ADS)

    Quezada de Luna, M.; Farthing, M.; Guermond, J. L.; Kees, C. E.; Popov, B.

    2017-12-01

    The Shallow Water Equations (SWEs) are popular for modeling non-dispersive incompressible water waves where the horizontal wavelength is much larger than the vertical scales. They can be derived from the incompressible Navier-Stokes equations assuming a constant vertical velocity. The SWEs are important in Geophysical Fluid Dynamics for modeling surface gravity waves in shallow regimes; e.g., in the deep ocean. Some common geophysical applications are the evolution of tsunamis, river flooding and dam breaks, storm surge simulations, atmospheric flows and others. This work is concerned with the approximation of the time-dependent Shallow Water Equations with friction using explicit time stepping and continuous finite elements. The objective is to construct a method that is at least second-order accurate in space and third or higher-order accurate in time, positivity preserving, well-balanced with respect to rest states, well-balanced with respect to steady sliding solutions on inclined planes and robust with respect to dry states. Methods fulfilling the desired goals are common within the finite volume literature. However, to the best of our knowledge, schemes with the above properties are not well developed in the context of continuous finite elements. We start this work based on a finite element method that is second-order accurate in space, positivity preserving and well-balanced with respect to rest states. We extend it by: modifying the artificial viscosity (via the entropy viscosity method) to deal with issues of loss of accuracy around local extrema, considering a singular Manning friction term handled via an explicit discretization under the usual CFL condition, considering a water height regularization that depends on the mesh size and is consistent with the polynomial approximation, reducing dispersive errors introduced by lumping the mass matrix and others. After presenting the details of the method we show numerical tests that demonstrate the well

  5. Strong-field approximation in a rotating frame: High-order harmonic emission from p states in bicircular fields

    NASA Astrophysics Data System (ADS)

    Pisanty, Emilio; Jiménez-Galán, Álvaro

    2017-12-01

    High-order harmonic generation with bicircular fields—the combination of counter-rotating circularly polarized pulses at different frequencies—results in a series of short-wavelength XUV harmonics with alternating circular polarizations, and experiments show that there is an asymmetry in the emission between the two helicities: a slight one in helium and a larger one in neon and argon, where the emission is carried out by p -shell electrons. Here we analyze this asymmetry by switching to a rotating frame in which the field is linearly polarized; this induces an effective magnetic field which lowers the ionization potential of the p + orbital that corotates with the lower-frequency driver, enhancing its harmonic emission and the overall helicity of the generated harmonics, while also introducing nontrivial effects from the transformation to a noninertial frame in complex time. In addition, this analysis directly relates the small asymmetry produced by s -shell emission to the imaginary part of the recollision velocity in the standard strong-field-approximation formalism.

  6. Real-time approximate optimal guidance laws for the advanced launch system

    NASA Technical Reports Server (NTRS)

    Speyer, Jason L.; Feeley, Timothy; Hull, David G.

    1989-01-01

    An approach to optimal ascent guidance for a launch vehicle is developed using an expansion technique. The problem is to maximize the payload put into orbit subject to the equations of motion of a rocket over a rotating spherical earth. It is assumed that the thrust and gravitational forces dominate over the aerodynamic forces. It is shown that these forces can be separated by a small parameter epsilon, where epsilon is the ratio of the atmospheric scale height to the radius of the earth. The Hamilton-Jacobi-Bellman or dynamic programming equation is expanded in a series where the zeroth-order term (epsilon = 0) can be obtained in closed form. The zeroth-order problem is that of putting maximum payload into orbit subject to the equations of motion of a rocket in a vacuum over a flat earth. The neglected inertial and aerodynamic terms are included in higher order terms of the expansion, which are determined from the solution of first-order linear partial differential equations requiring only quadrature integrations. These quadrature integrations can be performed rapidly, so that real-time approximate optimization can be used to construct the launch guidance law.

  7. Tunable rejection filters with ultra-wideband using zeroth shear mode plate wave resonators

    NASA Astrophysics Data System (ADS)

    Kadota, Michio; Sannomiya, Toshio; Tanaka, Shuji

    2017-07-01

    This paper reports wide band rejection filters and tunable rejection filters using ultra-wideband zeroth shear mode (SH0) plate wave resonators. The frequency range covers the digital TV band in Japan that runs from 470 to 710 MHz. This range has been chosen to meet the TV white space cognitive radio requirements of rejection filters. Wide rejection bands were obtained using several resonators with different frequencies. Tunable rejection filters were demonstrated using Si diodes connected to the band rejection filters. Wide tunable ranges as high as 31% were measured by applying a DC voltage to the Si diodes.

  8. Higher-order force moments of active particles

    NASA Astrophysics Data System (ADS)

    Nasouri, Babak; Elfring, Gwynn J.

    2018-04-01

    Active particles moving through fluids generate disturbance flows due to their activity. For simplicity, the induced flow field is often modeled by the leading terms in a far-field approximation of the Stokes equations, whose coefficients are the force, torque, and stresslet (zeroth- and first-order force moments) of the active particle. This level of approximation is quite useful, but may also fail to predict more complex behaviors that are observed experimentally. In this study, to provide a better approximation, we evaluate the contribution of the second-order force moments to the flow field and, by reciprocal theorem, present explicit formulas for the stresslet dipole, rotlet dipole, and potential dipole for an arbitrarily shaped active particle. As examples of this method, we derive modified Faxén laws for active spherical particles and resolve higher-order moments for active rod-like particles.

  9. Multiplicative noise removal through fractional order tv-based model and fast numerical schemes for its approximation

    NASA Astrophysics Data System (ADS)

    Ullah, Asmat; Chen, Wen; Khan, Mushtaq Ahmad

    2017-07-01

    This paper introduces a fractional order total variation (FOTV) based model with three different weights in the fractional order derivative definition for multiplicative noise removal purpose. The fractional-order Euler Lagrange equation which is a highly non-linear partial differential equation (PDE) is obtained by the minimization of the energy functional for image restoration. Two numerical schemes namely an iterative scheme based on the dual theory and majorization- minimization algorithm (MMA) are used. To improve the restoration results, we opt for an adaptive parameter selection procedure for the proposed model by applying the trial and error method. We report numerical simulations which show the validity and state of the art performance of the fractional-order model in visual improvement as well as an increase in the peak signal to noise ratio comparing to corresponding methods. Numerical experiments also demonstrate that MMAbased methodology is slightly better than that of an iterative scheme.

  10. An approximation to the temporal order in endogenous circadian rhythms of genes implicated in human adipose tissue metabolism

    USDA-ARS?s Scientific Manuscript database

    Although it is well established that human adipose tissue (AT) shows circadian rhythmicity, published studies have been discussed as if tissues or systems showed only one or few circadian rhythms at a time. To provide an overall view of the internal temporal order of circadian rhythms in human AT in...

  11. An Approximation to the Temporal Order in Endogenous Circadian Rhythms of Genes Implicated in Human Adipose Tissue Metabolism

    PubMed Central

    GARAULET, MARTA; ORDOVÁS, JOSÉ M.; GÓMEZ-ABELLÁN, PURIFICACIÓN; MARTÍNEZ, JOSE A.; MADRID, JUAN A.

    2015-01-01

    Although it is well established that human adipose tissue (AT) shows circadian rhythmicity, published studies have been discussed as if tissues or systems showed only one or few circadian rhythms at a time. To provide an overall view of the internal temporal order of circadian rhythms in human AT including genes implicated in metabolic processes such as energy intake and expenditure, insulin resistance, adipocyte differentiation, dyslipidemia, and body fat distribution. Visceral and subcutaneous abdominal AT biopsies (n = 6) were obtained from morbid obese women (BMI ≥ 40 kg/m2). To investigate rhythmic expression pattern, AT explants were cultured during 24-h and gene expression was analyzed at the following times: 08:00, 14:00, 20:00, 02:00 h using quantitative real-time PCR. Clock genes, glucocorticoid metabolism-related genes, leptin, adiponectin and their receptors were studied. Significant differences were found both in achrophases and relative-amplitude among genes (P <0.05). Amplitude of most genes rhythms was high (>30%). When interpreting the phase map of gene expression in both depots, data indicated that circadian rhythmicity of the genes studied followed a predictable physiological pattern, particularly for subcutaneous AT. Interesting are the relationships between adiponectin, leptin, and glucocorticoid metabolism-related genes circadian profiles. Their metabolic significance is discussed. Visceral AT behaved in a different way than subcutaneous for most of the genes studied. For every gene, protein mRNA levels fluctuated during the day in synchrony with its receptors. We have provided an overall view of the internal temporal order of circadian rhythms in human adipose tissue. PMID:21520059

  12. Convergent sum of gradient expansion of the kinetic-energy density functional up to the sixth order term using Padé approximant

    NASA Astrophysics Data System (ADS)

    Sergeev, A.; Alharbi, F. H.; Jovanovic, R.; Kais, S.

    2016-04-01

    The gradient expansion of the kinetic energy density functional, when applied to atoms or finite systems, usually grossly overestimates the energy in the fourth order and generally diverges in the sixth order. We avoid the divergence of the integral by replacing the asymptotic series including the sixth order term in the integrand by a rational function. Padé approximants show moderate improvements in accuracy in comparison with partial sums of the series. The results are discussed for atoms and Hooke’s law model for two-electron atoms.

  13. A first-order statistical smoothing approximation for the coherent wave field in random porous random media

    NASA Astrophysics Data System (ADS)

    Müller, Tobias M.; Gurevich, Boris

    2005-04-01

    An important dissipation mechanism for waves in randomly inhomogeneous poroelastic media is the effect of wave-induced fluid flow. In the framework of Biot's theory of poroelasticity, this mechanism can be understood as scattering from fast into slow compressional waves. To describe this conversion scattering effect in poroelastic random media, the dynamic characteristics of the coherent wavefield using the theory of statistical wave propagation are analyzed. In particular, the method of statistical smoothing is applied to Biot's equations of poroelasticity. Within the accuracy of the first-order statistical smoothing an effective wave number of the coherent field, which accounts for the effect of wave-induced flow, is derived. This wave number is complex and involves an integral over the correlation function of the medium's fluctuations. It is shown that the known one-dimensional (1-D) result can be obtained as a special case of the present 3-D theory. The expression for the effective wave number allows to derive a model for elastic attenuation and dispersion due to wave-induced fluid flow. These wavefield attributes are analyzed in a companion paper. .

  14. Deterministic time-reversible thermostats: chaos, ergodicity, and the zeroth law of thermodynamics

    NASA Astrophysics Data System (ADS)

    Patra, Puneet Kumar; Sprott, Julien Clinton; Hoover, William Graham; Griswold Hoover, Carol

    2015-09-01

    The relative stability and ergodicity of deterministic time-reversible thermostats, both singly and in coupled pairs, are assessed through their Lyapunov spectra. Five types of thermostat are coupled to one another through a single Hooke's-law harmonic spring. The resulting dynamics shows that three specific thermostat types, Hoover-Holian, Ju-Bulgac, and Martyna-Klein-Tuckerman, have very similar Lyapunov spectra in their equilibrium four-dimensional phase spaces and when coupled in equilibrium or nonequilibrium pairs. All three of these oscillator-based thermostats are shown to be ergodic, with smooth analytic Gaussian distributions in their extended phase spaces (coordinate, momentum, and two control variables). Evidently these three ergodic and time-reversible thermostat types are particularly useful as statistical-mechanical thermometers and thermostats. Each of them generates Gibbs' universal canonical distribution internally as well as for systems to which they are coupled. Thus they obey the zeroth law of thermodynamics, as a good heat bath should. They also provide dissipative heat flow with relatively small nonlinearity when two or more such temperature baths interact and provide useful deterministic replacements for the stochastic Langevin equation.

  15. A comprehensive comparison between thermodynamic perturbation theory and first-order mean spherical approximation: Based on discrete potentials with hard core

    NASA Astrophysics Data System (ADS)

    Zhou, Shiqi; Zhou, Run

    2017-08-01

    Using the TL (Tang and Lu, 1993) method, Ornstein-Zernike integral equation is solved perturbatively under the mean spherical approximation (MSA) for fluid with potential consisting of a hard sphere plus square-well plus square-shoulder (HS + SW + SS) to obtain first-order analytic expressions of radial distribution function (RDF), second-order direct correlation function, and semi-analytic expressions for common thermodynamic properties. A comprehensive comparison between the first-order MSA and high temperature series expansion (HTSE) to third-, fifth- and seventh-order is performed over a wide parameter range for both a HS + SW and the HS + SW + SS model fluids by using corresponding ;exact; Monte Carlo results as a reference; although the HTSE is carried out up to seventh-order, and not to the first order as the first-order MSA the comparison is considered fair from a calculation complexity perspective. It is found that the performance of the first-order MSA is dramatically model-dependent: as target potentials go from the HS + SW to the HS + SW + SS, (i) there is a dramatic dropping of performance of the first-order MSA expressions in calculating the thermodynamic properties, especially both the excess internal energy and constant volume excess heat capacity of the HS + SW + SS model cannot be predicted even qualitatively correctly. (ii) One tendency is noticed that the first-order MSA gets more reliable with increasing temperatures in dealing with the pressure, excess Helmholtz free energy, excess enthalpy and excess chemical potential. (iii) Concerning the RDF, the first-order MSA is not as disappointing as it displays in the cases of thermodynamics. (iv) In the case of the HS + SW model, the first-order MSA solution is shown to be quantitatively correct in calculating the pressure and excess chemical potential even if the reduced temperatures are as low as 0.8. On the other hand, the seventh-order HTSE is less model-dependent; in most cases of the HS + SW

  16. A Modified Formula of the First-order Approximation for Assessing the Contribution of Climate Change to Runoff Based on the Budyko Hypothesis

    NASA Astrophysics Data System (ADS)

    Liu, W.; Ning, T.; Han, X.

    2015-12-01

    The climate elasticity based on the Budyko curves has been widely used to evaluate the hydrological responses to climate change. The Mezentsev-Choudhury-Yang formula is one of the representative analytical equations for Budyko curves. Previous researches mostly used the variation of runoff (R) caused by the changes of annual precipitation (P) and potential evapotranspiration (ET0) as the hydrological response to climate change and evaluated it by a first-order approximation in a form of total differential, the major components of which include the partial derivatives of R to P and ET0, as well as climate elasticity on this basis. Based on analytic derivation and the characteristics of Budyko curves, this study proposed a modified formula of the first-order approximation to reduce the errors from the approximation. In the calculation of partial derivatives and climate elasticity, the values of P and ET0 were taken to the sum of their base values and half increments, respectively. The calculation was applied in 33 catchments of the Hai River basin in China and the results showed that the mean absolute value of relative error of approximated runoff change decreased from 8.4% to 0.4% and the maximum value, from 23.4% to 1.3%. Given the variation values of P, ET0 and the controlling parameter (n), the modified formula can exactly quantify the contributions of climate fluctuation and underlying surface change to runoff. Taking the Murray-Darling basin in Australia as an example of the contribution calculated by the modified formula, the reductions of mean annual runoff caused by changes of P, ET0 and n from 1895-1996 to 1997-2006 were 2.6, 0.6 and 2.9 mm, respectively, and the sum of them was 6.1 mm, which was completely consistent with the observed runoff. The modified formula of the first-order approximation proposed in this study can be not only used to assess the contributions of climate change to the runoff, but also widely used to analyze the effects of similar

  17. Speeding up spin-component-scaled third-order pertubation theory with the chain of spheres approximation: the COSX-SCS-MP3 method

    NASA Astrophysics Data System (ADS)

    Izsák, Róbert; Neese, Frank

    2013-07-01

    The 'chain of spheres' approximation, developed earlier for the efficient evaluation of the self-consistent field exchange term, is introduced here into the evaluation of the external exchange term of higher order correlation methods. Its performance is studied in the specific case of the spin-component-scaled third-order Møller--Plesset perturbation (SCS-MP3) theory. The results indicate that the approximation performs excellently in terms of both computer time and achievable accuracy. Significant speedups over a conventional method are obtained for larger systems and basis sets. Owing to this development, SCS-MP3 calculations on molecules of the size of penicillin (42 atoms) with a polarised triple-zeta basis set can be performed in ∼3 hours using 16 cores of an Intel Xeon E7-8837 processor with a 2.67 GHz clock speed, which represents a speedup by a factor of 8-9 compared to the previously most efficient algorithm. Thus, the increased accuracy offered by SCS-MP3 can now be explored for at least medium-sized molecules.

  18. A high-order time-parallel scheme for solving wave propagation problems via the direct construction of an approximate time-evolution operator

    SciTech Connect

    Haut, T. S.; Babb, T.; Martinsson, P. G.

    2015-06-16

    Our manuscript demonstrates a technique for efficiently solving the classical wave equation, the shallow water equations, and, more generally, equations of the form ∂u/∂t=Lu∂u/∂t=Lu, where LL is a skew-Hermitian differential operator. The idea is to explicitly construct an approximation to the time-evolution operator exp(τL)exp(τL) for a relatively large time-step ττ. Recently developed techniques for approximating oscillatory scalar functions by rational functions, and accelerated algorithms for computing functions of discretized differential operators are exploited. Principal advantages of the proposed method include: stability even for large time-steps, the possibility to parallelize in time over many characteristic wavelengths and large speed-ups over existingmore » methods in situations where simulation over long times are required. Numerical examples involving the 2D rotating shallow water equations and the 2D wave equation in an inhomogenous medium are presented, and the method is compared to the 4th order Runge–Kutta (RK4) method and to the use of Chebyshev polynomials. The new method achieved high accuracy over long-time intervals, and with speeds that are orders of magnitude faster than both RK4 and the use of Chebyshev polynomials.« less

  19. A neutral molecular-based layered magnet [Fe(C2O4)(CH3OH)]n exhibiting magnetic ordering at TN approximately 23 K.

    PubMed

    Zhang, Bin; Zhang, Yan; Zhang, Jinbiao; Li, Junchao; Zhu, Daoben

    2008-10-07

    Solvothermal synthesis of FeCl(2).4H2O and H2C2O(4).2H2O in methanol at 120 degrees C yielded yellow plate-like crystals of [Fe(C2O4)(CH3OH)]n. Each iron atom is in a distorted octahedral environment, being bonded to four oxygen atoms from two bisbidentate oxalate anions, one O atom of a chelating oxalate anion and one O atom from a methanol molecule as an oxalate group bridging ligand in a five-coordination mode. The neutral layer of [Fe(C2O4)(CH3OH)]n with a [4,4] net along the ac plane. There is no interaction between layers. A long range magnetic ordering with spin canting at TN approximately 23 K was observed and confirmed by AC susceptibility measurements.

  20. SU-E-J-238: First-Order Approximation of Time-Resolved 4DMRI From Cine 2DMRI and Respiratory-Correlated 4DMRI

    SciTech Connect

    Li, G; Tyagi, N; Deasy, J

    2015-06-15

    Purpose: Cine 2DMRI is useful in MR-guided radiotherapy but it lacks volumetric information. We explore the feasibility of estimating timeresolved (TR) 4DMRI based on cine 2DMRI and respiratory-correlated (RC) 4DMRI though simulation. Methods: We hypothesize that a volumetric image during free breathing can be approximated by interpolation among 3DMRI image sets generated from a RC-4DMRI. Two patients’ RC-4DMRI with 4 or 5 phases were used to generate additional 3DMRI by interpolation. For each patient, six libraries were created to have total 5-to-35 3DMRI images by 0–6 equi-spaced tri-linear interpolation between adjacent and full-inhalation/full-exhalation phases. Sagittal cine 2DMRI were generated frommore » reference 3DMRIs created from separate, unique interpolations from the original RC-4DMRI. To test if accurate 3DMRI could be generated through rigid registration of the cine 2DMRI to the 3DMRI libraries, each sagittal 2DMRI was registered to sagittal cuts in the same location in the 3DMRI within each library to identify the two best matches: one with greater lung volume and one with smaller. A final interpolation between the corresponding 3DMRI was then performed to produce the first-order-approximation (FOA) 3DMRI. The quality and performance of the FOA as a function of library size was assessed using both the difference in lung volume and average voxel intensity between the FOA and the reference 3DMRI. Results: The discrepancy between the FOA and reference 3DMRI decreases as the library size increases. The 3D lung volume difference decreases from 5–15% to 1–2% as the library size increases from 5 to 35 image sets. The average difference in lung voxel intensity decreases from 7–8 to 5–6 with the lung intensity being 0–135. Conclusion: This study indicates that the quality of FOA 3DMRI improves with increasing 3DMRI library size. On-going investigations will test this approach using actual cine 2DMRI and introduce a higher order

  1. CFHTLenS: a Gaussian likelihood is a sufficient approximation for a cosmological analysis of third-order cosmic shear statistics

    NASA Astrophysics Data System (ADS)

    Simon, P.; Semboloni, E.; van Waerbeke, L.; Hoekstra, H.; Erben, T.; Fu, L.; Harnois-Déraps, J.; Heymans, C.; Hildebrandt, H.; Kilbinger, M.; Kitching, T. D.; Miller, L.; Schrabback, T.

    2015-05-01

    We study the correlations of the shear signal between triplets of sources in the Canada-France-Hawaii Telescope Lensing Survey (CFHTLenS) to probe cosmological parameters via the matter bispectrum. In contrast to previous studies, we adopt a non-Gaussian model of the data likelihood which is supported by our simulations of the survey. We find that for state-of-the-art surveys, similar to CFHTLenS, a Gaussian likelihood analysis is a reasonable approximation, albeit small differences in the parameter constraints are already visible. For future surveys we expect that a Gaussian model becomes inaccurate. Our algorithm for a refined non-Gaussian analysis and data compression is then of great utility especially because it is not much more elaborate if simulated data are available. Applying this algorithm to the third-order correlations of shear alone in a blind analysis, we find a good agreement with the standard cosmological model: Σ _8=σ _8(Ω _m/0.27)^{0.64}=0.79^{+0.08}_{-0.11} for a flat Λ cold dark matter cosmology with h = 0.7 ± 0.04 (68 per cent credible interval). Nevertheless our models provide only moderately good fits as indicated by χ2/dof = 2.9, including a 20 per cent rms uncertainty in the predicted signal amplitude. The models cannot explain a signal drop on scales around 15 arcmin, which may be caused by systematics. It is unclear whether the discrepancy can be fully explained by residual point spread function systematics of which we find evidence at least on scales of a few arcmin. Therefore we need a better understanding of higher order correlations of cosmic shear and their systematics to confidently apply them as cosmological probes.

  2. How long do natural waters “remember” release incidents of Marcellus Shale waters: a first order approximation using reactive transport modeling

    DOE PAGES

    Cai, Zhang; Li, Li

    2016-12-13

    Natural gas production from the Marcellus Shale formation has significantly changed energy landscape in recent years. Accidental release, including spills, leakage, and seepage of the Marcellus Shale flow back and produced waters can impose risks on natural water resources. With many competing processes during the reactive transport of chemical species, it is not clear what processes are dominant and govern the impacts of accidental release of Marcellus Shale waters (MSW) into natural waters. Here we carry out numerical experiments to explore this largely unexploited aspect using cations from MSW as tracers with a focus on abiotic interactions between cations releasedmore » from MSW and natural water systems. Reactive transport models were set up using characteristics of natural water systems (aquifers and rivers) in Bradford County, Pennsylvania. Results show that in clay-rich sandstone aquifers, ion exchange plays a key role in determining the maximum concentration and the time scale of released cations in receiving natural waters. In contrast, mineral dissolution and precipitation play a relatively minor role. The relative time scales of recovery τ rr, a dimensionless number defined as the ratio of the time needed to return to background concentrations over the residence time of natural waters, vary between 5 and 10 for Na, Ca, and Mg, and between 10 and 20 for Sr and Ba. In rivers and sand and gravel aquifers with negligible clay, τrr values are close to 1 because cations are flushed out at approximately one residence time. These values can be used as first order estimates of time scales of released MSW in natural water systems. This work emphasizes the importance of clay content and suggests that it is more likely to detect contamination in clay-rich geological formations. As a result, this work highlights the use of reactive transport modeling in understanding natural attenuation, guiding monitoring, and predicting impacts of contamination for risk

  3. How long do natural waters “remember” release incidents of Marcellus Shale waters: a first order approximation using reactive transport modeling

    SciTech Connect

    Cai, Zhang; Li, Li

    Natural gas production from the Marcellus Shale formation has significantly changed energy landscape in recent years. Accidental release, including spills, leakage, and seepage of the Marcellus Shale flow back and produced waters can impose risks on natural water resources. With many competing processes during the reactive transport of chemical species, it is not clear what processes are dominant and govern the impacts of accidental release of Marcellus Shale waters (MSW) into natural waters. Here we carry out numerical experiments to explore this largely unexploited aspect using cations from MSW as tracers with a focus on abiotic interactions between cations releasedmore » from MSW and natural water systems. Reactive transport models were set up using characteristics of natural water systems (aquifers and rivers) in Bradford County, Pennsylvania. Results show that in clay-rich sandstone aquifers, ion exchange plays a key role in determining the maximum concentration and the time scale of released cations in receiving natural waters. In contrast, mineral dissolution and precipitation play a relatively minor role. The relative time scales of recovery τ rr, a dimensionless number defined as the ratio of the time needed to return to background concentrations over the residence time of natural waters, vary between 5 and 10 for Na, Ca, and Mg, and between 10 and 20 for Sr and Ba. In rivers and sand and gravel aquifers with negligible clay, τrr values are close to 1 because cations are flushed out at approximately one residence time. These values can be used as first order estimates of time scales of released MSW in natural water systems. This work emphasizes the importance of clay content and suggests that it is more likely to detect contamination in clay-rich geological formations. As a result, this work highlights the use of reactive transport modeling in understanding natural attenuation, guiding monitoring, and predicting impacts of contamination for risk

  4. Density-functional expansion methods: evaluation of LDA, GGA, and meta-GGA functionals and different integral approximations.

    PubMed

    Giese, Timothy J; York, Darrin M

    2010-12-28

    We extend the Kohn-Sham potential energy expansion (VE) to include variations of the kinetic energy density and use the VE formulation with a 6-31G* basis to perform a "Jacob's ladder" comparison of small molecule properties using density functionals classified as being either LDA, GGA, or meta-GGA. We show that the VE reproduces standard Kohn-Sham DFT results well if all integrals are performed without further approximation, and there is no substantial improvement in using meta-GGA functionals relative to GGA functionals. The advantages of using GGA versus LDA functionals becomes apparent when modeling hydrogen bonds. We furthermore examine the effect of using integral approximations to compute the zeroth-order energy and first-order matrix elements, and the results suggest that the origin of the short-range repulsive potential within self-consistent charge density-functional tight-binding methods mainly arises from the approximations made to the first-order matrix elements.

  5. Connection between the regular approximation and the normalized elimination of the small component in relativistic quantum theory

    NASA Astrophysics Data System (ADS)

    Filatov, Michael; Cremer, Dieter

    2005-02-01

    The regular approximation to the normalized elimination of the small component (NESC) in the modified Dirac equation has been developed and presented in matrix form. The matrix form of the infinite-order regular approximation (IORA) expressions, obtained in [Filatov and Cremer, J. Chem. Phys. 118, 6741 (2003)] using the resolution of the identity, is the exact matrix representation and corresponds to the zeroth-order regular approximation to NESC (NESC-ZORA). Because IORA (=NESC-ZORA) is a variationally stable method, it was used as a suitable starting point for the development of the second-order regular approximation to NESC (NESC-SORA). As shown for hydrogenlike ions, NESC-SORA energies are closer to the exact Dirac energies than the energies from the fifth-order Douglas-Kroll approximation, which is much more computationally demanding than NESC-SORA. For the application of IORA (=NESC-ZORA) and NESC-SORA to many-electron systems, the number of the two-electron integrals that need to be evaluated (identical to the number of the two-electron integrals of a full Dirac-Hartree-Fock calculation) was drastically reduced by using the resolution of the identity technique. An approximation was derived, which requires only the two-electron integrals of a nonrelativistic calculation. The accuracy of this approach was demonstrated for heliumlike ions. The total energy based on the approximate integrals deviates from the energy calculated with the exact integrals by less than 5×10-9hartree units. NESC-ZORA and NESC-SORA can easily be implemented in any nonrelativistic quantum chemical program. Their application is comparable in cost with that of nonrelativistic methods. The methods can be run with density functional theory and any wave function method. NESC-SORA has the advantage that it does not imply a picture change.

  6. A real-time approximate optimal guidance law for flight in a plane

    NASA Technical Reports Server (NTRS)

    Feeley, Timothy S.; Speyer, Jason L.

    1990-01-01

    A real-time guidance scheme is presented for the problem of maximizing the payload into orbit subject to the equations of motion of a rocket over a nonrotating spherical earth. The flight is constrained to a path in the equatorial plane while reaching an orbital altitude at orbital injection speeds. The dynamics of the problem can be separated into primary and perturbation effects by a small parameter, epsilon, which is the ratio of the atmospheric scale height to the radius of the earth. The Hamilton-Jacobi-Bellman or dynamic programming equation is expanded in an asymptotic series where the zeroth-order term (epsilon = 0) can be obtained in closed form. The neglected perturbation terms are included in the higher-order terms of the expansion, which are determined from the solution of first-order linear partial differential equations requiring only integrations which are quadratures. The quadratures can be performed rapidly with emerging computer capability, so that real-time approximate optimization can be used to construct the launch guidance law. The application of this technique to flight in three-dimensions is made apparent from the solution presented.

  7. A three-dimensional parabolic equation model of sound propagation using higher-order operator splitting and Padé approximants.

    PubMed

    Lin, Ying-Tsong; Collis, Jon M; Duda, Timothy F

    2012-11-01

    An alternating direction implicit (ADI) three-dimensional fluid parabolic equation solution method with enhanced accuracy is presented. The method uses a square-root Helmholtz operator splitting algorithm that retains cross-multiplied operator terms that have been previously neglected. With these higher-order cross terms, the valid angular range of the parabolic equation solution is improved. The method is tested for accuracy against an image solution in an idealized wedge problem. Computational efficiency improvements resulting from the ADI discretization are also discussed.

  8. Modeling Elastic Wave Propagation from an Underground Chemical Explosion Using Higher Order Finite Difference Approximation: Theory, Validation and Application to SPE

    NASA Astrophysics Data System (ADS)

    Hirakawa, E. T.; Ezzedine, S. M.; Petersson, A.; Sjogreen, B.; Vorobiev, O.; Pitarka, A.; Antoun, T.; Walter, W. R.

    2016-12-01

    Motions from underground explosions are governed by non-linear hydrodynamic response of material. However, the numerical calculation of this non-linear constitutive behavior is computationally intensive in contrast to the elastic and acoustic linear wave propagation solvers. Here, we develop a hybrid modeling approach with one-way hydrodynamic-to-elastic coupling in three dimensions in order to propagate explosion generated ground motions from the non-linear near-source region to the far-field. Near source motions are computed using GEODYN-L, a Lagrangian hydrodynamics code for high-energy loading of earth materials. Motions on a dense grid of points sampled on two nested shells located beyond the non-linear damaged zone are saved, and then passed to SW4, an anelastic anisotropic fourth order finite difference code for seismic wave modeling. Our coupling strategy is based on the decomposition and uniqueness theorems where motions are introduced into SW4 as a boundary source and continue to propagate as elastic waves at a much lower computational cost than by using GEODYN-L to cover the entire near- and the far-field domain. The accuracy of the numerical calculations and the coupling strategy is demonstrated in cases with a purely elastic medium as well as non-linear medium. Our hybrid modeling approach is applied to SPE-4' and SPE-5 which are the most recent underground chemical explosions conducted at the Nevada National Security Site (NNSS) where the Source Physics Experiments (SPE) are performed. Our strategy by design is capable of incorporating complex non-linear effects near the source as well as volumetric and topographic material heterogeneity along the propagation path to receiver, and provides new prospects for modeling and understanding explosion generated seismic waveforms. This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. LLNL-ABS-698608.

  9. Albany/FELIX: A parallel, scalable and robust, finite element, first-order Stokes approximation ice sheet solver built for advanced analysis

    DOE PAGES

    Tezaur, I. K.; Perego, M.; Salinger, A. G.; ...

    2015-04-27

    This paper describes a new parallel, scalable and robust finite element based solver for the first-order Stokes momentum balance equations for ice flow. The solver, known as Albany/FELIX, is constructed using the component-based approach to building application codes, in which mature, modular libraries developed as a part of the Trilinos project are combined using abstract interfaces and template-based generic programming, resulting in a final code with access to dozens of algorithmic and advanced analysis capabilities. Following an overview of the relevant partial differential equations and boundary conditions, the numerical methods chosen to discretize the ice flow equations are described, alongmore » with their implementation. The results of several verification studies of the model accuracy are presented using (1) new test cases for simplified two-dimensional (2-D) versions of the governing equations derived using the method of manufactured solutions, and (2) canonical ice sheet modeling benchmarks. Model accuracy and convergence with respect to mesh resolution are then studied on problems involving a realistic Greenland ice sheet geometry discretized using hexahedral and tetrahedral meshes. Also explored as a part of this study is the effect of vertical mesh resolution on the solution accuracy and solver performance. The robustness and scalability of our solver on these problems is demonstrated. Lastly, we show that good scalability can be achieved by preconditioning the iterative linear solver using a new algebraic multilevel preconditioner, constructed based on the idea of semi-coarsening.« less

  10. On the scalability of the Albany/FELIX first-order Stokes approximation ice sheet solver for large-scale simulations of the Greenland and Antarctic ice sheets

    DOE PAGES

    Tezaur, Irina K.; Tuminaro, Raymond S.; Perego, Mauro; ...

    2015-01-01

    We examine the scalability of the recently developed Albany/FELIX finite-element based code for the first-order Stokes momentum balance equations for ice flow. We focus our analysis on the performance of two possible preconditioners for the iterative solution of the sparse linear systems that arise from the discretization of the governing equations: (1) a preconditioner based on the incomplete LU (ILU) factorization, and (2) a recently-developed algebraic multigrid (AMG) preconditioner, constructed using the idea of semi-coarsening. A strong scalability study on a realistic, high resolution Greenland ice sheet problem reveals that, for a given number of processor cores, the AMG preconditionermore » results in faster linear solve times but the ILU preconditioner exhibits better scalability. In addition, a weak scalability study is performed on a realistic, moderate resolution Antarctic ice sheet problem, a substantial fraction of which contains floating ice shelves, making it fundamentally different from the Greenland ice sheet problem. We show that as the problem size increases, the performance of the ILU preconditioner deteriorates whereas the AMG preconditioner maintains scalability. This is because the linear systems are extremely ill-conditioned in the presence of floating ice shelves, and the ill-conditioning has a greater negative effect on the ILU preconditioner than on the AMG preconditioner.« less

  11. Stress stiffening and approximate equations in flexible multibody dynamics

    NASA Technical Reports Server (NTRS)

    Padilla, Carlos E.; Vonflotow, Andreas H.

    1993-01-01

    A useful model for open chains of flexible bodies undergoing large rigid body motions, but small elastic deformations, is one in which the equations of motion are linearized in the small elastic deformations and deformation rates. For slow rigid body motions, the correctly linearized, or consistent, set of equations can be compared to prematurely linearized, or inconsistent, equations and to 'oversimplified,' or ruthless, equations through the use of open loop dynamic simulations. It has been shown that the inconsistent model should never be used, while the ruthless model should be used whenever possible. The consistent and inconsistent models differ by stress stiffening terms. These are due to zeroth-order stresses effecting virtual work via nonlinear strain-displacement terms. In this paper we examine in detail the nature of these stress stiffening terms and conclude that they are significant only when the associated zeroth-order stresses approach 'buckling' stresses. Finally it is emphasized that when the stress stiffening terms are negligible the ruthlessly linearized equations should be used.

  12. Applied Routh approximation

    NASA Technical Reports Server (NTRS)

    Merrill, W. C.

    1978-01-01

    The Routh approximation technique for reducing the complexity of system models was applied in the frequency domain to a 16th order, state variable model of the F100 engine and to a 43d order, transfer function model of a launch vehicle boost pump pressure regulator. The results motivate extending the frequency domain formulation of the Routh method to the time domain in order to handle the state variable formulation directly. The time domain formulation was derived and a characterization that specifies all possible Routh similarity transformations was given. The characterization was computed by solving two eigenvalue-eigenvector problems. The application of the time domain Routh technique to the state variable engine model is described, and some results are given. Additional computational problems are discussed, including an optimization procedure that can improve the approximation accuracy by taking advantage of the transformation characterization.

  13. Enskog theory for polydisperse granular mixtures. I. Navier-Stokes order transport.

    PubMed

    Garzó, Vicente; Dufty, James W; Hrenya, Christine M

    2007-09-01

    A hydrodynamic description for an s -component mixture of inelastic, smooth hard disks (two dimensions) or spheres (three dimensions) is derived based on the revised Enskog theory for the single-particle velocity distribution functions. In this first part of the two-part series, the macroscopic balance equations for mass, momentum, and energy are derived. Constitutive equations are calculated from exact expressions for the fluxes by a Chapman-Enskog expansion carried out to first order in spatial gradients, thereby resulting in a Navier-Stokes order theory. Within this context of small gradients, the theory is applicable to a wide range of restitution coefficients and densities. The resulting integral-differential equations for the zeroth- and first-order approximations of the distribution functions are given in exact form. An approximate solution to these equations is required for practical purposes in order to cast the constitutive quantities as algebraic functions of the macroscopic variables; this task is described in the companion paper.

  14. Infinite-order sudden approximation for collisions involving molecules in Pi electronic states: a new derivation and calculations of rotationally inelastic cross sections for NO(x superscript 2 Pi) + He and Ar

    SciTech Connect

    Corey, G.C.; Alexander, M.H.

    1986-11-15

    A new derivation is presented of the infinite order sudden (IOS) approximation for rotationally inelastic collisions of a diatomic molecule in a Pi electronic state with a closed shell atom. This derivation clearly demonstrates the connection between the two sudden S functions for scattering off the adiabatic potential surface of A' and A symmetry, which would arise from an ab initio calculation on an atom + Pi-state molecule system, and the S matrix elements in diabatic basis, which are required in the quantum treatment of the collision dynamics. Coupled states and IOS calculations were carried out for collisions of NImore » X 2 Pi with helium and argon, based on a electron gas potential surface at total energies of 63, 150, and 300 meV. The IOS approximation is not reliable for collisions of NO with Ar, even at the highest collision energy considered here. However, for collisions with He at 150 and 300 meV, the IOS approximation is nearly quantitative for transitions both within and between the Omega = 1/2 and Omega = 3/2 manifolds.« less

  15. Extension of the KLI approximation toward the exact optimized effective potential.

    PubMed

    Iafrate, G J; Krieger, J B

    2013-03-07

    determined by iteration with the natural zeroth order correction given by the KLI exchange-correlation potential. Explicit analytic results are provided to illustrate the first order iterative correction beyond the KLI approximation. The derived correction term to the KLI potential explicitly involves spatially weighted products of occupied orbital densities in any assumed orbital-dependent exchange-correlation energy functional; as well, the correction term is obtained with no adjustable parameters. Moreover, if the equation for the exact optimized effective potential is further iterated, one can obtain the OEP as accurately as desired.

  16. Extension of the KLI approximation toward the exact optimized effective potential

    NASA Astrophysics Data System (ADS)

    Iafrate, G. J.; Krieger, J. B.

    2013-03-01

    determined by iteration with the natural zeroth order correction given by the KLI exchange-correlation potential. Explicit analytic results are provided to illustrate the first order iterative correction beyond the KLI approximation. The derived correction term to the KLI potential explicitly involves spatially weighted products of occupied orbital densities in any assumed orbital-dependent exchange-correlation energy functional; as well, the correction term is obtained with no adjustable parameters. Moreover, if the equation for the exact optimized effective potential is further iterated, one can obtain the OEP as accurately as desired.

  17. Tunneling effects in electromagnetic wave scattering by nonspherical particles: A comparison of the Debye series and physical-geometric optics approximations

    NASA Astrophysics Data System (ADS)

    Bi, Lei; Yang, Ping

    2016-07-01

    The accuracy of the physical-geometric optics (PG-O) approximation is examined for the simulation of electromagnetic scattering by nonspherical dielectric particles. This study seeks a better understanding of the tunneling effect on the phase matrix by employing the invariant imbedding method to rigorously compute the zeroth-order Debye series, from which the tunneling efficiency and the phase matrix corresponding to the diffraction and external reflection are obtained. The tunneling efficiency is shown to be a factor quantifying the relative importance of the tunneling effect over the Fraunhofer diffraction near the forward scattering direction. Due to the tunneling effect, different geometries with the same projected cross section might have different diffraction patterns, which are traditionally assumed to be identical according to the Babinet principle. For particles with a fixed orientation, the PG-O approximation yields the external reflection pattern with reasonable accuracy, but ordinarily fails to predict the locations of peaks and minima in the diffraction pattern. The larger the tunneling efficiency, the worse the PG-O accuracy is at scattering angles less than 90°. If the particles are assumed to be randomly oriented, the PG-O approximation yields the phase matrix close to the rigorous counterpart, primarily due to error cancellations in the orientation-average process. Furthermore, the PG-O approximation based on an electric field volume-integral equation is shown to usually be much more accurate than the Kirchhoff surface integral equation at side-scattering angles, particularly when the modulus of the complex refractive index is close to unity. Finally, tunneling efficiencies are tabulated for representative faceted particles.

  18. Approximate symmetries of Hamiltonians

    NASA Astrophysics Data System (ADS)

    Chubb, Christopher T.; Flammia, Steven T.

    2017-08-01

    We explore the relationship between approximate symmetries of a gapped Hamiltonian and the structure of its ground space. We start by considering approximate symmetry operators, defined as unitary operators whose commutators with the Hamiltonian have norms that are sufficiently small. We show that when approximate symmetry operators can be restricted to the ground space while approximately preserving certain mutual commutation relations. We generalize the Stone-von Neumann theorem to matrices that approximately satisfy the canonical (Heisenberg-Weyl-type) commutation relations and use this to show that approximate symmetry operators can certify the degeneracy of the ground space even though they only approximately form a group. Importantly, the notions of "approximate" and "small" are all independent of the dimension of the ambient Hilbert space and depend only on the degeneracy in the ground space. Our analysis additionally holds for any gapped band of sufficiently small width in the excited spectrum of the Hamiltonian, and we discuss applications of these ideas to topological quantum phases of matter and topological quantum error correcting codes. Finally, in our analysis, we also provide an exponential improvement upon bounds concerning the existence of shared approximate eigenvectors of approximately commuting operators under an added normality constraint, which may be of independent interest.

  19. CONTRIBUTIONS TO RATIONAL APPROXIMATION,

    DTIC Science & Technology

    Some of the key results of linear Chebyshev approximation theory are extended to generalized rational functions. Prominent among these is Haar’s...linear theorem which yields necessary and sufficient conditions for uniqueness. Some new results in the classic field of rational function Chebyshev...Furthermore a Weierstrass type theorem is proven for rational Chebyshev approximation. A characterization theorem for rational trigonometric Chebyshev approximation in terms of sign alternation is developed. (Author)

  20. Second-order small disturbance theory for hypersonic flow over power-law bodies. Ph.D. Thesis

    NASA Technical Reports Server (NTRS)

    Townsend, J. C.

    1974-01-01

    A mathematical method for determining the flow field about power-law bodies in hypersonic flow conditions is developed. The second-order solutions, which reflect the effects of the second-order terms in the equations, are obtained by applying the method of small perturbations in terms of body slenderness parameter to the zeroth-order solutions. The method is applied by writing each flow variable as the sum of a zeroth-order and a perturbation function, each multiplied by the axial variable raised to a power. The similarity solutions are developed for infinite Mach number. All results obtained are for no flow through the body surface (as a boundary condition), but the derivation indicates that small amounts of blowing or suction through the wall can be accommodated.

  1. Approximate spatial reasoning

    NASA Technical Reports Server (NTRS)

    Dutta, Soumitra

    1988-01-01

    A model for approximate spatial reasoning using fuzzy logic to represent the uncertainty in the environment is presented. Algorithms are developed which can be used to reason about spatial information expressed in the form of approximate linguistic descriptions similar to the kind of spatial information processed by humans. Particular attention is given to static spatial reasoning.

  2. Approximate spatial reasoning

    NASA Technical Reports Server (NTRS)

    Dutta, Soumitra

    1988-01-01

    Much of human reasoning is approximate in nature. Formal models of reasoning traditionally try to be precise and reject the fuzziness of concepts in natural use and replace them with non-fuzzy scientific explicata by a process of precisiation. As an alternate to this approach, it has been suggested that rather than regard human reasoning processes as themselves approximating to some more refined and exact logical process that can be carried out with mathematical precision, the essence and power of human reasoning is in its capability to grasp and use inexact concepts directly. This view is supported by the widespread fuzziness of simple everyday terms (e.g., near tall) and the complexity of ordinary tasks (e.g., cleaning a room). Spatial reasoning is an area where humans consistently reason approximately with demonstrably good results. Consider the case of crossing a traffic intersection. We have only an approximate idea of the locations and speeds of various obstacles (e.g., persons and vehicles), but we nevertheless manage to cross such traffic intersections without any harm. The details of our mental processes which enable us to carry out such intricate tasks in such apparently simple manner are not well understood. However, it is that we try to incorporate such approximate reasoning techniques in our computer systems. Approximate spatial reasoning is very important for intelligent mobile agents (e.g., robots), specially for those operating in uncertain or unknown or dynamic domains.

  3. Approximate kernel competitive learning.

    PubMed

    Wu, Jian-Sheng; Zheng, Wei-Shi; Lai, Jian-Huang

    2015-03-01

    Kernel competitive learning has been successfully used to achieve robust clustering. However, kernel competitive learning (KCL) is not scalable for large scale data processing, because (1) it has to calculate and store the full kernel matrix that is too large to be calculated and kept in the memory and (2) it cannot be computed in parallel. In this paper we develop a framework of approximate kernel competitive learning for processing large scale dataset. The proposed framework consists of two parts. First, it derives an approximate kernel competitive learning (AKCL), which learns kernel competitive learning in a subspace via sampling. We provide solid theoretical analysis on why the proposed approximation modelling would work for kernel competitive learning, and furthermore, we show that the computational complexity of AKCL is largely reduced. Second, we propose a pseudo-parallelled approximate kernel competitive learning (PAKCL) based on a set-based kernel competitive learning strategy, which overcomes the obstacle of using parallel programming in kernel competitive learning and significantly accelerates the approximate kernel competitive learning for large scale clustering. The empirical evaluation on publicly available datasets shows that the proposed AKCL and PAKCL can perform comparably as KCL, with a large reduction on computational cost. Also, the proposed methods achieve more effective clustering performance in terms of clustering precision against related approximate clustering approaches. Copyright © 2014 Elsevier Ltd. All rights reserved.

  4. Optical diffraction by ordered 2D arrays of silica microspheres

    NASA Astrophysics Data System (ADS)

    Shcherbakov, A. A.; Shavdina, O.; Tishchenko, A. V.; Veillas, C.; Verrier, I.; Dellea, O.; Jourlin, Y.

    2017-03-01

    The article presents experimental and theoretical studies of angular dependent diffraction properties of 2D monolayer arrays of silica microspheres. High-quality large area defect-free monolayers of 1 μm diameter silica microspheres were deposited by the Langmuir-Blodgett technique under an accurate optical control. Measured angular dependencies of zeroth and one of the first order diffraction efficiencies produced by deposited samples were simulated by the rigorous Generalized Source Method taking into account particle size dispersion and lattice nonideality.

  5. Approximating Integrals Using Probability

    ERIC Educational Resources Information Center

    Maruszewski, Richard F., Jr.; Caudle, Kyle A.

    2005-01-01

    As part of a discussion on Monte Carlo methods, which outlines how to use probability expectations to approximate the value of a definite integral. The purpose of this paper is to elaborate on this technique and then to show several examples using visual basic as a programming tool. It is an interesting method because it combines two branches of…

  6. Monotone Boolean approximation

    SciTech Connect

    Hulme, B.L.

    1982-12-01

    This report presents a theory of approximation of arbitrary Boolean functions by simpler, monotone functions. Monotone increasing functions can be expressed without the use of complements. Nonconstant monotone increasing functions are important in their own right since they model a special class of systems known as coherent systems. It is shown here that when Boolean expressions for noncoherent systems become too large to treat exactly, then monotone approximations are easily defined. The algorithms proposed here not only provide simpler formulas but also produce best possible upper and lower monotone bounds for any Boolean function. This theory has practical application formore » the analysis of noncoherent fault trees and event tree sequences.« less

  7. Variationally consistent approximation scheme for charge transfer

    NASA Technical Reports Server (NTRS)

    Halpern, A. M.

    1978-01-01

    The author has developed a technique for testing various charge-transfer approximation schemes for consistency with the requirements of the Kohn variational principle for the amplitude to guarantee that the amplitude is correct to second order in the scattering wave functions. Applied to Born-type approximations for charge transfer it allows the selection of particular groups of first-, second-, and higher-Born-type terms that obey the consistency requirement, and hence yield more reliable approximation to the amplitude.

  8. Uniform analytic approximation of Wigner rotation matrices

    NASA Astrophysics Data System (ADS)

    Hoffmann, Scott E.

    2018-02-01

    We derive the leading asymptotic approximation, for low angle θ, of the Wigner rotation matrix elements, dm1m2 j(θ ) , uniform in j, m1, and m2. The result is in terms of a Bessel function of integer order. We numerically investigate the error for a variety of cases and find that the approximation can be useful over a significant range of angles. This approximation has application in the partial wave analysis of wavepacket scattering.

  9. Computational aspects of pseudospectral Laguerre approximations

    NASA Technical Reports Server (NTRS)

    Funaro, Daniele

    1989-01-01

    Pseudospectral approximations in unbounded domains by Laguerre polynomials lead to ill-conditioned algorithms. Introduced are a scaling function and appropriate numerical procedures in order to limit these unpleasant phenomena.

  10. Approximate techniques of structural reanalysis

    NASA Technical Reports Server (NTRS)

    Noor, A. K.; Lowder, H. E.

    1974-01-01

    A study is made of two approximate techniques for structural reanalysis. These include Taylor series expansions for response variables in terms of design variables and the reduced-basis method. In addition, modifications to these techniques are proposed to overcome some of their major drawbacks. The modifications include a rational approach to the selection of the reduced-basis vectors and the use of Taylor series approximation in an iterative process. For the reduced basis a normalized set of vectors is chosen which consists of the original analyzed design and the first-order sensitivity analysis vectors. The use of the Taylor series approximation as a first (initial) estimate in an iterative process, can lead to significant improvements in accuracy, even with one iteration cycle. Therefore, the range of applicability of the reanalysis technique can be extended. Numerical examples are presented which demonstrate the gain in accuracy obtained by using the proposed modification techniques, for a wide range of variations in the design variables.

  11. Pumping approximately integrable systems

    PubMed Central

    Lange, Florian; Lenarčič, Zala; Rosch, Achim

    2017-01-01

    Weak perturbations can drive an interacting many-particle system far from its initial equilibrium state if one is able to pump into degrees of freedom approximately protected by conservation laws. This concept has for example been used to realize Bose–Einstein condensates of photons, magnons and excitons. Integrable quantum systems, like the one-dimensional Heisenberg model, are characterized by an infinite set of conservation laws. Here, we develop a theory of weakly driven integrable systems and show that pumping can induce large spin or heat currents even in the presence of integrability breaking perturbations, since it activates local and quasi-local approximate conserved quantities. The resulting steady state is qualitatively captured by a truncated generalized Gibbs ensemble with Lagrange parameters that depend on the structure but not on the overall amplitude of perturbations nor the initial state. We suggest to use spin-chain materials driven by terahertz radiation to realize integrability-based spin and heat pumps. PMID:28598444

  12. Approximating lens power.

    PubMed

    Kaye, Stephen B

    2009-04-01

    To provide a scalar measure of refractive error, based on geometric lens power through principal, orthogonal and oblique meridians, that is not limited to the paraxial and sag height approximations. A function is derived to model sections through the principal meridian of a lens, followed by rotation of the section through orthogonal and oblique meridians. Average focal length is determined using the definition for the average of a function. Average univariate power in the principal meridian (including spherical aberration), can be computed from the average of a function over the angle of incidence as determined by the parameters of the given lens, or adequately computed from an integrated series function. Average power through orthogonal and oblique meridians, can be similarly determined using the derived formulae. The widely used computation for measuring refractive error, the spherical equivalent, introduces non-constant approximations, leading to a systematic bias. The equations proposed provide a good univariate representation of average lens power and are not subject to a systematic bias. They are particularly useful for the analysis of aggregate data, correlating with biological treatment variables and for developing analyses, which require a scalar equivalent representation of refractive power.

  13. Approximate strip exchanging.

    PubMed

    Roy, Swapnoneel; Thakur, Ashok Kumar

    2008-01-01

    Genome rearrangements have been modelled by a variety of primitives such as reversals, transpositions, block moves and block interchanges. We consider such a genome rearrangement primitive Strip Exchanges. Given a permutation, the challenge is to sort it by using minimum number of strip exchanges. A strip exchanging move interchanges the positions of two chosen strips so that they merge with other strips. The strip exchange problem is to sort a permutation using minimum number of strip exchanges. We present here the first non-trivial 2-approximation algorithm to this problem. We also observe that sorting by strip-exchanges is fixed-parameter-tractable. Lastly we discuss the application of strip exchanges in a different area Optical Character Recognition (OCR) with an example.

  14. Cosmological applications of Padé approximant

    NASA Astrophysics Data System (ADS)

    Wei, Hao; Yan, Xiao-Peng; Zhou, Ya-Nan

    2014-01-01

    As is well known, in mathematics, any function could be approximated by the Padé approximant. The Padé approximant is the best approximation of a function by a rational function of given order. In fact, the Padé approximant often gives better approximation of the function than truncating its Taylor series, and it may still work where the Taylor series does not converge. In the present work, we consider the Padé approximant in two issues. First, we obtain the analytical approximation of the luminosity distance for the flat XCDM model, and find that the relative error is fairly small. Second, we propose several parameterizations for the equation-of-state parameter (EoS) of dark energy based on the Padé approximant. They are well motivated from the mathematical and physical points of view. We confront these EoS parameterizations with the latest observational data, and find that they can work well. In these practices, we show that the Padé approximant could be an useful tool in cosmology, and it deserves further investigation.

  15. Countably QC-Approximating Posets

    PubMed Central

    Mao, Xuxin; Xu, Luoshan

    2014-01-01

    As a generalization of countably C-approximating posets, the concept of countably QC-approximating posets is introduced. With the countably QC-approximating property, some characterizations of generalized completely distributive lattices and generalized countably approximating posets are given. The main results are as follows: (1) a complete lattice is generalized completely distributive if and only if it is countably QC-approximating and weakly generalized countably approximating; (2) a poset L having countably directed joins is generalized countably approximating if and only if the lattice σ c(L)op of all σ-Scott-closed subsets of L is weakly generalized countably approximating. PMID:25165730

  16. Plasma Physics Approximations in Ares

    SciTech Connect

    Managan, R. A.

    Lee & More derived analytic forms for the transport properties of a plasma. Many hydro-codes use their formulae for electrical and thermal conductivity. The coefficients are complex functions of Fermi-Dirac integrals, F n( μ/θ ), the chemical potential, μ or ζ = ln(1+e μ/θ ), and the temperature, θ = kT. Since these formulae are expensive to compute, rational function approximations were fit to them. Approximations are also used to find the chemical potential, either μ or ζ . The fits use ζ as the independent variable instead of μ/θ . New fits are provided for A α (ζmore » ),A β (ζ ), ζ, f(ζ ) = (1 + e -μ/θ)F 1/2(μ/θ), F 1/2'/F 1/2, F c α, and F c β. In each case the relative error of the fit is minimized since the functions can vary by many orders of magnitude. The new fits are designed to exactly preserve the limiting values in the non-degenerate and highly degenerate limits or as ζ→ 0 or ∞. The original fits due to Lee & More and George Zimmerman are presented for comparison.« less

  17. Cloze, Discourse, and Approximations to English.

    ERIC Educational Resources Information Center

    Oller, John W., Jr.

    Five orders of approximation to normal English prose were constructed; 5th, 10th, 25th, 50th, and 100th plus. Five cloze tests were then constructed by inserting blanks for deleted words in 5 word segments (5th order), 10 word segments (10th), 25 word segments (25th), 50 word segments (50th), and 100 word segments of five different passages of…

  18. Decision analysis with approximate probabilities

    NASA Technical Reports Server (NTRS)

    Whalen, Thomas

    1992-01-01

    This paper concerns decisions under uncertainty in which the probabilities of the states of nature are only approximately known. Decision problems involving three states of nature are studied. This is due to the fact that some key issues do not arise in two-state problems, while probability spaces with more than three states of nature are essentially impossible to graph. The primary focus is on two levels of probabilistic information. In one level, the three probabilities are separately rounded to the nearest tenth. This can lead to sets of rounded probabilities which add up to 0.9, 1.0, or 1.1. In the other level, probabilities are rounded to the nearest tenth in such a way that the rounded probabilities are forced to sum to 1.0. For comparison, six additional levels of probabilistic information, previously analyzed, were also included in the present analysis. A simulation experiment compared four criteria for decisionmaking using linearly constrained probabilities (Maximin, Midpoint, Standard Laplace, and Extended Laplace) under the eight different levels of information about probability. The Extended Laplace criterion, which uses a second order maximum entropy principle, performed best overall.

  19. Approximating genomic reliabilities for national genomic evaluation

    USDA-ARS?s Scientific Manuscript database

    With the introduction of standard methods for approximating effective daughter/data contribution by Interbull in 2001, conventional EDC or reliabilities contributed by daughter phenotypes are directly comparable across countries and used in routine conventional evaluations. In order to make publishe...

  20. Combining global and local approximations

    NASA Technical Reports Server (NTRS)

    Haftka, Raphael T.

    1991-01-01

    A method based on a linear approximation to a scaling factor, designated the 'global-local approximation' (GLA) method, is presented and shown capable of extending the range of usefulness of derivative-based approximations to a more refined model. The GLA approach refines the conventional scaling factor by means of a linearly varying, rather than constant, scaling factor. The capabilities of the method are demonstrated for a simple beam example with a crude and more refined FEM model.

  1. Limitations of shallow nets approximation.

    PubMed

    Lin, Shao-Bo

    2017-10-01

    In this paper, we aim at analyzing the approximation abilities of shallow networks in reproducing kernel Hilbert spaces (RKHSs). We prove that there is a probability measure such that the achievable lower bound for approximating by shallow nets can be realized for all functions in balls of reproducing kernel Hilbert space with high probability, which is different with the classical minimax approximation error estimates. This result together with the existing approximation results for deep nets shows the limitations for shallow nets and provides a theoretical explanation on why deep nets perform better than shallow nets. Copyright © 2017 Elsevier Ltd. All rights reserved.

  2. Structural optimization with approximate sensitivities

    NASA Technical Reports Server (NTRS)

    Patnaik, S. N.; Hopkins, D. A.; Coroneos, R.

    1994-01-01

    Computational efficiency in structural optimization can be enhanced if the intensive computations associated with the calculation of the sensitivities, that is, gradients of the behavior constraints, are reduced. Approximation to gradients of the behavior constraints that can be generated with small amount of numerical calculations is proposed. Structural optimization with these approximate sensitivities produced correct optimum solution. Approximate gradients performed well for different nonlinear programming methods, such as the sequence of unconstrained minimization technique, method of feasible directions, sequence of quadratic programming, and sequence of linear programming. Structural optimization with approximate gradients can reduce by one third the CPU time that would otherwise be required to solve the problem with explicit closed-form gradients. The proposed gradient approximation shows potential to reduce intensive computation that has been associated with traditional structural optimization.

  3. Approximate circuits for increased reliability

    SciTech Connect

    Hamlet, Jason R.; Mayo, Jackson R.

    2015-08-18

    Embodiments of the invention describe a Boolean circuit having a voter circuit and a plurality of approximate circuits each based, at least in part, on a reference circuit. The approximate circuits are each to generate one or more output signals based on values of received input signals. The voter circuit is to receive the one or more output signals generated by each of the approximate circuits, and is to output one or more signals corresponding to a majority value of the received signals. At least some of the approximate circuits are to generate an output value different than the referencemore » circuit for one or more input signal values; however, for each possible input signal value, the majority values of the one or more output signals generated by the approximate circuits and received by the voter circuit correspond to output signal result values of the reference circuit.« less

  4. Approximate Dynamic Programming: Combining Regional and Local State Following Approximations.

    PubMed

    Deptula, Patryk; Rosenfeld, Joel A; Kamalapurkar, Rushikesh; Dixon, Warren E

    2018-06-01

    An infinite-horizon optimal regulation problem for a control-affine deterministic system is solved online using a local state following (StaF) kernel and a regional model-based reinforcement learning (R-MBRL) method to approximate the value function. Unlike traditional methods such as R-MBRL that aim to approximate the value function over a large compact set, the StaF kernel approach aims to approximate the value function in a local neighborhood of the state that travels within a compact set. In this paper, the value function is approximated using a state-dependent convex combination of the StaF-based and the R-MBRL-based approximations. As the state enters a neighborhood containing the origin, the value function transitions from being approximated by the StaF approach to the R-MBRL approach. Semiglobal uniformly ultimately bounded (SGUUB) convergence of the system states to the origin is established using a Lyapunov-based analysis. Simulation results are provided for two, three, six, and ten-state dynamical systems to demonstrate the scalability and performance of the developed method.

  5. Cosmological collapse and the improved Zel'dovich approximation.

    NASA Astrophysics Data System (ADS)

    Salopek, D. S.; Stewart, J. M.; Croudace, K. M.; Parry, J.

    Using a general relativistic formulation, the authors show how to compute the higher order terms in the Zel'dovich approximation which describes cosmological collapse. They evolve the 3-metric in a spatial gradient expansion. Their method is an advance over earlier work because it is local at each order. Using the improved Zel'dovich approximation, they compute the epoch of collapse.

  6. Pawlak Algebra and Approximate Structure on Fuzzy Lattice

    PubMed Central

    Zhuang, Ying; Liu, Wenqi; Wu, Chin-Chia; Li, Jinhai

    2014-01-01

    The aim of this paper is to investigate the general approximation structure, weak approximation operators, and Pawlak algebra in the framework of fuzzy lattice, lattice topology, and auxiliary ordering. First, we prove that the weak approximation operator space forms a complete distributive lattice. Then we study the properties of transitive closure of approximation operators and apply them to rough set theory. We also investigate molecule Pawlak algebra and obtain some related properties. PMID:25152922

  7. Pawlak algebra and approximate structure on fuzzy lattice.

    PubMed

    Zhuang, Ying; Liu, Wenqi; Wu, Chin-Chia; Li, Jinhai

    2014-01-01

    The aim of this paper is to investigate the general approximation structure, weak approximation operators, and Pawlak algebra in the framework of fuzzy lattice, lattice topology, and auxiliary ordering. First, we prove that the weak approximation operator space forms a complete distributive lattice. Then we study the properties of transitive closure of approximation operators and apply them to rough set theory. We also investigate molecule Pawlak algebra and obtain some related properties.

  8. Approximate Genealogies Under Genetic Hitchhiking

    PubMed Central

    Pfaffelhuber, P.; Haubold, B.; Wakolbinger, A.

    2006-01-01

    The rapid fixation of an advantageous allele leads to a reduction in linked neutral variation around the target of selection. The genealogy at a neutral locus in such a selective sweep can be simulated by first generating a random path of the advantageous allele's frequency and then a structured coalescent in this background. Usually the frequency path is approximated by a logistic growth curve. We discuss an alternative method that approximates the genealogy by a random binary splitting tree, a so-called Yule tree that does not require first constructing a frequency path. Compared to the coalescent in a logistic background, this method gives a slightly better approximation for identity by descent during the selective phase and a much better approximation for the number of lineages that stem from the founder of the selective sweep. In applications such as the approximation of the distribution of Tajima's D, the two approximation methods perform equally well. For relevant parameter ranges, the Yule approximation is faster. PMID:17182733

  9. Exponential approximations in optimal design

    NASA Technical Reports Server (NTRS)

    Belegundu, A. D.; Rajan, S. D.; Rajgopal, J.

    1990-01-01

    One-point and two-point exponential functions have been developed and proved to be very effective approximations of structural response. The exponential has been compared to the linear, reciprocal and quadratic fit methods. Four test problems in structural analysis have been selected. The use of such approximations is attractive in structural optimization to reduce the numbers of exact analyses which involve computationally expensive finite element analysis.

  10. Topics in Multivariate Approximation Theory.

    DTIC Science & Technology

    1982-05-01

    once that a continuous function f can be approximated from Sa :o span (N3 )B63 to within *(f, 131 ), with 13 t- sup3 e3 dian PS The simple approximation...N(C) 3- U P s P3AC 0 0 ) . Then, as in Lebesgue’s inequality, we could conclude that f - Qf - f-p - Q(f-p) , for all p e k k therefore I(f-0f) JCI 4 I

  11. Approximate Model of Zone Sedimentation

    NASA Astrophysics Data System (ADS)

    Dzianik, František

    2011-12-01

    The process of zone sedimentation is affected by many factors that are not possible to express analytically. For this reason, the zone settling is evaluated in practice experimentally or by application of an empirical mathematical description of the process. The paper presents the development of approximate model of zone settling, i.e. the general function which should properly approximate the behaviour of the settling process within its entire range and at the various conditions. Furthermore, the specification of the model parameters by the regression analysis of settling test results is shown. The suitability of the model is reviewed by graphical dependencies and by statistical coefficients of correlation. The approximate model could by also useful on the simplification of process design of continual settling tanks and thickeners.

  12. Approximations to camera sensor noise

    NASA Astrophysics Data System (ADS)

    Jin, Xiaodan; Hirakawa, Keigo

    2013-02-01

    Noise is present in all image sensor data. Poisson distribution is said to model the stochastic nature of the photon arrival process, while it is common to approximate readout/thermal noise by additive white Gaussian noise (AWGN). Other sources of signal-dependent noise such as Fano and quantization also contribute to the overall noise profile. Question remains, however, about how best to model the combined sensor noise. Though additive Gaussian noise with signal-dependent noise variance (SD-AWGN) and Poisson corruption are two widely used models to approximate the actual sensor noise distribution, the justification given to these types of models are based on limited evidence. The goal of this paper is to provide a more comprehensive characterization of random noise. We concluded by presenting concrete evidence that Poisson model is a better approximation to real camera model than SD-AWGN. We suggest further modification to Poisson that may improve the noise model.

  13. Dynamical cluster approximation plus semiclassical approximation study for a Mott insulator and d-wave pairing

    NASA Astrophysics Data System (ADS)

    Kim, SungKun; Lee, Hunpyo

    2017-06-01

    Via a dynamical cluster approximation with N c = 4 in combination with a semiclassical approximation (DCA+SCA), we study the doped two-dimensional Hubbard model. We obtain a plaquette antiferromagnetic (AF) Mott insulator, a plaquette AF ordered metal, a pseudogap (or d-wave superconductor) and a paramagnetic metal by tuning the doping concentration. These features are similar to the behaviors observed in copper-oxide superconductors and are in qualitative agreement with the results calculated by the cluster dynamical mean field theory with the continuous-time quantum Monte Carlo (CDMFT+CTQMC) approach. The results of our DCA+SCA differ from those of the CDMFT+CTQMC approach in that the d-wave superconducting order parameters are shown even in the high doped region, unlike the results of the CDMFT+CTQMC approach. We think that the strong plaquette AF orderings in the dynamical cluster approximation (DCA) with N c = 4 suppress superconducting states with increasing doping up to strongly doped region, because frozen dynamical fluctuations in a semiclassical approximation (SCA) approach are unable to destroy those orderings. Our calculation with short-range spatial fluctuations is initial research, because the SCA can manage long-range spatial fluctuations in feasible computational times beyond the CDMFT+CTQMC tool. We believe that our future DCA+SCA calculations should supply information on the fully momentum-resolved physical properties, which could be compared with the results measured by angle-resolved photoemission spectroscopy experiments.

  14. Singularly Perturbed Lie Bracket Approximation

    DOE PAGES

    Durr, Hans-Bernd; Krstic, Miroslav; Scheinker, Alexander; ...

    2015-03-27

    Here, we consider the interconnection of two dynamical systems where one has an input-affine vector field. We show that by employing a singular perturbation analysis and the Lie bracket approximation technique, the stability of the overall system can be analyzed by regarding the stability properties of two reduced, uncoupled systems.

  15. Pythagorean Approximations and Continued Fractions

    ERIC Educational Resources Information Center

    Peralta, Javier

    2008-01-01

    In this article, we will show that the Pythagorean approximations of [the square root of] 2 coincide with those achieved in the 16th century by means of continued fractions. Assuming this fact and the known relation that connects the Fibonacci sequence with the golden section, we shall establish a procedure to obtain sequences of rational numbers…

  16. Lognormal Approximations of Fault Tree Uncertainty Distributions.

    PubMed

    El-Shanawany, Ashraf Ben; Ardron, Keith H; Walker, Simon P

    2018-01-26

    Fault trees are used in reliability modeling to create logical models of fault combinations that can lead to undesirable events. The output of a fault tree analysis (the top event probability) is expressed in terms of the failure probabilities of basic events that are input to the model. Typically, the basic event probabilities are not known exactly, but are modeled as probability distributions: therefore, the top event probability is also represented as an uncertainty distribution. Monte Carlo methods are generally used for evaluating the uncertainty distribution, but such calculations are computationally intensive and do not readily reveal the dominant contributors to the uncertainty. In this article, a closed-form approximation for the fault tree top event uncertainty distribution is developed, which is applicable when the uncertainties in the basic events of the model are lognormally distributed. The results of the approximate method are compared with results from two sampling-based methods: namely, the Monte Carlo method and the Wilks method based on order statistics. It is shown that the closed-form expression can provide a reasonable approximation to results obtained by Monte Carlo sampling, without incurring the computational expense. The Wilks method is found to be a useful means of providing an upper bound for the percentiles of the uncertainty distribution while being computationally inexpensive compared with full Monte Carlo sampling. The lognormal approximation method and Wilks's method appear attractive, practical alternatives for the evaluation of uncertainty in the output of fault trees and similar multilinear models. © 2018 Society for Risk Analysis.

  17. CMB-lensing beyond the Born approximation

    SciTech Connect

    Marozzi, Giovanni; Fanizza, Giuseppe; Durrer, Ruth

    2016-09-01

    We investigate the weak lensing corrections to the cosmic microwave background temperature anisotropies considering effects beyond the Born approximation. To this aim, we use the small deflection angle approximation, to connect the lensed and unlensed power spectra, via expressions for the deflection angles up to third order in the gravitational potential. While the small deflection angle approximation has the drawback to be reliable only for multipoles ℓ ∼< 2500, it allows us to consistently take into account the non-Gaussian nature of cosmological perturbation theory beyond the linear level. The contribution to the lensed temperature power spectrum coming from the non-Gaussianmore » nature of the deflection angle at higher order is a new effect which has not been taken into account in the literature so far. It turns out to be the leading contribution among the post-Born lensing corrections. On the other hand, the effect is smaller than corrections coming from non-linearities in the matter power spectrum, and its imprint on CMB lensing is too small to be seen in present experiments.« less

  18. Chemical Laws, Idealization and Approximation

    NASA Astrophysics Data System (ADS)

    Tobin, Emma

    2013-07-01

    This paper examines the notion of laws in chemistry. Vihalemm ( Found Chem 5(1):7-22, 2003) argues that the laws of chemistry are fundamentally the same as the laws of physics they are all ceteris paribus laws which are true "in ideal conditions". In contrast, Scerri (2000) contends that the laws of chemistry are fundamentally different to the laws of physics, because they involve approximations. Christie ( Stud Hist Philos Sci 25:613-629, 1994) and Christie and Christie ( Of minds and molecules. Oxford University Press, New York, pp. 34-50, 2000) agree that the laws of chemistry are operationally different to the laws of physics, but claim that the distinction between exact and approximate laws is too simplistic to taxonomise them. Approximations in chemistry involve diverse kinds of activity and often what counts as a scientific law in chemistry is dictated by the context of its use in scientific practice. This paper addresses the question of what makes chemical laws distinctive independently of the separate question as to how they are related to the laws of physics. From an analysis of some candidate ceteris paribus laws in chemistry, this paper argues that there are two distinct kinds of ceteris paribus laws in chemistry; idealized and approximate chemical laws. Thus, while Christie ( Stud Hist Philos Sci 25:613-629, 1994) and Christie and Christie ( Of minds and molecules. Oxford University Press, New York, pp. 34--50, 2000) are correct to point out that the candidate generalisations in chemistry are diverse and heterogeneous, a distinction between idealizations and approximations can nevertheless be used to successfully taxonomise them.

  19. A Varifold Approach to Surface Approximation

    NASA Astrophysics Data System (ADS)

    Buet, Blanche; Leonardi, Gian Paolo; Masnou, Simon

    2017-11-01

    We show that the theory of varifolds can be suitably enriched to open the way to applications in the field of discrete and computational geometry. Using appropriate regularizations of the mass and of the first variation of a varifold we introduce the notion of approximate mean curvature and show various convergence results that hold, in particular, for sequences of discrete varifolds associated with point clouds or pixel/voxel-type discretizations of d-surfaces in the Euclidean n-space, without restrictions on dimension and codimension. The variational nature of the approach also allows us to consider surfaces with singularities, and in that case the approximate mean curvature is consistent with the generalized mean curvature of the limit surface. A series of numerical tests are provided in order to illustrate the effectiveness and generality of the method.

  20. Second derivatives for approximate spin projection methods

    SciTech Connect

    Thompson, Lee M.; Hratchian, Hrant P., E-mail: hhratchian@ucmerced.edu

    2015-02-07

    The use of broken-symmetry electronic structure methods is required in order to obtain correct behavior of electronically strained open-shell systems, such as transition states, biradicals, and transition metals. This approach often has issues with spin contamination, which can lead to significant errors in predicted energies, geometries, and properties. Approximate projection schemes are able to correct for spin contamination and can often yield improved results. To fully make use of these methods and to carry out exploration of the potential energy surface, it is desirable to develop an efficient second energy derivative theory. In this paper, we formulate the analytical secondmore » derivatives for the Yamaguchi approximate projection scheme, building on recent work that has yielded an efficient implementation of the analytical first derivatives.« less

  1. Random-Phase Approximation Methods

    NASA Astrophysics Data System (ADS)

    Chen, Guo P.; Voora, Vamsee K.; Agee, Matthew M.; Balasubramani, Sree Ganesh; Furche, Filipp

    2017-05-01

    Random-phase approximation (RPA) methods are rapidly emerging as cost-effective validation tools for semilocal density functional computations. We present the theoretical background of RPA in an intuitive rather than formal fashion, focusing on the physical picture of screening and simple diagrammatic analysis. A new decomposition of the RPA correlation energy into plasmonic modes leads to an appealing visualization of electron correlation in terms of charge density fluctuations. Recent developments in the areas of beyond-RPA methods, RPA correlation potentials, and efficient algorithms for RPA energy and property calculations are reviewed. The ability of RPA to approximately capture static correlation in molecules is quantified by an analysis of RPA natural occupation numbers. We illustrate the use of RPA methods in applications to small-gap systems such as open-shell d- and f-element compounds, radicals, and weakly bound complexes, where semilocal density functional results exhibit strong functional dependence.

  2. Reliable Function Approximation and Estimation

    DTIC Science & Technology

    2016-08-16

    AUSTIN , TX 78712 08/16/2016 Final Report DISTRIBUTION A: Distribution approved for public release. Air Force Research Laboratory AF Office Of Scientific...UNIVERSITY OF TEXAS AT AUSTIN 101 EAST 27TH STREET STE 4308 AUSTIN , TX 78712 DISTRIBUTION A: Distribution approved for public release. INSTRUCTIONS...AFRL-AFOSR-VA-TR-2016-0293 Reliable Function Approximation and Estimation Rachel Ward UNIVERSITY OF TEXAS AT AUSTIN 101 EAST 27TH STREET STE 4308

  3. Approximate reasoning using terminological models

    NASA Technical Reports Server (NTRS)

    Yen, John; Vaidya, Nitin

    1992-01-01

    Term Subsumption Systems (TSS) form a knowledge-representation scheme in AI that can express the defining characteristics of concepts through a formal language that has a well-defined semantics and incorporates a reasoning mechanism that can deduce whether one concept subsumes another. However, TSS's have very limited ability to deal with the issue of uncertainty in knowledge bases. The objective of this research is to address issues in combining approximate reasoning with term subsumption systems. To do this, we have extended an existing AI architecture (CLASP) that is built on the top of a term subsumption system (LOOM). First, the assertional component of LOOM has been extended for asserting and representing uncertain propositions. Second, we have extended the pattern matcher of CLASP for plausible rule-based inferences. Third, an approximate reasoning model has been added to facilitate various kinds of approximate reasoning. And finally, the issue of inconsistency in truth values due to inheritance is addressed using justification of those values. This architecture enhances the reasoning capabilities of expert systems by providing support for reasoning under uncertainty using knowledge captured in TSS. Also, as definitional knowledge is explicit and separate from heuristic knowledge for plausible inferences, the maintainability of expert systems could be improved.

  4. zeldovich-PLT: Zel'dovich approximation initial conditions generator

    NASA Astrophysics Data System (ADS)

    Eisenstein, Daniel; Garrison, Lehman

    2016-05-01

    zeldovich-PLT generates Zel'dovich approximation (ZA) initial conditions (i.e. first-order Lagrangian perturbation theory) for cosmological N-body simulations, optionally applying particle linear theory (PLT) corrections.

  5. Order Up

    ERIC Educational Resources Information Center

    Gibeault, Michael

    2005-01-01

    Change orders. The words can turn the stomachs of administrators. Horror stories about change orders create fear and distrust among school officials, designers and builders. Can change orders be avoided? If car manufacturers can produce millions of intricately designed vehicles, why can't the same quality control be achieved on a construction…

  6. Series approximation to probability densities

    NASA Astrophysics Data System (ADS)

    Cohen, L.

    2018-04-01

    One of the historical and fundamental uses of the Edgeworth and Gram-Charlier series is to "correct" a Gaussian density when it is determined that the probability density under consideration has moments that do not correspond to the Gaussian [5, 6]. There is a fundamental difficulty with these methods in that if the series are truncated, then the resulting approximate density is not manifestly positive. The aim of this paper is to attempt to expand a probability density so that if it is truncated it will still be manifestly positive.

  7. Approximate solutions to Mathieu's equation

    NASA Astrophysics Data System (ADS)

    Wilkinson, Samuel A.; Vogt, Nicolas; Golubev, Dmitry S.; Cole, Jared H.

    2018-06-01

    Mathieu's equation has many applications throughout theoretical physics. It is especially important to the theory of Josephson junctions, where it is equivalent to Schrödinger's equation. Mathieu's equation can be easily solved numerically, however there exists no closed-form analytic solution. Here we collect various approximations which appear throughout the physics and mathematics literature and examine their accuracy and regimes of applicability. Particular attention is paid to quantities relevant to the physics of Josephson junctions, but the arguments and notation are kept general so as to be of use to the broader physics community.

  8. The unitary convolution approximation for heavy ions

    NASA Astrophysics Data System (ADS)

    Grande, P. L.; Schiwietz, G.

    2002-10-01

    The convolution approximation for the impact-parameter dependent energy loss is reviewed with emphasis on the determination of the stopping force for heavy projectiles. In this method, the energy loss in different impact-parameter regions is well determined and interpolated smoothly. The physical inputs of the model are the projectile-screening function (in the case of dressed ions), the electron density and oscillators strengths of the target atoms. Moreover, the convolution approximation, in the perturbative mode (called PCA), yields remarkable agreement with full semi-classical-approximation (SCA) results for bare as well as for screened ions at all impact parameters. In the unitary mode (called UCA), the method contains some higher-order effects (yielding in some cases rather good agreement with full coupled-channel calculations) and approaches the classical regime similar as the Bohr model for large perturbations ( Z/ v≫1). The results are then used to compare with experimental values of the non-equilibrium stopping force as a function of the projectile charge as well as with the equilibrium energy loss under non-aligned and channeling conditions.

  9. Approximation abilities of neuro-fuzzy networks

    NASA Astrophysics Data System (ADS)

    Mrówczyńska, Maria

    2010-01-01

    The paper presents the operation of two neuro-fuzzy systems of an adaptive type, intended for solving problems of the approximation of multi-variable functions in the domain of real numbers. Neuro-fuzzy systems being a combination of the methodology of artificial neural networks and fuzzy sets operate on the basis of a set of fuzzy rules "if-then", generated by means of the self-organization of data grouping and the estimation of relations between fuzzy experiment results. The article includes a description of neuro-fuzzy systems by Takaga-Sugeno-Kang (TSK) and Wang-Mendel (WM), and in order to complement the problem in question, a hierarchical structural self-organizing method of teaching a fuzzy network. A multi-layer structure of the systems is a structure analogous to the structure of "classic" neural networks. In its final part the article presents selected areas of application of neuro-fuzzy systems in the field of geodesy and surveying engineering. Numerical examples showing how the systems work concerned: the approximation of functions of several variables to be used as algorithms in the Geographic Information Systems (the approximation of a terrain model), the transformation of coordinates, and the prediction of a time series. The accuracy characteristics of the results obtained have been taken into consideration.

  10. Monotonically improving approximate answers to relational algebra queries

    NASA Technical Reports Server (NTRS)

    Smith, Kenneth P.; Liu, J. W. S.

    1989-01-01

    We present here a query processing method that produces approximate answers to queries posed in standard relational algebra. This method is monotone in the sense that the accuracy of the approximate result improves with the amount of time spent producing the result. This strategy enables us to trade the time to produce the result for the accuracy of the result. An approximate relational model that characterizes appromimate relations and a partial order for comparing them is developed. Relational operators which operate on and return approximate relations are defined.

  11. An approximation theory for the identification of linear thermoelastic systems

    NASA Technical Reports Server (NTRS)

    Rosen, I. G.; Su, Chien-Hua Frank

    1990-01-01

    An abstract approximation framework and convergence theory for the identification of thermoelastic systems is developed. Starting from an abstract operator formulation consisting of a coupled second order hyperbolic equation of elasticity and first order parabolic equation for heat conduction, well-posedness is established using linear semigroup theory in Hilbert space, and a class of parameter estimation problems is then defined involving mild solutions. The approximation framework is based upon generic Galerkin approximation of the mild solutions, and convergence of solutions of the resulting sequence of approximating finite dimensional parameter identification problems to a solution of the original infinite dimensional inverse problem is established using approximation results for operator semigroups. An example involving the basic equations of one dimensional linear thermoelasticity and a linear spline based scheme are discussed. Numerical results indicate how the approach might be used in a study of damping mechanisms in flexible structures.

  12. Linear approximations of nonlinear systems

    NASA Technical Reports Server (NTRS)

    Hunt, L. R.; Su, R.

    1983-01-01

    The development of a method for designing an automatic flight controller for short and vertical take off aircraft is discussed. This technique involves transformations of nonlinear systems to controllable linear systems and takes into account the nonlinearities of the aircraft. In general, the transformations cannot always be given in closed form. Using partial differential equations, an approximate linear system called the modified tangent model was introduced. A linear transformation of this tangent model to Brunovsky canonical form can be constructed, and from this the linear part (about a state space point x sub 0) of an exact transformation for the nonlinear system can be found. It is shown that a canonical expansion in Lie brackets about the point x sub 0 yields the same modified tangent model.

  13. Strong shock implosion, approximate solution

    NASA Astrophysics Data System (ADS)

    Fujimoto, Y.; Mishkin, E. A.; Alejaldre, C.

    1983-01-01

    The self-similar, center-bound motion of a strong spherical, or cylindrical, shock wave moving through an ideal gas with a constant, γ= cp/ cv, is considered and a linearized, approximate solution is derived. An X, Y phase plane of the self-similar solution is defined and the representative curved of the system behind the shock front is replaced by a straight line connecting the mappings of the shock front with that of its tail. The reduced pressure P(ξ), density R(ξ) and velocity U1(ξ) are found in closed, quite accurate, form. Comparison with numerically obtained results, for γ= {5}/{3} and γ= {7}/{5}, is shown.

  14. Analytical approximations for spiral waves

    SciTech Connect

    Löber, Jakob, E-mail: jakob@physik.tu-berlin.de; Engel, Harald

    2013-12-15

    We propose a non-perturbative attempt to solve the kinematic equations for spiral waves in excitable media. From the eikonal equation for the wave front we derive an implicit analytical relation between rotation frequency Ω and core radius R{sub 0}. For free, rigidly rotating spiral waves our analytical prediction is in good agreement with numerical solutions of the linear eikonal equation not only for very large but also for intermediate and small values of the core radius. An equivalent Ω(R{sub +}) dependence improves the result by Keener and Tyson for spiral waves pinned to a circular defect of radius R{sub +}more » with Neumann boundaries at the periphery. Simultaneously, analytical approximations for the shape of free and pinned spirals are given. We discuss the reasons why the ansatz fails to correctly describe the dependence of the rotation frequency on the excitability of the medium.« less

  15. Wavelets and distributed approximating functionals

    NASA Astrophysics Data System (ADS)

    Wei, G. W.; Kouri, D. J.; Hoffman, D. K.

    1998-07-01

    A general procedure is proposed for constructing father and mother wavelets that have excellent time-frequency localization and can be used to generate entire wavelet families for use as wavelet transforms. One interesting feature of our father wavelets (scaling functions) is that they belong to a class of generalized delta sequences, which we refer to as distributed approximating functionals (DAFs). We indicate this by the notation wavelet-DAFs. Correspondingly, the mother wavelets generated from these wavelet-DAFs are appropriately called DAF-wavelets. Wavelet-DAFs can be regarded as providing a pointwise (localized) spectral method, which furnishes a bridge between the traditional global methods and local methods for solving partial differential equations. They are shown to provide extremely accurate numerical solutions for a number of nonlinear partial differential equations, including the Korteweg-de Vries (KdV) equation, for which a previous method has encountered difficulties (J. Comput. Phys. 132 (1997) 233).

  16. Order-parameter model for unstable multilane traffic flow

    NASA Astrophysics Data System (ADS)

    Lubashevsky, Ihor A.; Mahnke, Reinhard

    2000-11-01

    We discuss a phenomenological approach to the description of unstable vehicle motion on multilane highways that explains in a simple way the observed sequence of the ``free flow <--> synchronized mode <--> jam'' phase transitions as well as the hysteresis in these transitions. We introduce a variable called an order parameter that accounts for possible correlations in the vehicle motion at different lanes. So, it is principally due to the ``many-body'' effects in the car interaction in contrast to such variables as the mean car density and velocity being actually the zeroth and first moments of the ``one-particle'' distribution function. Therefore, we regard the order parameter as an additional independent state variable of traffic flow. We assume that these correlations are due to a small group of ``fast'' drivers and by taking into account the general properties of the driver behavior we formulate a governing equation for the order parameter. In this context we analyze the instability of homogeneous traffic flow that manifested itself in the above-mentioned phase transitions and gave rise to the hysteresis in both of them. Besides, the jam is characterized by the vehicle flows at different lanes which are independent of one another. We specify a certain simplified model in order to study the general features of the car cluster self-formation under the ``free flow <--> synchronized motion'' phase transition. In particular, we show that the main local parameters of the developed cluster are determined by the state characteristics of vehicle motion only.

  17. Combination of the pair density approximation and the Takahashi–Imada approximation for path integral Monte Carlo simulations

    SciTech Connect

    Zillich, Robert E., E-mail: robert.zillich@jku.at

    2015-11-15

    We construct an accurate imaginary time propagator for path integral Monte Carlo simulations for heterogeneous systems consisting of a mixture of atoms and molecules. We combine the pair density approximation, which is highly accurate but feasible only for the isotropic interactions between atoms, with the Takahashi–Imada approximation for general interactions. We present finite temperature simulations results for energy and structure of molecules–helium clusters X{sup 4}He{sub 20} (X=HCCH and LiH) which show a marked improvement over the Trotter approximation which has a 2nd-order time step bias. We show that the 4th-order corrections of the Takahashi–Imada approximation can also be applied perturbativelymore » to a 2nd-order simulation.« less

  18. Approximate matching of regular expressions.

    PubMed

    Myers, E W; Miller, W

    1989-01-01

    Given a sequence A and regular expression R, the approximate regular expression matching problem is to find a sequence matching R whose optimal alignment with A is the highest scoring of all such sequences. This paper develops an algorithm to solve the problem in time O(MN), where M and N are the lengths of A and R. Thus, the time requirement is asymptotically no worse than for the simpler problem of aligning two fixed sequences. Our method is superior to an earlier algorithm by Wagner and Seiferas in several ways. First, it treats real-valued costs, in addition to integer costs, with no loss of asymptotic efficiency. Second, it requires only O(N) space to deliver just the score of the best alignment. Finally, its structure permits implementation techniques that make it extremely fast in practice. We extend the method to accommodate gap penalties, as required for typical applications in molecular biology, and further refine it to search for sub-strings of A that strongly align with a sequence in R, as required for typical data base searches. We also show how to deliver an optimal alignment between A and R in only O(N + log M) space using O(MN log M) time. Finally, an O(MN(M + N) + N2log N) time algorithm is presented for alignment scoring schemes where the cost of a gap is an arbitrary increasing function of its length.

  19. Comparison of dynamical approximation schemes for nonlinear gravitaional clustering

    NASA Technical Reports Server (NTRS)

    Melott, Adrian L.

    1994-01-01

    We have recently conducted a controlled comparison of a number of approximations for gravitational clustering against the same n-body simulations. These include ordinary linear perturbation theory (Eulerian), the lognormal approximation, the adhesion approximation, the frozen-flow approximation, the Zel'dovich approximation (describable as first-order Lagrangian perturbation theory), and its second-order generalization. In the last two cases we also created new versions of the approximation by truncation, i.e., by smoothing the initial conditions with various smoothing window shapes and varying their sizes. The primary tool for comparing simulations to approximation schemes was cross-correlation of the evolved mass density fields, testing the extent to which mass was moved to the right place. The Zel'dovich approximation, with initial convolution with a Gaussian e(exp -k(exp 2)/k(sub G(exp 2)), where k(sub G) is adjusted to be just into the nonlinear regime of the evolved model (details in text) worked extremely well. Its second-order generalization worked slightly better. We recommend either n-body simulations or our modified versions of the Zel'dovich approximation, depending upon the purpose. The theoretical implication is that pancaking is implicit in all cosmological gravitational clustering, at least from Gaussian initial conditions, even when subcondensations are present. This in turn provides a natural explanation for the presence of sheets and filaments in the observed galaxy distribution. Use of the approximation scheme can permit extremely rapid generation of large numbers of realizations of model universes with good accuracy down to galaxy group mass scales.

  20. Approximation methods for combined thermal/structural design

    NASA Technical Reports Server (NTRS)

    Haftka, R. T.; Shore, C. P.

    1979-01-01

    Two approximation concepts for combined thermal/structural design are evaluated. The first concept is an approximate thermal analysis based on the first derivatives of structural temperatures with respect to design variables. Two commonly used first-order Taylor series expansions are examined. The direct and reciprocal expansions are special members of a general family of approximations, and for some conditions other members of that family of approximations are more accurate. Several examples are used to compare the accuracy of the different expansions. The second approximation concept is the use of critical time points for combined thermal and stress analyses of structures with transient loading conditions. Significant time savings are realized by identifying critical time points and performing the stress analysis for those points only. The design of an insulated panel which is exposed to transient heating conditions is discussed.

  1. Analytic approximations to the modon dispersion relation. [in oceanography

    NASA Technical Reports Server (NTRS)

    Boyd, J. P.

    1981-01-01

    Three explicit analytic approximations are given to the modon dispersion relation developed by Flierl et al. (1980) to describe Gulf Stream rings and related phenomena in the oceans and atmosphere. The solutions are in the form of k(q), and are developed in the form of a power series in q for small q, an inverse power series in 1/q for large q, and a two-point Pade approximant. The low order Pade approximant is shown to yield a solution for the dispersion relation with a maximum relative error for the lowest branch of the function equal to one in 700 in the q interval zero to infinity.

  2. An approximation method for configuration optimization of trusses

    NASA Technical Reports Server (NTRS)

    Hansen, Scott R.; Vanderplaats, Garret N.

    1988-01-01

    Two- and three-dimensional elastic trusses are designed for minimum weight by varying the areas of the members and the location of the joints. Constraints on member stresses and Euler buckling are imposed and multiple static loading conditions are considered. The method presented here utilizes an approximate structural analysis based on first order Taylor series expansions of the member forces. A numerical optimizer minimizes the weight of the truss using information from the approximate structural analysis. Comparisons with results from other methods are made. It is shown that the method of forming an approximate structural analysis based on linearized member forces leads to a highly efficient method of truss configuration optimization.

  3. Self-consistent approximation beyond the CPA: Part II

    SciTech Connect

    Kaplan, T.; Gray, L.J.

    1981-08-01

    In Part I, Professor Leath has described the substantial efforts to generalize the CPA. In this second part, a particular self-consistent approximation for random alloys developed by Kaplan, Leath, Gray, and Diehl is described. This approximation is applicable to diagonal, off-diagonal and environmental disorder, includes cluster scattering, and yields a translationally invariant and analytic (Herglotz) average Green's function. Furthermore Gray and Kaplan have shown that an approximation for alloys with short-range order can be constructed from this theory.

  4. Structural Reliability Analysis and Optimization: Use of Approximations

    NASA Technical Reports Server (NTRS)

    Grandhi, Ramana V.; Wang, Liping

    1999-01-01

    This report is intended for the demonstration of function approximation concepts and their applicability in reliability analysis and design. Particularly, approximations in the calculation of the safety index, failure probability and structural optimization (modification of design variables) are developed. With this scope in mind, extensive details on probability theory are avoided. Definitions relevant to the stated objectives have been taken from standard text books. The idea of function approximations is to minimize the repetitive use of computationally intensive calculations by replacing them with simpler closed-form equations, which could be nonlinear. Typically, the approximations provide good accuracy around the points where they are constructed, and they need to be periodically updated to extend their utility. There are approximations in calculating the failure probability of a limit state function. The first one, which is most commonly discussed, is how the limit state is approximated at the design point. Most of the time this could be a first-order Taylor series expansion, also known as the First Order Reliability Method (FORM), or a second-order Taylor series expansion (paraboloid), also known as the Second Order Reliability Method (SORM). From the computational procedure point of view, this step comes after the design point identification; however, the order of approximation for the probability of failure calculation is discussed first, and it is denoted by either FORM or SORM. The other approximation of interest is how the design point, or the most probable failure point (MPP), is identified. For iteratively finding this point, again the limit state is approximated. The accuracy and efficiency of the approximations make the search process quite practical for analysis intensive approaches such as the finite element methods; therefore, the crux of this research is to develop excellent approximations for MPP identification and also different

  5. Producing approximate answers to database queries

    NASA Technical Reports Server (NTRS)

    Vrbsky, Susan V.; Liu, Jane W. S.

    1993-01-01

    We have designed and implemented a query processor, called APPROXIMATE, that makes approximate answers available if part of the database is unavailable or if there is not enough time to produce an exact answer. The accuracy of the approximate answers produced improves monotonically with the amount of data retrieved to produce the result. The exact answer is produced if all of the needed data are available and query processing is allowed to continue until completion. The monotone query processing algorithm of APPROXIMATE works within the standard relational algebra framework and can be implemented on a relational database system with little change to the relational architecture. We describe here the approximation semantics of APPROXIMATE that serves as the basis for meaningful approximations of both set-valued and single-valued queries. We show how APPROXIMATE is implemented to make effective use of semantic information, provided by an object-oriented view of the database, and describe the additional overhead required by APPROXIMATE.

  6. Energy conservation - A test for scattering approximations

    NASA Technical Reports Server (NTRS)

    Acquista, C.; Holland, A. C.

    1980-01-01

    The roles of the extinction theorem and energy conservation in obtaining the scattering and absorption cross sections for several light scattering approximations are explored. It is shown that the Rayleigh, Rayleigh-Gans, anomalous diffraction, geometrical optics, and Shifrin approximations all lead to reasonable values of the cross sections, while the modified Mie approximation does not. Further examination of the modified Mie approximation for the ensembles of nonspherical particles reveals additional problems with that method.

  7. Legendre-tau approximation for functional differential equations. Part 2: The linear quadratic optimal control problem

    NASA Technical Reports Server (NTRS)

    Ito, K.; Teglas, R.

    1984-01-01

    The numerical scheme based on the Legendre-tau approximation is proposed to approximate the feedback solution to the linear quadratic optimal control problem for hereditary differential systems. The convergence property is established using Trotter ideas. The method yields very good approximations at low orders and provides an approximation technique for computing closed-loop eigenvalues of the feedback system. A comparison with existing methods (based on averaging and spline approximations) is made.

  8. Legendre-tau approximation for functional differential equations. II - The linear quadratic optimal control problem

    NASA Technical Reports Server (NTRS)

    Ito, Kazufumi; Teglas, Russell

    1987-01-01

    The numerical scheme based on the Legendre-tau approximation is proposed to approximate the feedback solution to the linear quadratic optimal control problem for hereditary differential systems. The convergence property is established using Trotter ideas. The method yields very good approximations at low orders and provides an approximation technique for computing closed-loop eigenvalues of the feedback system. A comparison with existing methods (based on averaging and spline approximations) is made.

  9. Analysis of corrections to the eikonal approximation

    NASA Astrophysics Data System (ADS)

    Hebborn, C.; Capel, P.

    2017-11-01

    Various corrections to the eikonal approximations are studied for two- and three-body nuclear collisions with the goal to extend the range of validity of this approximation to beam energies of 10 MeV/nucleon. Wallace's correction does not improve much the elastic-scattering cross sections obtained at the usual eikonal approximation. On the contrary, a semiclassical approximation that substitutes the impact parameter by a complex distance of closest approach computed with the projectile-target optical potential efficiently corrects the eikonal approximation. This opens the possibility to analyze data measured down to 10 MeV/nucleon within eikonal-like reaction models.

  10. An Approximate Approach to Automatic Kernel Selection.

    PubMed

    Ding, Lizhong; Liao, Shizhong

    2016-02-02

    Kernel selection is a fundamental problem of kernel-based learning algorithms. In this paper, we propose an approximate approach to automatic kernel selection for regression from the perspective of kernel matrix approximation. We first introduce multilevel circulant matrices into automatic kernel selection, and develop two approximate kernel selection algorithms by exploiting the computational virtues of multilevel circulant matrices. The complexity of the proposed algorithms is quasi-linear in the number of data points. Then, we prove an approximation error bound to measure the effect of the approximation in kernel matrices by multilevel circulant matrices on the hypothesis and further show that the approximate hypothesis produced with multilevel circulant matrices converges to the accurate hypothesis produced with kernel matrices. Experimental evaluations on benchmark datasets demonstrate the effectiveness of approximate kernel selection.

  11. More on approximations of Poisson probabilities

    SciTech Connect

    Kao, C

    1980-05-01

    Calculation of Poisson probabilities frequently involves calculating high factorials, which becomes tedious and time-consuming with regular calculators. The usual way to overcome this difficulty has been to find approximations by making use of the table of the standard normal distribution. A new transformation proposed by Kao in 1978 appears to perform better for this purpose than traditional transformations. In the present paper several approximation methods are stated and compared numerically, including an approximation method that utilizes a modified version of Kao's transformation. An approximation based on a power transformation was found to outperform those based on the square-root type transformationsmore » as proposed in literature. The traditional Wilson-Hilferty approximation and Makabe-Morimura approximation are extremely poor compared with this approximation. 4 tables. (RWR)« less

  12. Bounded-Degree Approximations of Stochastic Networks

    SciTech Connect

    Quinn, Christopher J.; Pinar, Ali; Kiyavash, Negar

    2017-06-01

    We propose algorithms to approximate directed information graphs. Directed information graphs are probabilistic graphical models that depict causal dependencies between stochastic processes in a network. The proposed algorithms identify optimal and near-optimal approximations in terms of Kullback-Leibler divergence. The user-chosen sparsity trades off the quality of the approximation against visual conciseness and computational tractability. One class of approximations contains graphs with speci ed in-degrees. Another class additionally requires that the graph is connected. For both classes, we propose algorithms to identify the optimal approximations and also near-optimal approximations, using a novel relaxation of submodularity. We also propose algorithms to identifymore » the r-best approximations among these classes, enabling robust decision making.« less

  13. Approximating smooth functions using algebraic-trigonometric polynomials

    SciTech Connect

    Sharapudinov, Idris I

    2011-01-14

    The problem under consideration is that of approximating classes of smooth functions by algebraic-trigonometric polynomials of the form p{sub n}(t)+{tau}{sub m}(t), where p{sub n}(t) is an algebraic polynomial of degree n and {tau}{sub m}(t)=a{sub 0}+{Sigma}{sub k=1}{sup m}a{sub k} cos k{pi}t + b{sub k} sin k{pi}t is a trigonometric polynomial of order m. The precise order of approximation by such polynomials in the classes W{sup r}{sub {infinity}(}M) and an upper bound for similar approximations in the class W{sup r}{sub p}(M) with 4/3

  14. Dynamical Vertex Approximation for the Hubbard Model

    NASA Astrophysics Data System (ADS)

    Toschi, Alessandro

    A full understanding of correlated electron systems in the physically relevant situations of three and two dimensions represents a challenge for the contemporary condensed matter theory. However, in the last years considerable progress has been achieved by means of increasingly more powerful quantum many-body algorithms, applied to the basic model for correlated electrons, the Hubbard Hamiltonian. Here, I will review the physics emerging from studies performed with the dynamical vertex approximation, which includes diagrammatic corrections to the local description of the dynamical mean field theory (DMFT). In particular, I will first discuss the phase diagram in three dimensions with a special focus on the commensurate and incommensurate magnetic phases, their (quantum) critical properties, and the impact of fluctuations on electronic lifetimes and spectral functions. In two dimensions, the effects of non-local fluctuations beyond DMFT grow enormously, determining the appearance of a low-temperature insulating behavior for all values of the interaction in the unfrustrated model: Here the prototypical features of the Mott-Hubbard metal-insulator transition, as well as the existence of magnetically ordered phases, are completely overwhelmed by antiferromagnetic fluctuations of exponentially large extension, in accordance with the Mermin-Wagner theorem. Eventually, by a fluctuation diagnostics analysis of cluster DMFT self-energies, the same magnetic fluctuations are identified as responsible for the pseudogap regime in the holed-doped frustrated case, with important implications for the theoretical modeling of the cuprate physics.

  15. Approximated analytical solution to an Ebola optimal control problem

    NASA Astrophysics Data System (ADS)

    Hincapié-Palacio, Doracelly; Ospina, Juan; Torres, Delfim F. M.

    2016-11-01

    An analytical expression for the optimal control of an Ebola problem is obtained. The analytical solution is found as a first-order approximation to the Pontryagin Maximum Principle via the Euler-Lagrange equation. An implementation of the method is given using the computer algebra system Maple. Our analytical solutions confirm the results recently reported in the literature using numerical methods.

  16. On singlet s-wave electron-hydrogen scattering.

    NASA Technical Reports Server (NTRS)

    Madan, R. N.

    1973-01-01

    Discussion of various zeroth-order approximations to s-wave scattering of electrons by hydrogen atoms below the first excitation threshold. The formalism previously developed by the author (1967, 1968) is applied to Feshbach operators to derive integro-differential equations, with the optical-potential set equal to zero, for the singlet and triplet cases. Phase shifts of s-wave scattering are computed in the zeroth-order approximation of the Feshbach operator method and in the static-exchange approximation. It is found that the convergence of numerical computations is faster in the former approximation than in the latter.

  17. Approximate Dispersion Relations for Waves on Arbitrary Shear Flows

    NASA Astrophysics Data System (ADS)

    Ellingsen, S. À.; Li, Y.

    2017-12-01

    An approximate dispersion relation is derived and presented for linear surface waves atop a shear current whose magnitude and direction can vary arbitrarily with depth. The approximation, derived to first order of deviation from potential flow, is shown to produce good approximations at all wavelengths for a wide range of naturally occuring shear flows as well as widely used model flows. The relation reduces in many cases to a 3-D generalization of the much used approximation by Skop (1987), developed further by Kirby and Chen (1989), but is shown to be more robust, succeeding in situations where the Kirby and Chen model fails. The two approximations incur the same numerical cost and difficulty. While the Kirby and Chen approximation is excellent for a wide range of currents, the exact criteria for its applicability have not been known. We explain the apparently serendipitous success of the latter and derive proper conditions of applicability for both approximate dispersion relations. Our new model has a greater range of applicability. A second order approximation is also derived. It greatly improves accuracy, which is shown to be important in difficult cases. It has an advantage over the corresponding second-order expression proposed by Kirby and Chen that its criterion of accuracy is explicitly known, which is not currently the case for the latter to our knowledge. Our second-order term is also arguably significantly simpler to implement, and more physically transparent, than its sibling due to Kirby and Chen.Plain Language SummaryIn <span class="hlt">order</span> to answer key questions such as how the ocean surface affects the climate, erodes the coastline and transports nutrients, we must understand how waves move. This is not so easy when depth varying currents are present, as they often are in coastal waters. We have developed a modeling tool for accurately predicting wave properties in such situations, ready for use, for example, in the</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/ADA029655','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/ADA029655"><span><span class="hlt">Approximations</span> of Two-Attribute Utility Functions</span></a></p> <p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p> <p></p> <p>1976-09-01</p> <p>preferred to") be a bina-zy relation on the set • of simple probability measures or ’gambles’ defined on a set T of consequences. Throughout this study it...simplifying independence assumptions. Although there are several approaches to this problem, the21 present study will focus on <span class="hlt">approximations</span> of u... study will elicit additional interest in the topic. 2. REMARKS ON <span class="hlt">APPROXIMATION</span> THEORY This section outlines a few basic ideas of <span class="hlt">approximation</span> theory</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19850022413','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19850022413"><span>On the convergence of difference <span class="hlt">approximations</span> to scalar conservation laws</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Osher, S.; Tadmor, E.</p> <p>1985-01-01</p> <p>A unified treatment of explicit in time, two level, second <span class="hlt">order</span> resolution, total variation diminishing, <span class="hlt">approximations</span> to scalar conservation laws are presented. The schemes are assumed only to have conservation form and incremental form. A modified flux and a viscosity coefficient are introduced and results in terms of the latter are obtained. The existence of a cell entropy inequality is discussed and such an equality for all entropies is shown to imply that the scheme is an E scheme on monotone (actually more general) data, hence at most only first <span class="hlt">order</span> accurate in general. Convergence for total variation diminishing-second <span class="hlt">order</span> resolution schemes <span class="hlt">approximating</span> convex or concave conservation laws is shown by enforcing a single discrete entropy inequality.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19850021314','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19850021314"><span>Intermediate boundary conditions for LOD, ADI and <span class="hlt">approximate</span> factorization methods</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Leveque, R. J.</p> <p>1985-01-01</p> <p>A general approach to determining the correct intermediate boundary conditions for dimensional splitting methods is presented. The intermediate solution U is viewed as a second <span class="hlt">order</span> accurate <span class="hlt">approximation</span> to a modified equation. Deriving the modified equation and using the relationship between this equation and the original equation allows us to determine the correct boundary conditions for U*. This technique is illustrated by applying it to locally one dimensional (LOD) and alternating direction implicit (ADI) methods for the heat equation in two and three space dimensions. The <span class="hlt">approximate</span> factorization method is considered in slightly more generality.</p> </li> </ol> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_8");'>8</a></li> <li><a href="#" onclick='return showDiv("page_9");'>9</a></li> <li class="active"><span>10</span></li> <li><a href="#" onclick='return showDiv("page_11");'>11</a></li> <li><a href="#" onclick='return showDiv("page_12");'>12</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div><!-- col-sm-12 --> </div><!-- row --> </div><!-- page_10 --> <div id="page_11" class="hiddenDiv"> <div class="row"> <div class="col-sm-12"> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_9");'>9</a></li> <li><a href="#" onclick='return showDiv("page_10");'>10</a></li> <li class="active"><span>11</span></li> <li><a href="#" onclick='return showDiv("page_12");'>12</a></li> <li><a href="#" onclick='return showDiv("page_13");'>13</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div> </div> <div class="row"> <div class="col-sm-12"> <ol class="result-class" start="201"> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19830025319','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19830025319"><span>Spline <span class="hlt">approximations</span> for nonlinear hereditary control systems</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Daniel, P. L.</p> <p>1982-01-01</p> <p>A sline-based <span class="hlt">approximation</span> scheme is discussed for optimal control problems governed by nonlinear nonautonomous delay differential equations. The <span class="hlt">approximating</span> framework reduces the original control problem to a sequence of optimization problems governed by ordinary differential equations. Convergence proofs, which appeal directly to dissipative-type estimates for the underlying nonlinear operator, are given and numerical findings are summarized.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=boltzmann&pg=2&id=EJ1017061','ERIC'); return false;" href="https://eric.ed.gov/?q=boltzmann&pg=2&id=EJ1017061"><span>Quirks of Stirling's <span class="hlt">Approximation</span></span></a></p> <p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p> <p>Macrae, Roderick M.; Allgeier, Benjamin M.</p> <p>2013-01-01</p> <p>Stirling's <span class="hlt">approximation</span> to ln "n"! is typically introduced to physical chemistry students as a step in the derivation of the statistical expression for the entropy. However, naive application of this <span class="hlt">approximation</span> leads to incorrect conclusions. In this article, the problem is first illustrated using a familiar "toy…</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=calculus+AND+1&pg=2&id=EJ994590','ERIC'); return false;" href="https://eric.ed.gov/?q=calculus+AND+1&pg=2&id=EJ994590"><span>Computing Functions by <span class="hlt">Approximating</span> the Input</span></a></p> <p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p> <p>Goldberg, Mayer</p> <p>2012-01-01</p> <p>In computing real-valued functions, it is ordinarily assumed that the input to the function is known, and it is the output that we need to <span class="hlt">approximate</span>. In this work, we take the opposite approach: we show how to compute the values of some transcendental functions by <span class="hlt">approximating</span> the input to these functions, and obtaining exact answers for their…</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018JAGeo..12..139E','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018JAGeo..12..139E"><span>Spline <span class="hlt">approximation</span>, Part 1: Basic methodology</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Ezhov, Nikolaj; Neitzel, Frank; Petrovic, Svetozar</p> <p>2018-04-01</p> <p>In engineering geodesy point clouds derived from terrestrial laser scanning or from photogrammetric approaches are almost never used as final results. For further processing and analysis a curve or surface <span class="hlt">approximation</span> with a continuous mathematical function is required. In this paper the <span class="hlt">approximation</span> of 2D curves by means of splines is treated. Splines offer quite flexible and elegant solutions for interpolation or <span class="hlt">approximation</span> of "irregularly" distributed data. Depending on the problem they can be expressed as a function or as a set of equations that depend on some parameter. Many different types of splines can be used for spline <span class="hlt">approximation</span> and all of them have certain advantages and disadvantages depending on the <span class="hlt">approximation</span> problem. In a series of three articles spline <span class="hlt">approximation</span> is presented from a geodetic point of view. In this paper (Part 1) the basic methodology of spline <span class="hlt">approximation</span> is demonstrated using splines constructed from ordinary polynomials and splines constructed from truncated polynomials. In the forthcoming Part 2 the notion of B-spline will be explained in a unique way, namely by using the concept of convex combinations. The numerical stability of all spline <span class="hlt">approximation</span> approaches as well as the utilization of splines for deformation detection will be investigated on numerical examples in Part 3.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19800016563','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19800016563"><span>Inversion and <span class="hlt">approximation</span> of Laplace transforms</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Lear, W. M.</p> <p>1980-01-01</p> <p>A method of inverting Laplace transforms by using a set of orthonormal functions is reported. As a byproduct of the inversion, <span class="hlt">approximation</span> of complicated Laplace transforms by a transform with a series of simple poles along the left half plane real axis is shown. The inversion and <span class="hlt">approximation</span> process is simple enough to be put on a programmable hand calculator.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=Base&pg=3&id=EJ1158997','ERIC'); return false;" href="https://eric.ed.gov/?q=Base&pg=3&id=EJ1158997"><span><span class="hlt">Approximations</span> of e and ?: An Exploration</span></a></p> <p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p> <p>Brown, Philip R.</p> <p>2017-01-01</p> <p>Fractional <span class="hlt">approximations</span> of e and p are discovered by searching for repetitions or partial repetitions of digit strings in their expansions in different number bases. The discovery of such fractional <span class="hlt">approximations</span> is suggested for students and teachers as an entry point into mathematics research.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/servlets/purl/1083968','DOE-PATENT-XML'); return false;" href="https://www.osti.gov/servlets/purl/1083968"><span><span class="hlt">Approximate</span> error conjugation gradient minimization methods</span></a></p> <p><a target="_blank" href="http://www.osti.gov/doepatents">DOEpatents</a></p> <p>Kallman, Jeffrey S</p> <p>2013-05-21</p> <p>In one embodiment, a method includes selecting a subset of rays from a set of all rays to use in an error calculation for a constrained conjugate gradient minimization problem, calculating an <span class="hlt">approximate</span> error using the subset of rays, and calculating a minimum in a conjugate gradient direction based on the <span class="hlt">approximate</span> error. In another embodiment, a system includes a processor for executing logic, logic for selecting a subset of rays from a set of all rays to use in an error calculation for a constrained conjugate gradient minimization problem, logic for calculating an <span class="hlt">approximate</span> error using the subset of rays, and logic for calculating a minimum in a conjugate gradient direction based on the <span class="hlt">approximate</span> error. In other embodiments, computer program products, methods, and systems are described capable of using <span class="hlt">approximate</span> error in constrained conjugate gradient minimization problems.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19870011345','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19870011345"><span>Piecewise linear <span class="hlt">approximation</span> for hereditary control problems</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Propst, Georg</p> <p>1987-01-01</p> <p>Finite dimensional <span class="hlt">approximations</span> are presented for linear retarded functional differential equations by use of discontinuous piecewise linear functions. The <span class="hlt">approximation</span> scheme is applied to optimal control problems when a quadratic cost integral has to be minimized subject to the controlled retarded system. It is shown that the <span class="hlt">approximate</span> optimal feedback operators converge to the true ones both in case the cost integral ranges over a finite time interval as well as in the case it ranges over an infinite time interval. The arguments in the latter case rely on the fact that the piecewise linear <span class="hlt">approximations</span> to stable systems are stable in a uniform sense. This feature is established using a vector-component stability criterion in the state space R(n) x L(2) and the favorable eigenvalue behavior of the piecewise linear <span class="hlt">approximations</span>.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018JMP....59e1502G','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018JMP....59e1502G"><span><span class="hlt">Approximate</span> isotropic cloak for the Maxwell equations</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Ghosh, Tuhin; Tarikere, Ashwin</p> <p>2018-05-01</p> <p>We construct a regular isotropic <span class="hlt">approximate</span> cloak for the Maxwell system of equations. The method of transformation optics has enabled the design of electromagnetic parameters that cloak a region from external observation. However, these constructions are singular and anisotropic, making practical implementation difficult. Thus, regular <span class="hlt">approximations</span> to these cloaks have been constructed that cloak a given region to any desired degree of accuracy. In this paper, we show how to construct isotropic <span class="hlt">approximations</span> to these regularized cloaks using homogenization techniques so that one obtains cloaking of arbitrary accuracy with regular and isotropic parameters.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19870020664','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19870020664"><span>UNAERO: A package of FORTRAN subroutines for <span class="hlt">approximating</span> unsteady aerodynamics in the time domain</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Dunn, H. J.</p> <p>1985-01-01</p> <p>This report serves as an instruction and maintenance manual for a collection of CDC CYBER FORTRAN IV subroutines for <span class="hlt">approximating</span> the unsteady aerodynamic forces in the time domain. The result is a set of constant-coefficient first-<span class="hlt">order</span> differential equations that <span class="hlt">approximate</span> the dynamics of the vehicle. Provisions are included for adjusting the number of modes used for calculating the <span class="hlt">approximations</span> so that an accurate <span class="hlt">approximation</span> is generated. The number of data points at different values of reduced frequency can also be varied to adjust the accuracy of the <span class="hlt">approximation</span> over the reduced-frequency range. The denominator coefficients of the <span class="hlt">approximation</span> may be calculated by means of a gradient method or a least-squares <span class="hlt">approximation</span> technique. Both the <span class="hlt">approximation</span> methods use weights on the residual error. A new set of system equations, at a different dynamic pressure, can be generated without the <span class="hlt">approximations</span> being recalculated.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/1434885-approximate-computing-techniques-iterative-graph-algorithms','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/1434885-approximate-computing-techniques-iterative-graph-algorithms"><span><span class="hlt">Approximate</span> Computing Techniques for Iterative Graph Algorithms</span></a></p> <p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p> <p>Panyala, Ajay R.; Subasi, Omer; Halappanavar, Mahantesh</p> <p></p> <p><span class="hlt">Approximate</span> computing enables processing of large-scale graphs by trading off quality for performance. <span class="hlt">Approximate</span> computing techniques have become critical not only due to the emergence of parallel architectures but also the availability of large scale datasets enabling data-driven discovery. Using two prototypical graph algorithms, PageRank and community detection, we present several <span class="hlt">approximate</span> computing heuristics to scale the performance with minimal loss of accuracy. We present several heuristics including loop perforation, data caching, incomplete graph coloring and synchronization, and evaluate their efficiency. We demonstrate performance improvements of up to 83% for PageRank and up to 450x for community detection, with lowmore » impact of accuracy for both the algorithms. We expect the proposed <span class="hlt">approximate</span> techniques will enable scalable graph analytics on data of importance to several applications in science and their subsequent adoption to scale similar graph algorithms.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2012AIPC.1497..184Z','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2012AIPC.1497..184Z"><span><span class="hlt">Approximation</span> by the iterates of Bernstein operator</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Zapryanova, Teodora; Tachev, Gancho</p> <p>2012-11-01</p> <p>We study the degree of pointwise <span class="hlt">approximation</span> of the iterated Bernstein operators to its limiting operator. We obtain a quantitative estimates related to the conjecture of Gonska and Raşa from 2006.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/pages/biblio/1286958-survey-techniques-approximate-computing','SCIGOV-DOEP'); return false;" href="https://www.osti.gov/pages/biblio/1286958-survey-techniques-approximate-computing"><span>A Survey of Techniques for <span class="hlt">Approximate</span> Computing</span></a></p> <p><a target="_blank" href="http://www.osti.gov/pages">DOE PAGES</a></p> <p>Mittal, Sparsh</p> <p>2016-03-18</p> <p><span class="hlt">Approximate</span> computing trades off computation quality with the effort expended and as rising performance demands confront with plateauing resource budgets, <span class="hlt">approximate</span> computing has become, not merely attractive, but even imperative. Here, we present a survey of techniques for <span class="hlt">approximate</span> computing (AC). We discuss strategies for finding <span class="hlt">approximable</span> program portions and monitoring output quality, techniques for using AC in different processing units (e.g., CPU, GPU and FPGA), processor components, memory technologies etc., and programming frameworks for AC. Moreover, we classify these techniques based on several key characteristics to emphasize their similarities and differences. Finally, the aim of this paper is tomore » provide insights to researchers into working of AC techniques and inspire more efforts in this area to make AC the mainstream computing approach in future systems.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19760011449','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19760011449"><span><span class="hlt">Approximation</span> concepts for efficient structural synthesis</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Schmit, L. A., Jr.; Miura, H.</p> <p>1976-01-01</p> <p>It is shown that efficient structural synthesis capabilities can be created by using <span class="hlt">approximation</span> concepts to mesh finite element structural analysis methods with nonlinear mathematical programming techniques. The history of the application of mathematical programming techniques to structural design optimization problems is reviewed. Several rather general <span class="hlt">approximation</span> concepts are described along with the technical foundations of the ACCESS 1 computer program, which implements several <span class="hlt">approximation</span> concepts. A substantial collection of structural design problems involving truss and idealized wing structures is presented. It is concluded that since the basic ideas employed in creating the ACCESS 1 program are rather general, its successful development supports the contention that the introduction of <span class="hlt">approximation</span> concepts will lead to the emergence of a new generation of practical and efficient, large scale, structural synthesis capabilities in which finite element analysis methods and mathematical programming algorithms will play a central role.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19790023272','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19790023272"><span>Linear <span class="hlt">Approximation</span> SAR Azimuth Processing Study</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Lindquist, R. B.; Masnaghetti, R. K.; Belland, E.; Hance, H. V.; Weis, W. G.</p> <p>1979-01-01</p> <p>A segmented linear <span class="hlt">approximation</span> of the quadratic phase function that is used to focus the synthetic antenna of a SAR was studied. Ideal focusing, using a quadratic varying phase focusing function during the time radar target histories are gathered, requires a large number of complex multiplications. These can be largely eliminated by using linear <span class="hlt">approximation</span> techniques. The result is a reduced processor size and chip count relative to ideally focussed processing and a correspondingly increased feasibility for spaceworthy implementation. A preliminary design and sizing for a spaceworthy linear <span class="hlt">approximation</span> SAR azimuth processor meeting requirements similar to those of the SEASAT-A SAR was developed. The study resulted in a design with <span class="hlt">approximately</span> 1500 IC's, 1.2 cubic feet of volume, and 350 watts of power for a single look, 4000 range cell azimuth processor with 25 meters resolution.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19880037825&hterms=levels+law&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D30%26Ntt%3DA%2Blevels%2Blaw','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19880037825&hterms=levels+law&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D30%26Ntt%3DA%2Blevels%2Blaw"><span>On the convergence of difference <span class="hlt">approximations</span> to scalar conservation laws</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Osher, Stanley; Tadmor, Eitan</p> <p>1988-01-01</p> <p>A unified treatment is given for time-explicit, two-level, second-<span class="hlt">order</span>-resolution (SOR), total-variation-diminishing (TVD) <span class="hlt">approximations</span> to scalar conservation laws. The schemes are assumed only to have conservation form and incremental form. A modified flux and a viscosity coefficient are introduced to obtain results in terms of the latter. The existence of a cell entropy inequality is discussed, and such an equality for all entropies is shown to imply that the scheme is an E scheme on monotone (actually more general) data, hence at most only first-<span class="hlt">order</span> accurate in general. Convergence for TVD-SOR schemes <span class="hlt">approximating</span> convex or concave conservation laws is shown by enforcing a single discrete entropy inequality.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2012JChPh.137l5102O','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2012JChPh.137l5102O"><span>Counting statistics for genetic switches based on effective interaction <span class="hlt">approximation</span></span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Ohkubo, Jun</p> <p>2012-09-01</p> <p>Applicability of counting statistics for a system with an infinite number of states is investigated. The counting statistics has been studied a lot for a system with a finite number of states. While it is possible to use the scheme in <span class="hlt">order</span> to count specific transitions in a system with an infinite number of states in principle, we have non-closed equations in general. A simple genetic switch can be described by a master equation with an infinite number of states, and we use the counting statistics in <span class="hlt">order</span> to count the number of transitions from inactive to active states in the gene. To avoid having the non-closed equations, an effective interaction <span class="hlt">approximation</span> is employed. As a result, it is shown that the switching problem can be treated as a simple two-state model <span class="hlt">approximately</span>, which immediately indicates that the switching obeys non-Poisson statistics.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/1994ApJ...423...22C','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/1994ApJ...423...22C"><span>Applying the Zel'dovich <span class="hlt">approximation</span> to general relativity</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Croudace, K. M.; Parry, J.; Salopek, D. S.; Stewart, J. M.</p> <p>1994-03-01</p> <p>Starting from general relativity, we give a systematic derivation of the Zel'dovich <span class="hlt">approximation</span> describing the nonlinear evolution of collisionless dust. We begin by evolving dust along world lines, and we demonstrate that the Szekeres line element is an exact but apparently unstable solution of the evolution equations describing pancake collapse. Next, we solve the Einstein field equations by employing Hamilton-Jacobi techniques and a spatial gradient expansion. We give a prescription for evolving a primordial or 'seed' metric up to the formation of pancakes, and demonstrate its validity by rederiving the Szekeres solution <span class="hlt">approximately</span> at third <span class="hlt">order</span> and exactly at fifth <span class="hlt">order</span> in spatial gradients. Finally we show that the range of validity of the expansion can be improved quite significantly if one notes that the 3-metric must have nonnegative eigenvalues. With this improvement the exact Szekeres solution is obtained after only one iteration.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19720031591&hterms=hierarchy&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D40%26Ntt%3Dhierarchy','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19720031591&hterms=hierarchy&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D40%26Ntt%3Dhierarchy"><span>The closure <span class="hlt">approximation</span> in the hierarchy equations.</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Adomian, G.</p> <p>1971-01-01</p> <p>The expectation of the solution process in a stochastic operator equation can be obtained from averaged equations only under very special circumstances. Conditions for validity are given and the significance and validity of the <span class="hlt">approximation</span> in widely used hierarchy methods and the ?self-consistent field' <span class="hlt">approximation</span> in nonequilibrium statistical mechanics are clarified. The error at any level of the hierarchy can be given and can be avoided by the use of the iterative method.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/ADA089726','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/ADA089726"><span>A Discrete <span class="hlt">Approximation</span> Framework for Hereditary Systems.</span></a></p> <p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p> <p></p> <p>1980-05-01</p> <p>schemes which are included in the general framework and which may be implemented directly on high-speed computing machines are developed. A numerical...an appropriately chosen Hilbert space. We then proceed to develop general <span class="hlt">approximation</span> schemes for the solutions to the homogeneous AEE which in turn...rich classes of these schemes . In addition, two particular families of <span class="hlt">approximation</span> schemes included in the general framework are developed and</p> </li> </ol> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_9");'>9</a></li> <li><a href="#" onclick='return showDiv("page_10");'>10</a></li> <li class="active"><span>11</span></li> <li><a href="#" onclick='return showDiv("page_12");'>12</a></li> <li><a href="#" onclick='return showDiv("page_13");'>13</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div><!-- col-sm-12 --> </div><!-- row --> </div><!-- page_11 --> <div id="page_12" class="hiddenDiv"> <div class="row"> <div class="col-sm-12"> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_10");'>10</a></li> <li><a href="#" onclick='return showDiv("page_11");'>11</a></li> <li class="active"><span>12</span></li> <li><a href="#" onclick='return showDiv("page_13");'>13</a></li> <li><a href="#" onclick='return showDiv("page_14");'>14</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div> </div> <div class="row"> <div class="col-sm-12"> <ol class="result-class" start="221"> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018IJMPB..3250008G','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018IJMPB..3250008G"><span><span class="hlt">Approximate</span> analytic expression for the Skyrmions crystal</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Grandi, Nicolás; Sturla, Mauricio</p> <p>2018-01-01</p> <p>We find <span class="hlt">approximate</span> solutions for the two-dimensional nonlinear Σ-model with Dzyalioshinkii-Moriya term, representing magnetic Skyrmions. They are built in an analytic form, by pasting different <span class="hlt">approximate</span> solutions found in different regions of space. We verify that our construction reproduces the phenomenology known from numerical solutions and Monte Carlo simulations, giving rise to a Skyrmion lattice at an intermediate range of magnetic field, flanked by spiral and spin-polarized phases for low and high magnetic fields, respectively.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018JPhCS1003a2054A','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018JPhCS1003a2054A"><span>On direct theorems for best polynomial <span class="hlt">approximation</span></span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Auad, A. A.; AbdulJabbar, R. S.</p> <p>2018-05-01</p> <p>This paper is to obtain similarity for the best <span class="hlt">approximation</span> degree of functions, which are unbounded in L p,α (A = [0,1]), which called weighted space by algebraic polynomials. {E}nH{(f)}p,α and the best <span class="hlt">approximation</span> degree in the same space on the interval [0,2π] by trigonometric polynomials {E}nT{(f)}p,α of direct wellknown theorems in forms the average modules.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/ADA215217','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/ADA215217"><span>Rational <span class="hlt">Approximations</span> with Hankel-Norm Criterion</span></a></p> <p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p> <p></p> <p>1980-01-01</p> <p>REPORT TYPE ANDu DATES COVERED It) L. TITLE AND SLWUIlL Fi901 ia FUNDING NUMOIRS, RATIONAL <span class="hlt">APPROXIMATIONS</span> WITH HANKEL-NORM CRITERION PE61102F i...problem is proved to be reducible to obtain a two-variable all- pass ration 1 function, interpolating a set of parametric values at specified points inside...PAGES WHICH DO NOT REPRODUCE LEGIBLY. V" C - w RATIONAL <span class="hlt">APPROXIMATIONS</span> WITH HANKEL-NORM CRITERION* Y. Genin* Philips Research Lab. 2, avenue van</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017PhRvD..95h3526W','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017PhRvD..95h3526W"><span>Double power series method for <span class="hlt">approximating</span> cosmological perturbations</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Wren, Andrew J.; Malik, Karim A.</p> <p>2017-04-01</p> <p>We introduce a double power series method for finding <span class="hlt">approximate</span> analytical solutions for systems of differential equations commonly found in cosmological perturbation theory. The method was set out, in a noncosmological context, by Feshchenko, Shkil' and Nikolenko (FSN) in 1966, and is applicable to cases where perturbations are on subhorizon scales. The FSN method is essentially an extension of the well known Wentzel-Kramers-Brillouin (WKB) method for finding <span class="hlt">approximate</span> analytical solutions for ordinary differential equations. The FSN method we use is applicable well beyond perturbation theory to solve systems of ordinary differential equations, linear in the derivatives, that also depend on a small parameter, which here we take to be related to the inverse wave-number. We use the FSN method to find new <span class="hlt">approximate</span> oscillating solutions in linear <span class="hlt">order</span> cosmological perturbation theory for a flat radiation-matter universe. Together with this model's well-known growing and decaying Mészáros solutions, these oscillating modes provide a complete set of subhorizon <span class="hlt">approximations</span> for the metric potential, radiation and matter perturbations. Comparison with numerical solutions of the perturbation equations shows that our <span class="hlt">approximations</span> can be made accurate to within a typical error of 1%, or better. We also set out a heuristic method for error estimation. A Mathematica notebook which implements the double power series method is made available online.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/22472091-loop-approximation-nucleon-qcd-sum-rules','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/22472091-loop-approximation-nucleon-qcd-sum-rules"><span>On the loop <span class="hlt">approximation</span> in nucleon QCD sum rules</span></a></p> <p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p> <p>Drukarev, E. G., E-mail: drukarev@thd.pnpi.spb.ru; Ryskin, M. G.; Sadovnikova, V. A.</p> <p></p> <p>There was a general belief that the nucleon QCD sum rules which include only the quark loops and thus contain only the condensates of dimension d = 3 and d = 4 have only a trivial solution. We demonstrate that there is also a nontrivial solution. We show that it can be treated as the lowest <span class="hlt">order</span> <span class="hlt">approximation</span> to the solution which includes the higher terms of the Operator Product Expansion. Inclusion of the radiative corrections improves the convergence of the series.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20170010676','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20170010676"><span>Temporal Planning for Compilation of Quantum <span class="hlt">Approximate</span> Optimization Algorithm Circuits</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Venturelli, Davide; Do, Minh Binh; Rieffel, Eleanor Gilbert; Frank, Jeremy David</p> <p>2017-01-01</p> <p>We investigate the application of temporal planners to the problem of compiling quantum circuits to newly emerging quantum hardware. While our approach is general, we focus our initial experiments on Quantum <span class="hlt">Approximate</span> Optimization Algorithm (QAOA) circuits that have few <span class="hlt">ordering</span> constraints and allow highly parallel plans. We report on experiments using several temporal planners to compile circuits of various sizes to a realistic hardware. This early empirical evaluation suggests that temporal planning is a viable approach to quantum circuit compilation.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016JQSRT.179..132G','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016JQSRT.179..132G"><span>Comments on localized and integral localized <span class="hlt">approximations</span> in spherical coordinates</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Gouesbet, Gérard; Lock, James A.</p> <p>2016-08-01</p> <p>Localized <span class="hlt">approximation</span> procedures are efficient ways to evaluate beam shape coefficients of laser beams, and are particularly useful when other methods are ineffective or inefficient. Comments on these procedures are, however, required in <span class="hlt">order</span> to help researchers make correct decisions concerning their use. This paper has the flavor of a short review and takes the opportunity to attract the attention of the readers to a required refinement of terminology.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2003JCoAM.161..469F','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2003JCoAM.161..469F"><span>Numerical scheme <span class="hlt">approximating</span> solution and parameters in a beam equation</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Ferdinand, Robert R.</p> <p>2003-12-01</p> <p>We present a mathematical model which describes vibration in a metallic beam about its equilibrium position. This model takes the form of a nonlinear second-<span class="hlt">order</span> (in time) and fourth-<span class="hlt">order</span> (in space) partial differential equation with boundary and initial conditions. A finite-element Galerkin <span class="hlt">approximation</span> scheme is used to estimate model solution. Infinite-dimensional model parameters are then estimated numerically using an inverse method procedure which involves the minimization of a least-squares cost functional. Numerical results are presented and future work to be done is discussed.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19850002436','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19850002436"><span>Legendre-Tau <span class="hlt">approximation</span> for functional differential equations. Part 3: Eigenvalue <span class="hlt">approximations</span> and uniform stability</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Ito, K.</p> <p>1984-01-01</p> <p>The stability and convergence properties of the Legendre-tau <span class="hlt">approximation</span> for hereditary differential systems are analyzed. A charactristic equation is derived for the eigenvalues of the resulting <span class="hlt">approximate</span> system. As a result of this derivation the uniform exponential stability of the solution semigroup is preserved under <span class="hlt">approximation</span>. It is the key to obtaining the convergence of <span class="hlt">approximate</span> solutions of the algebraic Riccati equation in trace norm.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://rosap.ntl.bts.gov/view/dot/9918','DOTNTL'); return false;" href="https://rosap.ntl.bts.gov/view/dot/9918"><span>Derivation of a first <span class="hlt">order</span> <span class="hlt">approximation</span> of particulate matter from aircraft</span></a></p> <p><a target="_blank" href="http://ntlsearch.bts.gov/tris/index.do">DOT National Transportation Integrated Search</a></p> <p></p> <p>2003-06-22</p> <p>The mass of particulate matter (PM) emitted from aircraft must be predicted for major : actions at airports to comply with current federal regulations. However, this PM mass in : the jet exhaust has not been effectively quantified to permit accurate ...</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3511751','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3511751"><span>Averaging principle for second-<span class="hlt">order</span> <span class="hlt">approximation</span> of heterogeneous models with homogeneous models</span></a></p> <p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p> <p>Fibich, Gadi; Gavious, Arieh; Solan, Eilon</p> <p>2012-01-01</p> <p>Typically, models with a heterogeneous property are considerably harder to analyze than the corresponding homogeneous models, in which the heterogeneous property is replaced by its average value. In this study we show that any outcome of a heterogeneous model that satisfies the two properties of differentiability and symmetry is O(ɛ2) equivalent to the outcome of the corresponding homogeneous model, where ɛ is the level of heterogeneity. We then use this averaging principle to obtain new results in queuing theory, game theory (auctions), and social networks (marketing). PMID:23150569</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/23150569','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/23150569"><span>Averaging principle for second-<span class="hlt">order</span> <span class="hlt">approximation</span> of heterogeneous models with homogeneous models.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Fibich, Gadi; Gavious, Arieh; Solan, Eilon</p> <p>2012-11-27</p> <p>Typically, models with a heterogeneous property are considerably harder to analyze than the corresponding homogeneous models, in which the heterogeneous property is replaced by its average value. In this study we show that any outcome of a heterogeneous model that satisfies the two properties of differentiability and symmetry is O(ε(2)) equivalent to the outcome of the corresponding homogeneous model, where ε is the level of heterogeneity. We then use this averaging principle to obtain new results in queuing theory, game theory (auctions), and social networks (marketing).</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017PhRvB..95s5158B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017PhRvB..95s5158B"><span>Convergence behavior of the random phase <span class="hlt">approximation</span> renormalized correlation energy</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Bates, Jefferson E.; Sensenig, Jonathon; Ruzsinszky, Adrienn</p> <p>2017-05-01</p> <p>Based on the random phase <span class="hlt">approximation</span> (RPA), RPA renormalization [J. E. Bates and F. Furche, J. Chem. Phys. 139, 171103 (2013), 10.1063/1.4827254] is a robust many-body perturbation theory that works for molecules and materials because it does not diverge as the Kohn-Sham gap approaches zero. Additionally, RPA renormalization enables the simultaneous calculation of RPA and beyond-RPA correlation energies since the total correlation energy is the sum of a series of independent contributions. The first-<span class="hlt">order</span> <span class="hlt">approximation</span> (RPAr1) yields the dominant beyond-RPA contribution to the correlation energy for a given exchange-correlation kernel, but systematically underestimates the total beyond-RPA correction. For both the homogeneous electron gas model and real systems, we demonstrate numerically that RPA renormalization beyond first <span class="hlt">order</span> converges monotonically to the infinite-<span class="hlt">order</span> beyond-RPA correlation energy for several model exchange-correlation kernels and that the rate of convergence is principally determined by the choice of the kernel and spin polarization of the ground state. The monotonic convergence is rationalized from an analysis of the RPA renormalized correlation energy corrections, assuming the exchange-correlation kernel and response functions satisfy some reasonable conditions. For spin-unpolarized atoms, molecules, and bulk solids, we find that RPA renormalization is typically converged to 1 meV error or less by fourth <span class="hlt">order</span> regardless of the band gap or dimensionality. Most spin-polarized systems converge at a slightly slower rate, with errors on the <span class="hlt">order</span> of 10 meV at fourth <span class="hlt">order</span> and typically requiring up to sixth <span class="hlt">order</span> to reach 1 meV error or less. Slowest to converge, however, open-shell atoms present the most challenging case and require many higher <span class="hlt">orders</span> to converge.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/22181475','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/22181475"><span>Construction and accuracy of partial differential equation <span class="hlt">approximations</span> to the chemical master equation.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Grima, Ramon</p> <p>2011-11-01</p> <p>The mesoscopic description of chemical kinetics, the chemical master equation, can be exactly solved in only a few simple cases. The analytical intractability stems from the discrete character of the equation, and hence considerable effort has been invested in the development of Fokker-Planck equations, second-<span class="hlt">order</span> partial differential equation <span class="hlt">approximations</span> to the master equation. We here consider two different types of higher-<span class="hlt">order</span> partial differential <span class="hlt">approximations</span>, one derived from the system-size expansion and the other from the Kramers-Moyal expansion, and derive the accuracy of their predictions for chemical reactive networks composed of arbitrary numbers of unimolecular and bimolecular reactions. In particular, we show that the partial differential equation <span class="hlt">approximation</span> of <span class="hlt">order</span> Q from the Kramers-Moyal expansion leads to estimates of the mean number of molecules accurate to <span class="hlt">order</span> Ω(-(2Q-3)/2), of the variance of the fluctuations in the number of molecules accurate to <span class="hlt">order</span> Ω(-(2Q-5)/2), and of skewness accurate to <span class="hlt">order</span> Ω(-(Q-2)). We also show that for large Q, the accuracy in the estimates can be matched only by a partial differential equation <span class="hlt">approximation</span> from the system-size expansion of <span class="hlt">approximate</span> <span class="hlt">order</span> 2Q. Hence, we conclude that partial differential <span class="hlt">approximations</span> based on the Kramers-Moyal expansion generally lead to considerably more accurate estimates in the mean, variance, and skewness than <span class="hlt">approximations</span> of the same <span class="hlt">order</span> derived from the system-size expansion.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018Metro..55..147P','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018Metro..55..147P"><span><span class="hlt">Approximate</span> Bayesian evaluations of measurement uncertainty</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Possolo, Antonio; Bodnar, Olha</p> <p>2018-04-01</p> <p>The Guide to the Expression of Uncertainty in Measurement (GUM) includes formulas that produce an estimate of a scalar output quantity that is a function of several input quantities, and an <span class="hlt">approximate</span> evaluation of the associated standard uncertainty. This contribution presents <span class="hlt">approximate</span>, Bayesian counterparts of those formulas for the case where the output quantity is a parameter of the joint probability distribution of the input quantities, also taking into account any information about the value of the output quantity available prior to measurement expressed in the form of a probability distribution on the set of possible values for the measurand. The <span class="hlt">approximate</span> Bayesian estimates and uncertainty evaluations that we present have a long history and illustrious pedigree, and provide sufficiently accurate <span class="hlt">approximations</span> in many applications, yet are very easy to implement in practice. Differently from exact Bayesian estimates, which involve either (analytical or numerical) integrations, or Markov Chain Monte Carlo sampling, the <span class="hlt">approximations</span> that we describe involve only numerical optimization and simple algebra. Therefore, they make Bayesian methods widely accessible to metrologists. We illustrate the application of the proposed techniques in several instances of measurement: isotopic ratio of silver in a commercial silver nitrate; odds of cryptosporidiosis in AIDS patients; height of a manometer column; mass fraction of chromium in a reference material; and potential-difference in a Zener voltage standard.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19860048947&hterms=Gravitational+motion+system&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D30%26Ntt%3DGravitational%2Bmotion%2Bsystem','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19860048947&hterms=Gravitational+motion+system&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D30%26Ntt%3DGravitational%2Bmotion%2Bsystem"><span><span class="hlt">Approximation</span> methods in gravitational-radiation theory</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Will, C. M.</p> <p>1986-01-01</p> <p>The observation of gravitational-radiation damping in the binary pulsar PSR 1913 + 16 and the ongoing experimental search for gravitational waves of extraterrestrial origin have made the theory of gravitational radiation an active branch of classical general relativity. In calculations of gravitational radiation, <span class="hlt">approximation</span> methods play a crucial role. Recent developments are summarized in two areas in which <span class="hlt">approximations</span> are important: (a) the quadrupole approxiamtion, which determines the energy flux and the radiation reaction forces in weak-field, slow-motion, source-within-the-near-zone systems such as the binary pulsar; and (b) the normal modes of oscillation of black holes, where the Wentzel-Kramers-Brillouin <span class="hlt">approximation</span> gives accurate estimates of the complex frequencies of the modes.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2014JSMTE..02..009F','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2014JSMTE..02..009F"><span>Minimal entropy <span class="hlt">approximation</span> for cellular automata</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Fukś, Henryk</p> <p>2014-02-01</p> <p>We present a method for the construction of <span class="hlt">approximate</span> orbits of measures under the action of cellular automata which is complementary to the local structure theory. The local structure theory is based on the idea of Bayesian extension, that is, construction of a probability measure consistent with given block probabilities and maximizing entropy. If instead of maximizing entropy one minimizes it, one can develop another method for the construction of <span class="hlt">approximate</span> orbits, at the heart of which is the iteration of finite-dimensional maps, called minimal entropy maps. We present numerical evidence that the minimal entropy <span class="hlt">approximation</span> sometimes outperforms the local structure theory in characterizing the properties of cellular automata. The density response curve for elementary CA rule 26 is used to illustrate this claim.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/15732387','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/15732387"><span>Smooth function <span class="hlt">approximation</span> using neural networks.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Ferrari, Silvia; Stengel, Robert F</p> <p>2005-01-01</p> <p>An algebraic approach for representing multidimensional nonlinear functions by feedforward neural networks is presented. In this paper, the approach is implemented for the <span class="hlt">approximation</span> of smooth batch data containing the function's input, output, and possibly, gradient information. The training set is associated to the network adjustable parameters by nonlinear weight equations. The cascade structure of these equations reveals that they can be treated as sets of linear systems. Hence, the training process and the network <span class="hlt">approximation</span> properties can be investigated via linear algebra. Four algorithms are developed to achieve exact or <span class="hlt">approximate</span> matching of input-output and/or gradient-based training sets. Their application to the design of forward and feedback neurocontrollers shows that algebraic training is characterized by faster execution speeds and better generalization properties than contemporary optimization techniques.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2011EPJC...71.1831B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2011EPJC...71.1831B"><span>On Born <span class="hlt">approximation</span> in black hole scattering</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Batic, D.; Kelkar, N. G.; Nowakowski, M.</p> <p>2011-12-01</p> <p>A massless field propagating on spherically symmetric black hole metrics such as the Schwarzschild, Reissner-Nordström and Reissner-Nordström-de Sitter backgrounds is considered. In particular, explicit formulae in terms of transcendental functions for the scattering of massless scalar particles off black holes are derived within a Born <span class="hlt">approximation</span>. It is shown that the conditions on the existence of the Born integral forbid a straightforward extraction of the quasi normal modes using the Born <span class="hlt">approximation</span> for the scattering amplitude. Such a method has been used in literature. We suggest a novel, well defined method, to extract the large imaginary part of quasinormal modes via the Coulomb-like phase shift. Furthermore, we compare the numerically evaluated exact scattering amplitude with the Born one to find that the <span class="hlt">approximation</span> is not very useful for the scattering of massless scalar, electromagnetic as well as gravitational waves from black holes.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19970041606','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19970041606"><span>Exponential <span class="hlt">Approximations</span> Using Fourier Series Partial Sums</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Banerjee, Nana S.; Geer, James F.</p> <p>1997-01-01</p> <p>The problem of accurately reconstructing a piece-wise smooth, 2(pi)-periodic function f and its first few derivatives, given only a truncated Fourier series representation of f, is studied and solved. The reconstruction process is divided into two steps. In the first step, the first 2N + 1 Fourier coefficients of f are used to <span class="hlt">approximate</span> the locations and magnitudes of the discontinuities in f and its first M derivatives. This is accomplished by first finding initial estimates of these quantities based on certain properties of Gibbs phenomenon, and then refining these estimates by fitting the asymptotic form of the Fourier coefficients to the given coefficients using a least-squares approach. It is conjectured that the locations of the singularities are <span class="hlt">approximated</span> to within O(N(sup -M-2), and the associated jump of the k(sup th) derivative of f is <span class="hlt">approximated</span> to within O(N(sup -M-l+k), as N approaches infinity, and the method is robust. These estimates are then used with a class of singular basis functions, which have certain 'built-in' singularities, to construct a new sequence of <span class="hlt">approximations</span> to f. Each of these new <span class="hlt">approximations</span> is the sum of a piecewise smooth function and a new Fourier series partial sum. When N is proportional to M, it is shown that these new <span class="hlt">approximations</span>, and their derivatives, converge exponentially in the maximum norm to f, and its corresponding derivatives, except in the union of a finite number of small open intervals containing the points of singularity of f. The total measure of these intervals decreases exponentially to zero as M approaches infinity. The technique is illustrated with several examples.</p> </li> </ol> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_10");'>10</a></li> <li><a href="#" onclick='return showDiv("page_11");'>11</a></li> <li class="active"><span>12</span></li> <li><a href="#" onclick='return showDiv("page_13");'>13</a></li> <li><a href="#" onclick='return showDiv("page_14");'>14</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div><!-- col-sm-12 --> </div><!-- row --> </div><!-- page_12 --> <div id="page_13" class="hiddenDiv"> <div class="row"> <div class="col-sm-12"> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_11");'>11</a></li> <li><a href="#" onclick='return showDiv("page_12");'>12</a></li> <li class="active"><span>13</span></li> <li><a href="#" onclick='return showDiv("page_14");'>14</a></li> <li><a href="#" onclick='return showDiv("page_15");'>15</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div> </div> <div class="row"> <div class="col-sm-12"> <ol class="result-class" start="241"> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3844179','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3844179"><span>An Expansion Formula with Higher-<span class="hlt">Order</span> Derivatives for Fractional Operators of Variable <span class="hlt">Order</span></span></a></p> <p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p> <p>Almeida, Ricardo</p> <p>2013-01-01</p> <p>We obtain <span class="hlt">approximation</span> formulas for fractional integrals and derivatives of Riemann-Liouville and Marchaud types with a variable fractional <span class="hlt">order</span>. The <span class="hlt">approximations</span> involve integer-<span class="hlt">order</span> derivatives only. An estimation for the error is given. The efficiency of the <span class="hlt">approximation</span> method is illustrated with examples. As applications, we show how the obtained results are useful to solve differential equations, and problems of the calculus of variations that depend on fractional derivatives of Marchaud type. PMID:24319382</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19790054041&hterms=Balancing+equations&qs=N%3D0%26Ntk%3DAll%26Ntx%3Dmode%2Bmatchall%26Ntt%3DBalancing%2Bequations','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19790054041&hterms=Balancing+equations&qs=N%3D0%26Ntk%3DAll%26Ntx%3Dmode%2Bmatchall%26Ntt%3DBalancing%2Bequations"><span><span class="hlt">Approximate</span> convective heating equations for hypersonic flows</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Zoby, E. V.; Moss, J. N.; Sutton, K.</p> <p>1979-01-01</p> <p>Laminar and turbulent heating-rate equations appropriate for engineering predictions of the convective heating rates about blunt reentry spacecraft at hypersonic conditions are developed. The <span class="hlt">approximate</span> methods are applicable to both nonreacting and reacting gas mixtures for either constant or variable-entropy edge conditions. A procedure which accounts for variable-entropy effects and is not based on mass balancing is presented. Results of the <span class="hlt">approximate</span> heating methods are in good agreement with existing experimental results as well as boundary-layer and viscous-shock-layer solutions.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018LMaPh.108..185B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018LMaPh.108..185B"><span>On the dipole <span class="hlt">approximation</span> with error estimates</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Boßmann, Lea; Grummt, Robert; Kolb, Martin</p> <p>2018-01-01</p> <p>The dipole <span class="hlt">approximation</span> is employed to describe interactions between atoms and radiation. It essentially consists of neglecting the spatial variation of the external field over the atom. Heuristically, this is justified by arguing that the wavelength is considerably larger than the atomic length scale, which holds under usual experimental conditions. We prove the dipole <span class="hlt">approximation</span> in the limit of infinite wavelengths compared to the atomic length scale and estimate the rate of convergence. Our results include N-body Coulomb potentials and experimentally relevant electromagnetic fields such as plane waves and laser pulses.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=20020052603&hterms=congruence&qs=N%3D0%26Ntk%3DAll%26Ntx%3Dmode%2Bmatchall%26Ntt%3Dcongruence','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=20020052603&hterms=congruence&qs=N%3D0%26Ntk%3DAll%26Ntx%3Dmode%2Bmatchall%26Ntt%3Dcongruence"><span>Congruence <span class="hlt">Approximations</span> for Entrophy Endowed Hyperbolic Systems</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Barth, Timothy J.; Saini, Subhash (Technical Monitor)</p> <p>1998-01-01</p> <p>Building upon the standard symmetrization theory for hyperbolic systems of conservation laws, congruence properties of the symmetrized system are explored. These congruence properties suggest variants of several stabilized numerical discretization procedures for hyperbolic equations (upwind finite-volume, Galerkin least-squares, discontinuous Galerkin) that benefit computationally from congruence <span class="hlt">approximation</span>. Specifically, it becomes straightforward to construct the spatial discretization and Jacobian linearization for these schemes (given a small amount of derivative information) for possible use in Newton's method, discrete optimization, homotopy algorithms, etc. Some examples will be given for the compressible Euler equations and the nonrelativistic MHD equations using linear and quadratic spatial <span class="hlt">approximation</span>.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/957262-solving-infeasible-trust-region-problem-using-approximations','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/957262-solving-infeasible-trust-region-problem-using-approximations"><span>Solving the infeasible trust-region problem using <span class="hlt">approximations</span>.</span></a></p> <p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p> <p>Renaud, John E.; Perez, Victor M.; Eldred, Michael Scott</p> <p>2004-07-01</p> <p>The use of optimization in engineering design has fueled the development of algorithms for specific engineering needs. When the simulations are expensive to evaluate or the outputs present some noise, the direct use of nonlinear optimizers is not advisable, since the optimization process will be expensive and may result in premature convergence. The use of <span class="hlt">approximations</span> for both cases is an alternative investigated by many researchers including the authors. When <span class="hlt">approximations</span> are present, a model management is required for proper convergence of the algorithm. In nonlinear programming, the use of trust-regions for globalization of a local algorithm has been provenmore » effective. The same approach has been used to manage the local move limits in sequential <span class="hlt">approximate</span> optimization frameworks as in Alexandrov et al., Giunta and Eldred, Perez et al. , Rodriguez et al., etc. The experience in the mathematical community has shown that more effective algorithms can be obtained by the specific inclusion of the constraints (SQP type of algorithms) rather than by using a penalty function as in the augmented Lagrangian formulation. The presence of explicit constraints in the local problem bounded by the trust region, however, may have no feasible solution. In <span class="hlt">order</span> to remedy this problem the mathematical community has developed different versions of a composite steps approach. This approach consists of a normal step to reduce the amount of constraint violation and a tangential step to minimize the objective function maintaining the level of constraint violation attained at the normal step. Two of the authors have developed a different approach for a sequential <span class="hlt">approximate</span> optimization framework using homotopy ideas to relax the constraints. This algorithm called interior-point trust-region sequential <span class="hlt">approximate</span> optimization (IPTRSAO) presents some similarities to the two normal-tangential steps algorithms. In this paper, a description of the similarities is presented</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20090041751','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20090041751"><span>Quickly <span class="hlt">Approximating</span> the Distance Between Two Objects</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Hammen, David</p> <p>2009-01-01</p> <p>A method of quickly <span class="hlt">approximating</span> the distance between two objects (one smaller, regarded as a point; the other larger and complexly shaped) has been devised for use in computationally simulating motions of the objects for the purpose of planning the motions to prevent collisions.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4652602','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4652602"><span>Fostering Formal Commutativity Knowledge with <span class="hlt">Approximate</span> Arithmetic</span></a></p> <p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p> <p>Hansen, Sonja Maria; Haider, Hilde; Eichler, Alexandra; Godau, Claudia; Frensch, Peter A.; Gaschler, Robert</p> <p>2015-01-01</p> <p>How can we enhance the understanding of abstract mathematical principles in elementary school? Different studies found out that nonsymbolic estimation could foster subsequent exact number processing and simple arithmetic. Taking the commutativity principle as a test case, we investigated if the <span class="hlt">approximate</span> calculation of symbolic commutative quantities can also alter the access to procedural and conceptual knowledge of a more abstract arithmetic principle. Experiment 1 tested first graders who had not been instructed about commutativity in school yet. <span class="hlt">Approximate</span> calculation with symbolic quantities positively influenced the use of commutativity-based shortcuts in formal arithmetic. We replicated this finding with older first graders (Experiment 2) and third graders (Experiment 3). Despite the positive effect of <span class="hlt">approximation</span> on the spontaneous application of commutativity-based shortcuts in arithmetic problems, we found no comparable impact on the application of conceptual knowledge of the commutativity principle. Overall, our results show that the usage of a specific arithmetic principle can benefit from <span class="hlt">approximation</span>. However, the findings also suggest that the correct use of certain procedures does not always imply conceptual understanding. Rather, the conceptual understanding of commutativity seems to lag behind procedural proficiency during elementary school. PMID:26560311</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018InvPr..34b5005A','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018InvPr..34b5005A"><span>Variational Gaussian <span class="hlt">approximation</span> for Poisson data</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Arridge, Simon R.; Ito, Kazufumi; Jin, Bangti; Zhang, Chen</p> <p>2018-02-01</p> <p>The Poisson model is frequently employed to describe count data, but in a Bayesian context it leads to an analytically intractable posterior probability distribution. In this work, we analyze a variational Gaussian <span class="hlt">approximation</span> to the posterior distribution arising from the Poisson model with a Gaussian prior. This is achieved by seeking an optimal Gaussian distribution minimizing the Kullback-Leibler divergence from the posterior distribution to the <span class="hlt">approximation</span>, or equivalently maximizing the lower bound for the model evidence. We derive an explicit expression for the lower bound, and show the existence and uniqueness of the optimal Gaussian <span class="hlt">approximation</span>. The lower bound functional can be viewed as a variant of classical Tikhonov regularization that penalizes also the covariance. Then we develop an efficient alternating direction maximization algorithm for solving the optimization problem, and analyze its convergence. We discuss strategies for reducing the computational complexity via low rank structure of the forward operator and the sparsity of the covariance. Further, as an application of the lower bound, we discuss hierarchical Bayesian modeling for selecting the hyperparameter in the prior distribution, and propose a monotonically convergent algorithm for determining the hyperparameter. We present extensive numerical experiments to illustrate the Gaussian <span class="hlt">approximation</span> and the algorithms.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19760013825','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19760013825"><span>Multidimensional stochastic <span class="hlt">approximation</span> using locally contractive functions</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Lawton, W. M.</p> <p>1975-01-01</p> <p>A Robbins-Monro type multidimensional stochastic <span class="hlt">approximation</span> algorithm which converges in mean square and with probability one to the fixed point of a locally contractive regression function is developed. The algorithm is applied to obtain maximum likelihood estimates of the parameters for a mixture of multivariate normal distributions.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.fs.usda.gov/treesearch/pubs/32874','TREESEARCH'); return false;" href="https://www.fs.usda.gov/treesearch/pubs/32874"><span>Variance <span class="hlt">approximations</span> for assessments of classification accuracy</span></a></p> <p><a target="_blank" href="http://www.fs.usda.gov/treesearch/">Treesearch</a></p> <p>R. L. Czaplewski</p> <p>1994-01-01</p> <p>Variance <span class="hlt">approximations</span> are derived for the weighted and unweighted kappa statistics, the conditional kappa statistic, and conditional probabilities. These statistics are useful to assess classification accuracy, such as accuracy of remotely sensed classifications in thematic maps when compared to a sample of reference classifications made in the field. Published...</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19850012840','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19850012840"><span>Alternative <span class="hlt">approximation</span> concepts for space frame synthesis</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Lust, R. V.; Schmit, L. A.</p> <p>1985-01-01</p> <p>A structural synthesis methodology for the minimum mass design of 3-dimensionall frame-truss structures under multiple static loading conditions and subject to limits on displacements, rotations, stresses, local buckling, and element cross-sectional dimensions is presented. A variety of <span class="hlt">approximation</span> concept options are employed to yield near optimum designs after no more than 10 structural analyses. Available options include: (A) formulation of the nonlinear mathematcal programming problem in either reciprocal section property (RSP) or cross-sectional dimension (CSD) space; (B) two alternative <span class="hlt">approximate</span> problem structures in each design space; and (C) three distinct assumptions about element end-force variations. Fixed element, design element linking, and temporary constraint deletion features are also included. The solution of each <span class="hlt">approximate</span> problem, in either its primal or dual form, is obtained using CONMIN, a feasible directions program. The frame-truss synthesis methodology is implemented in the COMPASS computer program and is used to solve a variety of problems. These problems were chosen so that, in addition to exercising the various <span class="hlt">approximation</span> concepts options, the results could be compared with previously published work.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/AD0628888','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/AD0628888"><span>RATIONAL <span class="hlt">APPROXIMATIONS</span> TO GENERALIZED HYPERGEOMETRIC FUNCTIONS.</span></a></p> <p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p> <p></p> <p></p> <p>Under weak restrictions on the various free parameters, general theorems for rational representations of the generalized hypergeometric functions...and certain Meijer G-functions are developed. Upon specialization, these theorems yield a sequency of rational <span class="hlt">approximations</span> which converge to the</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/1992JChPh..96.6654Z','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/1992JChPh..96.6654Z"><span>An <span class="hlt">approximate</span> classical unimolecular reaction rate theory</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Zhao, Meishan; Rice, Stuart A.</p> <p>1992-05-01</p> <p>We describe a classical theory of unimolecular reaction rate which is derived from the analysis of Davis and Gray by use of simplifying <span class="hlt">approximations</span>. These <span class="hlt">approximations</span> concern the calculation of the locations of, and the fluxes of phase points across, the bottlenecks to fragmentation and to intramolecular energy transfer. The bottleneck to fragment separation is represented as a vibration-rotation state dependent separatrix, which <span class="hlt">approximation</span> is similar to but extends and improves the <span class="hlt">approximations</span> for the separatrix introduced by Gray, Rice, and Davis and by Zhao and Rice. The novel feature in our analysis is the representation of the bottlenecks to intramolecular energy transfer as dividing surfaces in phase space; the locations of these dividing surfaces are determined by the same conditions as locate the remnants of robust tori with frequency ratios related to the golden mean (in a two degree of freedom system these are the cantori). The flux of phase points across each dividing surface is calculated with an analytic representation instead of a stroboscopic mapping. The rate of unimolecular reaction is identified with the net rate at which phase points escape from the region of quasiperiodic bounded motion to the region of free fragment motion by consecutively crossing the dividing surfaces for intramolecular energy exchange and the separatrix. This new theory generates predictions of the rates of predissociation of the van der Waals molecules HeI2, NeI2 and ArI2 which are in very good agreement with available experimental data.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19900019726','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19900019726"><span><span class="hlt">Approximation</span> algorithms for planning and control</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Boddy, Mark; Dean, Thomas</p> <p>1989-01-01</p> <p>A control system operating in a complex environment will encounter a variety of different situations, with varying amounts of time available to respond to critical events. Ideally, such a control system will do the best possible with the time available. In other words, its responses should <span class="hlt">approximate</span> those that would result from having unlimited time for computation, where the degree of the <span class="hlt">approximation</span> depends on the amount of time it actually has. There exist <span class="hlt">approximation</span> algorithms for a wide variety of problems. Unfortunately, the solution to any reasonably complex control problem will require solving several computationally intensive problems. Algorithms for successive <span class="hlt">approximation</span> are a subclass of the class of anytime algorithms, algorithms that return answers for any amount of computation time, where the answers improve as more time is allotted. An architecture is described for allocating computation time to a set of anytime algorithms, based on expectations regarding the value of the answers they return. The architecture described is quite general, producing optimal schedules for a set of algorithms under widely varying conditions.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://files.eric.ed.gov/fulltext/EJ1128175.pdf','ERIC'); return false;" href="http://files.eric.ed.gov/fulltext/EJ1128175.pdf"><span>Tension and <span class="hlt">Approximation</span> in Poetic Translation</span></a></p> <p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p> <p>Al-Shabab, Omar A. S.; Baka, Farida H.</p> <p>2015-01-01</p> <p>Simple observation reveals that each language and each culture enjoys specific linguistic features and rhetorical traditions. In poetry translation difference and the resultant linguistic tension create a gap between Source Language and Target language, a gap that needs to be bridged by creating an <span class="hlt">approximation</span> processed through the translator's…</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=2137171','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=2137171"><span>Two-Term Asymptotic <span class="hlt">Approximation</span> of a Cardiac Restitution Curve*</span></a></p> <p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p> <p>Cain, John W.; Schaeffer, David G.</p> <p>2007-01-01</p> <p>If spatial extent is neglected, ionic models of cardiac cells consist of systems of ordinary differential equations (ODEs) which have the property of excitability, i.e., a brief stimulus produces a prolonged evolution (called an action potential in the cardiac context) before the eventual return to equilibrium. Under repeated stimulation, or pacing, cardiac tissue exhibits electrical restitution: the steady-state action potential duration (APD) at a given pacing period B shortens as B is decreased. Independent of ionic models, restitution is often modeled phenomenologically by a one-dimensional mapping of the form APDnext = f(B – APDprevious). Under some circumstances, a restitution function f can be derived as an asymptotic <span class="hlt">approximation</span> to the behavior of an ionic model. In this paper, extending previous work, we derive the next term in such an asymptotic <span class="hlt">approximation</span> for a particular ionic model consisting of two ODEs. The two-term <span class="hlt">approximation</span> exhibits excellent quantitative agreement with the actual restitution curve, whereas the leading-<span class="hlt">order</span> <span class="hlt">approximation</span> significantly underestimates actual APD values. PMID:18080006</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016PhRvB..94g5158D','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016PhRvB..94g5158D"><span>Revised Thomas-Fermi <span class="hlt">approximation</span> for singular potentials</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Dufty, James W.; Trickey, S. B.</p> <p>2016-08-01</p> <p><span class="hlt">Approximations</span> for the many-fermion free-energy density functional that include the Thomas-Fermi (TF) form for the noninteracting part lead to singular densities for singular external potentials (e.g., attractive Coulomb). This limitation of the TF <span class="hlt">approximation</span> is addressed here by a formal map of the exact Euler equation for the density onto an equivalent TF form characterized by a modified Kohn-Sham potential. It is shown to be a "regularized" version of the Kohn-Sham potential, tempered by convolution with a finite-temperature response function. The resulting density is nonsingular, with the equilibrium properties obtained from the total free-energy functional evaluated at this density. This new representation is formally exact. <span class="hlt">Approximate</span> expressions for the regularized potential are given to leading <span class="hlt">order</span> in a nonlocality parameter, and the limiting behavior at high and low temperatures is described. The noninteracting part of the free energy in this <span class="hlt">approximation</span> is the usual Thomas-Fermi functional. These results generalize and extend to finite temperatures the ground-state regularization by R. G. Parr and S. Ghosh [Proc. Natl. Acad. Sci. U.S.A. 83, 3577 (1986), 10.1073/pnas.83.11.3577] and by L. R. Pratt, G. G. Hoffman, and R. A. Harris [J. Chem. Phys. 88, 1818 (1988), 10.1063/1.454105] and formally systematize the finite-temperature regularization given by the latter authors.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016JPhCS.699a2017O','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016JPhCS.699a2017O"><span>Sparse <span class="hlt">approximation</span> problem: how rapid simulated annealing succeeds and fails</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Obuchi, Tomoyuki; Kabashima, Yoshiyuki</p> <p>2016-03-01</p> <p>Information processing techniques based on sparseness have been actively studied in several disciplines. Among them, a mathematical framework to <span class="hlt">approximately</span> express a given dataset by a combination of a small number of basis vectors of an overcomplete basis is termed the sparse <span class="hlt">approximation</span>. In this paper, we apply simulated annealing, a metaheuristic algorithm for general optimization problems, to sparse <span class="hlt">approximation</span> in the situation where the given data have a planted sparse representation and noise is present. The result in the noiseless case shows that our simulated annealing works well in a reasonable parameter region: the planted solution is found fairly rapidly. This is true even in the case where a common relaxation of the sparse <span class="hlt">approximation</span> problem, the G-relaxation, is ineffective. On the other hand, when the dimensionality of the data is close to the number of non-zero components, another metastable state emerges, and our algorithm fails to find the planted solution. This phenomenon is associated with a first-<span class="hlt">order</span> phase transition. In the case of very strong noise, it is no longer meaningful to search for the planted solution. In this situation, our algorithm determines a solution with close-to-minimum distortion fairly quickly.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2015JHyd..523..278B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2015JHyd..523..278B"><span><span class="hlt">Approximation</span> of the exponential integral (well function) using sampling methods</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Baalousha, Husam Musa</p> <p>2015-04-01</p> <p>Exponential integral (also known as well function) is often used in hydrogeology to solve Theis and Hantush equations. Many methods have been developed to <span class="hlt">approximate</span> the exponential integral. Most of these methods are based on numerical <span class="hlt">approximations</span> and are valid for a certain range of the argument value. This paper presents a new approach to <span class="hlt">approximate</span> the exponential integral. The new approach is based on sampling methods. Three different sampling methods; Latin Hypercube Sampling (LHS), Orthogonal Array (OA), and Orthogonal Array-based Latin Hypercube (OA-LH) have been used to <span class="hlt">approximate</span> the function. Different argument values, covering a wide range, have been used. The results of sampling methods were compared with results obtained by Mathematica software, which was used as a benchmark. All three sampling methods converge to the result obtained by Mathematica, at different rates. It was found that the orthogonal array (OA) method has the fastest convergence rate compared with LHS and OA-LH. The root mean square error RMSE of OA was in the <span class="hlt">order</span> of 1E-08. This method can be used with any argument value, and can be used to solve other integrals in hydrogeology such as the leaky aquifer integral.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/22390649-data-approximation-using-blending-type-spline-construction','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/22390649-data-approximation-using-blending-type-spline-construction"><span>Data <span class="hlt">approximation</span> using a blending type spline construction</span></a></p> <p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p> <p>Dalmo, Rune; Bratlie, Jostein</p> <p>2014-11-18</p> <p>Generalized expo-rational B-splines (GERBS) is a blending type spline construction where local functions at each knot are blended together by C{sup k}-smooth basis functions. One way of <span class="hlt">approximating</span> discrete regular data using GERBS is by partitioning the data set into subsets and fit a local function to each subset. Partitioning and fitting strategies can be devised such that important or interesting data points are interpolated in <span class="hlt">order</span> to preserve certain features. We present a method for fitting discrete data using a tensor product GERBS construction. The method is based on detection of feature points using differential geometry. Derivatives, which aremore » necessary for feature point detection and used to construct local surface patches, are <span class="hlt">approximated</span> from the discrete data using finite differences.« less</p> </li> </ol> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_11");'>11</a></li> <li><a href="#" onclick='return showDiv("page_12");'>12</a></li> <li class="active"><span>13</span></li> <li><a href="#" onclick='return showDiv("page_14");'>14</a></li> <li><a href="#" onclick='return showDiv("page_15");'>15</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div><!-- col-sm-12 --> </div><!-- row --> </div><!-- page_13 --> <div id="page_14" class="hiddenDiv"> <div class="row"> <div class="col-sm-12"> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_12");'>12</a></li> <li><a href="#" onclick='return showDiv("page_13");'>13</a></li> <li class="active"><span>14</span></li> <li><a href="#" onclick='return showDiv("page_15");'>15</a></li> <li><a href="#" onclick='return showDiv("page_16");'>16</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div> </div> <div class="row"> <div class="col-sm-12"> <ol class="result-class" start="261"> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/22667643-approximating-tunneling-rates-multi-dimensional-field-spaces','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/22667643-approximating-tunneling-rates-multi-dimensional-field-spaces"><span><span class="hlt">Approximating</span> tunneling rates in multi-dimensional field spaces</span></a></p> <p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p> <p>Masoumi, Ali; Olum, Ken D.; Wachter, Jeremy M., E-mail: ali@cosmos.phy.tufts.edu, E-mail: kdo@cosmos.phy.tufts.edu, E-mail: Jeremy.Wachter@tufts.edu</p> <p></p> <p>Quantum mechanics makes the otherwise stable vacua of a theory metastable through the nucleation of bubbles of the new vacuum. This in turn causes a first <span class="hlt">order</span> phase transition. These cosmological phase transitions may have played an important role in settling our universe into its current vacuum, and they may also happen in future. The most important frameworks where vacuum decay happens contain a large number of fields. Unfortunately, calculating the tunneling rates in these models is very time-consuming. In this paper we present a simple <span class="hlt">approximation</span> for the tunneling rate by reducing it to a one-field problem which ismore » easy to calculate. We demonstrate the validity of this <span class="hlt">approximation</span> using our recent code 'Anybubble' for several classes of potentials.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017JCAP...10..022M','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017JCAP...10..022M"><span><span class="hlt">Approximating</span> tunneling rates in multi-dimensional field spaces</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Masoumi, Ali; Olum, Ken D.; Wachter, Jeremy M.</p> <p>2017-10-01</p> <p>Quantum mechanics makes the otherwise stable vacua of a theory metastable through the nucleation of bubbles of the new vacuum. This in turn causes a first <span class="hlt">order</span> phase transition. These cosmological phase transitions may have played an important role in settling our universe into its current vacuum, and they may also happen in future. The most important frameworks where vacuum decay happens contain a large number of fields. Unfortunately, calculating the tunneling rates in these models is very time-consuming. In this paper we present a simple <span class="hlt">approximation</span> for the tunneling rate by reducing it to a one-field problem which is easy to calculate. We demonstrate the validity of this <span class="hlt">approximation</span> using our recent code "Anybubble" for several classes of potentials.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018JPhCS1015c2181P','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018JPhCS1015c2181P"><span>Investigation of <span class="hlt">approximate</span> models of experimental temperature characteristics of machines</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Parfenov, I. V.; Polyakov, A. N.</p> <p>2018-05-01</p> <p>This work is devoted to the investigation of various approaches to the <span class="hlt">approximation</span> of experimental data and the creation of simulation mathematical models of thermal processes in machines with the aim of finding ways to reduce the time of their field tests and reducing the temperature error of the treatments. The main methods of research which the authors used in this work are: the full-scale thermal testing of machines; realization of various approaches at <span class="hlt">approximation</span> of experimental temperature characteristics of machine tools by polynomial models; analysis and evaluation of modelling results (model quality) of the temperature characteristics of machines and their derivatives up to the third <span class="hlt">order</span> in time. As a result of the performed researches, rational methods, type, parameters and complexity of simulation mathematical models of thermal processes in machine tools are proposed.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/5296447-uniform-semiclassical-sudden-approximation-rotationally-inelastic-scattering','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/5296447-uniform-semiclassical-sudden-approximation-rotationally-inelastic-scattering"><span>Uniform semiclassical sudden <span class="hlt">approximation</span> for rotationally inelastic scattering</span></a></p> <p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p> <p>Korsch, H.J.; Schinke, R.</p> <p>1980-08-01</p> <p>The infinite-<span class="hlt">order</span>-sudden (IOS) <span class="hlt">approximation</span> is investigated in the semiclassical limit. A simplified IOS formula for rotationally inelastic differential cross sections is derived involving a uniform stationary phase <span class="hlt">approximation</span> for two-dimensional oscillatory integrals with two stationary points. The semiclassical analysis provides a quantitative description of the rotational rainbow structure in the differential cross section. The numerical calculation of semiclassical IOS cross sections is extremely fast compared to numerically exact IOS methods, especially if high ..delta..j transitions are involved. Rigid rotor results for He--Na/sub 2/ collisions with ..delta..j< or approx. =26 and for K--CO collisions with ..delta..j< or approx. =70 show satisfactorymore » agreement with quantal IOS calculations.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2003APS..DFD.KG004H','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2003APS..DFD.KG004H"><span>A Nonlinear Interactions <span class="hlt">Approximation</span> Model for Large-Eddy Simulation</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Haliloglu, Mehmet U.; Akhavan, Rayhaneh</p> <p>2003-11-01</p> <p>A new approach to LES modelling is proposed based on direct <span class="hlt">approximation</span> of the nonlinear terms \\overlineu_iuj in the filtered Navier-Stokes equations, instead of the subgrid-scale stress, τ_ij. The proposed model, which we call the Nonlinear Interactions <span class="hlt">Approximation</span> (NIA) model, uses graded filters and deconvolution to parameterize the local interactions across the LES cutoff, and a Smagorinsky eddy viscosity term to parameterize the distant interactions. A dynamic procedure is used to determine the unknown eddy viscosity coefficient, rendering the model free of adjustable parameters. The proposed NIA model has been applied to LES of turbulent channel flows at Re_τ ≈ 210 and Re_τ ≈ 570. The results show good agreement with DNS not only for the mean and resolved second-<span class="hlt">order</span> turbulence statistics but also for the full (resolved plus subgrid) Reynolds stress and turbulence intensities.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016JSMTE..11.3401T','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016JSMTE..11.3401T"><span>Typical performance of <span class="hlt">approximation</span> algorithms for NP-hard problems</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Takabe, Satoshi; Hukushima, Koji</p> <p>2016-11-01</p> <p>Typical performance of <span class="hlt">approximation</span> algorithms is studied for randomized minimum vertex cover problems. A wide class of random graph ensembles characterized by an arbitrary degree distribution is discussed with the presentation of a theoretical framework. Herein, three <span class="hlt">approximation</span> algorithms are examined: linear-programming relaxation, loopy-belief propagation, and the leaf-removal algorithm. The former two algorithms are analyzed using a statistical-mechanical technique, whereas the average-case analysis of the last one is conducted using the generating function method. These algorithms have a threshold in the typical performance with increasing average degree of the random graph, below which they find true optimal solutions with high probability. Our study reveals that there exist only three cases, determined by the <span class="hlt">order</span> of the typical performance thresholds. In addition, we provide some conditions for classification of the graph ensembles and demonstrate explicitly some examples for the difference in thresholds.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/22680072-slow-roll-approximation-loop-quantum-cosmology','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/22680072-slow-roll-approximation-loop-quantum-cosmology"><span>Slow-roll <span class="hlt">approximation</span> in loop quantum cosmology</span></a></p> <p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p> <p>Luc, Joanna; Mielczarek, Jakub, E-mail: joanna.luc@uj.edu.pl, E-mail: jakub.mielczarek@uj.edu.pl</p> <p></p> <p>The slow-roll <span class="hlt">approximation</span> is an analytical approach to study dynamical properties of the inflationary universe. In this article, systematic construction of the slow-roll expansion for effective loop quantum cosmology is presented. The analysis is performed up to the fourth <span class="hlt">order</span> in both slow-roll parameters and the parameter controlling the strength of deviation from the classical case. The expansion is performed for three types of the slow-roll parameters: Hubble slow-roll parameters, Hubble flow parameters and potential slow-roll parameters. An accuracy of the <span class="hlt">approximation</span> is verified by comparison with the numerical phase space trajectories for the case with a massive potential term.more » The results obtained in this article may be helpful in the search for the subtle quantum gravitational effects with use of the cosmological data.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2006JCoAM.189..494S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2006JCoAM.189..494S"><span>Capturing planar shapes by <span class="hlt">approximating</span> their outlines</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Sarfraz, M.; Riyazuddin, M.; Baig, M. H.</p> <p>2006-05-01</p> <p>A non-deterministic evolutionary approach for <span class="hlt">approximating</span> the outlines of planar shapes has been developed. Non-uniform Rational B-splines (NURBS) have been utilized as an underlying <span class="hlt">approximation</span> curve scheme. Simulated Annealing heuristic is used as an evolutionary methodology. In addition to independent studies of the optimization of weight and knot parameters of the NURBS, a separate scheme has also been developed for the optimization of weights and knots simultaneously. The optimized NURBS models have been fitted over the contour data of the planar shapes for the ultimate and automatic output. The output results are visually pleasing with respect to the threshold provided by the user. A web-based system has also been developed for the effective and worldwide utilization. The objective of this system is to provide the facility to visualize the output to the whole world through internet by providing the freedom to the user for various desired input parameters setting in the algorithm designed.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2008PhRvL.101n0503O','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2008PhRvL.101n0503O"><span><span class="hlt">Approximate</span> Locality for Quantum Systems on Graphs</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Osborne, Tobias J.</p> <p>2008-10-01</p> <p>In this Letter we make progress on a long-standing open problem of Aaronson and Ambainis [Theory Comput. 1, 47 (2005)1557-2862]: we show that if U is a sparse unitary operator with a gap Δ in its spectrum, then there exists an <span class="hlt">approximate</span> logarithm H of U which is also sparse. The sparsity pattern of H gets more dense as 1/Δ increases. This result can be interpreted as a way to convert between local continuous-time and local discrete-time quantum processes. As an example we show that the discrete-time coined quantum walk can be realized stroboscopically from an <span class="hlt">approximately</span> local continuous-time quantum walk.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/pages/biblio/1407456-incomplete-sparse-approximate-inverses-parallel-preconditioning','SCIGOV-DOEP'); return false;" href="https://www.osti.gov/pages/biblio/1407456-incomplete-sparse-approximate-inverses-parallel-preconditioning"><span>Incomplete Sparse <span class="hlt">Approximate</span> Inverses for Parallel Preconditioning</span></a></p> <p><a target="_blank" href="http://www.osti.gov/pages">DOE PAGES</a></p> <p>Anzt, Hartwig; Huckle, Thomas K.; Bräckle, Jürgen; ...</p> <p>2017-10-28</p> <p>In this study, we propose a new preconditioning method that can be seen as a generalization of block-Jacobi methods, or as a simplification of the sparse <span class="hlt">approximate</span> inverse (SAI) preconditioners. The “Incomplete Sparse <span class="hlt">Approximate</span> Inverses” (ISAI) is in particular efficient in the solution of sparse triangular linear systems of equations. Those arise, for example, in the context of incomplete factorization preconditioning. ISAI preconditioners can be generated via an algorithm providing fine-grained parallelism, which makes them attractive for hardware with a high concurrency level. Finally, in a study covering a large number of matrices, we identify the ISAI preconditioner as anmore » attractive alternative to exact triangular solves in the context of incomplete factorization preconditioning.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4865172','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4865172"><span>Solving Math Problems <span class="hlt">Approximately</span>: A Developmental Perspective</span></a></p> <p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p> <p>Ganor-Stern, Dana</p> <p>2016-01-01</p> <p>Although solving arithmetic problems <span class="hlt">approximately</span> is an important skill in everyday life, little is known about the development of this skill. Past research has shown that when children are asked to solve multi-digit multiplication problems <span class="hlt">approximately</span>, they provide estimates that are often very far from the exact answer. This is unfortunate as computation estimation is needed in many circumstances in daily life. The present study examined 4th graders, 6th graders and adults’ ability to estimate the results of arithmetic problems relative to a reference number. A developmental pattern was observed in accuracy, speed and strategy use. With age there was a general increase in speed, and an increase in accuracy mainly for trials in which the reference number was close to the exact answer. The children tended to use the sense of magnitude strategy, which does not involve any calculation but relies mainly on an intuitive coarse sense of magnitude, while the adults used the <span class="hlt">approximated</span> calculation strategy which involves rounding and multiplication procedures, and relies to a greater extent on calculation skills and working memory resources. Importantly, the children were less accurate than the adults, but were well above chance level. In all age groups performance was enhanced when the reference number was smaller (vs. larger) than the exact answer and when it was far (vs. close) from it, suggesting the involvement of an <span class="hlt">approximate</span> number system. The results suggest the existence of an intuitive sense of magnitude for the results of arithmetic problems that might help children and even adults with difficulties in math. The present findings are discussed in the context of past research reporting poor estimation skills among children, and the conditions that might allow using children estimation skills in an effective manner. PMID:27171224</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20110023853','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20110023853"><span>Capacitor-Chain Successive-<span class="hlt">Approximation</span> ADC</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Cunningham, Thomas</p> <p>2003-01-01</p> <p>A proposed successive-<span class="hlt">approximation</span> analog-to-digital converter (ADC) would contain a capacitively terminated chain of identical capacitor cells. Like a conventional successive-<span class="hlt">approximation</span> ADC containing a bank of binary-scaled capacitors, the proposed ADC would store an input voltage on a sample-and-hold capacitor and would digitize the stored input voltage by finding the closest match between this voltage and a capacitively generated sum of binary fractions of a reference voltage (Vref). However, the proposed capacitor-chain ADC would offer two major advantages over a conventional binary-scaled-capacitor ADC: (1) In a conventional ADC that digitizes to n bits, the largest capacitor (representing the most significant bit) must have 2(exp n-1) times as much capacitance, and hence, <span class="hlt">approximately</span> 2(exp n-1) times as much area as does the smallest capacitor (representing the least significant bit), so that the total capacitor area must be 2(exp n) times that of the smallest capacitor. In the proposed capacitor-chain ADC, there would be three capacitors per cell, each <span class="hlt">approximately</span> equal to the smallest capacitor in the conventional ADC, and there would be one cell per bit. Therefore, the total capacitor area would be only about 3(exp n) times that of the smallest capacitor. The net result would be that the proposed ADC could be considerably smaller than the conventional ADC. (2) Because of edge effects, parasitic capacitances, and manufacturing tolerances, it is difficult to make capacitor banks in which the values of capacitance are scaled by powers of 2 to the required precision. In contrast, because all the capacitors in the proposed ADC would be identical, the problem of precise binary scaling would not arise.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/ADA551784','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/ADA551784"><span>Optimal Budget Allocation for Sample Average <span class="hlt">Approximation</span></span></a></p> <p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p> <p></p> <p>2011-06-01</p> <p>an optimization algorithm applied to the sample average problem. We examine the convergence rate of the estimator as the computing budget tends to...regime for the optimization algorithm . 1 Introduction Sample average <span class="hlt">approximation</span> (SAA) is a frequently used approach to solving stochastic programs...appealing due to its simplicity and the fact that a large number of standard optimization algorithms are often available to optimize the resulting sample</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2015PhDT.......103N','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2015PhDT.......103N"><span><span class="hlt">Approximation</span> methods in relativistic eigenvalue perturbation theory</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Noble, Jonathan Howard</p> <p></p> <p>In this dissertation, three questions, concerning <span class="hlt">approximation</span> methods for the eigenvalues of quantum mechanical systems, are investigated: (i) What is a pseudo--Hermitian Hamiltonian, and how can its eigenvalues be <span class="hlt">approximated</span> via numerical calculations? This is a fairly broad topic, and the scope of the investigation is narrowed by focusing on a subgroup of pseudo--Hermitian operators, namely, PT--symmetric operators. Within a numerical approach, one projects a PT--symmetric Hamiltonian onto an appropriate basis, and uses a straightforward two--step algorithm to diagonalize the resulting matrix, leading to numerically <span class="hlt">approximated</span> eigenvalues. (ii) Within an analytic ansatz, how can a relativistic Dirac Hamiltonian be decoupled into particle and antiparticle degrees of freedom, in appropriate kinematic limits? One possible answer is the Foldy--Wouthuysen transform; however, there are alter- native methods which seem to have some advantages over the time--tested approach. One such method is investigated by applying both the traditional Foldy--Wouthuysen transform and the "chiral" Foldy--Wouthuysen transform to a number of Dirac Hamiltonians, including the central-field Hamiltonian for a gravitationally bound system; namely, the Dirac-(Einstein-)Schwarzschild Hamiltonian, which requires the formal- ism of general relativity. (iii) Are there are pseudo--Hermitian variants of Dirac Hamiltonians that can be <span class="hlt">approximated</span> using a decoupling transformation? The tachyonic Dirac Hamiltonian, which describes faster-than-light spin-1/2 particles, is gamma5--Hermitian, i.e., pseudo-Hermitian. Superluminal particles remain faster than light upon a Lorentz transformation, and hence, the Foldy--Wouthuysen program is unsuited for this case. Thus, inspired by the Foldy--Wouthuysen program, a decoupling transform in the ultrarelativistic limit is proposed, which is applicable to both sub- and superluminal particles.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19930007473','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19930007473"><span><span class="hlt">Approximation</span> methods for stochastic petri nets</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Jungnitz, Hauke Joerg</p> <p>1992-01-01</p> <p>Stochastic Marked Graphs are a concurrent decision free formalism provided with a powerful synchronization mechanism generalizing conventional Fork Join Queueing Networks. In some particular cases the analysis of the throughput can be done analytically. Otherwise the analysis suffers from the classical state explosion problem. Embedded in the divide and conquer paradigm, <span class="hlt">approximation</span> techniques are introduced for the analysis of stochastic marked graphs and Macroplace/Macrotransition-nets (MPMT-nets), a new subclass introduced herein. MPMT-nets are a subclass of Petri nets that allow limited choice, concurrency and sharing of resources. The modeling power of MPMT is much larger than that of marked graphs, e.g., MPMT-nets can model manufacturing flow lines with unreliable machines and dataflow graphs where choice and synchronization occur. The basic idea leads to the notion of a cut to split the original net system into two subnets. The cuts lead to two aggregated net systems where one of the subnets is reduced to a single transition. A further reduction leads to a basic skeleton. The generalization of the idea leads to multiple cuts, where single cuts can be applied recursively leading to a hierarchical decomposition. Based on the decomposition, a response time <span class="hlt">approximation</span> technique for the performance analysis is introduced. Also, delay equivalence, which has previously been introduced in the context of marked graphs by Woodside et al., Marie's method and flow equivalent aggregation are applied to the aggregated net systems. The experimental results show that response time <span class="hlt">approximation</span> converges quickly and shows reasonable accuracy in most cases. The convergence of Marie's method and flow equivalent aggregation are applied to the aggregated net systems. The experimental results show that response time <span class="hlt">approximation</span> converges quickly and shows reasonable accuracy in most cases. The convergence of Marie's is slower, but the accuracy is generally better. Delay</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/ADA049995','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/ADA049995"><span><span class="hlt">Approximate</span> Solutions for Certain Optimal Stopping Problems</span></a></p> <p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p> <p></p> <p>1978-01-05</p> <p>one-armed bandit problem) has arisen in a number of statistical applications (Chernoff and Ray (1965);, Chernoff (±9&]), Mallik (1971)): Let X(t... Mallik (1971) and Chernoff (1972). These previous <span class="hlt">approximations</span> were determined without the benefit of the "correction for continuity" given in (5.1...Vol. 1, 3rd edition, John Wiley and Sons, Inc., New York» 7. Mallik , A.K» (1971), "Sequential estimation of the common mean of two normal</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/17126801','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/17126801"><span><span class="hlt">Approximation</span> for limit cycles and their isochrons.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Demongeot, Jacques; Françoise, Jean-Pierre</p> <p>2006-12-01</p> <p>Local analysis of trajectories of dynamical systems near an attractive periodic orbit displays the notion of asymptotic phase and isochrons. These notions are quite useful in applications to biosciences. In this note, we give an expression for the first <span class="hlt">approximation</span> of equations of isochrons in the setting of perturbations of polynomial Hamiltonian systems. This method can be generalized to perturbations of systems that have a polynomial integral factor (like the Lotka-Volterra equation).</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/ADA225292','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/ADA225292"><span><span class="hlt">Approximations</span> of Thermoelastic and Viscoelastic Control Systems</span></a></p> <p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p> <p></p> <p>1990-06-01</p> <p>parabolic partial differential equations. The development of computational algorithms for designing controllers for such systems is an Immenselv complex...hereditary differential system on Rr , then <span class="hlt">approximate</span> the "’historv" or -’memory- term (i.e.. the integral term in i.S)). In this paper we will use a... variation introduced by Fabiano and Ito ([FI]) of the averaging scheme considered by Banks and Burns ([BB]) for the second stage. The idea of the "’AVE</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/24206291','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/24206291"><span>Block correlated second <span class="hlt">order</span> perturbation theory with a generalized valence bond reference function.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Xu, Enhua; Li, Shuhua</p> <p>2013-11-07</p> <p>The block correlated second-<span class="hlt">order</span> perturbation theory with a generalized valence bond (GVB) reference (GVB-BCPT2) is proposed. In this approach, each geminal in the GVB reference is considered as a "multi-orbital" block (a subset of spin orbitals), and each occupied or virtual spin orbital is also taken as a single block. The <span class="hlt">zeroth-order</span> Hamiltonian is set to be the summation of the individual Hamiltonians of all blocks (with explicit two-electron operators within each geminal) so that the GVB reference function and all excited configuration functions are its eigenfunctions. The GVB-BCPT2 energy can be directly obtained without iteration, just like the second <span class="hlt">order</span> Mo̸ller-Plesset perturbation method (MP2), both of which are size consistent. We have applied this GVB-BCPT2 method to investigate the equilibrium distances and spectroscopic constants of 7 diatomic molecules, conformational energy differences of 8 small molecules, and bond-breaking potential energy profiles in 3 systems. GVB-BCPT2 is demonstrated to have noticeably better performance than MP2 for systems with significant multi-reference character, and provide reasonably accurate results for some systems with large active spaces, which are beyond the capability of all CASSCF-based methods.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/29671043','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/29671043"><span>Local <span class="hlt">approximation</span> of a metapopulation's equilibrium.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Barbour, A D; McVinish, R; Pollett, P K</p> <p>2018-04-18</p> <p>We consider the <span class="hlt">approximation</span> of the equilibrium of a metapopulation model, in which a finite number of patches are randomly distributed over a bounded subset [Formula: see text] of Euclidean space. The <span class="hlt">approximation</span> is good when a large number of patches contribute to the colonization pressure on any given unoccupied patch, and when the quality of the patches varies little over the length scale determined by the colonization radius. If this is the case, the equilibrium probability of a patch at z being occupied is shown to be close to [Formula: see text], the equilibrium occupation probability in Levins's model, at any point [Formula: see text] not too close to the boundary, if the local colonization pressure and extinction rates appropriate to z are assumed. The <span class="hlt">approximation</span> is justified by giving explicit upper and lower bounds for the occupation probabilities, expressed in terms of the model parameters. Since the patches are distributed randomly, the occupation probabilities are also random, and we complement our bounds with explicit bounds on the probability that they are satisfied at all patches simultaneously.</p> </li> </ol> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_12");'>12</a></li> <li><a href="#" onclick='return showDiv("page_13");'>13</a></li> <li class="active"><span>14</span></li> <li><a href="#" onclick='return showDiv("page_15");'>15</a></li> <li><a href="#" onclick='return showDiv("page_16");'>16</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div><!-- col-sm-12 --> </div><!-- row --> </div><!-- page_14 --> <div id="page_15" class="hiddenDiv"> <div class="row"> <div class="col-sm-12"> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_13");'>13</a></li> <li><a href="#" onclick='return showDiv("page_14");'>14</a></li> <li class="active"><span>15</span></li> <li><a href="#" onclick='return showDiv("page_16");'>16</a></li> <li><a href="#" onclick='return showDiv("page_17");'>17</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div> </div> <div class="row"> <div class="col-sm-12"> <ol class="result-class" start="281"> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/9783204','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/9783204"><span>Compression of strings with <span class="hlt">approximate</span> repeats.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Allison, L; Edgoose, T; Dix, T I</p> <p>1998-01-01</p> <p>We describe a model for strings of characters that is loosely based on the Lempel Ziv model with the addition that a repeated substring can be an <span class="hlt">approximate</span> match to the original substring; this is close to the situation of DNA, for example. Typically there are many explanations for a given string under the model, some optimal and many suboptimal. Rather than commit to one optimal explanation, we sum the probabilities over all explanations under the model because this gives the probability of the data under the model. The model has a small number of parameters and these can be estimated from the given string by an expectation-maximization (EM) algorithm. Each iteration of the EM algorithm takes O(n2) time and a few iterations are typically sufficient. O(n2) complexity is impractical for strings of more than a few tens of thousands of characters and a faster <span class="hlt">approximation</span> algorithm is also given. The model is further extended to include <span class="hlt">approximate</span> reverse complementary repeats when analyzing DNA strings. Tests include the recovery of parameter estimates from known sources and applications to real DNA strings.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2009LNCS.5687...98E','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2009LNCS.5687...98E"><span>New Hardness Results for Diophantine <span class="hlt">Approximation</span></span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Eisenbrand, Friedrich; Rothvoß, Thomas</p> <p></p> <p>We revisit simultaneous Diophantine <span class="hlt">approximation</span>, a classical problem from the geometry of numbers which has many applications in algorithms and complexity. The input to the decision version of this problem consists of a rational vector α ∈ ℚ n , an error bound ɛ and a denominator bound N ∈ ℕ + . One has to decide whether there exists an integer, called the denominator Q with 1 ≤ Q ≤ N such that the distance of each number Q ·α i to its nearest integer is bounded by ɛ. Lagarias has shown that this problem is NP-complete and optimization versions have been shown to be hard to <span class="hlt">approximate</span> within a factor n c/ loglogn for some constant c > 0. We strengthen the existing hardness results and show that the optimization problem of finding the smallest denominator Q ∈ ℕ + such that the distances of Q·α i to the nearest integer are bounded by ɛ is hard to <span class="hlt">approximate</span> within a factor 2 n unless {textrm{P}} = NP.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19980232929','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19980232929"><span>Using <span class="hlt">Approximations</span> to Accelerate Engineering Design Optimization</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Torczon, Virginia; Trosset, Michael W.</p> <p>1998-01-01</p> <p>Optimization problems that arise in engineering design are often characterized by several features that hinder the use of standard nonlinear optimization techniques. Foremost among these features is that the functions used to define the engineering optimization problem often are computationally intensive. Within a standard nonlinear optimization algorithm, the computational expense of evaluating the functions that define the problem would necessarily be incurred for each iteration of the optimization algorithm. Faced with such prohibitive computational costs, an attractive alternative is to make use of surrogates within an optimization context since surrogates can be chosen or constructed so that they are typically much less expensive to compute. For the purposes of this paper, we will focus on the use of algebraic <span class="hlt">approximations</span> as surrogates for the objective. In this paper we introduce the use of so-called merit functions that explicitly recognize the desirability of improving the current <span class="hlt">approximation</span> to the objective during the course of the optimization. We define and experiment with the use of merit functions chosen to simultaneously improve both the solution to the optimization problem (the objective) and the quality of the <span class="hlt">approximation</span>. Our goal is to further improve the effectiveness of our general approach without sacrificing any of its rigor.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016JHyd..535..340A','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016JHyd..535..340A"><span>Green-Ampt <span class="hlt">approximations</span>: A comprehensive analysis</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Ali, Shakir; Islam, Adlul; Mishra, P. K.; Sikka, Alok K.</p> <p>2016-04-01</p> <p>Green-Ampt (GA) model and its modifications are widely used for simulating infiltration process. Several explicit <span class="hlt">approximate</span> solutions to the implicit GA model have been developed with varying degree of accuracy. In this study, performance of nine explicit <span class="hlt">approximations</span> to the GA model is compared with the implicit GA model using the published data for broad range of soil classes and infiltration time. The explicit GA models considered are Li et al. (1976) (LI), Stone et al. (1994) (ST), Salvucci and Entekhabi (1994) (SE), Parlange et al. (2002) (PA), Barry et al. (2005) (BA), Swamee et al. (2012) (SW), Ali et al. (2013) (AL), Almedeij and Esen (2014) (AE), and Vatankhah (2015) (VA). Six statistical indicators (e.g., percent relative error, maximum absolute percent relative error, average absolute percent relative errors, percent bias, index of agreement, and Nash-Sutcliffe efficiency) and relative computer computation time are used for assessing the model performance. Models are ranked based on the overall performance index (OPI). The BA model is found to be the most accurate followed by the PA and VA models for variety of soil classes and infiltration periods. The AE, SW, SE, and LI model also performed comparatively better. Based on the overall performance index, the explicit models are ranked as BA > PA > VA > LI > AE > SE > SW > ST > AL. Results of this study will be helpful in selection of accurate and simple explicit <span class="hlt">approximate</span> GA models for solving variety of hydrological problems.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017APS..MARC25005B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017APS..MARC25005B"><span>A Gaussian <span class="hlt">Approximation</span> Potential for Silicon</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Bernstein, Noam; Bartók, Albert; Kermode, James; Csányi, Gábor</p> <p></p> <p>We present an interatomic potential for silicon using the Gaussian <span class="hlt">Approximation</span> Potential (GAP) approach, which uses the Gaussian process regression method to <span class="hlt">approximate</span> the reference potential energy surface as a sum of atomic energies. Each atomic energy is <span class="hlt">approximated</span> as a function of the local environment around the atom, which is described with the smooth overlap of atomic environments (SOAP) descriptor. The potential is fit to a database of energies, forces, and stresses calculated using density functional theory (DFT) on a wide range of configurations from zero and finite temperature simulations. These include crystalline phases, liquid, amorphous, and low coordination structures, and diamond-structure point defects, dislocations, surfaces, and cracks. We compare the results of the potential to DFT calculations, as well as to previously published models including Stillinger-Weber, Tersoff, modified embedded atom method (MEAM), and ReaxFF. We show that it is very accurate as compared to the DFT reference results for a wide range of properties, including low energy bulk phases, liquid structure, as well as point, line, and plane defects in the diamond structure.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=5331172','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=5331172"><span>Ranking Support Vector Machine with Kernel <span class="hlt">Approximation</span></span></a></p> <p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p> <p>Dou, Yong</p> <p>2017-01-01</p> <p>Learning to rank algorithm has become important in recent years due to its successful application in information retrieval, recommender system, and computational biology, and so forth. Ranking support vector machine (RankSVM) is one of the state-of-art ranking models and has been favorably used. Nonlinear RankSVM (RankSVM with nonlinear kernels) can give higher accuracy than linear RankSVM (RankSVM with a linear kernel) for complex nonlinear ranking problem. However, the learning methods for nonlinear RankSVM are still time-consuming because of the calculation of kernel matrix. In this paper, we propose a fast ranking algorithm based on kernel <span class="hlt">approximation</span> to avoid computing the kernel matrix. We explore two types of kernel <span class="hlt">approximation</span> methods, namely, the Nyström method and random Fourier features. Primal truncated Newton method is used to optimize the pairwise L2-loss (squared Hinge-loss) objective function of the ranking model after the nonlinear kernel <span class="hlt">approximation</span>. Experimental results demonstrate that our proposed method gets a much faster training speed than kernel RankSVM and achieves comparable or better performance over state-of-the-art ranking algorithms. PMID:28293256</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/28293256','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/28293256"><span>Ranking Support Vector Machine with Kernel <span class="hlt">Approximation</span>.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Chen, Kai; Li, Rongchun; Dou, Yong; Liang, Zhengfa; Lv, Qi</p> <p>2017-01-01</p> <p>Learning to rank algorithm has become important in recent years due to its successful application in information retrieval, recommender system, and computational biology, and so forth. Ranking support vector machine (RankSVM) is one of the state-of-art ranking models and has been favorably used. Nonlinear RankSVM (RankSVM with nonlinear kernels) can give higher accuracy than linear RankSVM (RankSVM with a linear kernel) for complex nonlinear ranking problem. However, the learning methods for nonlinear RankSVM are still time-consuming because of the calculation of kernel matrix. In this paper, we propose a fast ranking algorithm based on kernel <span class="hlt">approximation</span> to avoid computing the kernel matrix. We explore two types of kernel <span class="hlt">approximation</span> methods, namely, the Nyström method and random Fourier features. Primal truncated Newton method is used to optimize the pairwise L2-loss (squared Hinge-loss) objective function of the ranking model after the nonlinear kernel <span class="hlt">approximation</span>. Experimental results demonstrate that our proposed method gets a much faster training speed than kernel RankSVM and achieves comparable or better performance over state-of-the-art ranking algorithms.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016IAUS..308...97N','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016IAUS..308...97N"><span>An Origami <span class="hlt">Approximation</span> to the Cosmic Web</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Neyrinck, Mark C.</p> <p>2016-10-01</p> <p>The powerful Lagrangian view of structure formation was essentially introduced to cosmology by Zel'dovich. In the current cosmological paradigm, a dark-matter-sheet 3D manifold, inhabiting 6D position-velocity phase space, was flat (with vanishing velocity) at the big bang. Afterward, gravity stretched and bunched the sheet together in different places, forming a cosmic web when projected to the position coordinates. Here, I explain some properties of an origami <span class="hlt">approximation</span>, in which the sheet does not stretch or contract (an assumption that is false in general), but is allowed to fold. Even without stretching, the sheet can form an idealized cosmic web, with convex polyhedral voids separated by straight walls and filaments, joined by convex polyhedral nodes. The nodes form in `polygonal' or `polyhedral' collapse, somewhat like spherical/ellipsoidal collapse, except incorporating simultaneous filament and wall formation. The origami <span class="hlt">approximation</span> allows phase-space geometries of nodes, filaments, and walls to be more easily understood, and may aid in understanding spin correlations between nearby galaxies. This contribution explores kinematic origami-<span class="hlt">approximation</span> models giving velocity fields for the first time.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/26587963','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/26587963"><span><span class="hlt">Approximate</span> number and <span class="hlt">approximate</span> time discrimination each correlate with school math abilities in young children.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Odic, Darko; Lisboa, Juan Valle; Eisinger, Robert; Olivera, Magdalena Gonzalez; Maiche, Alejandro; Halberda, Justin</p> <p>2016-01-01</p> <p>What is the relationship between our intuitive sense of number (e.g., when estimating how many marbles are in a jar), and our intuitive sense of other quantities, including time (e.g., when estimating how long it has been since we last ate breakfast)? Recent work in cognitive, developmental, comparative psychology, and computational neuroscience has suggested that our representations of <span class="hlt">approximate</span> number, time, and spatial extent are fundamentally linked and constitute a "generalized magnitude system". But, the shared behavioral and neural signatures between number, time, and space may alternatively be due to similar encoding and decision-making processes, rather than due to shared domain-general representations. In this study, we investigate the relationship between <span class="hlt">approximate</span> number and time in a large sample of 6-8 year-old children in Uruguay by examining how individual differences in the precision of number and time estimation correlate with school mathematics performance. Over four testing days, each child completed an <span class="hlt">approximate</span> number discrimination task, an <span class="hlt">approximate</span> time discrimination task, a digit span task, and a large battery of symbolic math tests. We replicate previous reports showing that symbolic math abilities correlate with <span class="hlt">approximate</span> number precision and extend those findings by showing that math abilities also correlate with <span class="hlt">approximate</span> time precision. But, contrary to <span class="hlt">approximate</span> number and time sharing common representations, we find that each of these dimensions uniquely correlates with formal math: <span class="hlt">approximate</span> number correlates more strongly with formal math compared to time and continues to correlate with math even when precision in time and individual differences in working memory are controlled for. These results suggest that there are important differences in the mental representations of <span class="hlt">approximate</span> number and <span class="hlt">approximate</span> time and further clarify the relationship between quantity representations and mathematics. Copyright </p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/19151097','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/19151097"><span>Biochemical simulations: stochastic, <span class="hlt">approximate</span> stochastic and hybrid approaches.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Pahle, Jürgen</p> <p>2009-01-01</p> <p>Computer simulations have become an invaluable tool to study the sometimes counterintuitive temporal dynamics of (bio-)chemical systems. In particular, stochastic simulation methods have attracted increasing interest recently. In contrast to the well-known deterministic approach based on ordinary differential equations, they can capture effects that occur due to the underlying discreteness of the systems and random fluctuations in molecular numbers. Numerous stochastic, <span class="hlt">approximate</span> stochastic and hybrid simulation methods have been proposed in the literature. In this article, they are systematically reviewed in <span class="hlt">order</span> to guide the researcher and help her find the appropriate method for a specific problem.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=2638628','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=2638628"><span>Biochemical simulations: stochastic, <span class="hlt">approximate</span> stochastic and hybrid approaches</span></a></p> <p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p> <p></p> <p>2009-01-01</p> <p>Computer simulations have become an invaluable tool to study the sometimes counterintuitive temporal dynamics of (bio-)chemical systems. In particular, stochastic simulation methods have attracted increasing interest recently. In contrast to the well-known deterministic approach based on ordinary differential equations, they can capture effects that occur due to the underlying discreteness of the systems and random fluctuations in molecular numbers. Numerous stochastic, <span class="hlt">approximate</span> stochastic and hybrid simulation methods have been proposed in the literature. In this article, they are systematically reviewed in <span class="hlt">order</span> to guide the researcher and help her find the appropriate method for a specific problem. PMID:19151097</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017ZaMP...68...82C','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017ZaMP...68...82C"><span>Asymptotic <span class="hlt">approximations</span> for pure bending of thin cylindrical shells</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Coman, Ciprian D.</p> <p>2017-08-01</p> <p>A simplified partial wrinkling scenario for in-plane bending of thin cylindrical shells is explored by using several asymptotic strategies. The eighth-<span class="hlt">order</span> boundary eigenvalue problem investigated here originates in the Donnel-Mushtari-Vlasov shallow shell theory coupled with a linear membrane pre-bifurcation state. It is shown that the corresponding neutral stability curve is amenable to a detailed asymptotic analysis based on the method of multiple scales. This is further complemented by an alternative WKB <span class="hlt">approximation</span> that provides comparable information with significantly less effort.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2012APS..MARB54007S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2012APS..MARB54007S"><span>Asymptotically inspired moment-closure <span class="hlt">approximation</span> for adaptive networks</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Shkarayev, Maxim; Shaw, Leah</p> <p>2012-02-01</p> <p>Adaptive social networks, in which nodes and network structure co-evolve, are often described using a mean-field system of equations for the density of node and link types. These equations constitute an open system due to dependence on higher <span class="hlt">order</span> topological structures. We propose a moment-closure <span class="hlt">approximation</span> based on the analytical description of the system in an asymptotic regime. We apply the proposed approach to two examples of adaptive networks: recruitment to a cause model and epidemic spread model. We show a good agreement between the improved mean-field prediction and simulations of the full network system.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2013APS..MARZ29005S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2013APS..MARZ29005S"><span>Asymptotically inspired moment-closure <span class="hlt">approximation</span> for adaptive networks</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Shkarayev, Maxim</p> <p>2013-03-01</p> <p>Dynamics of adaptive social networks, in which nodes and network structure co-evolve, are often described using a mean-field system of equations for the density of node and link types. These equations constitute an open system due to dependence on higher <span class="hlt">order</span> topological structures. We propose a systematic approach to moment closure <span class="hlt">approximation</span> based on the analytical description of the system in an asymptotic regime. We apply the proposed approach to two examples of adaptive networks: recruitment to a cause model and adaptive epidemic model. We show a good agreement between the mean-field prediction and simulations of the full network system.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016IAUS..308...69H','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016IAUS..308...69H"><span>The Zeldovich & Adhesion <span class="hlt">approximations</span> and applications to the local universe</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Hidding, Johan; van de Weygaert, Rien; Shandarin, Sergei</p> <p>2016-10-01</p> <p>The Zeldovich <span class="hlt">approximation</span> (ZA) predicts the formation of a web of singularities. While these singularities may only exist in the most formal interpretation of the ZA, they provide a powerful tool for the analysis of initial conditions. We present a novel method to find the skeleton of the resulting cosmic web based on singularities in the primordial deformation tensor and its higher <span class="hlt">order</span> derivatives. We show that the A 3 lines predict the formation of filaments in a two-dimensional model. We continue with applications of the adhesion model to visualise structures in the local (z < 0.03) universe.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19830044831&hterms=averaged+lagrangian&qs=N%3D0%26Ntk%3DAll%26Ntx%3Dmode%2Bmatchall%26Ntt%3Daveraged%2Blagrangian','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19830044831&hterms=averaged+lagrangian&qs=N%3D0%26Ntk%3DAll%26Ntx%3Dmode%2Bmatchall%26Ntt%3Daveraged%2Blagrangian"><span>Microscopic Lagrangian description of warm plasmas. IV - Macroscopic <span class="hlt">approximation</span></span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Kim, H.; Crawford, F. W.</p> <p>1983-01-01</p> <p>The averaged-Lagrangian method is applied to linear wave propagation and nonlinear three-wave interaction in a warm magnetoplasma, in the macroscopic <span class="hlt">approximation</span>. The microscopic Lagrangian treated by Kim and Crawford (1977) and by Galloway and Crawford (1977) is first expanded to third <span class="hlt">order</span> in perturbation. Velocity integration is then carried out, before applying Hamilton's principle to obtain a general description of wave propagation and coupling. The results are specialized to the case of interaction between two electron plasma waves and an Alfven wave. The method is shown to be more powerful than the alternative possibility of working from the beginning with a macroscopic Lagrangian density.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2015JPhB...48d5503A','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2015JPhB...48d5503A"><span><span class="hlt">Approximate</span> analytical solutions of a pair of coupled anharmonic oscillators</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Alam, Nasir; Mandal, Swapan; Öhberg, Patrik</p> <p>2015-02-01</p> <p>The Hamiltonian and the corresponding equations of motion involving the field operators of two quartic anharmonic oscillators indirectly coupled via a linear oscillator are constructed. The <span class="hlt">approximate</span> analytical solutions of the coupled differential equations involving the non-commuting field operators are solved up to the second <span class="hlt">order</span> in the anharmonic coupling. In the absence of nonlinearity these solutions are used to calculate the second <span class="hlt">order</span> variances and hence the squeezing in pure and in mixed modes. The higher <span class="hlt">order</span> quadrature squeezing and the amplitude squared squeezing of various field modes are also investigated where the squeezing in pure and in mixed modes are found to be suppressed. Moreover, the absence of a nonlinearity prohibits the higher <span class="hlt">order</span> quadrature and higher <span class="hlt">ordered</span> amplitude squeezing of the input coherent states. It is established that the mere coupling of two oscillators through a third one is unable to produce any squeezing effects of input coherent light, but the presence of a nonlinear interaction may provide squeezed states and other nonclassical phenomena.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017AGUFMMR34B..08B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017AGUFMMR34B..08B"><span>The Linear Mixing <span class="hlt">Approximation</span> for Planetary Ices</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Bethkenhagen, M.; Meyer, E. R.; Hamel, S.; Nettelmann, N.; French, M.; Scheibe, L.; Ticknor, C.; Collins, L. A.; Kress, J. D.; Fortney, J. J.; Redmer, R.</p> <p>2017-12-01</p> <p>We investigate the validity of the widely used linear mixing <span class="hlt">approximation</span> for the equations of state (EOS) of planetary ices, which are thought to dominate the interior of the ice giant planets Uranus and Neptune. For that purpose we perform density functional theory molecular dynamics simulations using the VASP code.[1] In particular, we compute 1:1 binary mixtures of water, ammonia, and methane, as well as their 2:1:4 ternary mixture at pressure-temperature conditions typical for the interior of Uranus and Neptune.[2,3] In addition, a new ab initio EOS for methane is presented. The linear mixing <span class="hlt">approximation</span> is verified for the conditions present inside Uranus ranging up to 10 Mbar based on the comprehensive EOS data set. We also calculate the diffusion coefficients for the ternary mixture along different Uranus interior profiles and compare them to the values of the pure compounds. We find that deviations of the linear mixing <span class="hlt">approximation</span> from the real mixture are generally small; for the EOS they fall within about 4% uncertainty while the diffusion coefficients deviate up to 20% . The EOS of planetary ices are applied to adiabatic models of Uranus. It turns out that a deep interior of almost pure ices is consistent with the gravity field data, in which case the planet becomes rather cold (T core ˜ 4000 K). [1] G. Kresse and J. Hafner, Physical Review B 47, 558 (1993). [2] R. Redmer, T.R. Mattsson, N. Nettelmann and M. French, Icarus 211, 798 (2011). [3] N. Nettelmann, K. Wang, J. J. Fortney, S. Hamel, S. Yellamilli, M. Bethkenhagen and R. Redmer, Icarus 275, 107 (2016).</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/22375874-strong-washout-approximation-resonant-leptogenesis','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/22375874-strong-washout-approximation-resonant-leptogenesis"><span>Strong washout <span class="hlt">approximation</span> to resonant leptogenesis</span></a></p> <p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p> <p>Garbrecht, Björn; Gautier, Florian; Klaric, Juraj, E-mail: garbrecht@tum.de, E-mail: florian.gautier@tum.de, E-mail: juraj.klaric@tum.de</p> <p></p> <p>We show that the effective decay asymmetry for resonant Leptogenesis in the strong washout regime with two sterile neutrinos and a single active flavour can in wide regions of parameter space be <span class="hlt">approximated</span> by its late-time limit ε=Xsin(2φ)/(X{sup 2}+sin{sup 2}φ), where X=8πΔ/(|Y{sub 1}|{sup 2}+|Y{sub 2}|{sup 2}), Δ=4(M{sub 1}-M{sub 2})/(M{sub 1}+M{sub 2}), φ=arg(Y{sub 2}/Y{sub 1}), and M{sub 1,2}, Y{sub 1,2} are the masses and Yukawa couplings of the sterile neutrinos. This <span class="hlt">approximation</span> in particular extends to parametric regions where |Y{sub 1,2}|{sup 2}>> Δ, i.e. where the width dominates the mass splitting. We generalise the formula for the effective decay asymmetry to themore » case of several flavours of active leptons and demonstrate how this quantity can be used to calculate the lepton asymmetry for phenomenological scenarios that are in agreement with the observed neutrino oscillations. We establish analytic criteria for the validity of the late-time <span class="hlt">approximation</span> for the decay asymmetry and compare these with numerical results that are obtained by solving for the mixing and the oscillations of the sterile neutrinos. For phenomenologically viable models with two sterile neutrinos, we find that the flavoured effective late-time decay asymmetry can be applied throughout parameter space.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016CMaPh.342...47B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016CMaPh.342...47B"><span>Product-State <span class="hlt">Approximations</span> to Quantum States</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Brandão, Fernando G. S. L.; Harrow, Aram W.</p> <p>2016-02-01</p> <p>We show that for any many-body quantum state there exists an unentangled quantum state such that most of the two-body reduced density matrices are close to those of the original state. This is a statement about the monogamy of entanglement, which cannot be shared without limit in the same way as classical correlation. Our main application is to Hamiltonians that are sums of two-body terms. For such Hamiltonians we show that there exist product states with energy that is close to the ground-state energy whenever the interaction graph of the Hamiltonian has high degree. This proves the validity of mean-field theory and gives an explicitly bounded <span class="hlt">approximation</span> error. If we allow states that are entangled within small clusters of systems but product across clusters then good <span class="hlt">approximations</span> exist when the Hamiltonian satisfies one or more of the following properties: (1) high degree, (2) small expansion, or (3) a ground state where the blocks in the partition have sublinear entanglement. Previously this was known only in the case of small expansion or in the regime where the entanglement was close to zero. Our <span class="hlt">approximations</span> allow an extensive error in energy, which is the scale considered by the quantum PCP (probabilistically checkable proof) and NLTS (no low-energy trivial-state) conjectures. Thus our results put restrictions on the possible Hamiltonians that could be used for a possible proof of the qPCP or NLTS conjectures. By contrast the classical PCP constructions are often based on constraint graphs with high degree. Likewise we show that the parallel repetition that is possible with classical constraint satisfaction problems cannot also be possible for quantum Hamiltonians, unless qPCP is false. The main technical tool behind our results is a collection of new classical and quantum de Finetti theorems which do not make any symmetry assumptions on the underlying states.</p> </li> </ol> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_13");'>13</a></li> <li><a href="#" onclick='return showDiv("page_14");'>14</a></li> <li class="active"><span>15</span></li> <li><a href="#" onclick='return showDiv("page_16");'>16</a></li> <li><a href="#" onclick='return showDiv("page_17");'>17</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div><!-- col-sm-12 --> </div><!-- row --> </div><!-- page_15 --> <div id="page_16" class="hiddenDiv"> <div class="row"> <div class="col-sm-12"> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_14");'>14</a></li> <li><a href="#" onclick='return showDiv("page_15");'>15</a></li> <li class="active"><span>16</span></li> <li><a href="#" onclick='return showDiv("page_17");'>17</a></li> <li><a href="#" onclick='return showDiv("page_18");'>18</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div> </div> <div class="row"> <div class="col-sm-12"> <ol class="result-class" start="301"> <li> <p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/ADA194685','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/ADA194685"><span><span class="hlt">Approximations</span> and Implementations of Nonlinear Filtering Schemes.</span></a></p> <p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p> <p></p> <p>1988-02-01</p> <p>17) 0 0 3) P(fn) - (pf)n 4) Pf v0 - (Po <-> dp - (p0 dm is invariant under f (i.e. for all measurable A: (f’l(A)) - p(A) Remark: The Perron - Frobenius ...invariant density of the map f is then nothing else than the fixed point of the Perron - Frobenius operator. The following theorem by Lasota and Yorke [8...transition matrix R is defined. With this construct, the Perron - Frobenius operator is effectively 39 A A . w7 <span class="hlt">approximated</span> (exact for Markov Maps)by</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017PhRvE..96d2139K','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017PhRvE..96d2139K"><span>Stochastic maps, continuous <span class="hlt">approximation</span>, and stable distribution</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Kessler, David A.; Burov, Stanislav</p> <p>2017-10-01</p> <p>A continuous <span class="hlt">approximation</span> framework for general nonlinear stochastic as well as deterministic discrete maps is developed. For the stochastic map with uncorelated Gaussian noise, by successively applying the Itô lemma, we obtain a Langevin type of equation. Specifically, we show how nonlinear maps give rise to a Langevin description that involves multiplicative noise. The multiplicative nature of the noise induces an additional effective force, not present in the absence of noise. We further exploit the continuum description and provide an explicit formula for the stable distribution of the stochastic map and conditions for its existence. Our results are in good agreement with numerical simulations of several maps.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017JMP....58h3506Z','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017JMP....58h3506Z"><span>Localization and stationary phase <span class="hlt">approximation</span> on supermanifolds</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Zakharevich, Valentin</p> <p>2017-08-01</p> <p>Given an odd vector field Q on a supermanifold M and a Q-invariant density μ on M, under certain compactness conditions on Q, the value of the integral ∫Mμ is determined by the value of μ on any neighborhood of the vanishing locus N of Q. We present a formula for the integral in the case where N is a subsupermanifold which is appropriately non-degenerate with respect to Q. In the process, we discuss the linear algebra necessary to express our result in a coordinate independent way. We also extend the stationary phase <span class="hlt">approximation</span> and the Morse-Bott lemma to supermanifolds.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018JMP....59a2107A','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018JMP....59a2107A"><span>Born-Oppenheimer <span class="hlt">approximation</span> for a singular system</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Akbas, Haci; Turgut, O. Teoman</p> <p>2018-01-01</p> <p>We discuss a simple singular system in one dimension, two heavy particles interacting with a light particle via an attractive contact interaction and not interacting among themselves. It is natural to apply the Born-Oppenheimer <span class="hlt">approximation</span> to this problem. We present a detailed discussion of this approach; the advantage of this simple model is that one can estimate the error terms self-consistently. Moreover, a Fock space approach to this problem is presented where an expansion can be proposed to get higher <span class="hlt">order</span> corrections. A slight modification of the same problem in which the light particle is relativistic is discussed in a later section by neglecting pair creation processes. Here, the second quantized description is more challenging, but with some care, one can recover the first <span class="hlt">order</span> expression exactly.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018EPJP..133...37A','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018EPJP..133...37A"><span>A new numerical <span class="hlt">approximation</span> of the fractal ordinary differential equation</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Atangana, Abdon; Jain, Sonal</p> <p>2018-02-01</p> <p>The concept of fractal medium is present in several real-world problems, for instance, in the geological formation that constitutes the well-known subsurface water called aquifers. However, attention has not been quite devoted to modeling for instance, the flow of a fluid within these media. We deem it important to remind the reader that the concept of fractal derivative is not to represent the fractal sharps but to describe the movement of the fluid within these media. Since this class of ordinary differential equations is highly complex to solve analytically, we present a novel numerical scheme that allows to solve fractal ordinary differential equations. Error analysis of the method is also presented. Application of the method and numerical <span class="hlt">approximation</span> are presented for fractal <span class="hlt">order</span> differential equation. The stability and the convergence of the numerical schemes are investigated in detail. Also some exact solutions of fractal <span class="hlt">order</span> differential equations are presented and finally some numerical simulations are presented.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/22611442-multi-level-methods-approximating-distribution-functions','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/22611442-multi-level-methods-approximating-distribution-functions"><span>Multi-level methods and <span class="hlt">approximating</span> distribution functions</span></a></p> <p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p> <p>Wilson, D., E-mail: daniel.wilson@dtc.ox.ac.uk; Baker, R. E.</p> <p>2016-07-15</p> <p>Biochemical reaction networks are often modelled using discrete-state, continuous-time Markov chains. System statistics of these Markov chains usually cannot be calculated analytically and therefore estimates must be generated via simulation techniques. There is a well documented class of simulation techniques known as exact stochastic simulation algorithms, an example of which is Gillespie’s direct method. These algorithms often come with high computational costs, therefore <span class="hlt">approximate</span> stochastic simulation algorithms such as the tau-leap method are used. However, in <span class="hlt">order</span> to minimise the bias in the estimates generated using them, a relatively small value of tau is needed, rendering the computational costs comparablemore » to Gillespie’s direct method. The multi-level Monte Carlo method (Anderson and Higham, Multiscale Model. Simul. 10:146–179, 2012) provides a reduction in computational costs whilst minimising or even eliminating the bias in the estimates of system statistics. This is achieved by first crudely <span class="hlt">approximating</span> required statistics with many sample paths of low accuracy. Then correction terms are added until a required level of accuracy is reached. Recent literature has primarily focussed on implementing the multi-level method efficiently to estimate a single system statistic. However, it is clearly also of interest to be able to <span class="hlt">approximate</span> entire probability distributions of species counts. We present two novel methods that combine known techniques for distribution reconstruction with the multi-level method. We demonstrate the potential of our methods using a number of examples.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/23005202','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/23005202"><span>Analytical <span class="hlt">approximations</span> for the collapse of an empty spherical bubble.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Obreschkow, D; Bruderer, M; Farhat, M</p> <p>2012-06-01</p> <p>The Rayleigh equation 3/2R+RR+pρ(-1)=0 with initial conditions R(0)=R(0), R(0)=0 models the collapse of an empty spherical bubble of radius R(T) in an ideal, infinite liquid with far-field pressure p and density ρ. The solution for r≡R/R(0) as a function of time t≡T/T(c), where R(T(c))≡0, is independent of R(0), p, and ρ. While no closed-form expression for r(t) is known, we find that r(0)(t)=(1-t(2))(2/5) <span class="hlt">approximates</span> r(t) with an error below 1%. A systematic development in <span class="hlt">orders</span> of t(2) further yields the 0.001% <span class="hlt">approximation</span> r(*)(t)=r(0)(t)[1-a(1)Li(2.21)(t(2))], where a(1)≈-0.01832099 is a constant and Li is the polylogarithm. The usefulness of these <span class="hlt">approximations</span> is demonstrated by comparison to high-precision cavitation data obtained in microgravity.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2013CoPhC.184.1045J','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2013CoPhC.184.1045J"><span>ASP: Automated symbolic computation of <span class="hlt">approximate</span> symmetries of differential equations</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Jefferson, G. F.; Carminati, J.</p> <p>2013-03-01</p> <p>A recent paper (Pakdemirli et al. (2004) [12]) compared three methods of determining <span class="hlt">approximate</span> symmetries of differential equations. Two of these methods are well known and involve either a perturbation of the classical Lie symmetry generator of the differential system (Baikov, Gazizov and Ibragimov (1988) [7], Ibragimov (1996) [6]) or a perturbation of the dependent variable/s and subsequent determination of the classical Lie point symmetries of the resulting coupled system (Fushchych and Shtelen (1989) [11]), both up to a specified <span class="hlt">order</span> in the perturbation parameter. The third method, proposed by Pakdemirli, Yürüsoy and Dolapçi (2004) [12], simplifies the calculations required by Fushchych and Shtelen's method through the assignment of arbitrary functions to the non-linear components prior to computing symmetries. All three methods have been implemented in the new MAPLE package ASP (Automated Symmetry Package) which is an add-on to the MAPLE symmetry package DESOLVII (Vu, Jefferson and Carminati (2012) [25]). To our knowledge, this is the first computer package to automate all three methods of determining <span class="hlt">approximate</span> symmetries for differential systems. Extensions to the theory have also been suggested for the third method and which generalise the first method to systems of differential equations. Finally, a number of <span class="hlt">approximate</span> symmetries and corresponding solutions are compared with results in the literature.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19870046420&hterms=rational+better&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D30%26Ntt%3Drational%2Bbetter','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19870046420&hterms=rational+better&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D30%26Ntt%3Drational%2Bbetter"><span>Nonlinear programming extensions to rational function <span class="hlt">approximations</span> of unsteady aerodynamics</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Tiffany, Sherwood H.; Adams, William M., Jr.</p> <p>1987-01-01</p> <p>This paper deals with <span class="hlt">approximating</span> unsteady generalized aerodynamic forces in the equations of motion of a flexible aircraft. Two methods of formulating these <span class="hlt">approximations</span> are extended to include both the same flexibility in constraining them and the same methodology in optimizing nonlinear parameters as another currently used 'extended least-squares' method. Optimal selection of 'nonlinear' parameters is made in each of the three methods by use of the same nonlinear (nongradient) optimizer. The objective of the nonlinear optimization is to obtain rational <span class="hlt">approximations</span> to the unsteady aerodynamics whose state-space realization is of lower <span class="hlt">order</span> than that required when no optimization of the nonlinear terms is performed. The free 'linear' parameters are determined using least-squares matrix techniques on a Lagrange multiplier formulation of an objective function which incorporates selected linear equality constraints. State-space mathematical models resulting from the different approaches are described, and results are presented which show comparative evaluations from application of each of the extended methods to a numerical example. The results obtained for the example problem show a significant (up to 63 percent) reduction in the number of differential equations used to represent the unsteady aerodynamic forces in linear time-invariant equations of motion as compared to a conventional method in which nonlinear terms are not optimized.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20020038859','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20020038859"><span>Variable <span class="hlt">Order</span> and Distributed <span class="hlt">Order</span> Fractional Operators</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Lorenzo, Carl F.; Hartley, Tom T.</p> <p>2002-01-01</p> <p>Many physical processes appear to exhibit fractional <span class="hlt">order</span> behavior that may vary with time or space. The continuum of <span class="hlt">order</span> in the fractional calculus allows the <span class="hlt">order</span> of the fractional operator to be considered as a variable. This paper develops the concept of variable and distributed <span class="hlt">order</span> fractional operators. Definitions based on the Riemann-Liouville definitions are introduced and behavior of the operators is studied. Several time domain definitions that assign different arguments to the <span class="hlt">order</span> q in the Riemann-Liouville definition are introduced. For each of these definitions various characteristics are determined. These include: time invariance of the operator, operator initialization, physical realization, linearity, operational transforms. and memory characteristics of the defining kernels. A measure (m2) for memory retentiveness of the <span class="hlt">order</span> history is introduced. A generalized linear argument for the <span class="hlt">order</span> q allows the concept of "tailored" variable <span class="hlt">order</span> fractional operators whose a, memory may be chosen for a particular application. Memory retentiveness (m2) and <span class="hlt">order</span> dynamic behavior are investigated and applications are shown. The concept of distributed <span class="hlt">order</span> operators where the <span class="hlt">order</span> of the time based operator depends on an additional independent (spatial) variable is also forwarded. Several definitions and their Laplace transforms are developed, analysis methods with these operators are demonstrated, and examples shown. Finally operators of multivariable and distributed <span class="hlt">order</span> are defined in their various applications are outlined.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20080040183','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20080040183"><span><span class="hlt">Approximation</span> of Failure Probability Using Conditional Sampling</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Giesy. Daniel P.; Crespo, Luis G.; Kenney, Sean P.</p> <p>2008-01-01</p> <p>In analyzing systems which depend on uncertain parameters, one technique is to partition the uncertain parameter domain into a failure set and its complement, and judge the quality of the system by estimating the probability of failure. If this is done by a sampling technique such as Monte Carlo and the probability of failure is small, accurate <span class="hlt">approximation</span> can require so many sample points that the computational expense is prohibitive. Previous work of the authors has shown how to bound the failure event by sets of such simple geometry that their probabilities can be calculated analytically. In this paper, it is shown how to make use of these failure bounding sets and conditional sampling within them to substantially reduce the computational burden of <span class="hlt">approximating</span> failure probability. It is also shown how the use of these sampling techniques improves the confidence intervals for the failure probability estimate for a given number of sample points and how they reduce the number of sample point analyses needed to achieve a given level of confidence.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=5375850','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=5375850"><span><span class="hlt">Approximate</span> Sensory Data Collection: A Survey</span></a></p> <p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p> <p>Cheng, Siyao; Cai, Zhipeng; Li, Jianzhong</p> <p>2017-01-01</p> <p>With the rapid development of the Internet of Things (IoTs), wireless sensor networks (WSNs) and related techniques, the amount of sensory data manifests an explosive growth. In some applications of IoTs and WSNs, the size of sensory data has already exceeded several petabytes annually, which brings too many troubles and challenges for the data collection, which is a primary operation in IoTs and WSNs. Since the exact data collection is not affordable for many WSN and IoT systems due to the limitations on bandwidth and energy, many <span class="hlt">approximate</span> data collection algorithms have been proposed in the last decade. This survey reviews the state of the art of <span class="hlt">approximate</span> data collection algorithms. We classify them into three categories: the model-based ones, the compressive sensing based ones, and the query-driven ones. For each category of algorithms, the advantages and disadvantages are elaborated, some challenges and unsolved problems are pointed out, and the research prospects are forecasted. PMID:28287440</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19910047246&hterms=1089&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D10%26Ntt%3D%2526%25231089','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19910047246&hterms=1089&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D10%26Ntt%3D%2526%25231089"><span>Eigenvalue and eigenvector sensitivity and <span class="hlt">approximate</span> analysis for repeated eigenvalue problems</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Hou, Gene J. W.; Kenny, Sean P.</p> <p>1991-01-01</p> <p>A set of computationally efficient equations for eigenvalue and eigenvector sensitivity analysis are derived, and a method for eigenvalue and eigenvector <span class="hlt">approximate</span> analysis in the presence of repeated eigenvalues is presented. The method developed for <span class="hlt">approximate</span> analysis involves a reparamaterization of the multivariable structural eigenvalue problem in terms of a single positive-valued parameter. The resulting equations yield first-<span class="hlt">order</span> <span class="hlt">approximations</span> of changes in both the eigenvalues and eigenvectors associated with the repeated eigenvalue problem. Examples are given to demonstrate the application of such equations for sensitivity and <span class="hlt">approximate</span> analysis.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2008JMAA..345..845C','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2008JMAA..345..845C"><span><span class="hlt">Approximate</span> controllability of a system of parabolic equations with delay</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Carrasco, Alexander; Leiva, Hugo</p> <p>2008-09-01</p> <p>In this paper we give necessary and sufficient conditions for the <span class="hlt">approximate</span> controllability of the following system of parabolic equations with delay: where [Omega] is a bounded domain in , D is an n×n nondiagonal matrix whose eigenvalues are semi-simple with nonnegative real part, the control and B[set membership, variant]L(U,Z) with , . The standard notation zt(x) defines a function from [-[tau],0] to (with x fixed) by zt(x)(s)=z(t+s,x), -[tau][less-than-or-equals, slant]s[less-than-or-equals, slant]0. Here [tau][greater-or-equal, slanted]0 is the maximum delay, which is supposed to be finite. We assume that the operator is linear and bounded, and [phi]0[set membership, variant]Z, [phi][set membership, variant]L2([-[tau],0];Z). To this end: First, we reformulate this system into a standard first-<span class="hlt">order</span> delay equation. Secondly, the semigroup associated with the first-<span class="hlt">order</span> delay equation on an appropriate product space is expressed as a series of strongly continuous semigroups and orthogonal projections related with the eigenvalues of the Laplacian operator (); this representation allows us to reduce the controllability of this partial differential equation with delay to a family of ordinary delay equations. Finally, we use the well-known result on the rank condition for the <span class="hlt">approximate</span> controllability of delay system to derive our main result.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19750022560','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19750022560"><span>Spline function <span class="hlt">approximation</span> techniques for image geometric distortion representation. [for registration of multitemporal remote sensor imagery</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Anuta, P. E.</p> <p>1975-01-01</p> <p>Least squares <span class="hlt">approximation</span> techniques were developed for use in computer aided correction of spatial image distortions for registration of multitemporal remote sensor imagery. Polynomials were first used to define image distortion over the entire two dimensional image space. Spline functions were then investigated to determine if the combination of lower <span class="hlt">order</span> polynomials could <span class="hlt">approximate</span> a higher <span class="hlt">order</span> distortion with less computational difficulty. Algorithms for generating <span class="hlt">approximating</span> functions were developed and applied to the description of image distortion in aircraft multispectral scanner imagery. Other applications of the techniques were suggested for earth resources data processing areas other than geometric distortion representation.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2005AGUSMGP22A..06G','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2005AGUSMGP22A..06G"><span>New Tests of the Fixed Hotspot <span class="hlt">Approximation</span></span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Gordon, R. G.; Andrews, D. L.; Horner-Johnson, B. C.; Kumar, R. R.</p> <p>2005-05-01</p> <p>We present new methods for estimating uncertainties in plate reconstructions relative to the hotspots and new tests of the fixed hotspot <span class="hlt">approximation</span>. We find no significant motion between Pacific hotspots, on the one hand, and Indo-Atlantic hotspots, on the other, for the past ~ 50 Myr, but large and significant apparent motion before 50 Ma. Whether this motion is truly due to motion between hotspots or alternatively due to flaws in the global plate motion circuit can be tested with paleomagnetic data. These tests give results consistent with the fixed hotspot <span class="hlt">approximation</span> and indicate significant misfits when a relative plate motion circuit through Antarctica is employed for times before 50 Ma. If all of the misfit to the global plate motion circuit is due to motion between East and West Antarctica, then that motion is 800 ± 500 km near the Ross Sea Embayment and progressively less along the Trans-Antarctic Mountains toward the Weddell Sea. Further paleomagnetic tests of the fixed hotspot <span class="hlt">approximation</span> can be made. Cenozoic and Cretaceous paleomagnetic data from the Pacific plate, along with reconstructions of the Pacific plate relative to the hotspots, can be used to estimate an apparent polar wander (APW) path of Pacific hotspots. An APW path of Indo-Atlantic hotspots can be similarly estimated (e.g. Besse & Courtillot 2002). If both paths diverge in similar ways from the north pole of the hotspot reference frame, it would indicate that the hotspots have moved in unison relative to the spin axis, which may be attributed to true polar wander. If the two paths diverge from one another, motion between Pacific hotspots and Indo-Atlantic hotspots would be indicated. The general agreement of the two paths shows that the former is more important than the latter. The data require little or no motion between groups of hotspots, but up to ~10 mm/yr of motion is allowed within uncertainties. The results disagree, in particular, with the recent extreme interpretation of</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2015PhRvE..92b3012F','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2015PhRvE..92b3012F"><span>Projection-free <span class="hlt">approximate</span> balanced truncation of large unstable systems</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Flinois, Thibault L. B.; Morgans, Aimee S.; Schmid, Peter J.</p> <p>2015-08-01</p> <p>In this article, we show that the projection-free, snapshot-based, balanced truncation method can be applied directly to unstable systems. We prove that even for unstable systems, the unmodified balanced proper orthogonal decomposition algorithm theoretically yields a converged transformation that balances the Gramians (including the unstable subspace). We then apply the method to a spatially developing unstable system and show that it results in reduced-<span class="hlt">order</span> models of similar quality to the ones obtained with existing methods. Due to the unbounded growth of unstable modes, a practical restriction on the final impulse response simulation time appears, which can be adjusted depending on the desired <span class="hlt">order</span> of the reduced-<span class="hlt">order</span> model. Recommendations are given to further reduce the cost of the method if the system is large and to improve the performance of the method if it does not yield acceptable results in its unmodified form. Finally, the method is applied to the linearized flow around a cylinder at Re = 100 to show that it actually is able to accurately reproduce impulse responses for more realistic unstable large-scale systems in practice. The well-established <span class="hlt">approximate</span> balanced truncation numerical framework therefore can be safely applied to unstable systems without any modifications. Additionally, balanced reduced-<span class="hlt">order</span> models can readily be obtained even for large systems, where the computational cost of existing methods is prohibitive.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/28287440','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/28287440"><span><span class="hlt">Approximate</span> Sensory Data Collection: A Survey.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Cheng, Siyao; Cai, Zhipeng; Li, Jianzhong</p> <p>2017-03-10</p> <p>With the rapid development of the Internet of Things (IoTs), wireless sensor networks (WSNs) and related techniques, the amount of sensory data manifests an explosive growth. In some applications of IoTs and WSNs, the size of sensory data has already exceeded several petabytes annually, which brings too many troubles and challenges for the data collection, which is a primary operation in IoTs and WSNs. Since the exact data collection is not affordable for many WSN and IoT systems due to the limitations on bandwidth and energy, many <span class="hlt">approximate</span> data collection algorithms have been proposed in the last decade. This survey reviews the state of the art of approximatedatacollectionalgorithms. Weclassifythemintothreecategories: themodel-basedones, the compressive sensing based ones, and the query-driven ones. For each category of algorithms, the advantages and disadvantages are elaborated, some challenges and unsolved problems are pointed out, and the research prospects are forecasted.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/21067438-bloch-approximation-periodically-perforated-media','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/21067438-bloch-approximation-periodically-perforated-media"><span>The Bloch <span class="hlt">Approximation</span> in Periodically Perforated Media</span></a></p> <p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p> <p>Conca, C.; Gomez, D., E-mail: gomezdel@unican.es; Lobo, M.</p> <p>2005-06-15</p> <p>We consider a periodically heterogeneous and perforated medium filling an open domain {omega} of R{sup N}. Assuming that the size of the periodicity of the structure and of the holes is O({epsilon}),we study the asymptotic behavior, as {epsilon} {sup {yields}} 0, of the solution of an elliptic boundary value problem with strongly oscillating coefficients posed in {omega}{sup {epsilon}}({omega}{sup {epsilon}} being {omega} minus the holes) with a Neumann condition on the boundary of the holes. We use Bloch wave decomposition to introduce an <span class="hlt">approximation</span> of the solution in the energy norm which can be computed from the homogenized solution and themore » first Bloch eigenfunction. We first consider the case where {omega}is R{sup N} and then localize the problem for abounded domain {omega}, considering a homogeneous Dirichlet condition on the boundary of {omega}.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017AIPC.1905c0017I','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017AIPC.1905c0017I"><span><span class="hlt">Approximate</span> analytical modeling of leptospirosis infection</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Ismail, Nur Atikah; Azmi, Amirah; Yusof, Fauzi Mohamed; Ismail, Ahmad Izani</p> <p>2017-11-01</p> <p>Leptospirosis is an infectious disease carried by rodents which can cause death in humans. The disease spreads directly through contact with feces, urine or through bites of infected rodents and indirectly via water contaminated with urine and droppings from them. Significant increase in the number of leptospirosis cases in Malaysia caused by the recent severe floods were recorded during heavy rainfall season. Therefore, to understand the dynamics of leptospirosis infection, a mathematical model based on fractional differential equations have been developed and analyzed. In this paper an <span class="hlt">approximate</span> analytical method, the multi-step Laplace Adomian decomposition method, has been used to conduct numerical simulations so as to gain insight on the spread of leptospirosis infection.</p> </li> </ol> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_14");'>14</a></li> <li><a href="#" onclick='return showDiv("page_15");'>15</a></li> <li class="active"><span>16</span></li> <li><a href="#" onclick='return showDiv("page_17");'>17</a></li> <li><a href="#" onclick='return showDiv("page_18");'>18</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div><!-- col-sm-12 --> </div><!-- row --> </div><!-- page_16 --> <div id="page_17" class="hiddenDiv"> <div class="row"> <div class="col-sm-12"> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_15");'>15</a></li> <li><a href="#" onclick='return showDiv("page_16");'>16</a></li> <li class="active"><span>17</span></li> <li><a href="#" onclick='return showDiv("page_18");'>18</a></li> <li><a href="#" onclick='return showDiv("page_19");'>19</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div> </div> <div class="row"> <div class="col-sm-12"> <ol class="result-class" start="321"> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3257063','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3257063"><span>Is probabilistic bias analysis <span class="hlt">approximately</span> Bayesian?</span></a></p> <p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p> <p>MacLehose, Richard F.; Gustafson, Paul</p> <p>2011-01-01</p> <p>Case-control studies are particularly susceptible to differential exposure misclassification when exposure status is determined following incident case status. Probabilistic bias analysis methods have been developed as ways to adjust standard effect estimates based on the sensitivity and specificity of exposure misclassification. The iterative sampling method advocated in probabilistic bias analysis bears a distinct resemblance to a Bayesian adjustment; however, it is not identical. Furthermore, without a formal theoretical framework (Bayesian or frequentist), the results of a probabilistic bias analysis remain somewhat difficult to interpret. We describe, both theoretically and empirically, the extent to which probabilistic bias analysis can be viewed as <span class="hlt">approximately</span> Bayesian. While the differences between probabilistic bias analysis and Bayesian approaches to misclassification can be substantial, these situations often involve unrealistic prior specifications and are relatively easy to detect. Outside of these special cases, probabilistic bias analysis and Bayesian approaches to exposure misclassification in case-control studies appear to perform equally well. PMID:22157311</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016APS..APR.L1045A','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016APS..APR.L1045A"><span>Comparing numerical and analytic <span class="hlt">approximate</span> gravitational waveforms</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Afshari, Nousha; Lovelace, Geoffrey; SXS Collaboration</p> <p>2016-03-01</p> <p>A direct observation of gravitational waves will test Einstein's theory of general relativity under the most extreme conditions. The Laser Interferometer Gravitational-Wave Observatory, or LIGO, began searching for gravitational waves in September 2015 with three times the sensitivity of initial LIGO. To help Advanced LIGO detect as many gravitational waves as possible, a major research effort is underway to accurately predict the expected waves. In this poster, I will explore how the gravitational waveform produced by a long binary-black-hole inspiral, merger, and ringdown is affected by how fast the larger black hole spins. In particular, I will present results from simulations of merging black holes, completed using the Spectral Einstein Code (black-holes.org/SpEC.html), including some new, long simulations designed to mimic black hole-neutron star mergers. I will present comparisons of the numerical waveforms with analytic <span class="hlt">approximations</span>.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017JMP....58j2104F','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017JMP....58j2104F"><span>Some remarks concerning the centrifugal term <span class="hlt">approximation</span></span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Ferreira, F. J. S.; Bezerra, V. B.</p> <p>2017-10-01</p> <p>We generalize the Pekeris <span class="hlt">approximation</span> for the centrifugal term potential, l/(l +1 ) r2 , and use this to obtain the solutions of the radial Schrödinger equation for the arbitrary angular quantum number, l, of the Hulthén potential. We also obtain the expressions for the bound state energies corresponding to this potential and calculate their values for different states and compare with other results presented in the literature. We also consider some models of physical potentials, namely, the Eckart potential, the Poschl-Teller potentials, the Rosen-Morse potential, the Woods-Saxon potential, and the Manning-Rosen potential. Thus, following straightforward the example corresponding to the Hulthén potential, we show what the radial solutions and the energy spectra for these potentials are.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/5287255-approximate-flavor-symmetries-lepton-sector','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/5287255-approximate-flavor-symmetries-lepton-sector"><span><span class="hlt">Approximate</span> flavor symmetries in the lepton sector</span></a></p> <p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p> <p>Rasin, A.; Silva, J.P.</p> <p>1994-01-01</p> <p><span class="hlt">Approximate</span> flavor symmetries in the quark sector have been used as a handle on physics beyond the standard model. Because of the great interest in neutrino masses and mixings and the wealth of existing and proposed neutrino experiments it is important to extend this analysis to the leptonic sector. We show that in the seesaw mechanism the neutrino masses and mixing angles do not depend on the details of the right-handed neutrino flavor symmetry breaking, and are related by a simple formula. We propose several [ital Ansa]$[ital uml]---[ital tze] which relate different flavor symmetry-breaking parameters and find that the MSWmore » solution to the solar neutrino problem is always easily fit. Further, the [nu][sub [mu]-][nu][sub [tau</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19890009103&hterms=Agreement&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D30%26Ntt%3DAgreement','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19890009103&hterms=Agreement&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D30%26Ntt%3DAgreement"><span>Accelerated convergence for synchronous <span class="hlt">approximate</span> agreement</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Kearns, J. P.; Park, S. K.; Sjogren, J. A.</p> <p>1988-01-01</p> <p>The protocol for synchronous <span class="hlt">approximate</span> agreement presented by Dolev et. al. exhibits the undesirable property that a faulty processor, by the dissemination of a value arbitrarily far removed from the values held by good processors, may delay the termination of the protocol by an arbitrary amount of time. Such behavior is clearly undesirable in a fault tolerant dynamic system subject to hard real-time constraints. A mechanism is presented by which editing data suspected of being from Byzantine-failed processors can lead to quicker, predictable, convergence to an agreement value. Under specific assumptions about the nature of values transmitted by failed processors relative to those transmitted by good processors, a Monte Carlo simulation is presented whose qualitative results illustrate the trade-off between accelerated convergence and the accuracy of the value agreed upon.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2014JPO....44.2433B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2014JPO....44.2433B"><span><span class="hlt">Approximate</span> Stokes Drift Profiles in Deep Water</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Breivik, Øyvind; Janssen, Peter A. E. M.; Bidlot, Jean-Raymond</p> <p>2014-09-01</p> <p>A deep-water <span class="hlt">approximation</span> to the Stokes drift velocity profile is explored as an alternative to the monochromatic profile. The alternative profile investigated relies on the same two quantities required for the monochromatic profile, viz the Stokes transport and the surface Stokes drift velocity. Comparisons with parametric spectra and profiles under wave spectra from the ERA-Interim reanalysis and buoy observations reveal much better agreement than the monochromatic profile even for complex sea states. That the profile gives a closer match and a more correct shear has implications for ocean circulation models since the Coriolis-Stokes force depends on the magnitude and direction of the Stokes drift profile and Langmuir turbulence parameterizations depend sensitively on the shear of the profile. The alternative profile comes at no added numerical cost compared to the monochromatic profile.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018JEI....27a3011L','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018JEI....27a3011L"><span>Local facet <span class="hlt">approximation</span> for image stitching</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Li, Jing; Lai, Shiming; Liu, Yu; Wang, Zhengming; Zhang, Maojun</p> <p>2018-01-01</p> <p>Image stitching aims at eliminating multiview parallax and generating a seamless panorama given a set of input images. This paper proposes a local adaptive stitching method, which could achieve both accurate and robust image alignments across the whole panorama. A transformation estimation model is introduced by <span class="hlt">approximating</span> the scene as a combination of neighboring facets. Then, the local adaptive stitching field is constructed using a series of linear systems of the facet parameters, which enables the parallax handling in three-dimensional space. We also provide a concise but effective global projectivity preserving technique that smoothly varies the transformations from local adaptive to global planar. The proposed model is capable of stitching both normal images and fisheye images. The efficiency of our method is quantitatively demonstrated in the comparative experiments on several challenging cases.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20150000116','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20150000116"><span>Exact and <span class="hlt">Approximate</span> Probabilistic Symbolic Execution</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Luckow, Kasper; Pasareanu, Corina S.; Dwyer, Matthew B.; Filieri, Antonio; Visser, Willem</p> <p>2014-01-01</p> <p>Probabilistic software analysis seeks to quantify the likelihood of reaching a target event under uncertain environments. Recent approaches compute probabilities of execution paths using symbolic execution, but do not support nondeterminism. Nondeterminism arises naturally when no suitable probabilistic model can capture a program behavior, e.g., for multithreading or distributed systems. In this work, we propose a technique, based on symbolic execution, to synthesize schedulers that resolve nondeterminism to maximize the probability of reaching a target event. To scale to large systems, we also introduce <span class="hlt">approximate</span> algorithms to search for good schedulers, speeding up established random sampling and reinforcement learning results through the quantification of path probabilities based on symbolic execution. We implemented the techniques in Symbolic PathFinder and evaluated them on nondeterministic Java programs. We show that our algorithms significantly improve upon a state-of- the-art statistical model checking algorithm, originally developed for Markov Decision Processes.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19960022271','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19960022271"><span>Spline <span class="hlt">Approximation</span> of Thin Shell Dynamics</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>delRosario, R. C. H.; Smith, R. C.</p> <p>1996-01-01</p> <p>A spline-based method for <span class="hlt">approximating</span> thin shell dynamics is presented here. While the method is developed in the context of the Donnell-Mushtari thin shell equations, it can be easily extended to the Byrne-Flugge-Lur'ye equations or other models for shells of revolution as warranted by applications. The primary requirements for the method include accuracy, flexibility and efficiency in smart material applications. To accomplish this, the method was designed to be flexible with regard to boundary conditions, material nonhomogeneities due to sensors and actuators, and inputs from smart material actuators such as piezoceramic patches. The accuracy of the method was also of primary concern, both to guarantee full resolution of structural dynamics and to facilitate the development of PDE-based controllers which ultimately require real-time implementation. Several numerical examples provide initial evidence demonstrating the efficacy of the method.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19910023149','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19910023149"><span>An <span class="hlt">approximate</span> Riemann solver for hypervelocity flows</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Jacobs, Peter A.</p> <p>1991-01-01</p> <p>We describe an <span class="hlt">approximate</span> Riemann solver for the computation of hypervelocity flows in which there are strong shocks and viscous interactions. The scheme has three stages, the first of which computes the intermediate states assuming isentropic waves. A second stage, based on the strong shock relations, may then be invoked if the pressure jump across either wave is large. The third stage interpolates the interface state from the two initial states and the intermediate states. The solver is used as part of a finite-volume code and is demonstrated on two test cases. The first is a high Mach number flow over a sphere while the second is a flow over a slender cone with an adiabatic boundary layer. In both cases the solver performs well.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19900042199&hterms=1060&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D20%26Ntt%3D%2526%25231060','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19900042199&hterms=1060&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D20%26Ntt%3D%2526%25231060"><span>Improved <span class="hlt">approximations</span> for control augmented structural synthesis</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Thomas, H. L.; Schmit, L. A.</p> <p>1990-01-01</p> <p>A methodology for control-augmented structural synthesis is presented for structure-control systems which can be modeled as an assemblage of beam, truss, and nonstructural mass elements augmented by a noncollocated direct output feedback control system. Truss areas, beam cross sectional dimensions, nonstructural masses and rotary inertias, and controller position and velocity gains are treated simultaneously as design variables. The structural mass and a control-system performance index can be minimized simultaneously, with design constraints placed on static stresses and displacements, dynamic harmonic displacements and forces, structural frequencies, and closed-loop eigenvalues and damping ratios. Intermediate design-variable and response-quantity concepts are used to generate new <span class="hlt">approximations</span> for displacements and actuator forces under harmonic dynamic loads and for system complex eigenvalues. This improves the overall efficiency of the procedure by reducing the number of complete analyses required for convergence. Numerical results which illustrate the effectiveness of the method are given.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19720053619&hterms=habibi&qs=N%3D0%26Ntk%3DAll%26Ntx%3Dmode%2Bmatchall%26Ntt%3Dhabibi','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19720053619&hterms=habibi&qs=N%3D0%26Ntk%3DAll%26Ntx%3Dmode%2Bmatchall%26Ntt%3Dhabibi"><span>Estimation of correlation functions by stochastic <span class="hlt">approximation</span>.</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Habibi, A.; Wintz, P. A.</p> <p>1972-01-01</p> <p>Consideration of the autocorrelation function of a zero-mean stationary random process. The techniques are applicable to processes with nonzero mean provided the mean is estimated first and subtracted. Two recursive techniques are proposed, both of which are based on the method of stochastic <span class="hlt">approximation</span> and assume a functional form for the correlation function that depends on a number of parameters that are recursively estimated from successive records. One technique uses a standard point estimator of the correlation function to provide estimates of the parameters that minimize the mean-square error between the point estimates and the parametric function. The other technique provides estimates of the parameters that maximize a likelihood function relating the parameters of the function to the random process. Examples are presented.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19790012619','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19790012619"><span><span class="hlt">Approximate</span> maximum likelihood decoding of block codes</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Greenberger, H. J.</p> <p>1979-01-01</p> <p><span class="hlt">Approximate</span> maximum likelihood decoding algorithms, based upon selecting a small set of candidate code words with the aid of the estimated probability of error of each received symbol, can give performance close to optimum with a reasonable amount of computation. By combining the best features of various algorithms and taking care to perform each step as efficiently as possible, a decoding scheme was developed which can decode codes which have better performance than those presently in use and yet not require an unreasonable amount of computation. The discussion of the details and tradeoffs of presently known efficient optimum and near optimum decoding algorithms leads, naturally, to the one which embodies the best features of all of them.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2000PhyA..280..566B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2000PhyA..280..566B"><span>Markovian <span class="hlt">approximation</span> in foreign exchange markets</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Baviera, Roberto; Vergni, Davide; Vulpiani, Angelo</p> <p>2000-06-01</p> <p>In this paper, using the exit-time statistic, we study the structure of the price variations for the high-frequency data set of the bid-ask Deutschemark/US dollar exchange rate quotes registered by the inter-bank Reuters network over the period October 1, 1992 to September 30, 1993. Having rejected random-walk models for the returns, we propose a Markovian model which reproduce the available information of the financial series. Besides the usual correlation analysis we have verified the validity of this model by means of other tools all inspired by information theory. These techniques are not only severe tests of the <span class="hlt">approximation</span> but also evidence of some aspects of the data series which have a clear financial relevance.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2009IEITI..92..149H','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2009IEITI..92..149H"><span><span class="hlt">Approximation</span> Preserving Reductions among Item Pricing Problems</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Hamane, Ryoso; Itoh, Toshiya; Tomita, Kouhei</p> <p></p> <p>When a store sells items to customers, the store wishes to determine the prices of the items to maximize its profit. Intuitively, if the store sells the items with low (resp. high) prices, the customers buy more (resp. less) items, which provides less profit to the store. So it would be hard for the store to decide the prices of items. Assume that the store has a set V of n items and there is a set E of m customers who wish to buy those items, and also assume that each item i ∈ V has the production cost di and each customer ej ∈ E has the valuation vj on the bundle ej ⊆ V of items. When the store sells an item i ∈ V at the price ri, the profit for the item i is pi = ri - di. The goal of the store is to decide the price of each item to maximize its total profit. We refer to this maximization problem as the item pricing problem. In most of the previous works, the item pricing problem was considered under the assumption that pi ≥ 0 for each i ∈ V, however, Balcan, et al. [In Proc. of WINE, LNCS 4858, 2007] introduced the notion of “loss-leader, ” and showed that the seller can get more total profit in the case that pi < 0 is allowed than in the case that pi < 0 is not allowed. In this paper, we derive <span class="hlt">approximation</span> preserving reductions among several item pricing problems and show that all of them have algorithms with good <span class="hlt">approximation</span> ratio.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2006JPRS...60..239L','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2006JPRS...60..239L"><span><span class="hlt">Approximate</span> direct georeferencing in national coordinates</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Legat, Klaus</p> <p></p> <p>Direct georeferencing has gained an increasing importance in photogrammetry and remote sensing. Thereby, the parameters of exterior orientation (EO) of an image sensor are determined by GPS/INS, yielding results in a global geocentric reference frame. Photogrammetric products like digital terrain models or orthoimages, however, are often required in national geodetic datums and mapped by national map projections, i.e., in "national coordinates". As the fundamental mathematics of photogrammetry is based on Cartesian coordinates, the scene restitution is often performed in a Cartesian frame located at some central position of the image block. The subsequent transformation to national coordinates is a standard problem in geodesy and can be done in a rigorous manner-at least if the formulas of the map projection are rigorous. Drawbacks of this procedure include practical deficiencies related to the photogrammetric processing as well as the computational cost of transforming the whole scene. To avoid these problems, the paper pursues an alternative processing strategy where the EO parameters are transformed prior to the restitution. If only this transition was done, however, the scene would be systematically distorted. The reason is that the national coordinates are not Cartesian due to the earth curvature and the unavoidable length distortion of map projections. To settle these distortions, several corrections need to be applied. These are treated in detail for both passive and active imaging. Since all these corrections are <span class="hlt">approximations</span> only, the resulting technique is termed "<span class="hlt">approximate</span> direct georeferencing". Still, the residual distortions are usually very low as is demonstrated by simulations, rendering the technique an attractive approach to direct georeferencing.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19810002914','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19810002914"><span>An analytical technique for <span class="hlt">approximating</span> unsteady aerodynamics in the time domain</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Dunn, H. J.</p> <p>1980-01-01</p> <p>An analytical technique is presented for <span class="hlt">approximating</span> unsteady aerodynamic forces in the time domain. The <span class="hlt">order</span> of elements of a matrix Pade <span class="hlt">approximation</span> was postulated, and the resulting polynomial coefficients were determined through a combination of least squares estimates for the numerator coefficients and a constrained gradient search for the denominator coefficients which insures stable <span class="hlt">approximating</span> functions. The number of differential equations required to represent the aerodynamic forces to a given accuracy tends to be smaller than that employed in certain existing techniques where the denominator coefficients are chosen a priori. Results are shown for an aeroelastic, cantilevered, semispan wing which indicate a good fit to the aerodynamic forces for oscillatory motion can be achieved with a matrix Pade <span class="hlt">approximation</span> having fourth <span class="hlt">order</span> numerator and second <span class="hlt">order</span> denominator polynomials.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19950023722&hterms=time-dependent&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DTitle%26N%3D0%26No%3D80%26Ntt%3Dtime-dependent','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19950023722&hterms=time-dependent&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DTitle%26N%3D0%26No%3D80%26Ntt%3Dtime-dependent"><span>Second-<span class="hlt">order</span> numerical solution of time-dependent, first-<span class="hlt">order</span> hyperbolic equations</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Shah, Patricia L.; Hardin, Jay</p> <p>1995-01-01</p> <p>A finite difference scheme is developed to find an <span class="hlt">approximate</span> solution of two similar hyperbolic equations, namely a first-<span class="hlt">order</span> plane wave and spherical wave problem. Finite difference <span class="hlt">approximations</span> are made for both the space and time derivatives. The result is a conditionally stable equation yielding an exact solution when the Courant number is set to one.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2011ThCFD..25..357V','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2011ThCFD..25..357V"><span>Stratified wakes, the high Froude number <span class="hlt">approximation</span>, and potential flow</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Vasholz, David P.</p> <p>2011-12-01</p> <p>Properties of a steady wake generated by a body moving uniformly at constant depth through a stratified fluid are studied as a function of two parameters inserted into the linearized equations of motion. The first parameter, μ, multiplies the along-track gradient term in the source equation. When formal solutions for an arbitrary buoyancy frequency profile are written as eigenfunction expansions, one finds that the limit μ → 0 corresponds to a high Froude number <span class="hlt">approximation</span> accompanied by a substantial reduction in the complexity of the calculation. For μ = 1, upstream effects are present and the eigenvalues correspond to critical speeds above which transverse waves disappear for any given mode. For sufficiently high modes, the high Froude number <span class="hlt">approximation</span> is valid. The second tracer multiplies the square of the buoyancy frequency term in the linearized conservation of mass equation and enables direct comparisons with the limit of potential flow. Detailed results are given for the simplest possible profile, in which the buoyancy frequency is independent of depth; emphasis is placed upon quantities that can, in principle, be experimentally measured in a laboratory experiment. The vertical displacement field is written in terms of a stratified wake form factor {{H}} , which is the sum of a wavelike contribution that is non-zero downstream and an evanescent contribution that appears symmetrically upstream and downstream. First- and second-<span class="hlt">order</span> cross-track moments of {{H}} are analyzed. First-<span class="hlt">order</span> results predict enhanced upstream vertical displacements. Second-<span class="hlt">order</span> results expand upon previous predictions of wavelike resonances and also predict evanescent resonance effects.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/29357078','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/29357078"><span>Risk <span class="hlt">approximation</span> in decision making: <span class="hlt">approximative</span> numeric abilities predict advantageous decisions under objective risk.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Mueller, Silke M; Schiebener, Johannes; Delazer, Margarete; Brand, Matthias</p> <p>2018-01-22</p> <p>Many decision situations in everyday life involve mathematical considerations. In decisions under objective risk, i.e., when explicit numeric information is available, executive functions and abilities to handle exact numbers and ratios are predictors of objectively advantageous choices. Although still debated, exact numeric abilities, e.g., normative calculation skills, are assumed to be related to <span class="hlt">approximate</span> number processing skills. The current study investigates the effects of <span class="hlt">approximative</span> numeric abilities on decision making under objective risk. Participants (N = 153) performed a paradigm measuring number-comparison, quantity-estimation, risk-estimation, and decision-making skills on the basis of rapid dot comparisons. Additionally, a risky decision-making task with exact numeric information was administered, as well as tasks measuring executive functions and exact numeric abilities, e.g., mental calculation and ratio processing skills, were conducted. <span class="hlt">Approximative</span> numeric abilities significantly predicted advantageous decision making, even beyond the effects of executive functions and exact numeric skills. Especially being able to make accurate risk estimations seemed to contribute to superior choices. We recommend <span class="hlt">approximation</span> skills and <span class="hlt">approximate</span> number processing to be subject of future investigations on decision making under risk.</p> </li> </ol> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_15");'>15</a></li> <li><a href="#" onclick='return showDiv("page_16");'>16</a></li> <li class="active"><span>17</span></li> <li><a href="#" onclick='return showDiv("page_18");'>18</a></li> <li><a href="#" onclick='return showDiv("page_19");'>19</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div><!-- col-sm-12 --> </div><!-- row --> </div><!-- page_17 --> <div id="page_18" class="hiddenDiv"> <div class="row"> <div class="col-sm-12"> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_16");'>16</a></li> <li><a href="#" onclick='return showDiv("page_17");'>17</a></li> <li class="active"><span>18</span></li> <li><a href="#" onclick='return showDiv("page_19");'>19</a></li> <li><a href="#" onclick='return showDiv("page_20");'>20</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div> </div> <div class="row"> <div class="col-sm-12"> <ol class="result-class" start="341"> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/23546248','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/23546248"><span>Limitations of the paraxial Debye <span class="hlt">approximation</span>.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Sheppard, Colin J R</p> <p>2013-04-01</p> <p>In the paraxial form of the Debye integral for focusing, higher <span class="hlt">order</span> defocus terms are ignored, which can result in errors in dealing with aberrations, even for low numerical aperture. These errors can be avoided by using a different integration variable. The aberrations of a glass slab, such as a coverslip, are expanded in terms of the new variable, and expressed in terms of Zernike polynomials to assist with aberration balancing. Tube length error is also discussed.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017AIPC.1895k0003K','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017AIPC.1895k0003K"><span>Numerical investigation of sixth <span class="hlt">order</span> Boussinesq equation</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Kolkovska, N.; Vucheva, V.</p> <p>2017-10-01</p> <p>We propose a family of conservative finite difference schemes for the Boussinesq equation with sixth <span class="hlt">order</span> dispersion terms. The schemes are of second <span class="hlt">order</span> of <span class="hlt">approximation</span>. The method is conditionally stable with a mild restriction τ = O(h) on the step sizes. Numerical tests are performed for quadratic and cubic nonlinearities. The numerical experiments show second <span class="hlt">order</span> of convergence of the discrete solution to the exact one.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2008NuPhB.789..525B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2008NuPhB.789..525B"><span>Analytical <span class="hlt">approximation</span> schemes for solving exact renormalization group equations in the local potential <span class="hlt">approximation</span></span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Bervillier, C.; Boisseau, B.; Giacomini, H.</p> <p>2008-02-01</p> <p>The relation between the Wilson-Polchinski and the Litim optimized ERGEs in the local potential <span class="hlt">approximation</span> is studied with high accuracy using two different analytical approaches based on a field expansion: a recently proposed genuine analytical <span class="hlt">approximation</span> scheme to two-point boundary value problems of ordinary differential equations, and a new one based on <span class="hlt">approximating</span> the solution by generalized hypergeometric functions. A comparison with the numerical results obtained with the shooting method is made. A similar accuracy is reached in each case. Both two methods appear to be more efficient than the usual field expansions frequently used in the current studies of ERGEs (in particular for the Wilson-Polchinski case in the study of which they fail).</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/servlets/purl/826901','SCIGOV-STC'); return false;" href="https://www.osti.gov/servlets/purl/826901"><span>Consistent Yokoya-Chen <span class="hlt">Approximation</span> to Beamstrahlung(LCC-0010)</span></a></p> <p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p> <p>Peskin, M</p> <p>2004-04-22</p> <p>I reconsider the Yokoya-Chen <span class="hlt">approximate</span> evolution equation for beamstrahlung and modify it slightly to generate simple, consistent analytical <span class="hlt">approximations</span> for the electron and photon energy spectra. I compare these <span class="hlt">approximations</span> to previous ones, and to simulation data.I reconsider the Yokoya-Chen <span class="hlt">approximate</span> evolution equation for beamstrahlung and modify it slightly to generate simple, consistent analytical <span class="hlt">approximations</span> for the electron and photon energy spectra. I compare these <span class="hlt">approximations</span> to previous ones, and to simulation data.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/servlets/purl/1235211','SCIGOV-STC'); return false;" href="https://www.osti.gov/servlets/purl/1235211"><span><span class="hlt">Approximate</span> Model for Turbulent Stagnation Point Flow.</span></a></p> <p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p> <p>Dechant, Lawrence</p> <p>2016-01-01</p> <p>Here we derive an <span class="hlt">approximate</span> turbulent self-similar model for a class of favorable pressure gradient wedge-like flows, focusing on the stagnation point limit. While the self-similar model provides a useful gross flow field estimate this approach must be combined with a near wall model is to determine skin friction and by Reynolds analogy the heat transfer coefficient. The combined approach is developed in detail for the stagnation point flow problem where turbulent skin friction and Nusselt number results are obtained. Comparison to the classical Van Driest (1958) result suggests overall reasonable agreement. Though the model is only valid near themore » stagnation region of cylinders and spheres it nonetheless provides a reasonable model for overall cylinder and sphere heat transfer. The enhancement effect of free stream turbulence upon the laminar flow is used to derive a similar expression which is valid for turbulent flow. Examination of free stream enhanced laminar flow suggests that the rather than enhancement of a laminar flow behavior free stream disturbance results in early transition to turbulent stagnation point behavior. Excellent agreement is shown between enhanced laminar flow and turbulent flow behavior for high levels, e.g. 5% of free stream turbulence. Finally the blunt body turbulent stagnation results are shown to provide realistic heat transfer results for turbulent jet impingement problems.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/10104879-simple-approximate-model-parachute-inflation','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/10104879-simple-approximate-model-parachute-inflation"><span>A simple, <span class="hlt">approximate</span> model of parachute inflation</span></a></p> <p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p> <p>Macha, J.M.</p> <p>1992-11-01</p> <p>A simple, <span class="hlt">approximate</span> model of parachute inflation is described. The model is based on the traditional, practical treatment of the fluid resistance of rigid bodies in nonsteady flow, with appropriate extensions to accommodate the change in canopy inflated shape. Correlations for the steady drag and steady radial force as functions of the inflated radius are required as input to the dynamic model. In a novel approach, the radial force is expressed in terms of easily obtainable drag and reefing fine tension measurements. A series of wind tunnel experiments provides the needed correlations. Coefficients associated with the added mass of fluidmore » are evaluated by calibrating the model against an extensive and reliable set of flight data. A parameter is introduced which appears to universally govern the strong dependence of the axial added mass coefficient on motion history. Through comparisons with flight data, the model is shown to realistically predict inflation forces for ribbon and ringslot canopies over a wide range of sizes and deployment conditions.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/7068077-simple-approximate-model-parachute-inflation','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/7068077-simple-approximate-model-parachute-inflation"><span>A simple, <span class="hlt">approximate</span> model of parachute inflation</span></a></p> <p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p> <p>Macha, J.M.</p> <p>1992-01-01</p> <p>A simple, <span class="hlt">approximate</span> model of parachute inflation is described. The model is based on the traditional, practical treatment of the fluid resistance of rigid bodies in nonsteady flow, with appropriate extensions to accommodate the change in canopy inflated shape. Correlations for the steady drag and steady radial force as functions of the inflated radius are required as input to the dynamic model. In a novel approach, the radial force is expressed in terms of easily obtainable drag and reefing fine tension measurements. A series of wind tunnel experiments provides the needed correlations. Coefficients associated with the added mass of fluidmore » are evaluated by calibrating the model against an extensive and reliable set of flight data. A parameter is introduced which appears to universally govern the strong dependence of the axial added mass coefficient on motion history. Through comparisons with flight data, the model is shown to realistically predict inflation forces for ribbon and ringslot canopies over a wide range of sizes and deployment conditions.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/9541868','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/9541868"><span><span class="hlt">Approximation</span> algorithms for a genetic diagnostics problem.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Kosaraju, S R; Schäffer, A A; Biesecker, L G</p> <p>1998-01-01</p> <p>We define and study a combinatorial problem called WEIGHTED DIAGNOSTIC COVER (WDC) that models the use of a laboratory technique called genotyping in the diagnosis of an important class of chromosomal aberrations. An optimal solution to WDC would enable us to define a genetic assay that maximizes the diagnostic power for a specified cost of laboratory work. We develop <span class="hlt">approximation</span> algorithms for WDC by making use of the well-known problem SET COVER for which the greedy heuristic has been extensively studied. We prove worst-case performance bounds on the greedy heuristic for WDC and for another heuristic we call directional greedy. We implemented both heuristics. We also implemented a local search heuristic that takes the solutions obtained by greedy and dir-greedy and applies swaps until they are locally optimal. We report their performance on a real data set that is representative of the options that a clinical geneticist faces for the real diagnostic problem. Many open problems related to WDC remain, both of theoretical interest and practical importance.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/pages/biblio/1420280-planetary-ices-linear-mixing-approximation','SCIGOV-DOEP'); return false;" href="https://www.osti.gov/pages/biblio/1420280-planetary-ices-linear-mixing-approximation"><span>Planetary Ices and the Linear Mixing <span class="hlt">Approximation</span></span></a></p> <p><a target="_blank" href="http://www.osti.gov/pages">DOE PAGES</a></p> <p>Bethkenhagen, M.; Meyer, Edmund Richard; Hamel, S.; ...</p> <p>2017-10-10</p> <p>Here, the validity of the widely used linear mixing <span class="hlt">approximation</span> (LMA) for the equations of state (EOSs) of planetary ices is investigated at pressure–temperature conditions typical for the interiors of Uranus and Neptune. The basis of this study is ab initio data ranging up to 1000 GPa and 20,000 K, calculated via density functional theory molecular dynamics simulations. In particular, we determine a new EOS for methane and EOS data for the 1:1 binary mixtures of methane, ammonia, and water, as well as their 2:1:4 ternary mixture. Additionally, the self-diffusion coefficients in the ternary mixture are calculated along three different Uranus interior profiles and compared to the values of the pure compounds. We find that deviations of the LMA from the results of the real mixture are generally small; for the thermal EOSs they amount to 4% or less. The diffusion coefficients in the mixture agree with those of the pure compounds within 20% or better. Finally, a new adiabatic model of Uranus with an inner layer of almost pure ices is developed. The model is consistent with the gravity field data and results in a rather cold interior (more » $${T}_{\\mathrm{core}}\\sim 4000$$ K).« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/servlets/purl/1376569','SCIGOV-STC'); return false;" href="https://www.osti.gov/servlets/purl/1376569"><span>Stopping power beyond the adiabatic <span class="hlt">approximation</span></span></a></p> <p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p> <p>Caro, M.; Correa, A. A.; Artacho, E.</p> <p>2017-06-01</p> <p>Energetic ions traveling in solids deposit energy in a variety of ways, being nuclear and electronic stopping the two avenues in which dissipation is usually treated. This separation between electrons and ions relies on the adiabatic <span class="hlt">approximation</span> in which ions interact via forces derived from the instantaneous electronic ground state. In a more detailed view, in which non-adiabatic effects are explicitly considered, electronic excitations alter the atomic bonding, which translates into changes in the interatomic forces. In this work, we use time dependent density functional theory and forces derived from the equations of Ehrenfest dynamics that depend instantaneously on themore » time-dependent electronic density. With them we analyze how the inter-ionic forces are affected by electronic excitations in a model of a Ni projectile interacting with a Ni target, a metallic system with strong electronic stopping and shallow core level states. We find that the electronic excitations induce substantial modifications to the inter-ionic forces, which translate into nuclear stopping power well above the adiabatic prediction. Particularly, we observe that most of the alteration of the adiabatic potential in early times comes from the ionization of the core levels of the target ions, not readily screened by the valence electrons.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/27532639','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/27532639"><span>When Density Functional <span class="hlt">Approximations</span> Meet Iron Oxides.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Meng, Yu; Liu, Xing-Wu; Huo, Chun-Fang; Guo, Wen-Ping; Cao, Dong-Bo; Peng, Qing; Dearden, Albert; Gonze, Xavier; Yang, Yong; Wang, Jianguo; Jiao, Haijun; Li, Yongwang; Wen, Xiao-Dong</p> <p>2016-10-11</p> <p>Three density functional <span class="hlt">approximations</span> (DFAs), PBE, PBE+U, and Heyd-Scuseria-Ernzerhof screened hybrid functional (HSE), were employed to investigate the geometric, electronic, magnetic, and thermodynamic properties of four iron oxides, namely, α-FeOOH, α-Fe 2 O 3 , Fe 3 O 4 , and FeO. Comparing our calculated results with available experimental data, we found that HSE (a = 0.15) (containing 15% "screened" Hartree-Fock exchange) can provide reliable values of lattice constants, Fe magnetic moments, band gaps, and formation energies of all four iron oxides, while standard HSE (a = 0.25) seriously overestimates the band gaps and formation energies. For PBE+U, a suitable U value can give quite good results for the electronic properties of each iron oxide, but it is challenging to accurately get other properties of the four iron oxides using the same U value. Subsequently, we calculated the Gibbs free energies of transformation reactions among iron oxides using the HSE (a = 0.15) functional and plotted the equilibrium phase diagrams of the iron oxide system under various conditions, which provide reliable theoretical insight into the phase transformations of iron oxides.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20100036788','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20100036788"><span>Configuring Airspace Sectors with <span class="hlt">Approximate</span> Dynamic Programming</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Bloem, Michael; Gupta, Pramod</p> <p>2010-01-01</p> <p>In response to changing traffic and staffing conditions, supervisors dynamically configure airspace sectors by assigning them to control positions. A finite horizon airspace sector configuration problem models this supervisor decision. The problem is to select an airspace configuration at each time step while considering a workload cost, a reconfiguration cost, and a constraint on the number of control positions at each time step. Three algorithms for this problem are proposed and evaluated: a myopic heuristic, an exact dynamic programming algorithm, and a rollouts <span class="hlt">approximate</span> dynamic programming algorithm. On problem instances from current operations with only dozens of possible configurations, an exact dynamic programming solution gives the optimal cost value. The rollouts algorithm achieves costs within 2% of optimal for these instances, on average. For larger problem instances that are representative of future operations and have thousands of possible configurations, excessive computation time prohibits the use of exact dynamic programming. On such problem instances, the rollouts algorithm reduces the cost achieved by the heuristic by more than 15% on average with an acceptable computation time.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20110016648','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20110016648"><span>Improved Discrete <span class="hlt">Approximation</span> of Laplacian of Gaussian</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Shuler, Robert L., Jr.</p> <p>2004-01-01</p> <p>An improved method of computing a discrete <span class="hlt">approximation</span> of the Laplacian of a Gaussian convolution of an image has been devised. The primary advantage of the method is that without substantially degrading the accuracy of the end result, it reduces the amount of information that must be processed and thus reduces the amount of circuitry needed to perform the Laplacian-of- Gaussian (LOG) operation. Some background information is necessary to place the method in context. The method is intended for application to the LOG part of a process of real-time digital filtering of digitized video data that represent brightnesses in pixels in a square array. The particular filtering process of interest is one that converts pixel brightnesses to binary form, thereby reducing the amount of information that must be performed in subsequent correlation processing (e.g., correlations between images in a stereoscopic pair for determining distances or correlations between successive frames of the same image for detecting motions). The Laplacian is often included in the filtering process because it emphasizes edges and textures, while the Gaussian is often included because it smooths out noise that might not be consistent between left and right images or between successive frames of the same image.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/6637686-approximate-stoichiometry-rich-hydrocarbon-mixtures','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/6637686-approximate-stoichiometry-rich-hydrocarbon-mixtures"><span><span class="hlt">Approximate</span> stoichiometry for rich hydrocarbon mixtures</span></a></p> <p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p> <p>Beans, E.W.</p> <p>1993-03-01</p> <p>The stoichiometry of lean mixtures can readily and accurately be determined from the assumption that all the carbon oxidizes to carbon dioxide and all the hydrogen oxidizes to water. This assumption is valid up to an equivalence ratio ([sigma]) of 0.8 and can be used with little error up to [sigma] = 1. The composition of the products of a hydrocarbon burnt in air under the foregoing assumption can be obtained from simple carbon, hydrogen, oxygen and nitrogen balances. Given the composition, one can determine the energy released and/or the adiabatic flame temperature. For rich mixtures, the foregoing assumption, ofmore » course, is not valid. Hence, there is no easy way to determine the stoichiometry of the products of a rich mixture. The objective of this note is to present an equation' which will allow one to readily determine the composition of the products of rich hydrocarbon mixtures. The equation is based on equilibrium composition calculations and some assumptions regarding the characteristics of hydrocarbons. The equation gives <span class="hlt">approximate</span> results. However, the results are sufficiently accurate for many situations. If more accuracy is wanted, one should use an equilibrium combustion program like the one by Gordon and McBride.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2014SPIE.9093E..0BM','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2014SPIE.9093E..0BM"><span>Born <span class="hlt">approximation</span>, multiple scattering, and butterfly algorithm</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Martinez, Alex; Qiao, Zhijun</p> <p>2014-06-01</p> <p>Many imaging algorithms have been designed assuming the absence of multiple scattering. In the 2013 SPIE proceeding, we discussed an algorithm for removing high <span class="hlt">order</span> scattering components from collected data. In this paper, our goal is to continue this work. First, we survey the current state of multiple scattering in SAR. Then, we revise our method and test it. Given an estimate of our target reflectivity, we compute the multi scattering effects in our target region for various frequencies. Furthermore, we propagate this energy through free space towards our antenna, and remove it from the collected data.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19950012948','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19950012948"><span>Comparison of dynamical <span class="hlt">approximation</span> schemes for non-linear gravitational clustering</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Melott, Adrian L.</p> <p>1994-01-01</p> <p>We have recently conducted a controlled comparison of a number of <span class="hlt">approximations</span> for gravitational clustering against the same n-body simulations. These include ordinary linear perturbation theory (Eulerian), the adhesion <span class="hlt">approximation</span>, the frozen-flow <span class="hlt">approximation</span>, the Zel'dovich <span class="hlt">approximation</span> (describable as first-<span class="hlt">order</span> Lagrangian perturbation theory), and its second-<span class="hlt">order</span> generalization. In the last two cases we also created new versions of <span class="hlt">approximation</span> by truncation, i.e., smoothing the initial conditions by various smoothing window shapes and varying their sizes. The primary tool for comparing simulations to <span class="hlt">approximation</span> schemes was crosscorrelation of the evolved mass density fields, testing the extent to which mass was moved to the right place. The Zel'dovich <span class="hlt">approximation</span>, with initial convolution with a Gaussian e(exp -k(exp 2)/k(exp 2, sub G)) where k(sub G) is adjusted to be just into the nonlinear regime of the evolved model (details in text) worked extremely well. Its second-<span class="hlt">order</span> generalization worked slightly better. All other schemes, including those proposed as generalizations of the Zel'dovich <span class="hlt">approximation</span> created by adding forces, were in fact generally worse by this measure. By explicitly checking, we verified that the success of our best-choice was a result of the best treatment of the phases of nonlinear Fourier components. Of all schemes tested, the adhesion <span class="hlt">approximation</span> produced the most accurate nonlinear power spectrum and density distribution, but its phase errors suggest mass condensations were moved to slightly the wrong location. Due to its better reproduction of the mass density distribution function and power spectrum, it might be preferred for some uses. We recommend either n-body simulations or our modified versions of the Zel'dovich <span class="hlt">approximation</span>, depending upon the purpose. The theoretical implication is that pancaking is implicit in all cosmological gravitational clustering, at least from Gaussian initial conditions, even</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/27679480','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/27679480"><span>Finding <span class="hlt">approximate</span> gene clusters with Gecko 3.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Winter, Sascha; Jahn, Katharina; Wehner, Stefanie; Kuchenbecker, Leon; Marz, Manja; Stoye, Jens; Böcker, Sebastian</p> <p>2016-11-16</p> <p>Gene-<span class="hlt">order</span>-based comparison of multiple genomes provides signals for functional analysis of genes and the evolutionary process of genome organization. Gene clusters are regions of co-localized genes on genomes of different species. The rapid increase in sequenced genomes necessitates bioinformatics tools for finding gene clusters in hundreds of genomes. Existing tools are often restricted to few (in many cases, only two) genomes, and often make restrictive assumptions such as short perfect conservation, conserved gene <span class="hlt">order</span> or monophyletic gene clusters. We present Gecko 3, an open-source software for finding gene clusters in hundreds of bacterial genomes, that comes with an easy-to-use graphical user interface. The underlying gene cluster model is intuitive, can cope with low degrees of conservation as well as misannotations and is complemented by a sound statistical evaluation. To evaluate the biological benefit of Gecko 3 and to exemplify our method, we search for gene clusters in a dataset of 678 bacterial genomes using Synechocystis sp. PCC 6803 as a reference. We confirm detected gene clusters reviewing the literature and comparing them to a database of operons; we detect two novel clusters, which were confirmed by publicly available experimental RNA-Seq data. The computational analysis is carried out on a laptop computer in <40 min. © The Author(s) 2016. Published by Oxford University Press on behalf of Nucleic Acids Research.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19770007977','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19770007977"><span>Test particle propagation in magnetostatic turbulence. 2: The local <span class="hlt">approximation</span> method</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Klimas, A. J.; Sandri, G.; Scudder, J. D.; Howell, D. R.</p> <p>1976-01-01</p> <p>An <span class="hlt">approximation</span> method for statistical mechanics is presented and applied to a class of problems which contains a test particle propagation problem. All of the available basic equations used in statistical mechanics are cast in the form of a single equation which is integrodifferential in time and which is then used as the starting point for the construction of the local <span class="hlt">approximation</span> method. Simplification of the integrodifferential equation is achieved through <span class="hlt">approximation</span> to the Laplace transform of its kernel. The <span class="hlt">approximation</span> is valid near the origin in the Laplace space and is based on the assumption of small Laplace variable. No other small parameter is necessary for the construction of this <span class="hlt">approximation</span> method. The n'th level of <span class="hlt">approximation</span> is constructed formally, and the first five levels of <span class="hlt">approximation</span> are calculated explicitly. It is shown that each level of <span class="hlt">approximation</span> is governed by an inhomogeneous partial differential equation in time with time independent operator coefficients. The <span class="hlt">order</span> in time of these partial differential equations is found to increase as n does. At n = 0 the most local first <span class="hlt">order</span> partial differential equation which governs the Markovian limit is regained.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016Galax...4....4Z','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016Galax...4....4Z"><span>Padé <span class="hlt">Approximant</span> and Minimax Rational <span class="hlt">Approximation</span> in Standard Cosmology</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Zaninetti, Lorenzo</p> <p>2016-02-01</p> <p>The luminosity distance in the standard cosmology as given by $\\Lambda$CDM and consequently the distance modulus for supernovae can be defined by the Pad\\'e <span class="hlt">approximant</span>. A comparison with a known analytical solution shows that the Pad\\'e <span class="hlt">approximant</span> for the luminosity distance has an error of $4\\%$ at redshift $= 10$. A similar procedure for the Taylor expansion of the luminosity distance gives an error of $4\\%$ at redshift $=0.7 $; this means that for the luminosity distance, the Pad\\'e <span class="hlt">approximation</span> is superior to the Taylor series. The availability of an analytical expression for the distance modulus allows applying the Levenberg--Marquardt method to derive the fundamental parameters from the available compilations for supernovae. A new luminosity function for galaxies derived from the truncated gamma probability density function models the observed luminosity function for galaxies when the observed range in absolute magnitude is modeled by the Pad\\'e <span class="hlt">approximant</span>. A comparison of $\\Lambda$CDM with other cosmologies is done adopting a statistical point of view.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://pubs.er.usgs.gov/publication/70020693','USGSPUBS'); return false;" href="https://pubs.er.usgs.gov/publication/70020693"><span>A comparison of zero-<span class="hlt">order</span>, first-<span class="hlt">order</span>, and monod biotransformation models</span></a></p> <p><a target="_blank" href="http://pubs.er.usgs.gov/pubs/index.jsp?view=adv">USGS Publications Warehouse</a></p> <p>Bekins, B.A.; Warren, E.; Godsy, E.M.</p> <p>1998-01-01</p> <p>Under some conditions, a first-<span class="hlt">order</span> kinetic model is a poor representation of biodegradation in contaminated aquifers. Although it is well known that the assumption of first-<span class="hlt">order</span> kinetics is valid only when substrate concentration, S, is much less than the half-saturation constant, K(s), this assumption is often made without verification of this condition. We present a formal error analysis showing that the relative error in the first-<span class="hlt">order</span> <span class="hlt">approximation</span> is S/K(S) and in the zero-<span class="hlt">order</span> <span class="hlt">approximation</span> the error is K(s)/S. We then examine the problems that arise when the first-<span class="hlt">order</span> <span class="hlt">approximation</span> is used outside the range for which it is valid. A series of numerical simulations comparing results of first- and zero-<span class="hlt">order</span> rate <span class="hlt">approximations</span> to Monod kinetics for a real data set illustrates that if concentrations observed in the field are higher than K(s), it may better to model degradation using a zero-<span class="hlt">order</span> rate expression. Compared with Monod kinetics, extrapolation of a first-<span class="hlt">order</span> rate to lower concentrations under-predicts the biotransformation potential, while extrapolation to higher concentrations may grossly over-predict the transformation rate. A summary of solubilities and Monod parameters for aerobic benzene, toluene, and xylene (BTX) degradation shows that the a priori assumption of first-<span class="hlt">order</span> degradation kinetics at sites contaminated with these compounds is not valid. In particular, out of six published values of KS for toluene, only one is greater than 2 mg/L, indicating that when toluene is present in concentrations greater than about a part per million, the assumption of first-<span class="hlt">order</span> kinetics may be invalid. Finally, we apply an existing analytical solution for steady-state one-dimensional advective transport with Monod degradation kinetics to a field data set.A formal error analysis is presented showing that the relative error in the first-<span class="hlt">order</span> <span class="hlt">approximation</span> is S/KS and in the zero-<span class="hlt">order</span> <span class="hlt">approximation</span> the error is KS/S where S is the substrate</p> </li> </ol> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_16");'>16</a></li> <li><a href="#" onclick='return showDiv("page_17");'>17</a></li> <li class="active"><span>18</span></li> <li><a href="#" onclick='return showDiv("page_19");'>19</a></li> <li><a href="#" onclick='return showDiv("page_20");'>20</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div><!-- col-sm-12 --> </div><!-- row --> </div><!-- page_18 --> <div id="page_19" class="hiddenDiv"> <div class="row"> <div class="col-sm-12"> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_17");'>17</a></li> <li><a href="#" onclick='return showDiv("page_18");'>18</a></li> <li class="active"><span>19</span></li> <li><a href="#" onclick='return showDiv("page_20");'>20</a></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div> </div> <div class="row"> <div class="col-sm-12"> <ol class="result-class" start="361"> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/28793122','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/28793122"><span>Scenario <span class="hlt">approximation</span> in a phenomenological study in Mexico: experience report.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Guerrero-Castañeda, Raúl Fernando; Menezes, Tânia Maria de Oliva; Vargas, Ma Guadalupe Ojeda</p> <p>2017-01-01</p> <p>To report our experience using scenario <span class="hlt">approximation</span> in a phenomenological study of nursing in Mexico. Experience report on scenario <span class="hlt">approximation</span> to coexist with elderly in <span class="hlt">order</span> to select the participants of a phenomenological study. During a four-month period in 2016, visits were carried out two groups of elderly individuals where several activities were carried out. Coexistence with the elderly throughout accompaniment in the groups' activities together with joint dialogue allowed selection of those who corresponded to the characteristics of the study objective. Scenario <span class="hlt">approximation</span> is necessary in phenomenological studies, not only for creating empathy among the participants but also for the researchers to immerse themselves in the phenomenon under study, as shown by the first approaches of the researcher. Relatar la experiencia del acercamiento al escenario de un estudio fenomenológico en enfermería en México. Relato de experiencia sobre el acercamiento al escenario de estudio para convivir con adultos mayores con la finalidad de seleccionar a los participantes de un estudio fenomenológico. Se llevaron a cabo visitas durante el año 2016, en un periodo de cuatro meses a dos grupos de adultos mayores en donde se realizaron diversas actividades. La convivencia con los adultos mayores a través del acompañamiento en las actividades que realizaban en los grupos y el diálogo conjunto permitió seleccionar a aquellos que respondían a las características del objeto de estudio. Es necesaria la aproximación al escenario de estudios fenomenológicos, no sólo con la finalidad de ganar empatía de los participantes sino para sumergirse en el fenómeno de estudio, mismo que se va mostrando desde los primeros acercamientos del investigador.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/1987SPIE..754..117B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/1987SPIE..754..117B"><span>Differential Geometry Applied To Least-Square Error Surface <span class="hlt">Approximations</span></span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Bolle, Ruud M.; Sabbah, Daniel</p> <p>1987-08-01</p> <p>This paper focuses on extraction of the parameters of individual surfaces from noisy depth maps. The basis for this are least-square error polynomial <span class="hlt">approximations</span> to the range data and the curvature properties that can be computed from these <span class="hlt">approximations</span>. The curvature properties are derived using the invariants of the Weingarten Map evaluated at the origin of local coordinate systems centered at the range points. The Weingarten Map is a well-known concept in differential geometry; a brief treatment of the differential geometry pertinent to surface curvature is given. We use the curvature properties for extracting certain surface parameters from the curvature properties of the <span class="hlt">approximations</span>. Then we show that curvature properties alone are not enough to obtain all the parameters of the surfaces; higher <span class="hlt">order</span> properties (information about change of curvature) are needed to obtain full parametric descriptions. This surface parameter estimation problem arises in the design of a vision system to recognize 3D objects whose surfaces are composed of planar patches and patches of quadrics of revolution. (Quadrics of revolution are quadrics that are surfaces of revolution.) A significant portion of man-made objects can be modeled using these surfaces. The actual process of recognition and parameter extraction is framed as a set of stacked parameter space transforms. The transforms are "stacked" in the sense that any one transform computes only a partial geometric description that forms the input to the next transform. Those who are interested in the organization and control of the recognition and parameter recognition process are referred to [Sabbah86], this paper briefly touches upon the organization, but concentrates mainly on geometrical aspects of the parameter extraction.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2012IJSyS..43.1642L','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2012IJSyS..43.1642L"><span>New realisation of Preisach model using adaptive polynomial <span class="hlt">approximation</span></span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Liu, Van-Tsai; Lin, Chun-Liang; Wing, Home-Young</p> <p>2012-09-01</p> <p>Modelling system with hysteresis has received considerable attention recently due to the increasing accurate requirement in engineering applications. The classical Preisach model (CPM) is the most popular model to demonstrate hysteresis which can be represented by infinite but countable first-<span class="hlt">order</span> reversal curves (FORCs). The usage of look-up tables is one way to approach the CPM in actual practice. The data in those tables correspond with the samples of a finite number of FORCs. This approach, however, faces two major problems: firstly, it requires a large amount of memory space to obtain an accurate prediction of hysteresis; secondly, it is difficult to derive efficient ways to modify the data table to reflect the timing effect of elements with hysteresis. To overcome, this article proposes the idea of using a set of polynomials to emulate the CPM instead of table look-up. The polynomial <span class="hlt">approximation</span> requires less memory space for data storage. Furthermore, the polynomial coefficients can be obtained accurately by using the least-square <span class="hlt">approximation</span> or adaptive identification algorithm, such as the possibility of accurate tracking of hysteresis model parameters.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/1337238-efficient-approximation-algorithms-weighted-matching','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/1337238-efficient-approximation-algorithms-weighted-matching"><span>Efficient <span class="hlt">Approximation</span> Algorithms for Weighted $b$-Matching</span></a></p> <p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p> <p>Khan, Arif; Pothen, Alex; Mostofa Ali Patwary, Md.</p> <p>2016-01-01</p> <p>We describe a half-<span class="hlt">approximation</span> algorithm, b-Suitor, for computing a b-Matching of maximum weight in a graph with weights on the edges. b-Matching is a generalization of the well-known Matching problem in graphs, where the objective is to choose a subset of M edges in the graph such that at most a specified number b(v) of edges in M are incident on each vertex v. Subject to this restriction we maximize the sum of the weights of the edges in M. We prove that the b-Suitor algorithm computes the same b-Matching as the one obtained by the greedy algorithm for themore » problem. We implement the algorithm on serial and shared-memory parallel processors, and compare its performance against a collection of <span class="hlt">approximation</span> algorithms that have been proposed for the Matching problem. Our results show that the b-Suitor algorithm outperforms the Greedy and Locally Dominant edge algorithms by one to two <span class="hlt">orders</span> of magnitude on a serial processor. The b-Suitor algorithm has a high degree of concurrency, and it scales well up to 240 threads on a shared memory multiprocessor. The b-Suitor algorithm outperforms the Locally Dominant edge algorithm by a factor of fourteen on 16 cores of an Intel Xeon multiprocessor.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19960008825','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19960008825"><span>On High-<span class="hlt">Order</span> Radiation Boundary Conditions</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Hagstrom, Thomas</p> <p>1995-01-01</p> <p>In this paper we develop the theory of high-<span class="hlt">order</span> radiation boundary conditions for wave propagation problems. In particular, we study the convergence of sequences of time-local <span class="hlt">approximate</span> conditions to the exact boundary condition, and subsequently estimate the error in the solutions obtained using these <span class="hlt">approximations</span>. We show that for finite times the Pade <span class="hlt">approximants</span> proposed by Engquist and Majda lead to exponential convergence if the solution is smooth, but that good long-time error estimates cannot hold for spatially local conditions. Applications in fluid dynamics are also discussed.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018NIMPB.418...87S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018NIMPB.418...87S"><span>SEE rate estimation based on diffusion <span class="hlt">approximation</span> of charge collection</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Sogoyan, Armen V.; Chumakov, Alexander I.; Smolin, Anatoly A.</p> <p>2018-03-01</p> <p>The integral rectangular parallelepiped (IRPP) method remains the main approach to single event rate (SER) prediction for aerospace systems, despite the growing number of issues impairing method's validity when applied to scaled technology nodes. One of such issues is uncertainty in parameters extraction in the IRPP method, which can lead to a spread of several <span class="hlt">orders</span> of magnitude in the subsequently calculated SER. The paper presents an alternative approach to SER estimation based on diffusion <span class="hlt">approximation</span> of the charge collection by an IC element and geometrical interpretation of SEE cross-section. In contrast to the IRPP method, the proposed model includes only two parameters which are uniquely determined from the experimental data for normal incidence irradiation at an ion accelerator. This approach eliminates the necessity of arbitrary decisions during parameter extraction and, thus, greatly simplifies calculation procedure and increases the robustness of the forecast.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/27118889','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/27118889"><span>Universal recovery map for <span class="hlt">approximate</span> Markov chains.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Sutter, David; Fawzi, Omar; Renner, Renato</p> <p>2016-02-01</p> <p>A central question in quantum information theory is to determine how well lost information can be reconstructed. Crucially, the corresponding recovery operation should perform well without knowing the information to be reconstructed. In this work, we show that the quantum conditional mutual information measures the performance of such recovery operations. More precisely, we prove that the conditional mutual information I ( A : C | B ) of a tripartite quantum state ρ ABC can be bounded from below by its distance to the closest recovered state [Formula: see text], where the C -part is reconstructed from the B -part only and the recovery map [Formula: see text] merely depends on ρ BC . One particular application of this result implies the equivalence between two different approaches to define topological <span class="hlt">order</span> in quantum systems.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/2058450','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/2058450"><span>[Bladder endometriosis. Diagnostic and therapeutic <span class="hlt">approximation</span>].</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Monllor Gisbert, J; Merino Hernaez, C; Olivier Gómez, C; Carballido Rodríguez, J</p> <p>1991-01-01</p> <p>Endometriosis is defined by the presence of functionally active endometria in ectopic position; so when its clinical behaviour adopts the characteristics of tumoration, endometrioma is accepted as an alternative name. Location of this pathology in the urinary apparatus is uncommon in terms of incidence, since the bladder is a selected site. This condition implies a high morbidity rate because it requires a high level of suspicion in <span class="hlt">order</span> to be diagnosed. It is acknowledged that no image study (CAT, Echography, MNR, etc.) is pathognomonic for endometriosis and it is necessary to perform endoscopy and biopsy to achieve a correct diagnosis. The paper includes the strategy to follow, both with regard to diagnosis and the current therapeutic approaches.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4841654','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4841654"><span>Universal recovery map for <span class="hlt">approximate</span> Markov chains</span></a></p> <p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p> <p>Sutter, David; Fawzi, Omar; Renner, Renato</p> <p>2016-01-01</p> <p>A central question in quantum information theory is to determine how well lost information can be reconstructed. Crucially, the corresponding recovery operation should perform well without knowing the information to be reconstructed. In this work, we show that the quantum conditional mutual information measures the performance of such recovery operations. More precisely, we prove that the conditional mutual information I(A:C|B) of a tripartite quantum state ρABC can be bounded from below by its distance to the closest recovered state RB→BC(ρAB), where the C-part is reconstructed from the B-part only and the recovery map RB→BC merely depends on ρBC. One particular application of this result implies the equivalence between two different approaches to define topological <span class="hlt">order</span> in quantum systems. PMID:27118889</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2006CoPhC.174..447W','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2006CoPhC.174..447W"><span>A Mathematica program for the <span class="hlt">approximate</span> analytical solution to a nonlinear undamped Duffing equation by a new <span class="hlt">approximate</span> approach</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Wu, Dongmei; Wang, Zhongcheng</p> <p>2006-03-01</p> <p>According to Mickens [R.E. Mickens, Comments on a Generalized Galerkin's method for non-linear oscillators, J. Sound Vib. 118 (1987) 563], the general HB (harmonic balance) method is an <span class="hlt">approximation</span> to the convergent Fourier series representation of the periodic solution of a nonlinear oscillator and not an <span class="hlt">approximation</span> to an expansion in terms of a small parameter. Consequently, for a nonlinear undamped Duffing equation with a driving force Bcos(ωx), to find a periodic solution when the fundamental frequency is identical to ω, the corresponding Fourier series can be written as y˜(x)=∑n=1m acos[(2n-1)ωx]. How to calculate the coefficients of the Fourier series efficiently with a computer program is still an open problem. For HB method, by substituting <span class="hlt">approximation</span> y˜(x) into force equation, expanding the resulting expression into a trigonometric series, then letting the coefficients of the resulting lowest-<span class="hlt">order</span> harmonic be zero, one can obtain <span class="hlt">approximate</span> coefficients of <span class="hlt">approximation</span> y˜(x) [R.E. Mickens, Comments on a Generalized Galerkin's method for non-linear oscillators, J. Sound Vib. 118 (1987) 563]. But for nonlinear differential equations such as Duffing equation, it is very difficult to construct higher-<span class="hlt">order</span> analytical <span class="hlt">approximations</span>, because the HB method requires solving a set of algebraic equations for a large number of unknowns with very complex nonlinearities. To overcome the difficulty, forty years ago, Urabe derived a computational method for Duffing equation based on Galerkin procedure [M. Urabe, A. Reiter, Numerical computation of nonlinear forced oscillations by Galerkin's procedure, J. Math. Anal. Appl. 14 (1966) 107-140]. Dooren obtained an <span class="hlt">approximate</span> solution of the Duffing oscillator with a special set of parameters by using Urabe's method [R. van Dooren, Stabilization of Cowell's classic finite difference method for numerical integration, J. Comput. Phys. 16 (1974) 186-192]. In this paper, in the frame of the general HB method</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018PhRvD..97a6016B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018PhRvD..97a6016B"><span>Born-Oppenheimer <span class="hlt">approximation</span> in an effective field theory language</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Brambilla, Nora; Krein, Gastão; Tarrús Castellà, Jaume; Vairo, Antonio</p> <p>2018-01-01</p> <p>The Born-Oppenheimer <span class="hlt">approximation</span> is the standard tool for the study of molecular systems. It is founded on the observation that the energy scale of the electron dynamics in a molecule is larger than that of the nuclei. A very similar physical picture can be used to describe QCD states containing heavy quarks as well as light-quarks or gluonic excitations. In this work, we derive the Born-Oppenheimer <span class="hlt">approximation</span> for QED molecular systems in an effective field theory framework by sequentially integrating out degrees of freedom living at energies above the typical energy scale where the dynamics of the heavy degrees of freedom occurs. In particular, we compute the matching coefficients of the effective field theory for the case of the H2+ diatomic molecule that are relevant to compute its spectrum up to O (m α5). Ultrasoft photon loops contribute at this <span class="hlt">order</span>, being ultimately responsible for the molecular Lamb shift. In the effective field theory the scaling of all the operators is homogeneous, which facilitates the determination of all the relevant contributions, an observation that may become useful for high-precision calculations. Using the above case as a guidance, we construct under some conditions an effective field theory for QCD states formed by a color-octet heavy quark-antiquark pair bound with a color-octet light-quark pair or excited gluonic state, highlighting the similarities and differences between the QED and QCD systems. Assuming that the multipole expansion is applicable, we construct the heavy-quark potential up to next-to-leading <span class="hlt">order</span> in the multipole expansion in terms of nonperturbative matching coefficients to be obtained from lattice QCD.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19890002382','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19890002382"><span><span class="hlt">Approximation</span> theory for LQG (Linear-Quadratic-Gaussian) optimal control of flexible structures</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Gibson, J. S.; Adamian, A.</p> <p>1988-01-01</p> <p>An <span class="hlt">approximation</span> theory is presented for the LQG (Linear-Quadratic-Gaussian) optimal control problem for flexible structures whose distributed models have bounded input and output operators. The main purpose of the theory is to guide the design of finite dimensional compensators that <span class="hlt">approximate</span> closely the optimal compensator. The optimal LQG problem separates into an optimal linear-quadratic regulator problem and an optimal state estimation problem. The solution of the former problem lies in the solution to an infinite dimensional Riccati operator equation. The <span class="hlt">approximation</span> scheme <span class="hlt">approximates</span> the infinite dimensional LQG problem with a sequence of finite dimensional LQG problems defined for a sequence of finite dimensional, usually finite element or modal, <span class="hlt">approximations</span> of the distributed model of the structure. Two Riccati matrix equations determine the solution to each <span class="hlt">approximating</span> problem. The finite dimensional equations for numerical <span class="hlt">approximation</span> are developed, including formulas for converting matrix control and estimator gains to their functional representation to allow comparison of gains based on different <span class="hlt">orders</span> of <span class="hlt">approximation</span>. Convergence of the <span class="hlt">approximating</span> control and estimator gains and of the corresponding finite dimensional compensators is studied. Also, convergence and stability of the closed-loop systems produced with the finite dimensional compensators are discussed. The convergence theory is based on the convergence of the solutions of the finite dimensional Riccati equations to the solutions of the infinite dimensional Riccati equations. A numerical example with a flexible beam, a rotating rigid body, and a lumped mass is given.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19930043559&hterms=clustering&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D70%26Ntt%3Dclustering','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19930043559&hterms=clustering&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D70%26Ntt%3Dclustering"><span>Testing <span class="hlt">approximations</span> for non-linear gravitational clustering</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Coles, Peter; Melott, Adrian L.; Shandarin, Sergei F.</p> <p>1993-01-01</p> <p>The accuracy of various analytic <span class="hlt">approximations</span> for following the evolution of cosmological density fluctuations into the nonlinear regime is investigated. The Zel'dovich <span class="hlt">approximation</span> is found to be consistently the best <span class="hlt">approximation</span> scheme. It is extremely accurate for power spectra characterized by n = -1 or less; when the <span class="hlt">approximation</span> is 'enhanced' by truncating highly nonlinear Fourier modes the <span class="hlt">approximation</span> is excellent even for n = +1. The performance of linear theory is less spectrum-dependent, but this <span class="hlt">approximation</span> is less accurate than the Zel'dovich one for all cases because of the failure to treat dynamics. The lognormal <span class="hlt">approximation</span> generally provides a very poor fit to the spatial pattern.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19890028795&hterms=discrete+mathematics&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D50%26Ntt%3Ddiscrete%2Bmathematics','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19890028795&hterms=discrete+mathematics&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D50%26Ntt%3Ddiscrete%2Bmathematics"><span>Singular perturbation and time scale approaches in discrete control systems</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Naidu, D. S.; Price, D. B.</p> <p>1988-01-01</p> <p>After considering a singularly perturbed discrete control system, a singular perturbation approach is used to obtain outer and correction subsystems. A time scale approach is then applied via block diagonalization transformations to decouple the system into slow and fast subsystems. To a <span class="hlt">zeroth-order</span> <span class="hlt">approximation</span>, the singular perturbation and time-scale approaches are found to yield equivalent results.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19940032991','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19940032991"><span><span class="hlt">Approximate</span> analysis for repeated eigenvalue problems with applications to controls-structure integrated design</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Kenny, Sean P.; Hou, Gene J. W.</p> <p>1994-01-01</p> <p>A method for eigenvalue and eigenvector <span class="hlt">approximate</span> analysis for the case of repeated eigenvalues with distinct first derivatives is presented. The <span class="hlt">approximate</span> analysis method developed involves a reparameterization of the multivariable structural eigenvalue problem in terms of a single positive-valued parameter. The resulting equations yield first-<span class="hlt">order</span> <span class="hlt">approximations</span> to changes in the eigenvalues and the eigenvectors associated with the repeated eigenvalue problem. This work also presents a numerical technique that facilitates the definition of an eigenvector derivative for the case of repeated eigenvalues with repeated eigenvalue derivatives (of all <span class="hlt">orders</span>). Examples are given which demonstrate the application of such equations for sensitivity and <span class="hlt">approximate</span> analysis. Emphasis is placed on the application of sensitivity analysis to large-scale structural and controls-structures optimization problems.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/22676078-testing-approximate-predictions-displacements-cosmological-dark-matter-halos','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/22676078-testing-approximate-predictions-displacements-cosmological-dark-matter-halos"><span>Testing <span class="hlt">approximate</span> predictions of displacements of cosmological dark matter halos</span></a></p> <p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p> <p>Munari, Emiliano; Monaco, Pierluigi; Borgani, Stefano</p> <p></p> <p>We present a test to quantify how well some <span class="hlt">approximate</span> methods, designed to reproduce the mildly non-linear evolution of perturbations, are able to reproduce the clustering of DM halos once the grouping of particles into halos is defined and kept fixed. The following methods have been considered: Lagrangian Perturbation Theory (LPT) up to third <span class="hlt">order</span>, Truncated LPT, Augmented LPT, MUSCLE and COLA. The test runs as follows: halos are defined by applying a friends-of-friends (FoF) halo finder to the output of an N-body simulation. The <span class="hlt">approximate</span> methods are then applied to the same initial conditions of the simulation, producing formore » all particles displacements from their starting position and velocities. The position and velocity of each halo are computed by averaging over the particles that belong to that halo, according to the FoF halo finder. This procedure allows us to perform a well-posed test of how clustering of the matter density and halo density fields are recovered, without asking to the <span class="hlt">approximate</span> method an accurate reconstruction of halos. We have considered the results at z =0,0.5,1, and we have analysed power spectrum in real and redshift space, object-by-object difference in position and velocity, density Probability Distribution Function (PDF) and its moments, phase difference of Fourier modes. We find that higher LPT <span class="hlt">orders</span> are generally able to better reproduce the clustering of halos, while little or no improvement is found for the matter density field when going to 2LPT and 3LPT. Augmentation provides some improvement when coupled with 2LPT, while its effect is limited when coupled with 3LPT. Little improvement is brought by MUSCLE with respect to Augmentation. The more expensive particle-mesh code COLA outperforms all LPT methods, and this is true even for mesh sizes as large as the inter-particle distance. This test sets an upper limit on the ability of these methods to reproduce the clustering of halos, for the cases when these</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017JCAP...07..050M','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017JCAP...07..050M"><span>Testing <span class="hlt">approximate</span> predictions of displacements of cosmological dark matter halos</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Munari, Emiliano; Monaco, Pierluigi; Koda, Jun; Kitaura, Francisco-Shu; Sefusatti, Emiliano; Borgani, Stefano</p> <p>2017-07-01</p> <p>We present a test to quantify how well some <span class="hlt">approximate</span> methods, designed to reproduce the mildly non-linear evolution of perturbations, are able to reproduce the clustering of DM halos once the grouping of particles into halos is defined and kept fixed. The following methods have been considered: Lagrangian Perturbation Theory (LPT) up to third <span class="hlt">order</span>, Truncated LPT, Augmented LPT, MUSCLE and COLA. The test runs as follows: halos are defined by applying a friends-of-friends (FoF) halo finder to the output of an N-body simulation. The <span class="hlt">approximate</span> methods are then applied to the same initial conditions of the simulation, producing for all particles displacements from their starting position and velocities. The position and velocity of each halo are computed by averaging over the particles that belong to that halo, according to the FoF halo finder. This procedure allows us to perform a well-posed test of how clustering of the matter density and halo density fields are recovered, without asking to the <span class="hlt">approximate</span> method an accurate reconstruction of halos. We have considered the results at z=0,0.5,1, and we have analysed power spectrum in real and redshift space, object-by-object difference in position and velocity, density Probability Distribution Function (PDF) and its moments, phase difference of Fourier modes. We find that higher LPT <span class="hlt">orders</span> are generally able to better reproduce the clustering of halos, while little or no improvement is found for the matter density field when going to 2LPT and 3LPT. Augmentation provides some improvement when coupled with 2LPT, while its effect is limited when coupled with 3LPT. Little improvement is brought by MUSCLE with respect to Augmentation. The more expensive particle-mesh code COLA outperforms all LPT methods, and this is true even for mesh sizes as large as the inter-particle distance. This test sets an upper limit on the ability of these methods to reproduce the clustering of halos, for the cases when these objects are</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/21502460-analytic-derivation-approximate-su-symmetry-inside-symmetry-triangle-interacting-boson-approximation-model','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/21502460-analytic-derivation-approximate-su-symmetry-inside-symmetry-triangle-interacting-boson-approximation-model"><span>Analytic derivation of an <span class="hlt">approximate</span> SU(3) symmetry inside the symmetry triangle of the interacting boson <span class="hlt">approximation</span> model</span></a></p> <p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p> <p>Bonatsos, Dennis; Karampagia, S.; Casten, R. F.</p> <p>2011-05-15</p> <p>Using a contraction of the SU(3) algebra to the algebra of the rigid rotator in the large-boson-number limit of the interacting boson <span class="hlt">approximation</span> (IBA) model, a line is found inside the symmetry triangle of the IBA, along which the SU(3) symmetry is preserved. The line extends from the SU(3) vertex to near the critical line of the first-<span class="hlt">order</span> shape/phase transition separating the spherical and prolate deformed phases, and it lies within the Alhassid-Whelan arc of regularity, the unique valley of regularity connecting the SU(3) and U(5) vertices in the midst of chaotic regions. In addition to providing an explanation formore » the existence of the arc of regularity, the present line represents an example of an analytically determined <span class="hlt">approximate</span> symmetry in the interior of the symmetry triangle of the IBA. The method is applicable to algebraic models possessing subalgebras amenable to contraction. This condition is equivalent to algebras in which the equilibrium ground state and its rotational band become energetically isolated from intrinsic excitations, as typified by deformed solutions to the IBA for large numbers of valence nucleons.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/839007-corrections-one-photon-approximation-gt-transition','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/839007-corrections-one-photon-approximation-gt-transition"><span>Corrections to the one-photon <span class="hlt">approximation</span> in the 0+-->2+ transition of 12C</span></a></p> <p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p> <p>Paul Gueye; Madeleine Bernheim; J. F. Danel</p> <p>2001-04-18</p> <p>Contribution of higher <span class="hlt">order</span> effects to the one-photon exchange <span class="hlt">approximation</span> were studied in the first excited state of 12C by comparing inclusive inelastic scattering cross sections of electrons and positrons obtained at the Saclay Linear Accelerator. The data were compared to a distorted wave Born <span class="hlt">approximation</span> (DWBA)calculation. The results indicate an effect less than 2% within 2sigma, compatible with what was observed in recent elastic scattering measurements.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/20856423','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/20856423"><span><span class="hlt">Approximate</span> effective nonlinear coefficient of second-harmonic generation in KTiOPO(4).</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Asaumi, K</p> <p>1993-10-20</p> <p>A simplified <span class="hlt">approximate</span> expression for the effective nonlinear coefficient of type-II second-harmonicgeneration in KTiOPO(4) was obtained by observing that the difference between the refractive indices n(x) and n(y) is 1 <span class="hlt">order</span> of magnitude smaller than the difference between n(z) and n(y) (or n(x)). The agreement of this <span class="hlt">approximate</span> equation with the true definition is good, with a maximum discrepancy of 4%.</p> </li> </ol> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_17");'>17</a></li> <li><a href="#" onclick='return showDiv("page_18");'>18</a></li> <li class="active"><span>19</span></li> <li><a href="#" onclick='return showDiv("page_20");'>20</a></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div><!-- col-sm-12 --> </div><!-- row --> </div><!-- page_19 --> <div id="page_20" class="hiddenDiv"> <div class="row"> <div class="col-sm-12"> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_18");'>18</a></li> <li><a href="#" onclick='return showDiv("page_19");'>19</a></li> <li class="active"><span>20</span></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li><a href="#" onclick='return showDiv("page_22");'>22</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div> </div> <div class="row"> <div class="col-sm-12"> <ol class="result-class" start="381"> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19890014937','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19890014937"><span>Aeroservoelastic modeling and applications using minimum-state <span class="hlt">approximations</span> of the unsteady aerodynamics</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Tiffany, Sherwood H.; Karpel, Mordechay</p> <p>1989-01-01</p> <p>Various control analysis, design, and simulation techniques for aeroelastic applications require the equations of motion to be cast in a linear time-invariant state-space form. Unsteady aerodynamics forces have to be <span class="hlt">approximated</span> as rational functions of the Laplace variable in <span class="hlt">order</span> to put them in this framework. For the minimum-state method, the number of denominator roots in the rational <span class="hlt">approximation</span>. Results are shown of applying various <span class="hlt">approximation</span> enhancements (including optimization, frequency dependent weighting of the tabular data, and constraint selection) with the minimum-state formulation to the active flexible wing wind-tunnel model. The results demonstrate that good models can be developed which have an <span class="hlt">order</span> of magnitude fewer augmenting aerodynamic equations more than traditional approaches. This reduction facilitates the design of lower <span class="hlt">order</span> control systems, analysis of control system performance, and near real-time simulation of aeroservoelastic phenomena.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017ZNatA..72..673W','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017ZNatA..72..673W"><span>Highly Accurate Analytical <span class="hlt">Approximate</span> Solution to a Nonlinear Pseudo-Oscillator</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Wu, Baisheng; Liu, Weijia; Lim, C. W.</p> <p>2017-07-01</p> <p>A second-<span class="hlt">order</span> Newton method is presented to construct analytical <span class="hlt">approximate</span> solutions to a nonlinear pseudo-oscillator in which the restoring force is inversely proportional to the dependent variable. The nonlinear equation is first expressed in a specific form, and it is then solved in two steps, a predictor and a corrector step. In each step, the harmonic balance method is used in an appropriate manner to obtain a set of linear algebraic equations. With only one simple second-<span class="hlt">order</span> Newton iteration step, a short, explicit, and highly accurate analytical <span class="hlt">approximate</span> solution can be derived. The <span class="hlt">approximate</span> solutions are valid for all amplitudes of the pseudo-oscillator. Furthermore, the method incorporates second-<span class="hlt">order</span> Taylor expansion in a natural way, and it is of significant faster convergence rate.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19900013242','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19900013242"><span>Differential equation based method for accurate <span class="hlt">approximations</span> in optimization</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Pritchard, Jocelyn I.; Adelman, Howard M.</p> <p>1990-01-01</p> <p>A method to efficiently and accurately <span class="hlt">approximate</span> the effect of design changes on structural response is described. The key to this method is to interpret sensitivity equations as differential equations that may be solved explicitly for closed form <span class="hlt">approximations</span>, hence, the method is denoted the Differential Equation Based (DEB) method. <span class="hlt">Approximations</span> were developed for vibration frequencies, mode shapes and static displacements. The DEB <span class="hlt">approximation</span> method was applied to a cantilever beam and results compared with the commonly-used linear Taylor series <span class="hlt">approximations</span> and exact solutions. The test calculations involved perturbing the height, width, cross-sectional area, tip mass, and bending inertia of the beam. The DEB method proved to be very accurate, and in most cases, was more accurate than the linear Taylor series <span class="hlt">approximation</span>. The method is applicable to simultaneous perturbation of several design variables. Also, the <span class="hlt">approximations</span> may be used to calculate other system response quantities. For example, the <span class="hlt">approximations</span> for displacements are used to <span class="hlt">approximate</span> bending stresses.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19950044221&hterms=kolb&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D50%26Ntt%3Dkolb','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19950044221&hterms=kolb&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D50%26Ntt%3Dkolb"><span>First-<span class="hlt">order</span> inflation</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Kolb, Edward W.</p> <p>1991-01-01</p> <p>In the original proposal, inflation occurred in the process of a strongly first-<span class="hlt">order</span> phase transition. This model was soon demonstrated to be fatally flawed. Subsequent models for inflation involved phase transitions that were second-<span class="hlt">order</span>, or perhaps weakly first-<span class="hlt">order</span>; some even involved no phase transition at all. Recently the possibility of inflation during a strongly first-<span class="hlt">order</span> phase transition has been revived. In this talk I will discuss some models for first-<span class="hlt">order</span> inflation, and emphasize unique signatures that result if inflation is realized in a first-<span class="hlt">order</span> transition. Before discussing first-<span class="hlt">order</span> inflation, I will briefly review some of the history of inflation to demonstrate how first-<span class="hlt">order</span> inflation differs from other models.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4330682','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4330682"><span>The measurement of psychological literacy: a first <span class="hlt">approximation</span></span></a></p> <p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p> <p>Roberts, Lynne D.; Heritage, Brody; Gasson, Natalie</p> <p>2015-01-01</p> <p>Psychological literacy, the ability to apply psychological knowledge to personal, family, occupational, community and societal challenges, is promoted as the primary outcome of an undergraduate education in psychology. As the concept of psychological literacy becomes increasingly adopted as the core business of undergraduate psychology training courses world-wide, there is urgent need for the construct to be accurately measured so that student and institutional level progress can be assessed and monitored. Key to the measurement of psychological literacy is determining the underlying factor-structure of psychological literacy. In this paper we provide a first <span class="hlt">approximation</span> of the measurement of psychological literacy by identifying and evaluating self-report measures for psychological literacy. Multi-item and single-item self-report measures of each of the proposed nine dimensions of psychological literacy were completed by two samples (N = 218 and N = 381) of undergraduate psychology students at an Australian university. Single and multi-item measures of each dimension were weakly to moderately correlated. Exploratory and confirmatory factor analyses of multi-item measures indicated a higher <span class="hlt">order</span> three factor solution best represented the construct of psychological literacy. The three factors were reflective processes, generic graduate attributes, and psychology as a helping profession. For the measurement of psychological literacy to progress there is a need to further develop self-report measures and to identify/develop and evaluate objective measures of psychological literacy. Further <span class="hlt">approximations</span> of the measurement of psychological literacy remain an imperative, given the construct's ties to measuring institutional efficacy in teaching psychology to an undergraduate audience. PMID:25741300</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016GeoJI.204...59V','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016GeoJI.204...59V"><span>The impact of <span class="hlt">approximations</span> and arbitrary choices on geophysical images</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Valentine, Andrew P.; Trampert, Jeannot</p> <p>2016-01-01</p> <p>Whenever a geophysical image is to be constructed, a variety of choices must be made. Some, such as those governing data selection and processing, or model parametrization, are somewhat arbitrary: there may be little reason to prefer one choice over another. Others, such as defining the theoretical framework within which the data are to be explained, may be more straightforward: typically, an `exact' theory exists, but various <span class="hlt">approximations</span> may need to be adopted in <span class="hlt">order</span> to make the imaging problem computationally tractable. Differences between any two images of the same system can be explained in terms of differences between these choices. Understanding the impact of each particular decision is essential if images are to be interpreted properly-but little progress has been made towards a quantitative treatment of this effect. In this paper, we consider a general linearized inverse problem, applicable to a wide range of imaging situations. We write down an expression for the difference between two images produced using similar inversion strategies, but where different choices have been made. This provides a framework within which inversion algorithms may be analysed, and allows us to consider how image effects may arise. In this paper, we take a general view, and do not specialize our discussion to any specific imaging problem or setup (beyond the restrictions implied by the use of linearized inversion techniques). In particular, we look at the concept of `hybrid inversion', in which highly accurate synthetic data (typically the result of an expensive numerical simulation) is combined with an inverse operator constructed based on theoretical <span class="hlt">approximations</span>. It is generally supposed that this offers the benefits of using the more complete theory, without the full computational costs. We argue that the inverse operator is as important as the forward calculation in determining the accuracy of results. We illustrate this using a simple example, based on imaging the</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20150003441','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20150003441"><span>Atmospheric Turbulence Modeling for Aerospace Vehicles: Fractional <span class="hlt">Order</span> Fit</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Kopasakis, George (Inventor)</p> <p>2015-01-01</p> <p>An improved model for simulating atmospheric disturbances is disclosed. A scale Kolmogorov spectral may be scaled to convert the Kolmogorov spectral into a finite energy von Karman spectral and a fractional <span class="hlt">order</span> pole-zero transfer function (TF) may be derived from the von Karman spectral. Fractional <span class="hlt">order</span> atmospheric turbulence may be <span class="hlt">approximated</span> with an integer <span class="hlt">order</span> pole-zero TF fit, and the <span class="hlt">approximation</span> may be stored in memory.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2001PhRvD..63i6003M','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2001PhRvD..63i6003M"><span>Resumming the large-N <span class="hlt">approximation</span> for time evolving quantum systems</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Mihaila, Bogdan; Dawson, John F.; Cooper, Fred</p> <p>2001-05-01</p> <p>In this paper we discuss two methods of resumming the leading and next to leading <span class="hlt">order</span> in 1/N diagrams for the quartic O(N) model. These two approaches have the property that they preserve both boundedness and positivity for expectation values of operators in our numerical simulations. These <span class="hlt">approximations</span> can be understood either in terms of a truncation to the infinitely coupled Schwinger-Dyson hierarchy of equations, or by choosing a particular two-particle irreducible vacuum energy graph in the effective action of the Cornwall-Jackiw-Tomboulis formalism. We confine our discussion to the case of quantum mechanics where the Lagrangian is L(x,ẋ)=(12)∑Ni=1x˙2i-(g/8N)[∑Ni=1x2i- r20]2. The key to these <span class="hlt">approximations</span> is to treat both the x propagator and the x2 propagator on similar footing which leads to a theory whose graphs have the same topology as QED with the x2 propagator playing the role of the photon. The bare vertex <span class="hlt">approximation</span> is obtained by replacing the exact vertex function by the bare one in the exact Schwinger-Dyson equations for the one and two point functions. The second <span class="hlt">approximation</span>, which we call the dynamic Debye screening <span class="hlt">approximation</span>, makes the further <span class="hlt">approximation</span> of replacing the exact x2 propagator by its value at leading <span class="hlt">order</span> in the 1/N expansion. These two <span class="hlt">approximations</span> are compared with exact numerical simulations for the quantum roll problem. The bare vertex <span class="hlt">approximation</span> captures the physics at large and modest N better than the dynamic Debye screening <span class="hlt">approximation</span>.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19840042981&hterms=Nonlinear+equations&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D60%26Ntt%3DNonlinear%2Bequations','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19840042981&hterms=Nonlinear+equations&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D60%26Ntt%3DNonlinear%2Bequations"><span>Difference equation state <span class="hlt">approximations</span> for nonlinear hereditary control problems</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Rosen, I. G.</p> <p>1984-01-01</p> <p>Discrete <span class="hlt">approximation</span> schemes for the solution of nonlinear hereditary control problems are constructed. The methods involve <span class="hlt">approximation</span> by a sequence of optimal control problems in which the original infinite dimensional state equation has been <span class="hlt">approximated</span> by a finite dimensional discrete difference equation. Convergence of the state <span class="hlt">approximations</span> is argued using linear semigroup theory and is then used to demonstrate that solutions to the <span class="hlt">approximating</span> optimal control problems in some sense <span class="hlt">approximate</span> solutions to the original control problem. Two schemes, one based upon piecewise constant <span class="hlt">approximation</span>, and the other involving spline functions are discussed. Numerical results are presented, analyzed and used to compare the schemes to other available <span class="hlt">approximation</span> methods for the solution of hereditary control problems. Previously announced in STAR as N83-33589</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19830025318','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19830025318"><span>Difference equation state <span class="hlt">approximations</span> for nonlinear hereditary control problems</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Rosen, I. G.</p> <p>1982-01-01</p> <p>Discrete <span class="hlt">approximation</span> schemes for the solution of nonlinear hereditary control problems are constructed. The methods involve <span class="hlt">approximation</span> by a sequence of optimal control problems in which the original infinite dimensional state equation has been <span class="hlt">approximated</span> by a finite dimensional discrete difference equation. Convergence of the state <span class="hlt">approximations</span> is argued using linear semigroup theory and is then used to demonstrate that solutions to the <span class="hlt">approximating</span> optimal control problems in some sense <span class="hlt">approximate</span> solutions to the original control problem. Two schemes, one based upon piecewise constant <span class="hlt">approximation</span>, and the other involving spline functions are discussed. Numerical results are presented, analyzed and used to compare the schemes to other available <span class="hlt">approximation</span> methods for the solution of hereditary control problems.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19740011605','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19740011605"><span>Periodicity of high-<span class="hlt">order</span> neural functions</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Kellaway, P.; Borda, R. P.; Frost, J. D.; Carrie, J. R. G.; Coats, A. C.</p> <p>1973-01-01</p> <p>The results of recent studies on higher <span class="hlt">order</span>, integrative processes in the central nervous system are reported. Attempts were made to determine whether these processes exhibit any ongoing rhythmicity which might manifest itself in alterations of attention and alertness. Experiments were also designed to determine if a periodicity <span class="hlt">approximating</span> that of the REM could be detected in various parameters of brain electrical activity.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017JPhCS.901a2173G','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017JPhCS.901a2173G"><span>Born <span class="hlt">approximation</span> in linear-time invariant system</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Gumjudpai, Burin</p> <p>2017-09-01</p> <p>An alternative way of finding the LTI’s solution with the Born <span class="hlt">approximation</span>, is investigated. We use Born <span class="hlt">approximation</span> in the LTI and in the transformed LTI in form of Helmholtz equation. General solution are considered as infinite series or Feynman graph. Slow-roll <span class="hlt">approximation</span> are explored. Transforming the LTI system into Helmholtz equation, <span class="hlt">approximated</span> general solution can be found for any given forms of force with its initial value.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018MNRAS.475..244A','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018MNRAS.475..244A"><span>Second-<span class="hlt">order</span> singular pertubative theory for gravitational lenses</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Alard, C.</p> <p>2018-03-01</p> <p>The extension of the singular perturbative approach to the second <span class="hlt">order</span> is presented in this paper. The general expansion to the second <span class="hlt">order</span> is derived. The second-<span class="hlt">order</span> expansion is considered as a small correction to the first-<span class="hlt">order</span> expansion. Using this approach, it is demonstrated that in practice the second-<span class="hlt">order</span> expansion is reducible to a first <span class="hlt">order</span> expansion via a re-definition of the first-<span class="hlt">order</span> pertubative fields. Even if in usual applications the second-<span class="hlt">order</span> correction is small the reducibility of the second-<span class="hlt">order</span> expansion to the first-<span class="hlt">order</span> expansion indicates a potential degeneracy issue. In general, this degeneracy is hard to break. A useful and simple second-<span class="hlt">order</span> <span class="hlt">approximation</span> is the thin source <span class="hlt">approximation</span>, which offers a direct estimation of the correction. The practical application of the corrections derived in this paper is illustrated by using an elliptical NFW lens model. The second-<span class="hlt">order</span> pertubative expansion provides a noticeable improvement, even for the simplest case of thin source <span class="hlt">approximation</span>. To conclude, it is clear that for accurate modelization of gravitational lenses using the perturbative method the second-<span class="hlt">order</span> perturbative expansion should be considered. In particular, an evaluation of the degeneracy due to the second-<span class="hlt">order</span> term should be performed, for which the thin source <span class="hlt">approximation</span> is particularly useful.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19900014691','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19900014691"><span>Explicitly solvable complex Chebyshev <span class="hlt">approximation</span> problems related to sine polynomials</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Freund, Roland</p> <p>1989-01-01</p> <p>Explicitly solvable real Chebyshev <span class="hlt">approximation</span> problems on the unit interval are typically characterized by simple error curves. A similar principle is presented for complex <span class="hlt">approximation</span> problems with error curves induced by sine polynomials. As an application, some new explicit formulae for complex best <span class="hlt">approximations</span> are derived.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=estimator&pg=6&id=EJ1109033','ERIC'); return false;" href="https://eric.ed.gov/?q=estimator&pg=6&id=EJ1109033"><span>Meta-Regression <span class="hlt">Approximations</span> to Reduce Publication Selection Bias</span></a></p> <p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p> <p>Stanley, T. D.; Doucouliagos, Hristos</p> <p>2014-01-01</p> <p>Publication selection bias is a serious challenge to the integrity of all empirical sciences. We derive meta-regression <span class="hlt">approximations</span> to reduce this bias. Our approach employs Taylor polynomial <span class="hlt">approximations</span> to the conditional mean of a truncated distribution. A quadratic <span class="hlt">approximation</span> without a linear term, precision-effect estimate with…</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19810006883','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19810006883"><span>ACCESS 3. <span class="hlt">Approximation</span> concepts code for efficient structural synthesis: User's guide</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Fleury, C.; Schmit, L. A., Jr.</p> <p>1980-01-01</p> <p>A user's guide is presented for ACCESS-3, a research oriented program which combines dual methods and a collection of <span class="hlt">approximation</span> concepts to achieve excellent efficiency in structural synthesis. The finite element method is used for structural analysis and dual algorithms of mathematical programming are applied in the design optimization procedure. This program retains all of the ACCESS-2 capabilities and the data preparation formats are fully compatible. Four distinct optimizer options were added: interior point penalty function method (NEWSUMT); second <span class="hlt">order</span> primal projection method (PRIMAL2); second <span class="hlt">order</span> Newton-type dual method (DUAL2); and first <span class="hlt">order</span> gradient projection-type dual method (DUAL1). A pure discrete and mixed continuous-discrete design variable capability, and zero <span class="hlt">order</span> <span class="hlt">approximation</span> of the stress constraints are also included.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/26731788','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/26731788"><span>Minimal-<span class="hlt">Approximation</span>-Based Decentralized Backstepping Control of Interconnected Time-Delay Systems.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Choi, Yun Ho; Yoo, Sung Jin</p> <p>2016-12-01</p> <p>A decentralized adaptive backstepping control design using minimal function <span class="hlt">approximators</span> is proposed for nonlinear large-scale systems with unknown unmatched time-varying delayed interactions and unknown backlash-like hysteresis nonlinearities. Compared with existing decentralized backstepping methods, the contribution of this paper is to design a simple local control law for each subsystem, consisting of an actual control with one adaptive function <span class="hlt">approximator</span>, without requiring the use of multiple function <span class="hlt">approximators</span> and regardless of the <span class="hlt">order</span> of each subsystem. The virtual controllers for each subsystem are used as intermediate signals for designing a local actual control at the last step. For each subsystem, a lumped unknown function including the unknown nonlinear terms and the hysteresis nonlinearities is derived at the last step and is estimated by one function <span class="hlt">approximator</span>. Thus, the proposed approach only uses one function <span class="hlt">approximator</span> to implement each local controller, while existing decentralized backstepping control methods require the number of function <span class="hlt">approximators</span> equal to the <span class="hlt">order</span> of each subsystem and a calculation of virtual controllers to implement each local actual controller. The stability of the total controlled closed-loop system is analyzed using the Lyapunov stability theorem.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2014PhDT........34R','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2014PhDT........34R"><span>Dynamic Analyses of Result Quality in Energy-Aware <span class="hlt">Approximate</span> Programs</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>RIngenburg, Michael F.</p> <p></p> <p>Energy efficiency is a key concern in the design of modern computer systems. One promising approach to energy-efficient computation, <span class="hlt">approximate</span> computing, trades off output precision for energy efficiency. However, this tradeoff can have unexpected effects on computation quality. This thesis presents dynamic analysis tools to study, debug, and monitor the quality and energy efficiency of <span class="hlt">approximate</span> computations. We propose three styles of tools: prototyping tools that allow developers to experiment with <span class="hlt">approximation</span> in their applications, online tools that instrument code to determine the key sources of error, and online tools that monitor the quality of deployed applications in real time. Our prototyping tool is based on an extension to the functional language OCaml. We add <span class="hlt">approximation</span> constructs to the language, an <span class="hlt">approximation</span> simulator to the runtime, and profiling and auto-tuning tools for studying and experimenting with energy-quality tradeoffs. We also present two online debugging tools and three online monitoring tools. The first online tool identifies correlations between output quality and the total number of executions of, and errors in, individual <span class="hlt">approximate</span> operations. The second tracks the number of <span class="hlt">approximate</span> operations that flow into a particular value. Our online tools comprise three low-cost approaches to dynamic quality monitoring. They are designed to monitor quality in deployed applications without spending more energy than is saved by <span class="hlt">approximation</span>. Online monitors can be used to perform real time adjustments to energy usage in <span class="hlt">order</span> to meet specific quality goals. We present prototype implementations of all of these tools and describe their usage with several applications. Our prototyping, profiling, and autotuning tools allow us to experiment with <span class="hlt">approximation</span> strategies and identify new strategies, our online tools succeed in providing new insights into the effects of <span class="hlt">approximation</span> on output quality, and our monitors succeed in</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19980232922','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19980232922"><span><span class="hlt">Approximate</span> Analytical Solutions for Hypersonic Flow Over Slender Power Law Bodies</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Mirels, Harold</p> <p>1959-01-01</p> <p><span class="hlt">Approximate</span> analytical solutions are presented for two-dimensional and axisymmetric hypersonic flow over slender power law bodies. Both zero <span class="hlt">order</span> (M approaches infinity) and first <span class="hlt">order</span> (small but nonvanishing values of 1/(M(Delta)(sup 2) solutions are presented, where M is free-stream Mach number and Delta is a characteristic slope. These solutions are compared with exact numerical integration of the equations of motion and appear to be accurate particularly when the shock is relatively close to the body.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2000NuAlg..23..263A','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2000NuAlg..23..263A"><span>A result about scale transformation families in <span class="hlt">approximation</span></span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Apprato, Dominique; Gout, Christian</p> <p>2000-06-01</p> <p>Scale transformations are common in <span class="hlt">approximation</span>. In surface <span class="hlt">approximation</span> from rapidly varying data, one wants to suppress, or at least dampen the oscillations of the <span class="hlt">approximation</span> near steep gradients implied by the data. In that case, scale transformations can be used to give some control over overshoot when the surface has large variations of its gradient. Conversely, in image analysis, scale transformations are used in preprocessing to enhance some features present on the image or to increase jumps of grey levels before segmentation of the image. In this paper, we establish the convergence of an <span class="hlt">approximation</span> method which allows some control over the behavior of the <span class="hlt">approximation</span>. More precisely, we study the convergence of an <span class="hlt">approximation</span> from a data set of , while using scale transformations on the values before and after classical <span class="hlt">approximation</span>. In addition, the construction of scale transformations is also given. The algorithm is presented with some numerical examples.</p> </li> </ol> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_18");'>18</a></li> <li><a href="#" onclick='return showDiv("page_19");'>19</a></li> <li class="active"><span>20</span></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li><a href="#" onclick='return showDiv("page_22");'>22</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div><!-- col-sm-12 --> </div><!-- row --> </div><!-- page_20 --> <div id="page_21" class="hiddenDiv"> <div class="row"> <div class="col-sm-12"> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_19");'>19</a></li> <li><a href="#" onclick='return showDiv("page_20");'>20</a></li> <li class="active"><span>21</span></li> <li><a href="#" onclick='return showDiv("page_22");'>22</a></li> <li><a href="#" onclick='return showDiv("page_23");'>23</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div> </div> <div class="row"> <div class="col-sm-12"> <ol class="result-class" start="401"> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/21867342','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/21867342"><span><span class="hlt">Approximation</span> for discrete Fourier transform and application in study of three-dimensional interacting electron gas.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Yan, Xin-Zhong</p> <p>2011-07-01</p> <p>The discrete Fourier transform is <span class="hlt">approximated</span> by summing over part of the terms with corresponding weights. The <span class="hlt">approximation</span> reduces significantly the requirement for computer memory storage and enhances the numerical computation efficiency with several <span class="hlt">orders</span> without losing accuracy. As an example, we apply the algorithm to study the three-dimensional interacting electron gas under the renormalized-ring-diagram <span class="hlt">approximation</span> where the Green's function needs to be self-consistently solved. We present the results for the chemical potential, compressibility, free energy, entropy, and specific heat of the system. The ground-state energy obtained by the present calculation is compared with the existing results of Monte Carlo simulation and random-phase <span class="hlt">approximation</span>.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017imap.book.....V','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017imap.book.....V"><span>Introduction to Methods of <span class="hlt">Approximation</span> in Physics and Astronomy</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>van Putten, Maurice H. P. M.</p> <p>2017-04-01</p> <p> secular behavior. For instance, secular evolution of orbital parameters may derive from averaging over essentially periodic behavior on relatively short, orbital periods. When the original number of degrees of freedom is large, averaging over dynamical time scales may lead to a formulation in terms of a system in <span class="hlt">approximately</span> thermodynamic equilibrium subject to evolution on a secular time scale by a regular or singular perturbation. In modern astrophysics and cosmology, gravitation is being probed across an increasingly broad range of scales and more accurately so than ever before. These observations probe weak gravitational interactions below what is encountered in our solar system by many <span class="hlt">orders</span> of magnitude. These observations hereby probe (curved) spacetime at low energy scales that may reveal novel properties hitherto unanticipated in the classical vacuum of Newtonian mechanics and Minkowski spacetime. Dark energy and dark matter encountered on the scales of galaxies and beyond, therefore, may be, in part, revealing our ignorance of the vacuum at the lowest energy scales encountered in cosmology. In this context, our application of Newtonian mechanics to globular clusters, galaxies and cosmology is an <span class="hlt">approximation</span> assuming a classical vacuum, ignoring the potential for hidden low energy scales emerging on cosmological scales. Given our ignorance of the latter, this poses a challenge in the potential for unknown systematic deviations. If of quantum mechanical origin, such deviations are often referred to as anomalies. While they are small in traditional, macroscopic Newtonian experiments in the laboratory, they same is not a given in the limit of arbitrarily weak gravitational interactions. We hope this selection of introductory material is useful and kindles the reader's interest to become a creative member of modern astrophysics and cosmology.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018PhRvB..97l5428P','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018PhRvB..97l5428P"><span>Surface waves on multilayer hyperbolic metamaterials: Operator approach to effective medium <span class="hlt">approximation</span></span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Popov, Vladislav; Lavrinenko, Andrei V.; Novitsky, Andrey</p> <p>2018-03-01</p> <p>In this paper, we elaborate on the operator effective medium <span class="hlt">approximation</span> developed recently in Popov et al. [Phys. Rev. B 94, 085428 (2016), 10.1103/PhysRevB.94.085428] to get insight into the surface polariton excitation at the interface of a multilayer hyperbolic metamaterial (HMM). In particular, we find that HMMs with bilayer unit cells support the TE- and TM-polarized surface waves beyond the Maxwell Garnett <span class="hlt">approximation</span> due to the spatial dispersion interpreted as effective magnetoelectric coupling. The latter is also responsible for the dependence of surface wave propagation on the <span class="hlt">order</span> of layers in the unit cell. Elimination of the magnetoelectric coupling in three-layer unit cells complying with inversion symmetry restores the qualitative regularity of the Maxwell Garnett <span class="hlt">approximation</span>, as well as strongly suppresses the influence of the <span class="hlt">order</span> of layers in the unit cell.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4134886','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4134886"><span>Roughness in Lattice <span class="hlt">Ordered</span> Effect Algebras</span></a></p> <p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p> <p>Xin, Xiao Long; Hua, Xiu Juan; Zhu, Xi</p> <p>2014-01-01</p> <p>Many authors have studied roughness on various algebraic systems. In this paper, we consider a lattice <span class="hlt">ordered</span> effect algebra and discuss its roughness in this context. Moreover, we introduce the notions of the interior and the closure of a subset and give some of their properties in effect algebras. Finally, we use a Riesz ideal induced congruence and define a function e(a, b) in a lattice <span class="hlt">ordered</span> effect algebra E and build a relationship between it and congruence classes. Then we study some properties about <span class="hlt">approximation</span> of lattice <span class="hlt">ordered</span> effect algebras. PMID:25170523</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/24320847','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/24320847"><span><span class="hlt">Approximate</span>, computationally efficient online learning in Bayesian spiking neurons.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Kuhlmann, Levin; Hauser-Raspe, Michael; Manton, Jonathan H; Grayden, David B; Tapson, Jonathan; van Schaik, André</p> <p>2014-03-01</p> <p>Bayesian spiking neurons (BSNs) provide a probabilistic interpretation of how neurons perform inference and learning. Online learning in BSNs typically involves parameter estimation based on maximum-likelihood expectation-maximization (ML-EM) which is computationally slow and limits the potential of studying networks of BSNs. An online learning algorithm, fast learning (FL), is presented that is more computationally efficient than the benchmark ML-EM for a fixed number of time steps as the number of inputs to a BSN increases (e.g., 16.5 times faster run times for 20 inputs). Although ML-EM appears to converge 2.0 to 3.6 times faster than FL, the computational cost of ML-EM means that ML-EM takes longer to simulate to convergence than FL. FL also provides reasonable convergence performance that is robust to initialization of parameter estimates that are far from the true parameter values. However, parameter estimation depends on the range of true parameter values. Nevertheless, for a physiologically meaningful range of parameter values, FL gives very good average estimation accuracy, despite its <span class="hlt">approximate</span> nature. The FL algorithm therefore provides an efficient tool, complementary to ML-EM, for exploring BSN networks in more detail in <span class="hlt">order</span> to better understand their biological relevance. Moreover, the simplicity of the FL algorithm means it can be easily implemented in neuromorphic VLSI such that one can take advantage of the energy-efficient spike coding of BSNs.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2015JPS...278..274U','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2015JPS...278..274U"><span>Battery state-of-charge estimation using <span class="hlt">approximate</span> least squares</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Unterrieder, C.; Zhang, C.; Lunglmayr, M.; Priewasser, R.; Marsili, S.; Huemer, M.</p> <p>2015-03-01</p> <p>In recent years, much effort has been spent to extend the runtime of battery-powered electronic applications. In <span class="hlt">order</span> to improve the utilization of the available cell capacity, high precision estimation approaches for battery-specific parameters are needed. In this work, an <span class="hlt">approximate</span> least squares estimation scheme is proposed for the estimation of the battery state-of-charge (SoC). The SoC is determined based on the prediction of the battery's electromotive force. The proposed approach allows for an improved re-initialization of the Coulomb counting (CC) based SoC estimation method. Experimental results for an implementation of the estimation scheme on a fuel gauge system on chip are illustrated. Implementation details and design guidelines are presented. The performance of the presented concept is evaluated for realistic operating conditions (temperature effects, aging, standby current, etc.). For the considered test case of a GSM/UMTS load current pattern of a mobile phone, the proposed method is able to re-initialize the CC-method with a high accuracy, while state-of-the-art methods fail to perform a re-initialization.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017APS..DFDL31003D','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017APS..DFDL31003D"><span>Autonomic Closure for Turbulent Flows Using <span class="hlt">Approximate</span> Bayesian Computation</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Doronina, Olga; Christopher, Jason; Hamlington, Peter; Dahm, Werner</p> <p>2017-11-01</p> <p>Autonomic closure is a new technique for achieving fully adaptive and physically accurate closure of coarse-grained turbulent flow governing equations, such as those solved in large eddy simulations (LES). Although autonomic closure has been shown in recent a priori tests to more accurately represent unclosed terms than do dynamic versions of traditional LES models, the computational cost of the approach makes it challenging to implement for simulations of practical turbulent flows at realistically high Reynolds numbers. The optimization step used in the approach introduces large matrices that must be inverted and is highly memory intensive. In <span class="hlt">order</span> to reduce memory requirements, here we propose to use <span class="hlt">approximate</span> Bayesian computation (ABC) in place of the optimization step, thereby yielding a computationally-efficient implementation of autonomic closure that trades memory-intensive for processor-intensive computations. The latter challenge can be overcome as co-processors such as general purpose graphical processing units become increasingly available on current generation petascale and exascale supercomputers. In this work, we outline the formulation of ABC-enabled autonomic closure and present initial results demonstrating the accuracy and computational cost of the approach.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2001MNRAS.325.1463F','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2001MNRAS.325.1463F"><span>Axisymmetric modes of rotating relativistic stars in the Cowling <span class="hlt">approximation</span></span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Font, José A.; Dimmelmeier, Harald; Gupta, Anshu; Stergioulas, Nikolaos</p> <p>2001-08-01</p> <p>Axisymmetric pulsations of rotating neutron stars can be excited in several scenarios, such as core collapse, crust- and core-quakes or binary mergers, and could become detectable in either gravitational waves or high-energy radiation. Here, we present a comprehensive study of all low-<span class="hlt">order</span> axisymmetric modes of uniformly and rapidly rotating relativistic stars. Initial stationary configurations are appropriately perturbed and are numerically evolved using an axisymmetric, non-linear relativistic hydrodynamics code, assuming time-independence of the gravitational field (Cowling <span class="hlt">approximation</span>). The simulations are performed using a high-resolution shock-capturing finite-difference scheme accurate enough to maintain the initial rotation law for a large number of rotational periods, even for stars at the mass-shedding limit. Through Fourier transforms of the time evolution of selected fluid variables, we compute the frequencies of quasi-radial and non-radial modes with spherical harmonic indices l=0, 1, 2 and 3, for a sequence of rotating stars from the non-rotating limit to the mass-shedding limit. The frequencies of the axisymmetric modes are affected significantly by rotation only when the rotation rate exceeds about 50 per cent of the maximum allowed. As expected, at large rotation rates, apparent mode crossings between different modes appear. In addition to the above modes, several axisymmetric inertial modes are also excited in our numerical evolutions.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017APS..DPPG11031S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017APS..DPPG11031S"><span>Convergence of the Ponderomotive Guiding Center <span class="hlt">approximation</span> in the LWFA</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Silva, Thales; Vieira, Jorge; Helm, Anton; Fonseca, Ricardo; Silva, Luis</p> <p>2017-10-01</p> <p>Plasma accelerators arose as potential candidates for future accelerator technology in the last few decades because of its predicted compactness and low cost. One of the proposed designs for plasma accelerators is based on Laser Wakefield Acceleration (LWFA). However, simulations performed for such systems have to solve the laser wavelength which is <span class="hlt">orders</span> of magnitude lower than the plasma wavelength. In this context, the Ponderomotive Guiding Center (PGC) algorithm for particle-in-cell (PIC) simulations is a potent tool. The laser is <span class="hlt">approximated</span> by its envelope which leads to a speed-up of around 100 times because the laser wavelength is not solved. The plasma response is well understood, and comparison with the full PIC code show an excellent agreement. However, for LWFA, the convergence of the self-injected beam parameters, such as energy and charge, was not studied before and has vital importance for the use of the algorithm in predicting the beam parameters. Our goal is to do a thorough investigation of the stability and convergence of the algorithm in situations of experimental relevance for LWFA. To this end, we perform simulations using the PGC algorithm implemented in the PIC code OSIRIS. To verify the PGC predictions, we compare the results with full PIC simulations. This project has received funding from the European Union's Horizon 2020 research and innovation programme under Grant agreement No 653782.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/17568146','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/17568146"><span><span class="hlt">Approximate</span> labeling via graph cuts based on linear programming.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Komodakis, Nikos; Tziritas, Georgios</p> <p>2007-08-01</p> <p>A new framework is presented for both understanding and developing graph-cut-based combinatorial algorithms suitable for the <span class="hlt">approximate</span> optimization of a very wide class of Markov Random Fields (MRFs) that are frequently encountered in computer vision. The proposed framework utilizes tools from the duality theory of linear programming in <span class="hlt">order</span> to provide an alternative and more general view of state-of-the-art techniques like the \\alpha-expansion algorithm, which is included merely as a special case. Moreover, contrary to \\alpha-expansion, the derived algorithms generate solutions with guaranteed optimality properties for a much wider class of problems, for example, even for MRFs with nonmetric potentials. In addition, they are capable of providing per-instance suboptimality bounds in all occasions, including discrete MRFs with an arbitrary potential function. These bounds prove to be very tight in practice (that is, very close to 1), which means that the resulting solutions are almost optimal. Our algorithms' effectiveness is demonstrated by presenting experimental results on a variety of low-level vision tasks, such as stereo matching, image restoration, image completion, and optical flow estimation, as well as on synthetic problems.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016ChPhB..25l4307D','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016ChPhB..25l4307D"><span>Higher-<span class="hlt">order</span> harmonics of general limited diffraction Bessel beams</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Ding, De-Sheng; Huang, Jin-Huang</p> <p>2016-12-01</p> <p>In this paper, we extensively study the higher-<span class="hlt">order</span> harmonic generation of the general limited diffraction m-th-<span class="hlt">order</span> Bessel beam. The analysis is based on successive <span class="hlt">approximations</span> of the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation. Asymptotic expansions are presented for higher-<span class="hlt">order</span> harmonic Bessel beams in near and far fields. The validity of asymptotic <span class="hlt">approximation</span> is also analyzed. The higher-<span class="hlt">order</span> harmonic of the Bessel beam with the lowest zero-<span class="hlt">order</span> is taken as a special example. Project supported by the National Natural Science Foundation of China (Grant Nos. 11074038 and 11374051).</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=What+AND+soil+AND+conservation&pg=3&id=ED260969','ERIC'); return false;" href="https://eric.ed.gov/?q=What+AND+soil+AND+conservation&pg=3&id=ED260969"><span>Bioregions and World <span class="hlt">Order</span>.</span></a></p> <p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p> <p>Breakthrough, 1985</p> <p>1985-01-01</p> <p>What bioregions can do to contribute to world <span class="hlt">order</span> and security is discussed in this newsletter. A bioregion is defined as an identifiable geographical area of interacting life-systems that is relatively self-sustaining in the ever-renewing processes of nature. Articles included are: "Bioregionalism and World <span class="hlt">Order</span>" (Gerald Mische);…</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/29173456','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/29173456"><span>Recognition of computerized facial <span class="hlt">approximations</span> by familiar assessors.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Richard, Adam H; Monson, Keith L</p> <p>2017-11-01</p> <p>Studies testing the effectiveness of facial <span class="hlt">approximations</span> typically involve groups of participants who are unfamiliar with the <span class="hlt">approximated</span> individual(s). This limitation requires the use of photograph arrays including a picture of the subject for comparison to the facial <span class="hlt">approximation</span>. While this practice is often necessary due to the difficulty in obtaining a group of assessors who are familiar with the <span class="hlt">approximated</span> subject, it may not accurately simulate the thought process of the target audience (friends and family members) in comparing a mental image of the <span class="hlt">approximated</span> subject to the facial <span class="hlt">approximation</span>. As part of a larger process to evaluate the effectiveness and best implementation of the ReFace facial <span class="hlt">approximation</span> software program, the rare opportunity arose to conduct a recognition study using assessors who were personally acquainted with the subjects of the <span class="hlt">approximations</span>. ReFace facial <span class="hlt">approximations</span> were generated based on preexisting medical scans, and co-workers of the scan donors were tested on whether they could accurately pick out the <span class="hlt">approximation</span> of their colleague from arrays of facial <span class="hlt">approximations</span>. Results from the study demonstrated an overall poor recognition performance (i.e., where a single choice within a pool is not enforced) for individuals who were familiar with the <span class="hlt">approximated</span> subjects. Out of 220 recognition tests only 10.5% resulted in the assessor selecting the correct <span class="hlt">approximation</span> (or correctly choosing not to make a selection when the array consisted only of foils), an outcome that was not significantly different from the 9% random chance rate. When allowed to select multiple <span class="hlt">approximations</span> the assessors felt resembled the target individual, the overall sensitivity for ReFace <span class="hlt">approximations</span> was 16.0% and the overall specificity was 81.8%. These results differ markedly from the results of a previous study using assessors who were unfamiliar with the <span class="hlt">approximated</span> subjects. Some possible explanations for this disparity in</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4142747','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4142747"><span><span class="hlt">Approximation</span> Set of the Interval Set in Pawlak's Space</span></a></p> <p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p> <p>Wang, Jin; Wang, Guoyin</p> <p>2014-01-01</p> <p>The interval set is a special set, which describes uncertainty of an uncertain concept or set Z with its two crisp boundaries named upper-bound set and lower-bound set. In this paper, the concept of similarity degree between two interval sets is defined at first, and then the similarity degrees between an interval set and its two <span class="hlt">approximations</span> (i.e., upper <span class="hlt">approximation</span> set R¯(Z) and lower <span class="hlt">approximation</span> set R_(Z)) are presented, respectively. The disadvantages of using upper-<span class="hlt">approximation</span> set R¯(Z) or lower-<span class="hlt">approximation</span> set R_(Z) as <span class="hlt">approximation</span> sets of the uncertain set (uncertain concept) Z are analyzed, and a new method for looking for a better <span class="hlt">approximation</span> set of the interval set Z is proposed. The conclusion that the <span class="hlt">approximation</span> set R 0.5(Z) is an optimal <span class="hlt">approximation</span> set of interval set Z is drawn and proved successfully. The change rules of R 0.5(Z) with different binary relations are analyzed in detail. Finally, a kind of crisp <span class="hlt">approximation</span> set of the interval set Z is constructed. We hope this research work will promote the development of both the interval set model and granular computing theory. PMID:25177721</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19950017128','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19950017128"><span>Fourth <span class="hlt">order</span> difference methods for hyperbolic IBVP's</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Gustafsson, Bertil; Olsson, Pelle</p> <p>1994-01-01</p> <p>Fourth <span class="hlt">order</span> difference <span class="hlt">approximations</span> of initial-boundary value problems for hyperbolic partial differential equations are considered. We use the method of lines approach with both explicit and compact implicit difference operators in space. The explicit operator satisfies an energy estimate leading to strict stability. For the implicit operator we develop boundary conditions and give a complete proof of strong stability using the Laplace transform technique. We also present numerical experiments for the linear advection equation and Burgers' equation with discontinuities in the solution or in its derivative. The first equation is used for modeling contact discontinuities in fluid dynamics, the second one for modeling shocks and rarefaction waves. The time discretization is done with a third <span class="hlt">order</span> Runge-Kutta TVD method. For solutions with discontinuities in the solution itself we add a filter based on second <span class="hlt">order</span> viscosity. In case of the non-linear Burger's equation we use a flux splitting technique that results in an energy estimate for certain different <span class="hlt">approximations</span>, in which case also an entropy condition is fulfilled. In particular we shall demonstrate that the unsplit conservative form produces a non-physical shock instead of the physically correct rarefaction wave. In the numerical experiments we compare our fourth <span class="hlt">order</span> methods with a standard second <span class="hlt">order</span> one and with a third <span class="hlt">order</span> TVD-method. The results show that the fourth <span class="hlt">order</span> methods are the only ones that give good results for all the considered test problems.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2011AGUFM.A33B0195M','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2011AGUFM.A33B0195M"><span><span class="hlt">Approximating</span> Reflectance and Transmittance of Vegetation Using Multiple Spectral Invariants</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Mottus, M.</p> <p>2011-12-01</p> <p>Canopy spectral invariants, eigenvalues of the radiative transfer equation and photon recollision probability are some of the new theoretical tools that have been applied in remote sensing of vegetation and atmosphere. The theoretical approach based on spectral invariants, informally also referred to as the p-theory, owns its attractivity to several factors. Firstly, it provides a rapid and physically-based way of describing canopy scattering. Secondly, the p-theory aims at parameterizing canopy structure in reflectance models using a simple and intuitive concept which can be applied at various structural levels, from shoot to tree crown. The theory has already been applied at scales from the molecular level to forest stands. The most important shortcoming of the p-theory lies in its inability to predict the directionality of scattering. The theory is currently based on only one physical parameter, the photon recollision probability p. It is evident that one parameter cannot contain enough information to reasonably predict the observed complex reflectance patterns produced by natural vegetation canopies. Without estimating scattering directionality, however, the theory cannot be compared with even the most simple (and well-tested) two-stream vegetation reflectance models. In this study, we evaluate the possibility to use additional parameters to fit the measured reflectance and transmittance of a vegetation stand. As a first step, the parameters are applied to separate canopy scattering into reflectance and transmittance. New parameters are introduced following the general approach of eigenvector expansion. Thus, the new parameters are coined higher-<span class="hlt">order</span> spectral invariants. Calculation of higher-<span class="hlt">order</span> invariants is based on separating first-<span class="hlt">order</span> scattering from total scattering. Thus, the method explicitly accounts for different view geometries with different fractions of visible sunlit canopy (e.g., hot-spot). It additionally allows to produce different</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/21913750','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/21913750"><span>Second-<span class="hlt">order</span> perturbation theory with a density matrix renormalization group self-consistent field reference function: theory and application to the study of chromium dimer.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Kurashige, Yuki; Yanai, Takeshi</p> <p>2011-09-07</p> <p>We present a second-<span class="hlt">order</span> perturbation theory based on a density matrix renormalization group self-consistent field (DMRG-SCF) reference function. The method reproduces the solution of the complete active space with second-<span class="hlt">order</span> perturbation theory (CASPT2) when the DMRG reference function is represented by a sufficiently large number of renormalized many-body basis, thereby being named DMRG-CASPT2 method. The DMRG-SCF is able to describe non-dynamical correlation with large active space that is insurmountable to the conventional CASSCF method, while the second-<span class="hlt">order</span> perturbation theory provides an efficient description of dynamical correlation effects. The capability of our implementation is demonstrated for an application to the potential energy curve of the chromium dimer, which is one of the most demanding multireference systems that require best electronic structure treatment for non-dynamical and dynamical correlation as well as large basis sets. The DMRG-CASPT2/cc-pwCV5Z calculations were performed with a large (3d double-shell) active space consisting of 28 orbitals. Our approach using large-size DMRG reference addressed the problems of why the dissociation energy is largely overestimated by CASPT2 with the small active space consisting of 12 orbitals (3d4s), and also is oversensitive to the choice of the <span class="hlt">zeroth-order</span> Hamiltonian. © 2011 American Institute of Physics</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/16268689','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/16268689"><span>On the origins of <span class="hlt">approximations</span> for stochastic chemical kinetics.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Haseltine, Eric L; Rawlings, James B</p> <p>2005-10-22</p> <p>This paper considers the derivation of <span class="hlt">approximations</span> for stochastic chemical kinetics governed by the discrete master equation. Here, the concepts of (1) partitioning on the basis of fast and slow reactions as opposed to fast and slow species and (2) conditional probability densities are used to derive <span class="hlt">approximate</span>, partitioned master equations, which are Markovian in nature, from the original master equation. Under different conditions dictated by relaxation time arguments, such <span class="hlt">approximations</span> give rise to both the equilibrium and hybrid (deterministic or Langevin equations coupled with discrete stochastic simulation) <span class="hlt">approximations</span> previously reported. In addition, the derivation points out several weaknesses in previous justifications of both the hybrid and equilibrium systems and demonstrates the connection between the original and <span class="hlt">approximate</span> master equations. Two simple examples illustrate situations in which these two <span class="hlt">approximate</span> methods are applicable and demonstrate the two methods' efficiencies.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018ResPh...8.1234M','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018ResPh...8.1234M"><span>Precise analytic <span class="hlt">approximations</span> for the Bessel function J1 (x)</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Maass, Fernando; Martin, Pablo</p> <p>2018-03-01</p> <p>Precise and straightforward analytic <span class="hlt">approximations</span> for the Bessel function J1 (x) have been found. Power series and asymptotic expansions have been used to determine the parameters of the <span class="hlt">approximation</span>, which is as a bridge between both expansions, and it is a combination of rational and trigonometric functions multiplied with fractional powers of x. Here, several improvements with respect to the so called Multipoint Quasirational <span class="hlt">Approximation</span> technique have been performed. Two procedures have been used to determine the parameters of the <span class="hlt">approximations</span>. The maximum absolute errors are in both cases smaller than 0.01. The zeros of the <span class="hlt">approximation</span> are also very precise with less than 0.04 per cent for the first one. A second <span class="hlt">approximation</span> has been also determined using two more parameters, and in this way the accuracy has been increased to less than 0.001.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/AD1023882','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/AD1023882"><span>Convergence Rates of Finite Difference Stochastic <span class="hlt">Approximation</span> Algorithms</span></a></p> <p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p> <p></p> <p>2016-06-01</p> <p>dfferences as gradient <span class="hlt">approximations</span>. It is shown that the convergence of these algorithms can be accelerated by controlling the implementation of the...descent algorithm, under various updating schemes using finite dfferences as gradient <span class="hlt">approximations</span>. It is shown that the convergence of these...the Kiefer-Wolfowitz algorithm and the mirror descent algorithm, under various updating schemes using finite differences as gradient <span class="hlt">approximations</span>. It</p> </li> </ol> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_19");'>19</a></li> <li><a href="#" onclick='return showDiv("page_20");'>20</a></li> <li class="active"><span>21</span></li> <li><a href="#" onclick='return showDiv("page_22");'>22</a></li> <li><a href="#" onclick='return showDiv("page_23");'>23</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div><!-- col-sm-12 --> </div><!-- row --> </div><!-- page_21 --> <div id="page_22" class="hiddenDiv"> <div class="row"> <div class="col-sm-12"> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_20");'>20</a></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li class="active"><span>22</span></li> <li><a href="#" onclick='return showDiv("page_23");'>23</a></li> <li><a href="#" onclick='return showDiv("page_24");'>24</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div> </div> <div class="row"> <div class="col-sm-12"> <ol class="result-class" start="421"> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.gpo.gov/fdsys/pkg/FR-2010-09-28/pdf/2010-24285.pdf','FEDREG'); return false;" href="https://www.gpo.gov/fdsys/pkg/FR-2010-09-28/pdf/2010-24285.pdf"><span>75 FR 59743 - Public Land <span class="hlt">Order</span> No. 7750; Partial Revocation of Secretarial <span class="hlt">Order</span> dated January 20, 1910; Oregon</span></a></p> <p><a target="_blank" href="http://www.gpo.gov/fdsys/browse/collection.action?collectionCode=FR">Federal Register 2010, 2011, 2012, 2013, 2014</a></p> <p></p> <p>2010-09-28</p> <p>... partially revokes a Secretarial <span class="hlt">Order</span> dated January 20, 1910, insofar as it affects <span class="hlt">approximately</span> 9,001.84... is no longer needed for the purpose for which it was withdrawn. DATES: Effective Date: September 28..., it is <span class="hlt">ordered</span> as follows: The Secretarial <span class="hlt">Order</span> dated January 20, 1910, which withdrew land on behalf...</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=flight&pg=6&id=EJ726148','ERIC'); return false;" href="https://eric.ed.gov/?q=flight&pg=6&id=EJ726148"><span>Court <span class="hlt">Ordered</span> Desegregation</span></a></p> <p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p> <p>Reber, Sarah J.</p> <p>2005-01-01</p> <p>The effect of the court <span class="hlt">ordered</span> desegregation plans, on trends in segregation and white flight, are estimated. The effect of availability of school districts and other factors on the white flight across districts is also mentioned.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.epa.gov/laws-regulations/laws-and-executive-orders','PESTICIDES'); return false;" href="https://www.epa.gov/laws-regulations/laws-and-executive-orders"><span>Laws and Executive <span class="hlt">Orders</span></span></a></p> <p><a target="_blank" href="http://www.epa.gov/pesticides/search.htm">EPA Pesticide Factsheets</a></p> <p></p> <p></p> <p>Lists and links to descriptions of the major laws and executive <span class="hlt">orders</span> that EPA administers and/or that guide EPA rulemakings, including the Clean Air Act, the Toxic Substance Control Act, CERCLA or Superfund, and the Clean Water Act.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=computers+AND+optical&pg=4&id=EJ245729','ERIC'); return false;" href="https://eric.ed.gov/?q=computers+AND+optical&pg=4&id=EJ245729"><span>Automated <span class="hlt">Ordering</span> System.</span></a></p> <p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p> <p>Jones, Richard M.</p> <p>1981-01-01</p> <p>A computer program that utilizes an optical scanning machine is used for <span class="hlt">ordering</span> supplies in a Louisiana school system. The program provides savings in time and labor, more accurate data, and easy-to-use reports. (Author/MLF)</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/15065716','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/15065716"><span>Geometrical-optics <span class="hlt">approximation</span> of forward scattering by coated particles.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Xu, Feng; Cai, Xiaoshu; Ren, Kuanfang</p> <p>2004-03-20</p> <p>By means of geometrical optics we present an <span class="hlt">approximation</span> algorithm with which to accelerate the computation of scattering intensity distribution within a forward angular range (0 degrees-60 degrees) for coated particles illuminated by a collimated incident beam. Phases of emerging rays are exactly calculated to improve the <span class="hlt">approximation</span> precision. This method proves effective for transparent and tiny absorbent particles with size parameters larger than 75 but fails to give good <span class="hlt">approximation</span> results at scattering angles at which refractive rays are absent. When the absorption coefficient of a particle is greater than 0.01, the geometrical optics <span class="hlt">approximation</span> is effective only for forward small angles, typically less than 10 degrees or so.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/476036-better-approximation-guarantees-job-shop-scheduling','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/476036-better-approximation-guarantees-job-shop-scheduling"><span>Better <span class="hlt">approximation</span> guarantees for job-shop scheduling</span></a></p> <p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p> <p>Goldberg, L.A.; Paterson, M.; Srinivasan, A.</p> <p>1997-06-01</p> <p>Job-shop scheduling is a classical NP-hard problem. Shmoys, Stein & Wein presented the first polynomial-time <span class="hlt">approximation</span> algorithm for this problem that has a good (polylogarithmic) <span class="hlt">approximation</span> guarantee. We improve the <span class="hlt">approximation</span> guarantee of their work, and present further improvements for some important NP-hard special cases of this problem (e.g., in the preemptive case where machines can suspend work on operations and later resume). We also present NC algorithms with improved <span class="hlt">approximation</span> guarantees for some NP-hard special cases.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19860059247&hterms=Legendre+polynomials&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D10%26Ntt%3DLegendre%2Bpolynomials','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19860059247&hterms=Legendre+polynomials&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D10%26Ntt%3DLegendre%2Bpolynomials"><span>Legendre-tau <span class="hlt">approximations</span> for functional differential equations</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Ito, K.; Teglas, R.</p> <p>1986-01-01</p> <p>The numerical <span class="hlt">approximation</span> of solutions to linear retarded functional differential equations are considered using the so-called Legendre-tau method. The functional differential equation is first reformulated as a partial differential equation with a nonlocal boundary condition involving time-differentiation. The <span class="hlt">approximate</span> solution is then represented as a truncated Legendre series with time-varying coefficients which satisfy a certain system of ordinary differential equations. The method is very easy to code and yields very accurate <span class="hlt">approximations</span>. Convergence is established, various numerical examples are presented, and comparison between the latter and cubic spline <span class="hlt">approximation</span> is made.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19870002588','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19870002588"><span>Sensitivity analysis and <span class="hlt">approximation</span> methods for general eigenvalue problems</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Murthy, D. V.; Haftka, R. T.</p> <p>1986-01-01</p> <p>Optimization of dynamic systems involving complex non-hermitian matrices is often computationally expensive. Major contributors to the computational expense are the sensitivity analysis and reanalysis of a modified design. The present work seeks to alleviate this computational burden by identifying efficient sensitivity analysis and <span class="hlt">approximate</span> reanalysis methods. For the algebraic eigenvalue problem involving non-hermitian matrices, algorithms for sensitivity analysis and <span class="hlt">approximate</span> reanalysis are classified, compared and evaluated for efficiency and accuracy. Proper eigenvector normalization is discussed. An improved method for calculating derivatives of eigenvectors is proposed based on a more rational normalization condition and taking advantage of matrix sparsity. Important numerical aspects of this method are also discussed. To alleviate the problem of reanalysis, various <span class="hlt">approximation</span> methods for eigenvalues are proposed and evaluated. Linear and quadratic <span class="hlt">approximations</span> are based directly on the Taylor series. Several <span class="hlt">approximation</span> methods are developed based on the generalized Rayleigh quotient for the eigenvalue problem. <span class="hlt">Approximation</span> methods based on trace theorem give high accuracy without needing any derivatives. Operation counts for the computation of the <span class="hlt">approximations</span> are given. General recommendations are made for the selection of appropriate <span class="hlt">approximation</span> technique as a function of the matrix size, number of design variables, number of eigenvalues of interest and the number of design points at which <span class="hlt">approximation</span> is sought.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19830021837','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19830021837"><span>Legendre-Tau <span class="hlt">approximations</span> for functional differential equations</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Ito, K.; Teglas, R.</p> <p>1983-01-01</p> <p>The numerical <span class="hlt">approximation</span> of solutions to linear functional differential equations are considered using the so called Legendre tau method. The functional differential equation is first reformulated as a partial differential equation with a nonlocal boundary condition involving time differentiation. The <span class="hlt">approximate</span> solution is then represented as a truncated Legendre series with time varying coefficients which satisfy a certain system of ordinary differential equations. The method is very easy to code and yields very accurate <span class="hlt">approximations</span>. Convergence is established, various numerical examples are presented, and comparison between the latter and cubic spline <span class="hlt">approximations</span> is made.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/AD0704567','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/AD0704567"><span>FINITE-STATE <span class="hlt">APPROXIMATIONS</span> TO DENUMERABLE-STATE DYNAMIC PROGRAMS,</span></a></p> <p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p> <p></p> <p></p> <p>AIR FORCE OPERATIONS, LOGISTICS), (*INVENTORY CONTROL, DYNAMIC PROGRAMMING), (*DYNAMIC PROGRAMMING, <span class="hlt">APPROXIMATION</span>(MATHEMATICS)), INVENTORY CONTROL, DECISION MAKING, STOCHASTIC PROCESSES, GAME THEORY, ALGORITHMS, CONVERGENCE</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19900042212&hterms=equations+differential&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D90%26Ntt%3Dequations%2Bdifferential','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19900042212&hterms=equations+differential&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D90%26Ntt%3Dequations%2Bdifferential"><span>Differential equation based method for accurate <span class="hlt">approximations</span> in optimization</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Pritchard, Jocelyn I.; Adelman, Howard M.</p> <p>1990-01-01</p> <p>This paper describes a method to efficiently and accurately <span class="hlt">approximate</span> the effect of design changes on structural response. The key to this new method is to interpret sensitivity equations as differential equations that may be solved explicitly for closed form <span class="hlt">approximations</span>, hence, the method is denoted the Differential Equation Based (DEB) method. <span class="hlt">Approximations</span> were developed for vibration frequencies, mode shapes and static displacements. The DEB <span class="hlt">approximation</span> method was applied to a cantilever beam and results compared with the commonly-used linear Taylor series <span class="hlt">approximations</span> and exact solutions. The test calculations involved perturbing the height, width, cross-sectional area, tip mass, and bending inertia of the beam. The DEB method proved to be very accurate, and in msot cases, was more accurate than the linear Taylor series <span class="hlt">approximation</span>. The method is applicable to simultaneous perturbation of several design variables. Also, the <span class="hlt">approximations</span> may be used to calculate other system response quantities. For example, the <span class="hlt">approximations</span> for displacement are used to <span class="hlt">approximate</span> bending stresses.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018PhRvD..97j5002B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018PhRvD..97j5002B"><span>Thin-wall <span class="hlt">approximation</span> in vacuum decay: A lemma</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Brown, Adam R.</p> <p>2018-05-01</p> <p>The "thin-wall <span class="hlt">approximation</span>" gives a simple estimate of the decay rate of an unstable quantum field. Unfortunately, the <span class="hlt">approximation</span> is uncontrolled. In this paper I show that there are actually two different thin-wall <span class="hlt">approximations</span> and that they bracket the true decay rate: I prove that one is an upper bound and the other a lower bound. In the thin-wall limit, the two <span class="hlt">approximations</span> converge. In the presence of gravity, a generalization of this lemma provides a simple sufficient condition for nonperturbative vacuum instability.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/6174646-convergence-strong-potential-born-approximation-sub-less-than-sub-greater-than','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/6174646-convergence-strong-potential-born-approximation-sub-less-than-sub-greater-than"><span>Convergence of the strong-potential-Born <span class="hlt">approximation</span> in Z/sub less-than//Z/sub greater-than/</span></a></p> <p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p> <p>McGuire, J.H.; Sil, N.C.</p> <p>1986-01-01</p> <p>Convergence of the strong-potential Born (SPB) <span class="hlt">approximation</span> as a function of the charges of the projectile and target is studied numerically. Time-reversal invariance (or detailed balance) is satisfied at sufficiently high velocities even when the charges are asymmetric. This demonstarates that the SPB <span class="hlt">approximation</span> converges to the correct result even when the charge of the ''weak'' potential, which is kept to first <span class="hlt">order</span>, is larger than the charge of the ''strong'' potential, which is retained to all <span class="hlt">orders</span>. Consequently, the SPB <span class="hlt">approximation</span> is valid for systems of arbitrary charge symmetry (including symmetric systems) at sufficiently high velocities.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19930018571','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19930018571"><span>Embedding impedance <span class="hlt">approximations</span> in the analysis of SIS mixers</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Kerr, A. R.; Pan, S.-K.; Withington, S.</p> <p>1992-01-01</p> <p>Future millimeter-wave radio astronomy instruments will use arrays of many SIS receivers, either as focal plane arrays on individual radio telescopes, or as individual receivers on the many antennas of radio interferometers. Such applications will require broadband integrated mixers without mechanical tuners. To produce such mixers, it will be necessary to improve present mixer design techniques, most of which use the three-frequency <span class="hlt">approximation</span> to Tucker's quantum mixer theory. This paper examines the adequacy of three <span class="hlt">approximations</span> to Tucker's theory: (1) the usual three-frequency <span class="hlt">approximation</span> which assumes a sinusoidal LO voltage at the junction, and a short-circuit at all frequencies above the upper sideband; (2) a five-frequency <span class="hlt">approximation</span> which allows two LO voltage harmonics and five small-signal sidebands; and (3) a quasi five-frequency <span class="hlt">approximation</span> in which five small-signal sidebands are allowed, but the LO voltage is assumed sinusoidal. These are compared with a full harmonic-Newton solution of Tucker's equations, including eight LO harmonics and their corresponding sidebands, for realistic SIS mixer circuits. It is shown that the accuracy of the three <span class="hlt">approximations</span> depends strongly on the value of omega R(sub N)C for the SIS junctions used. For large omega R(sub N)C, all three <span class="hlt">approximations</span> approach the eight-harmonic solution. For omega R(sub N)C values in the range 0.5 to 10, the range of most practical interest, the quasi five-frequency <span class="hlt">approximation</span> is a considerable improvement over the three-frequency <span class="hlt">approximation</span>, and should be suitable for much design work. For the realistic SIS mixers considered here, the five-frequency <span class="hlt">approximation</span> gives results very close to those of the eight-harmonic solution. Use of these <span class="hlt">approximations</span>, where appropriate, considerably reduces the computational effort needed to analyze an SIS mixer, and allows the design and optimization of mixers using a personal computer.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/29197243','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/29197243"><span>Automated facial recognition of manually generated clay facial <span class="hlt">approximations</span>: Potential application in unidentified persons data repositories.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Parks, Connie L; Monson, Keith L</p> <p>2018-01-01</p> <p>This research examined how accurately 2D images (i.e., photographs) of 3D clay facial <span class="hlt">approximations</span> were matched to corresponding photographs of the <span class="hlt">approximated</span> individuals using an objective automated facial recognition system. Irrespective of search filter (i.e., blind, sex, or ancestry) or rank class (R 1 , R 10 , R 25 , and R 50 ) employed, few operationally informative results were observed. In only a single instance of 48 potential match opportunities was a clay <span class="hlt">approximation</span> matched to a corresponding life photograph within the top 50 images (R 50 ) of a candidate list, even with relatively small gallery sizes created from the application of search filters (e.g., sex or ancestry search restrictions). Increasing the candidate lists to include the top 100 images (R 100 ) resulted in only two additional instances of correct match. Although other untested variables (e.g., <span class="hlt">approximation</span> method, 2D photographic process, and practitioner skill level) may have impacted the observed results, this study suggests that 2D images of manually generated clay <span class="hlt">approximations</span> are not readily matched to life photos by automated facial recognition systems. Further investigation is necessary in <span class="hlt">order</span> to identify the underlying cause(s), if any, of the poor recognition results observed in this study (e.g., potential inferior facial feature detection and extraction). Additional inquiry exploring prospective remedial measures (e.g., stronger feature differentiation) is also warranted, particularly given the prominent use of clay <span class="hlt">approximations</span> in unidentified persons casework. Copyright © 2017. Published by Elsevier B.V.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/29882849','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/29882849"><span>Time and Memory Efficient Online Piecewise Linear <span class="hlt">Approximation</span> of Sensor Signals.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Grützmacher, Florian; Beichler, Benjamin; Hein, Albert; Kirste, Thomas; Haubelt, Christian</p> <p>2018-05-23</p> <p>Piecewise linear <span class="hlt">approximation</span> of sensor signals is a well-known technique in the fields of Data Mining and Activity Recognition. In this context, several algorithms have been developed, some of them with the purpose to be performed on resource constrained microcontroller architectures of wireless sensor nodes. While microcontrollers are usually constrained in computational power and memory resources, all state-of-the-art piecewise linear <span class="hlt">approximation</span> techniques either need to buffer sensor data or have an execution time depending on the segment’s length. In the paper at hand, we propose a novel piecewise linear <span class="hlt">approximation</span> algorithm, with a constant computational complexity as well as a constant memory complexity. Our proposed algorithm’s worst-case execution time is one to three <span class="hlt">orders</span> of magnitude smaller and its average execution time is three to seventy times smaller compared to the state-of-the-art Piecewise Linear <span class="hlt">Approximation</span> (PLA) algorithms in our experiments. In our evaluations, we show that our algorithm is time and memory efficient without sacrificing the <span class="hlt">approximation</span> quality compared to other state-of-the-art piecewise linear <span class="hlt">approximation</span> techniques, while providing a maximum error guarantee per segment, a small parameter space of only one parameter, and a maximum latency of one sample period plus its worst-case execution time.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/1999NIMPB.153....1S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/1999NIMPB.153....1S"><span>A unitary convolution <span class="hlt">approximation</span> for the impact-parameter dependent electronic energy loss</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Schiwietz, G.; Grande, P. L.</p> <p>1999-06-01</p> <p>In this work, we propose a simple method to calculate the impact-parameter dependence of the electronic energy loss of bare ions for all impact parameters. This perturbative convolution <span class="hlt">approximation</span> (PCA) is based on first-<span class="hlt">order</span> perturbation theory, and thus, it is only valid for fast particles with low projectile charges. Using Bloch's stopping-power result and a simple scaling, we get rid of the restriction to low charge states and derive the unitary convolution <span class="hlt">approximation</span> (UCA). Results of the UCA are then compared with full quantum-mechanical coupled-channel calculations for the impact-parameter dependent electronic energy loss.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/27006884','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/27006884"><span>Laplace transform homotopy perturbation method for the <span class="hlt">approximation</span> of variational problems.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Filobello-Nino, U; Vazquez-Leal, H; Rashidi, M M; Sedighi, H M; Perez-Sesma, A; Sandoval-Hernandez, M; Sarmiento-Reyes, A; Contreras-Hernandez, A D; Pereyra-Diaz, D; Hoyos-Reyes, C; Jimenez-Fernandez, V M; Huerta-Chua, J; Castro-Gonzalez, F; Laguna-Camacho, J R</p> <p>2016-01-01</p> <p>This article proposes the application of Laplace Transform-Homotopy Perturbation Method and some of its modifications in <span class="hlt">order</span> to find analytical <span class="hlt">approximate</span> solutions for the linear and nonlinear differential equations which arise from some variational problems. As case study we will solve four ordinary differential equations, and we will show that the proposed solutions have good accuracy, even we will obtain an exact solution. In the sequel, we will see that the square residual error for the <span class="hlt">approximate</span> solutions, belongs to the interval [0.001918936920, 0.06334882582], which confirms the accuracy of the proposed methods, taking into account the complexity and difficulty of variational problems.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19880014221','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19880014221"><span>Complex space monofilar <span class="hlt">approximation</span> of diffraction currents on a conducting half plane</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Lindell, I. V.</p> <p>1987-01-01</p> <p>Simple <span class="hlt">approximation</span> of diffraction surface currents on a conducting half plane, due to an incoming plane wave, is obtained with a line current (monofile) in complex space. When compared to an <span class="hlt">approximating</span> current at the edge, the diffraction pattern is seen to improve by an <span class="hlt">order</span> of magnitude for a minimal increase of computation effort. Thus, the inconvient Fresnel integral functions can be avoided for quick calculations of diffracted fields and the accuracy is good in other directions than along the half plane. The method can be applied to general problems involving planar metal edges.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017CPL...690...20K','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017CPL...690...20K"><span>Restoring the Pauli principle in the random phase <span class="hlt">approximation</span> ground state</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Kosov, D. S.</p> <p>2017-12-01</p> <p>Random phase <span class="hlt">approximation</span> ground state contains electronic configurations where two (and more) identical electrons can occupy the same molecular spin-orbital violating the Pauli exclusion principle. This overcounting of electronic configurations happens due to quasiboson <span class="hlt">approximation</span> in the treatment of electron-hole pair operators. We describe the method to restore the Pauli principle in the RPA wavefunction. The proposed theory is illustrated by the calculations of molecular dipole moments and electronic kinetic energies. The Hartree-Fock based RPA, which is corrected for the Pauli principle, gives the results of comparable accuracy with Møller-Plesset second <span class="hlt">order</span> perturbation theory and coupled-cluster singles and doubles method.</p> </li> </ol> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_20");'>20</a></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li class="active"><span>22</span></li> <li><a href="#" onclick='return showDiv("page_23");'>23</a></li> <li><a href="#" onclick='return showDiv("page_24");'>24</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div><!-- col-sm-12 --> </div><!-- row --> </div><!-- page_22 --> <div id="page_23" class="hiddenDiv"> <div class="row"> <div class="col-sm-12"> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li><a href="#" onclick='return showDiv("page_22");'>22</a></li> <li class="active"><span>23</span></li> <li><a href="#" onclick='return showDiv("page_24");'>24</a></li> <li><a href="#" onclick='return showDiv("page_25");'>25</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div> </div> <div class="row"> <div class="col-sm-12"> <ol class="result-class" start="441"> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/24479023','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/24479023"><span>Birth <span class="hlt">order</span> and psychopathology.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Risal, Ajay; Tharoor, Hema</p> <p>2012-07-01</p> <p>Ordinal position the child holds within the sibling ranking of a family is related to intellectual functioning, personality, behavior, and development of psychopathology. To study the association between birth <span class="hlt">order</span> and development of psychopathology in patients attending psychiatry services in a teaching hospital. Hospital-based cross-sectional study. Retrospective file review of three groups of patients was carried out. Patient-related variables like age of onset, birth <span class="hlt">order</span>, family type, and family history of mental illness were compared with psychiatry diagnosis (ICD-10) generated. SPSS 13; descriptive statistics and one-way analysis of variance (ANOVA) were used. Mean age of onset of mental illness among the adult general psychiatry patients (group I, n = 527) was found to be 33.01 ± 15.073, while it was 11.68 ± 4.764 among the child cases (group II, n = 47) and 26.74 ± 7.529 among substance abuse cases (group III, n = 110). Among group I patients, commonest diagnosis was depression followed by anxiety and somatoform disorders irrespective of birth <span class="hlt">order</span>. Dissociative disorders were most prevalent in the first born child (36.7%) among group II patients. Among group III patients, alcohol dependence was maximum diagnosis in all birth <span class="hlt">orders</span>. Depression and alcohol dependence was the commonest diagnosis in adult group irrespective of birth <span class="hlt">order</span>.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/27242648','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/27242648"><span>Arguments from Developmental <span class="hlt">Order</span>.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Stöckle-Schobel, Richard</p> <p>2016-01-01</p> <p>In this article, I investigate a special type of argument regarding the role of development in theorizing about psychological processes and cognitive capacities. Among the issues that developmental psychologists study, discovering the ontogenetic trajectory of mechanisms or capacities underpinning our cognitive functions ranks highly. The <span class="hlt">order</span> in which functions are developed or capacities are acquired is a matter of debate between competing psychological theories, and also philosophical conceptions of the mind - getting the role and the significance of the different steps in this <span class="hlt">order</span> right could be seen as an important virtue of such theories. Thus, a special kind of strategy in arguments between competing philosophical or psychological theories is using developmental <span class="hlt">order</span> in arguing for or against a given psychological claim. In this article, I will introduce an analysis of arguments from developmental <span class="hlt">order</span>, which come in two general types: arguments emphasizing the importance of the early cognitive processes and arguments emphasizing the late cognitive processes. I will discuss their role in one of the central tools for evaluating scientific theories, namely in making inferences to the best explanation. I will argue that appeal to developmental <span class="hlt">order</span> is, by itself, an insufficient criterion for theory choice and has to be part of an argument based on other core explanatory or empirical virtues. I will end by proposing a more concerted study of philosophical issues concerning (cognitive) development, and I will present some topics that also pertain to a full-fledged 'philosophy of development.'</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/501518','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/501518"><span>The birth <span class="hlt">order</span> puzzle.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Zajonc, R B; Markus, H; Markus, G B</p> <p>1979-08-01</p> <p>Studies relating intellectual performance to birth <span class="hlt">order</span> report conflicting results, some finding intellectual scores to increase, others to decrease with birth <span class="hlt">order</span>. In contrast, the relationship between intellectual performance and family size is stable and consistently replicable. Why do these two highly related variables generate such divergent results? This birth <span class="hlt">order</span> puzzle is resolved by means of the confluence model that quantifies the influences upon intellectual growth arising within the family context. At the time of a new birth, two opposing influences act upon intellectual growth of the elder sibling: (a) his or her intellectual environment is "diluted" and (b) he or she loses the "last-born's handicap" and begins serving as an intellectual resource to the younger sibling. Since these opposite effects are not equal in magnitude, the differences in intellectual performance among birth ranks are shown to be age dependent. While elder children may surpass their younger siblings in intellectual performance at some ages, they may be overtaken by them at others. Thus when age is taken into consideration, the birth <span class="hlt">order</span> literature loses its chaotic character and an <span class="hlt">orderly</span> pattern of results emerges.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/22572343-samba-sparse-approximation-moment-based-arbitrary-polynomial-chaos','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/22572343-samba-sparse-approximation-moment-based-arbitrary-polynomial-chaos"><span>SAMBA: Sparse <span class="hlt">Approximation</span> of Moment-Based Arbitrary Polynomial Chaos</span></a></p> <p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p> <p>Ahlfeld, R., E-mail: r.ahlfeld14@imperial.ac.uk; Belkouchi, B.; Montomoli, F.</p> <p>2016-09-01</p> <p>A new arbitrary Polynomial Chaos (aPC) method is presented for moderately high-dimensional problems characterised by limited input data availability. The proposed methodology improves the algorithm of aPC and extends the method, that was previously only introduced as tensor product expansion, to moderately high-dimensional stochastic problems. The fundamental idea of aPC is to use the statistical moments of the input random variables to develop the polynomial chaos expansion. This approach provides the possibility to propagate continuous or discrete probability density functions and also histograms (data sets) as long as their moments exist, are finite and the determinant of the moment matrixmore » is strictly positive. For cases with limited data availability, this approach avoids bias and fitting errors caused by wrong assumptions. In this work, an alternative way to calculate the aPC is suggested, which provides the optimal polynomials, Gaussian quadrature collocation points and weights from the moments using only a handful of matrix operations on the Hankel matrix of moments. It can therefore be implemented without requiring prior knowledge about statistical data analysis or a detailed understanding of the mathematics of polynomial chaos expansions. The extension to more input variables suggested in this work, is an anisotropic and adaptive version of Smolyak's algorithm that is solely based on the moments of the input probability distributions. It is referred to as SAMBA (PC), which is short for Sparse <span class="hlt">Approximation</span> of Moment-Based Arbitrary Polynomial Chaos. It is illustrated that for moderately high-dimensional problems (up to 20 different input variables or histograms) SAMBA can significantly simplify the calculation of sparse Gaussian quadrature rules. SAMBA's efficiency for multivariate functions with regard to data availability is further demonstrated by analysing higher <span class="hlt">order</span> convergence and accuracy for a set of nonlinear test functions with 2, 5 and</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3663730','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3663730"><span>Cluster and propensity based <span class="hlt">approximation</span> of a network</span></a></p> <p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p> <p></p> <p>2013-01-01</p> <p>Background The models in this article generalize current models for both correlation networks and multigraph networks. Correlation networks are widely applied in genomics research. In contrast to general networks, it is straightforward to test the statistical significance of an edge in a correlation network. It is also easy to decompose the underlying correlation matrix and generate informative network statistics such as the module eigenvector. However, correlation networks only capture the connections between numeric variables. An open question is whether one can find suitable decompositions of the similarity measures employed in constructing general networks. Multigraph networks are attractive because they support likelihood based inference. Unfortunately, it is unclear how to adjust current statistical methods to detect the clusters inherent in many data sets. Results Here we present an intuitive and parsimonious parametrization of a general similarity measure such as a network adjacency matrix. The cluster and propensity based <span class="hlt">approximation</span> (CPBA) of a network not only generalizes correlation network methods but also multigraph methods. In particular, it gives rise to a novel and more realistic multigraph model that accounts for clustering and provides likelihood based tests for assessing the significance of an edge after controlling for clustering. We present a novel Majorization-Minimization (MM) algorithm for estimating the parameters of the CPBA. To illustrate the practical utility of the CPBA of a network, we apply it to gene expression data and to a bi-partite network model for diseases and disease genes from the Online Mendelian Inheritance in Man (OMIM). Conclusions The CPBA of a network is theoretically appealing since a) it generalizes correlation and multigraph network methods, b) it improves likelihood based significance tests for edge counts, c) it directly models higher-<span class="hlt">order</span> relationships between clusters, and d) it suggests novel clustering</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016CMaPh.346..397B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016CMaPh.346..397B"><span>Local Random Quantum Circuits are <span class="hlt">Approximate</span> Polynomial-Designs</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Brandão, Fernando G. S. L.; Harrow, Aram W.; Horodecki, Michał</p> <p>2016-09-01</p> <p>We prove that local random quantum circuits acting on n qubits composed of O( t 10 n 2) many nearest neighbor two-qubit gates form an <span class="hlt">approximate</span> unitary t-design. Previously it was unknown whether random quantum circuits were a t-design for any t > 3. The proof is based on an interplay of techniques from quantum many-body theory, representation theory, and the theory of Markov chains. In particular we employ a result of Nachtergaele for lower bounding the spectral gap of frustration-free quantum local Hamiltonians; a quasi-orthogonality property of permutation matrices; a result of Oliveira which extends to the unitary group the path-coupling method for bounding the mixing time of random walks; and a result of Bourgain and Gamburd showing that dense subgroups of the special unitary group, composed of elements with algebraic entries, are ∞-copy tensor-product expanders. We also consider pseudo-randomness properties of local random quantum circuits of small depth and prove that circuits of depth O( t 10 n) constitute a quantum t-copy tensor-product expander. The proof also rests on techniques from quantum many-body theory, in particular on the detectability lemma of Aharonov, Arad, Landau, and Vazirani. We give applications of the results to cryptography, equilibration of closed quantum dynamics, and the generation of topological <span class="hlt">order</span>. In particular we show the following pseudo-randomness property of generic quantum circuits: Almost every circuit U of size O( n k ) on n qubits cannot be distinguished from a Haar uniform unitary by circuits of size O( n ( k-9)/11) that are given oracle access to U.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2012SPIE.8499E..1IS','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2012SPIE.8499E..1IS"><span>OCR enhancement through neighbor embedding and fast <span class="hlt">approximate</span> nearest neighbors</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Smith, D. C.</p> <p>2012-10-01</p> <p>Generic optical character recognition (OCR) engines often perform very poorly in transcribing scanned low resolution (LR) text documents. To improve OCR performance, we apply the Neighbor Embedding (NE) single-image super-resolution (SISR) technique to LR scanned text documents to obtain high resolution (HR) versions, which we subsequently process with OCR. For comparison, we repeat this procedure using bicubic interpolation (BI). We demonstrate that mean-square errors (MSE) in NE HR estimates do not increase substantially when NE is trained in one Latin font style and tested in another, provided both styles belong to the same font category (serif or sans serif). This is very important in practice, since for each font size, the number of training sets required for each category may be reduced from dozens to just one. We also incorporate randomized k-d trees into our NE implementation to perform <span class="hlt">approximate</span> nearest neighbor search, and obtain a 1000x speed up of our original NE implementation, with negligible MSE degradation. This acceleration also made it practical to combine all of our size-specific NE Latin models into a single Universal Latin Model (ULM). The ULM eliminates the need to determine the unknown font category and size of an input LR text document and match it to an appropriate model, a very challenging task, since the dpi (pixels per inch) of the input LR image is generally unknown. Our experiments show that OCR character error rates (CER) were over 90% when we applied the Tesseract OCR engine to LR text documents (scanned at 75 dpi and 100 dpi) in the 6-10 pt range. By contrast, using k-d trees and the ULM, CER after NE preprocessing averaged less than 7% at 3x (100 dpi LR scanning) and 4x (75 dpi LR scanning) magnification, over an <span class="hlt">order</span> of magnitude improvement. Moreover, CER after NE preprocessing was more that 6 times lower on average than after BI preprocessing.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2015PhRvE..92f2806S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2015PhRvE..92f2806S"><span><span class="hlt">Approximating</span> frustration scores in complex networks via perturbed Laplacian spectra</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Savol, Andrej J.; Chennubhotla, Chakra S.</p> <p>2015-12-01</p> <p>Systems of many interacting components, as found in physics, biology, infrastructure, and the social sciences, are often modeled by simple networks of nodes and edges. The real-world systems frequently confront outside intervention or internal damage whose impact must be predicted or minimized, and such perturbations are then mimicked in the models by altering nodes or edges. This leads to the broad issue of how to best quantify changes in a model network after some type of perturbation. In the case of node removal there are many centrality metrics which associate a scalar quantity with the removed node, but it can be difficult to associate the quantities with some intuitive aspect of physical behavior in the network. This presents a serious hurdle to the application of network theory: real-world utility networks are rarely altered according to theoretic principles unless the kinetic impact on the network's users are fully appreciated beforehand. In pursuit of a kinetically interpretable centrality score, we discuss the f-score, or frustration score. Each f-score quantifies whether a selected node accelerates or inhibits global mean first passage times to a second, independently selected target node. We show that this is a natural way of revealing the dynamical importance of a node in some networks. After discussing merits of the f-score metric, we combine spectral and Laplacian matrix theory in <span class="hlt">order</span> to quickly <span class="hlt">approximate</span> the exact f-score values, which can otherwise be expensive to compute. Following tests on both synthetic and real medium-sized networks, we report f-score runtime improvements over exact brute force approaches in the range of 0 to 400 % with low error (<3 % ).</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016JCoPh.320....1A','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016JCoPh.320....1A"><span>SAMBA: Sparse <span class="hlt">Approximation</span> of Moment-Based Arbitrary Polynomial Chaos</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Ahlfeld, R.; Belkouchi, B.; Montomoli, F.</p> <p>2016-09-01</p> <p>A new arbitrary Polynomial Chaos (aPC) method is presented for moderately high-dimensional problems characterised by limited input data availability. The proposed methodology improves the algorithm of aPC and extends the method, that was previously only introduced as tensor product expansion, to moderately high-dimensional stochastic problems. The fundamental idea of aPC is to use the statistical moments of the input random variables to develop the polynomial chaos expansion. This approach provides the possibility to propagate continuous or discrete probability density functions and also histograms (data sets) as long as their moments exist, are finite and the determinant of the moment matrix is strictly positive. For cases with limited data availability, this approach avoids bias and fitting errors caused by wrong assumptions. In this work, an alternative way to calculate the aPC is suggested, which provides the optimal polynomials, Gaussian quadrature collocation points and weights from the moments using only a handful of matrix operations on the Hankel matrix of moments. It can therefore be implemented without requiring prior knowledge about statistical data analysis or a detailed understanding of the mathematics of polynomial chaos expansions. The extension to more input variables suggested in this work, is an anisotropic and adaptive version of Smolyak's algorithm that is solely based on the moments of the input probability distributions. It is referred to as SAMBA (PC), which is short for Sparse <span class="hlt">Approximation</span> of Moment-Based Arbitrary Polynomial Chaos. It is illustrated that for moderately high-dimensional problems (up to 20 different input variables or histograms) SAMBA can significantly simplify the calculation of sparse Gaussian quadrature rules. SAMBA's efficiency for multivariate functions with regard to data availability is further demonstrated by analysing higher <span class="hlt">order</span> convergence and accuracy for a set of nonlinear test functions with 2, 5 and 10</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19920016496','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19920016496"><span>The blind leading the blind: Mutual refinement of <span class="hlt">approximate</span> theories</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Kedar, Smadar T.; Bresina, John L.; Dent, C. Lisa</p> <p>1991-01-01</p> <p>The mutual refinement theory, a method for refining world models in a reactive system, is described. The method detects failures, explains their causes, and repairs the <span class="hlt">approximate</span> models which cause the failures. The approach focuses on using one <span class="hlt">approximate</span> model to refine another.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=precalculus&id=EJ1127893','ERIC'); return false;" href="https://eric.ed.gov/?q=precalculus&id=EJ1127893"><span><span class="hlt">Approximating</span> Exponential and Logarithmic Functions Using Polynomial Interpolation</span></a></p> <p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p> <p>Gordon, Sheldon P.; Yang, Yajun</p> <p>2017-01-01</p> <p>This article takes a closer look at the problem of <span class="hlt">approximating</span> the exponential and logarithmic functions using polynomials. Either as an alternative to or a precursor to Taylor polynomial <span class="hlt">approximations</span> at the precalculus level, interpolating polynomials are considered. A measure of error is given and the behaviour of the error function is…</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=calculus+AND+8&pg=7&id=EJ952945','ERIC'); return false;" href="https://eric.ed.gov/?q=calculus+AND+8&pg=7&id=EJ952945"><span>Finding the Best Quadratic <span class="hlt">Approximation</span> of a Function</span></a></p> <p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p> <p>Yang, Yajun; Gordon, Sheldon P.</p> <p>2011-01-01</p> <p>This article examines the question of finding the best quadratic function to <span class="hlt">approximate</span> a given function on an interval. The prototypical function considered is f(x) = e[superscript x]. Two approaches are considered, one based on Taylor polynomial <span class="hlt">approximations</span> at various points in the interval under consideration, the other based on the fact…</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/24807446','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/24807446"><span>Function <span class="hlt">approximation</span> using combined unsupervised and supervised learning.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Andras, Peter</p> <p>2014-03-01</p> <p>Function <span class="hlt">approximation</span> is one of the core tasks that are solved using neural networks in the context of many engineering problems. However, good <span class="hlt">approximation</span> results need good sampling of the data space, which usually requires exponentially increasing volume of data as the dimensionality of the data increases. At the same time, often the high-dimensional data is arranged around a much lower dimensional manifold. Here we propose the breaking of the function <span class="hlt">approximation</span> task for high-dimensional data into two steps: (1) the mapping of the high-dimensional data onto a lower dimensional space corresponding to the manifold on which the data resides and (2) the <span class="hlt">approximation</span> of the function using the mapped lower dimensional data. We use over-complete self-organizing maps (SOMs) for the mapping through unsupervised learning, and single hidden layer neural networks for the function <span class="hlt">approximation</span> through supervised learning. We also extend the two-step procedure by considering support vector machines and Bayesian SOMs for the determination of the best parameters for the nonlinear neurons in the hidden layer of the neural networks used for the function <span class="hlt">approximation</span>. We compare the <span class="hlt">approximation</span> performance of the proposed neural networks using a set of functions and show that indeed the neural networks using combined unsupervised and supervised learning outperform in most cases the neural networks that learn the function <span class="hlt">approximation</span> using the original high-dimensional data.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/22471898-use-neural-networks-approximation-nuclear-data','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/22471898-use-neural-networks-approximation-nuclear-data"><span>The use of neural networks for <span class="hlt">approximation</span> of nuclear data</span></a></p> <p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p> <p>Korovin, Yu. A.; Maksimushkina, A. V., E-mail: AVMaksimushkina@mephi.ru</p> <p>2015-12-15</p> <p>The article discusses the possibility of using neural networks for <span class="hlt">approximation</span> or reconstruction of data such as the reaction cross sections. The quality of the <span class="hlt">approximation</span> using fitting criteria is also evaluated. The activity of materials under irradiation is calculated from data obtained using neural networks.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=filters&pg=5&id=EJ950130','ERIC'); return false;" href="https://eric.ed.gov/?q=filters&pg=5&id=EJ950130"><span>Reply to Steele & Ferrer: Modeling Oscillation, <span class="hlt">Approximately</span> or Exactly?</span></a></p> <p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p> <p>Oud, Johan H. L.; Folmer, Henk</p> <p>2011-01-01</p> <p>This article addresses modeling oscillation in continuous time. It criticizes Steele and Ferrer's article "Latent Differential Equation Modeling of Self-Regulatory and Coregulatory Affective Processes" (2011), particularly the <span class="hlt">approximate</span> estimation procedure applied. This procedure is the latent version of the local linear <span class="hlt">approximation</span> procedure…</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.gpo.gov/fdsys/pkg/CFR-2010-title36-vol2/pdf/CFR-2010-title36-vol2-sec254-11.pdf','CFR'); return false;" href="https://www.gpo.gov/fdsys/pkg/CFR-2010-title36-vol2/pdf/CFR-2010-title36-vol2-sec254-11.pdf"><span>36 CFR 254.11 - Exchanges at <span class="hlt">approximately</span> equal value.</span></a></p> <p><a target="_blank" href="http://www.gpo.gov/fdsys/browse/collectionCfr.action?selectedYearFrom=2010&page.go=Go">Code of Federal Regulations, 2010 CFR</a></p> <p></p> <p>2010-07-01</p> <p>... equal value. 254.11 Section 254.11 Parks, Forests, and Public Property FOREST SERVICE, DEPARTMENT OF AGRICULTURE LANDOWNERSHIP ADJUSTMENTS Land Exchanges § 254.11 Exchanges at <span class="hlt">approximately</span> equal value. (a) The authorized officer may exchange lands which are of <span class="hlt">approximately</span> equal value upon a determination that: (1...</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=electroquimica+AND+en+AND+baterias+AND+libros%7d&id=EJ1060255','ERIC'); return false;" href="https://eric.ed.gov/?q=electroquimica+AND+en+AND+baterias+AND+libros%7d&id=EJ1060255"><span>The Role of Intuitive <span class="hlt">Approximation</span> Skills for School Math Abilities</span></a></p> <p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p> <p>Libertus, Melissa E.</p> <p>2015-01-01</p> <p>Research has shown that educated children and adults have access to two ways of representing numerical information: an <span class="hlt">approximate</span> number system (ANS) that is present from birth and allows for quick <span class="hlt">approximations</span> of numbers of objects encountered in one's environment, and an exact number system (ENS) that is acquired through experience and…</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2007JMAA..334..909Z','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2007JMAA..334..909Z"><span>Best uniform <span class="hlt">approximation</span> to a class of rational functions</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Zheng, Zhitong; Yong, Jun-Hai</p> <p>2007-10-01</p> <p>We explicitly determine the best uniform polynomial <span class="hlt">approximation</span> to a class of rational functions of the form 1/(x-c)2+K(a,b,c,n)/(x-c) on [a,b] represented by their Chebyshev expansion, where a, b, and c are real numbers, n-1 denotes the degree of the best <span class="hlt">approximating</span> polynomial, and K is a constant determined by a, b, c, and n. Our result is based on the explicit determination of a phase angle [eta] in the representation of the <span class="hlt">approximation</span> error by a trigonometric function. Moreover, we formulate an ansatz which offers a heuristic strategies to determine the best <span class="hlt">approximating</span> polynomial to a function represented by its Chebyshev expansion. Combined with the phase angle method, this ansatz can be used to find the best uniform <span class="hlt">approximation</span> to some more functions.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2003JCoAM.155..359B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2003JCoAM.155..359B"><span>Laplace <span class="hlt">approximation</span> for Bessel functions of matrix argument</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Butler, Ronald W.; Wood, Andrew T. A.</p> <p>2003-06-01</p> <p>We derive Laplace <span class="hlt">approximations</span> to three functions of matrix argument which arise in statistics and elsewhere: matrix Bessel A[nu]; matrix Bessel B[nu]; and the type II confluent hypergeometric function of matrix argument, [Psi]. We examine the theoretical and numerical properties of the <span class="hlt">approximations</span>. On the theoretical side, it is shown that the Laplace <span class="hlt">approximations</span> to A[nu], B[nu] and [Psi] given here, together with the Laplace <span class="hlt">approximations</span> to the matrix argument functions 1F1 and 2F1 presented in Butler and Wood (Laplace <span class="hlt">approximations</span> to hyper-geometric functions with matrix argument, Ann. Statist. (2002)), satisfy all the important confluence relations and symmetry relations enjoyed by the original functions.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/pages/biblio/1188218-properties-boltzmann-equation-classical-approximation','SCIGOV-DOEP'); return false;" href="https://www.osti.gov/pages/biblio/1188218-properties-boltzmann-equation-classical-approximation"><span>Properties of the Boltzmann equation in the classical <span class="hlt">approximation</span></span></a></p> <p><a target="_blank" href="http://www.osti.gov/pages">DOE PAGES</a></p> <p>Epelbaum, Thomas; Gelis, François; Tanji, Naoto; ...</p> <p>2014-12-30</p> <p>We examine the Boltzmann equation with elastic point-like scalar interactions in two different versions of the the classical <span class="hlt">approximation</span>. Although solving numerically the Boltzmann equation with the unapproximated collision term poses no problem, this allows one to study the effect of the ultraviolet cutoff in these <span class="hlt">approximations</span>. This cutoff dependence in the classical <span class="hlt">approximations</span> of the Boltzmann equation is closely related to the non-renormalizability of the classical statistical <span class="hlt">approximation</span> of the underlying quantum field theory. The kinetic theory setup that we consider here allows one to study in a much simpler way the dependence on the ultraviolet cutoff, since onemore » has also access to the non-<span class="hlt">approximated</span> result for comparison.« less</p> </li> </ol> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li><a href="#" onclick='return showDiv("page_22");'>22</a></li> <li class="active"><span>23</span></li> <li><a href="#" onclick='return showDiv("page_24");'>24</a></li> <li><a href="#" onclick='return showDiv("page_25");'>25</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div><!-- col-sm-12 --> </div><!-- row --> </div><!-- page_23 --> <div id="page_24" class="hiddenDiv"> <div class="row"> <div class="col-sm-12"> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li><a href="#" onclick='return showDiv("page_22");'>22</a></li> <li><a href="#" onclick='return showDiv("page_23");'>23</a></li> <li class="active"><span>24</span></li> <li><a href="#" onclick='return showDiv("page_25");'>25</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div> </div> <div class="row"> <div class="col-sm-12"> <ol class="result-class" start="461"> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19910012800','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19910012800"><span>Recent advances in <span class="hlt">approximation</span> concepts for optimum structural design</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Barthelemy, Jean-Francois M.; Haftka, Raphael T.</p> <p>1991-01-01</p> <p>The basic <span class="hlt">approximation</span> concepts used in structural optimization are reviewed. Some of the most recent developments in that area since the introduction of the concept in the mid-seventies are discussed. The paper distinguishes between local, medium-range, and global <span class="hlt">approximations</span>; it covers functions <span class="hlt">approximations</span> and problem <span class="hlt">approximations</span>. It shows that, although the lack of comparative data established on reference test cases prevents an accurate assessment, there have been significant improvements. The largest number of developments have been in the areas of local function <span class="hlt">approximations</span> and use of intermediate variable and response quantities. It also appears that some new methodologies are emerging which could greatly benefit from the introduction of new computer architecture.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=ionic&pg=5&id=EJ826728','ERIC'); return false;" href="https://eric.ed.gov/?q=ionic&pg=5&id=EJ826728"><span>Concomitant <span class="hlt">Ordering</span> and Symmetry Lowering</span></a></p> <p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p> <p>Boo, William O. J.; Mattern, Daniell L.</p> <p>2008-01-01</p> <p>Examples of concomitant <span class="hlt">ordering</span> include magnetic <span class="hlt">ordering</span>, Jahn-Teller cooperative <span class="hlt">ordering</span>, electronic <span class="hlt">ordering</span>, ionic <span class="hlt">ordering</span>, and <span class="hlt">ordering</span> of partially-filled sites. Concomitant <span class="hlt">ordering</span> sets in when a crystal is cooled and always lowers the degree of symmetry of the crystal. Concomitant <span class="hlt">ordering</span> concepts can also be productively applied to…</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2003CPL...371..433P','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2003CPL...371..433P"><span>Hierarchically <span class="hlt">ordered</span> carbon tubes</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Pan, Zheng-Wei; Zhu, Hao-Guo; Zhang, Zong-Tao; Im, Hee-Jung; Dai, Sheng; Beach, David B.; Lowndes, Douglas H.</p> <p>2003-04-01</p> <p>Micropatterns of hierarchically <span class="hlt">ordered</span> carbon tubes (i.e., <span class="hlt">ordered</span> carbon microtubes composed of aligned carbon nanotubes) were grown on a film-like iron/silica substrate consisting of ring-like catalyst patterns. The substrates were prepared by a combined technique, in which the sol-gel method was used to prepare catalyst film and transmission electron microscope grids were used as a shadow mask. In comparison with other techniques that involve sophisticated lithography, this approach represents a simple and low-cost way to the micropatterning of aligned carbon nanotubes.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017PhDT........87C','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017PhDT........87C"><span>Nonlinear Schroedinger <span class="hlt">Approximations</span> for Partial Differential Equations with Quadratic and Quasilinear Terms</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Cummings, Patrick</p> <p></p> <p>We consider the <span class="hlt">approximation</span> of solutions of two complicated, physical systems via the nonlinear Schrodinger equation (NLS). In particular, we discuss the evolution of wave packets and long waves in two physical models. Due to the complicated nature of the equations governing many physical systems and the in-depth knowledge we have for solutions of the nonlinear Schrodinger equation, it is advantageous to use <span class="hlt">approximation</span> results of this kind to model these physical systems. The <span class="hlt">approximations</span> are simple enough that we can use them to understand the qualitative and quantitative behavior of the solutions, and by justifying them we can show that the behavior of the <span class="hlt">approximation</span> captures the behavior of solutions to the original equation, at least for long, but finite time. We first consider a model of the water wave equations which can be <span class="hlt">approximated</span> by wave packets using the NLS equation. We discuss a new proof that both simplifies and strengthens previous justification results of Schneider and Wayne. Rather than using analytic norms, as was done by Schneider and Wayne, we construct a modified energy functional so that the <span class="hlt">approximation</span> holds for the full interval of existence of the <span class="hlt">approximate</span> NLS solution as opposed to a subinterval (as is seen in the analytic case). Furthermore, the proof avoids problems associated with inverting the normal form transform by working with a modified energy functional motivated by Craig and Hunter et al. We then consider the Klein-Gordon-Zakharov system and prove a long wave <span class="hlt">approximation</span> result. In this case there is a non-trivial resonance that cannot be eliminated via a normal form transform. By combining the normal form transform for small Fourier modes and using analytic norms elsewhere, we can get a justification result on the <span class="hlt">order</span> 1 over epsilon squared time scale.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/25462030','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/25462030"><span><span class="hlt">Approximate</span> number word knowledge before the cardinal principle.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Gunderson, Elizabeth A; Spaepen, Elizabet; Levine, Susan C</p> <p>2015-02-01</p> <p><span class="hlt">Approximate</span> number word knowledge-understanding the relation between the count words and the <span class="hlt">approximate</span> magnitudes of sets-is a critical piece of knowledge that predicts later math achievement. However, researchers disagree about when children first show evidence of <span class="hlt">approximate</span> number word knowledge-before, or only after, they have learned the cardinal principle. In two studies, children who had not yet learned the cardinal principle (subset-knowers) produced sets in response to number words (verbal comprehension task) and produced number words in response to set sizes (verbal production task). As evidence of <span class="hlt">approximate</span> number word knowledge, we examined whether children's numerical responses increased with increasing numerosity of the stimulus. In Study 1, subset-knowers (ages 3.0-4.2 years) showed <span class="hlt">approximate</span> number word knowledge above their knower-level on both tasks, but this effect did not extend to numbers above 4. In Study 2, we collected data from a broader age range of subset-knowers (ages 3.1-5.6 years). In this sample, children showed <span class="hlt">approximate</span> number word knowledge on the verbal production task even when only examining set sizes above 4. Across studies, children's age predicted <span class="hlt">approximate</span> number word knowledge (above 4) on the verbal production task when controlling for their knower-level, study (1 or 2), and parents' education, none of which predicted <span class="hlt">approximation</span> ability. Thus, children can develop <span class="hlt">approximate</span> knowledge of number words up to 10 before learning the cardinal principle. Furthermore, <span class="hlt">approximate</span> number word knowledge increases with age and might not be closely related to the development of exact number word knowledge. Copyright © 2014 Elsevier Inc. All rights reserved.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=automation+AND+large+AND+data&pg=4&id=ED077516','ERIC'); return false;" href="https://eric.ed.gov/?q=automation+AND+large+AND+data&pg=4&id=ED077516"><span><span class="hlt">Order</span> Division Automated System.</span></a></p> <p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p> <p>Kniemeyer, Justin M.; And Others</p> <p></p> <p>This publication was prepared by the <span class="hlt">Order</span> Division Automation Project staff to fulfill the Library of Congress' requirement to document all automation efforts. The report was originally intended for internal use only and not for distribution outside the Library. It is now felt that the library community at-large may have an interest in the…</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=psychology+AND+birth+AND+order&pg=5&id=EJ219982','ERIC'); return false;" href="https://eric.ed.gov/?q=psychology+AND+birth+AND+order&pg=5&id=EJ219982"><span>The Birth <span class="hlt">Order</span> Puzzle.</span></a></p> <p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p> <p>Zajonc, R. B.; And Others</p> <p>1979-01-01</p> <p>Discusses the controversy of the relationship between birth <span class="hlt">order</span> and intellectual performance through a detailed evaluation of the confluence model which assumes that the rate of intellectual growth is a function of the intellectual environment within the family and associated with the special circumstances of last children. (CM)</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=global+AND+Governance&pg=4&id=EJ774301','ERIC'); return false;" href="https://eric.ed.gov/?q=global+AND+Governance&pg=4&id=EJ774301"><span>Education and World <span class="hlt">Order</span></span></a></p> <p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p> <p>Jones, Phillip W.</p> <p>2007-01-01</p> <p>The impact on educational analysis of mainstream international relations (IR) theories is yet to realize its full potential. The problem of education in relation to the construction of world <span class="hlt">order</span> is considered in relation to core developments in IR theory since the Second World War. In particular, the global architecture of education is seen as a…</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=proactive+AND+interference&pg=7&id=EJ396036','ERIC'); return false;" href="https://eric.ed.gov/?q=proactive+AND+interference&pg=7&id=EJ396036"><span>Memory for Serial <span class="hlt">Order</span>.</span></a></p> <p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p> <p>Lewandowsky, Stephan; Murdock, Bennet B., Jr.</p> <p>1989-01-01</p> <p>An extension to Murdock's Theory of Distributed Associative Memory, based on associative chaining between items, is presented. The extended theory is applied to several serial <span class="hlt">order</span> phenomena, including serial list learning, delayed recall effects, partial report effects, and buildup and release from proactive interference. (TJH)</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19730004865','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19730004865"><span><span class="hlt">Order</span>, topology and preference</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Sertel, M. R.</p> <p>1971-01-01</p> <p>Some standard <span class="hlt">order</span>-related and topological notions, facts, and methods are brought to bear on central topics in the theory of preference and the theory of optimization. Consequences of connectivity are considered, especially from the viewpoint of normally preordered spaces. Examples are given showing how the theory of preference, or utility theory, can be applied to social analysis.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=rodgers&pg=4&id=EJ628843','ERIC'); return false;" href="https://eric.ed.gov/?q=rodgers&pg=4&id=EJ628843"><span>Birth <span class="hlt">Order</span> Debate Resolved?</span></a></p> <p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p> <p>Zajonc, R. B.</p> <p>2001-01-01</p> <p>Critiques Rodgers et al.'s June 2000 research on the relation between birth <span class="hlt">order</span> and intelligence, which suggests that it is a methodological illusion. Explains how the intellectual environment and the teaching function (whereby older children tutor younger ones) contribute to the growth of intellectual maturity, the first negatively and the…</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=deforestation&pg=6&id=ED219316','ERIC'); return false;" href="https://eric.ed.gov/?q=deforestation&pg=6&id=ED219316"><span>Land and World <span class="hlt">Order</span>.</span></a></p> <p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p> <p>Mische, Patricia, Ed.; And Others</p> <p>1982-01-01</p> <p>The papers in this publication discuss the land and how what happens to the land affects us. The publication is one in a series of monographs that examine the linkages between local and global concerns and explore alternative world futures. Examples of topics discussed in the papers follow. The paper "Land and World <span class="hlt">Order</span>" examines…</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/22415855-similarity-transformed-perturbation-theory-top-truncated-local-coupled-cluster-solutions-theory-applications-intermolecular-interactions','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/22415855-similarity-transformed-perturbation-theory-top-truncated-local-coupled-cluster-solutions-theory-applications-intermolecular-interactions"><span>Similarity-transformed perturbation theory on top of truncated local coupled cluster solutions: Theory and applications to intermolecular interactions</span></a></p> <p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p> <p>Azar, Richard Julian, E-mail: julianazar2323@berkeley.edu; Head-Gordon, Martin, E-mail: mhg@cchem.berkeley.edu</p> <p>2015-05-28</p> <p>Your correspondents develop and apply fully nonorthogonal, local-reference perturbation theories describing non-covalent interactions. Our formulations are based on a Löwdin partitioning of the similarity-transformed Hamiltonian into a <span class="hlt">zeroth-order</span> intramonomer piece (taking local CCSD solutions as its <span class="hlt">zeroth-order</span> eigenfunction) plus a first-<span class="hlt">order</span> piece coupling the fragments. If considerations are limited to a single molecule, the proposed intermolecular similarity-transformed perturbation theory represents a frozen-orbital variant of the “(2)”-type theories shown to be competitive with CCSD(T) and of similar cost if all terms are retained. Different restrictions on the <span class="hlt">zeroth</span>- and first-<span class="hlt">order</span> amplitudes are explored in the context of large-computation tractability and elucidationmore » of non-local effects in the space of singles and doubles. To accurately <span class="hlt">approximate</span> CCSD intermolecular interaction energies, a quadratically growing number of variables must be included at <span class="hlt">zeroth-order</span>.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19950012943','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19950012943"><span>A test of the adhesion <span class="hlt">approximation</span> for gravitational clustering</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Melott, Adrian L.; Shandarin, Sergei; Weinberg, David H.</p> <p>1993-01-01</p> <p>We quantitatively compare a particle implementation of the adhesion <span class="hlt">approximation</span> to fully non-linear, numerical 'N-body' simulations. Our primary tool, cross-correlation of N-body simulations with the adhesion <span class="hlt">approximation</span>, indicates good agreement, better than that found by the same test performed with the Zel-dovich <span class="hlt">approximation</span> (hereafter ZA). However, the cross-correlation is not as good as that of the truncated Zel-dovich <span class="hlt">approximation</span> (TZA), obtained by applying the Zel'dovich <span class="hlt">approximation</span> after smoothing the initial density field with a Gaussian filter. We confirm that the adhesion <span class="hlt">approximation</span> produces an excessively filamentary distribution. Relative to the N-body results, we also find that: (a) the power spectrum obtained from the adhesion <span class="hlt">approximation</span> is more accurate than that from ZA or TZA, (b) the error in the phase angle of Fourier components is worse than that from TZA, and (c) the mass distribution function is more accurate than that from ZA or TZA. It appears that adhesion performs well statistically, but that TZA is more accurate dynamically, in the sense of moving mass to the right place.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19950048281&hterms=weinberg&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAuthor-Name%26N%3D0%26No%3D70%26Ntt%3Dweinberg','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19950048281&hterms=weinberg&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAuthor-Name%26N%3D0%26No%3D70%26Ntt%3Dweinberg"><span>A test of the adhesion <span class="hlt">approximation</span> for gravitational clustering</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Melott, Adrian L.; Shandarin, Sergei F.; Weinberg, David H.</p> <p>1994-01-01</p> <p>We quantitatively compare a particle implementation of the adhesion <span class="hlt">approximation</span> to fully nonlinear, numerical 'N-body' simulations. Our primary tool, cross-correlation of N-body simulations with the adhesion <span class="hlt">approximation</span>, indicates good agreement, better than that found by the same test performed with the Zel'dovich <span class="hlt">approximation</span> (hereafter ZA). However, the cross-correlation is not as good as that of the truncated Zel'dovich <span class="hlt">approximation</span> (TZA), obtained by applying the Zel'dovich <span class="hlt">approximation</span> after smoothing the initial density field with a Gaussian filter. We confirm that the adhesion <span class="hlt">approximation</span> produces an excessively filamentary distribution. Relative to the N-body results, we also find that: (a) the power spectrum obtained from the adhesion <span class="hlt">approximation</span> is more accurate that that from ZA to TZA, (b) the error in the phase angle of Fourier components is worse that that from TZA, and (c) the mass distribution function is more accurate than that from ZA or TZA. It appears that adhesion performs well statistically, but that TZA is more accurate dynamically, in the sense of moving mass to the right place.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19910021552','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19910021552"><span>The convergence rate of <span class="hlt">approximate</span> solutions for nonlinear scalar conservation laws</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Nessyahu, Haim; Tadmor, Eitan</p> <p>1991-01-01</p> <p>The convergence rate is discussed of <span class="hlt">approximate</span> solutions for the nonlinear scalar conservation law. The linear convergence theory is extended into a weak regime. The extension is based on the usual two ingredients of stability and consistency. On the one hand, the counterexamples show that one must strengthen the linearized L(sup 2)-stability requirement. It is assumed that the <span class="hlt">approximate</span> solutions are Lip(sup +)-stable in the sense that they satisfy a one-sided Lipschitz condition, in agreement with Oleinik's E-condition for the entropy solution. On the other hand, the lack of smoothness requires to weaken the consistency requirement, which is measured in the Lip'-(semi)norm. It is proved for Lip(sup +)-stable <span class="hlt">approximate</span> solutions, that their Lip'convergence rate to the entropy solution is of the same <span class="hlt">order</span> as their Lip'-consistency. The Lip'-convergence rate is then converted into stronger L(sup p) convergence rate estimates.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018CG....113...23B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018CG....113...23B"><span>Valid <span class="hlt">approximation</span> of spatially distributed grain size distributions - A priori information encoded to a feedforward network</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Berthold, T.; Milbradt, P.; Berkhahn, V.</p> <p>2018-04-01</p> <p>This paper presents a model for the <span class="hlt">approximation</span> of multiple, spatially distributed grain size distributions based on a feedforward neural network. Since a classical feedforward network does not guarantee to produce valid cumulative distribution functions, a priori information is incor porated into the model by applying weight and architecture constraints. The model is derived in two steps. First, a model is presented that is able to produce a valid distribution function for a single sediment sample. Although initially developed for sediment samples, the model is not limited in its application; it can also be used to <span class="hlt">approximate</span> any other multimodal continuous distribution function. In the second part, the network is extended in <span class="hlt">order</span> to capture the spatial variation of the sediment samples that have been obtained from 48 locations in the investigation area. Results show that the model provides an adequate <span class="hlt">approximation</span> of grain size distributions, satisfying the requirements of a cumulative distribution function.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018JChPh.148j2316R','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018JChPh.148j2316R"><span>Harmonic-phase path-integral <span class="hlt">approximation</span> of thermal quantum correlation functions</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Robertson, Christopher; Habershon, Scott</p> <p>2018-03-01</p> <p>We present an <span class="hlt">approximation</span> to the thermal symmetric form of the quantum time-correlation function in the standard position path-integral representation. By transforming to a sum-and-difference position representation and then Taylor-expanding the potential energy surface of the system to second <span class="hlt">order</span>, the resulting expression provides a harmonic weighting function that <span class="hlt">approximately</span> recovers the contribution of the phase to the time-correlation function. This method is readily implemented in a Monte Carlo sampling scheme and provides exact results for harmonic potentials (for both linear and non-linear operators) and near-quantitative results for anharmonic systems for low temperatures and times that are likely to be relevant to condensed phase experiments. This article focuses on one-dimensional examples to provide insights into convergence and sampling properties, and we also discuss how this <span class="hlt">approximation</span> method may be extended to many-dimensional systems.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19840040799&hterms=Butterfly&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D70%26Ntt%3DButterfly','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19840040799&hterms=Butterfly&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D70%26Ntt%3DButterfly"><span>The uniform asymptotic swallowtail <span class="hlt">approximation</span> - Practical methods for oscillating integrals with four coalescing saddle points</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Connor, J. N. L.; Curtis, P. R.; Farrelly, D.</p> <p>1984-01-01</p> <p>Methods that can be used in the numerical implementation of the uniform swallowtail <span class="hlt">approximation</span> are described. An explicit expression for that <span class="hlt">approximation</span> is presented to the lowest <span class="hlt">order</span>, showing that there are three problems which must be overcome in practice before the <span class="hlt">approximation</span> can be applied to any given problem. It is shown that a recently developed quadrature method can be used for the accurate numerical evaluation of the swallowtail canonical integral and its partial derivatives. Isometric plots of these are presented to illustrate some of their properties. The problem of obtaining the arguments of the swallowtail integral from an analytical function of its argument is considered, describing two methods of solving this problem. The asymptotic evaluation of the butterfly canonical integral is addressed.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/7047033-boson-expansions-based-random-phase-approximation-representation','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/7047033-boson-expansions-based-random-phase-approximation-representation"><span>Boson expansions based on the random phase <span class="hlt">approximation</span> representation</span></a></p> <p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p> <p>Pedrocchi, V.G.; Tamura, T.</p> <p>1984-04-01</p> <p>A new boson expansion theory based on the random phase <span class="hlt">approximation</span> is presented. The boson expansions are derived here directly in the random phase <span class="hlt">approximation</span> representation with the help of a technique that combines the use of the Usui operator with that of a new bosonization procedure, called the term-by-term bosonization method. The present boson expansion theory is constructed by retaining a single collective quadrupole random phase <span class="hlt">approximation</span> component, a truncation that allows for a perturbative treatment of the whole problem. Both Hermitian, as well as non-Hermitian boson expansions, valid for even nuclei, are obtained.</p> </li> </ol> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li><a href="#" onclick='return showDiv("page_22");'>22</a></li> <li><a href="#" onclick='return showDiv("page_23");'>23</a></li> <li class="active"><span>24</span></li> <li><a href="#" onclick='return showDiv("page_25");'>25</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div><!-- col-sm-12 --> </div><!-- row --> </div><!-- page_24 --> <div id="page_25" class="hiddenDiv"> <div class="row"> <div class="col-sm-12"> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li><a href="#" onclick='return showDiv("page_22");'>22</a></li> <li><a href="#" onclick='return showDiv("page_23");'>23</a></li> <li><a href="#" onclick='return showDiv("page_24");'>24</a></li> <li class="active"><span>25</span></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div> </div> <div class="row"> <div class="col-sm-12"> <ol class="result-class" start="481"> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/27802627','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/27802627"><span>Minimax rational <span class="hlt">approximation</span> of the Fermi-Dirac distribution.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Moussa, Jonathan E</p> <p>2016-10-28</p> <p>Accurate rational <span class="hlt">approximations</span> of the Fermi-Dirac distribution are a useful component in many numerical algorithms for electronic structure calculations. The best known <span class="hlt">approximations</span> use O(log(βΔ)log(ϵ -1 )) poles to achieve an error tolerance ϵ at temperature β -1 over an energy interval Δ. We apply minimax <span class="hlt">approximation</span> to reduce the number of poles by a factor of four and replace Δ with Δ occ , the occupied energy interval. This is particularly beneficial when Δ ≫ Δ occ , such as in electronic structure calculations that use a large basis set.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018MNRAS.476.2040D','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018MNRAS.476.2040D"><span>Polynomial <span class="hlt">approximation</span> of the Lense-Thirring rigid precession frequency</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>De Falco, Vittorio; Motta, Sara</p> <p>2018-05-01</p> <p>We propose a polynomial <span class="hlt">approximation</span> of the global Lense-Thirring rigid precession frequency to study low-frequency quasi-periodic oscillations around spinning black holes. This high-performing <span class="hlt">approximation</span> allows to determine the expected frequencies of a precessing thick accretion disc with fixed inner radius and variable outer radius around a black hole with given mass and spin. We discuss the accuracy and the applicability regions of our polynomial <span class="hlt">approximation</span>, showing that the computational times are reduced by a factor of ≈70 in the range of minutes.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/pages/biblio/1335130-minimax-rational-approximation-fermi-dirac-distribution','SCIGOV-DOEP'); return false;" href="https://www.osti.gov/pages/biblio/1335130-minimax-rational-approximation-fermi-dirac-distribution"><span>Minimax rational <span class="hlt">approximation</span> of the Fermi-Dirac distribution</span></a></p> <p><a target="_blank" href="http://www.osti.gov/pages">DOE PAGES</a></p> <p>Moussa, Jonathan E.</p> <p>2016-10-27</p> <p>Accurate rational <span class="hlt">approximations</span> of the Fermi-Dirac distribution are a useful component in many numerical algorithms for electronic structure calculations. The best known <span class="hlt">approximations</span> use O(log(βΔ)log(ϵ –1)) poles to achieve an error tolerance ϵ at temperature β –1 over an energy interval Δ. We apply minimax <span class="hlt">approximation</span> to reduce the number of poles by a factor of four and replace Δ with Δ occ, the occupied energy interval. Furthermore, this is particularly beneficial when Δ >> Δ occ, such as in electronic structure calculations that use a large basis set.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016JChPh.145p4108M','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016JChPh.145p4108M"><span>Minimax rational <span class="hlt">approximation</span> of the Fermi-Dirac distribution</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Moussa, Jonathan E.</p> <p>2016-10-01</p> <p>Accurate rational <span class="hlt">approximations</span> of the Fermi-Dirac distribution are a useful component in many numerical algorithms for electronic structure calculations. The best known <span class="hlt">approximations</span> use O(log(βΔ)log(ɛ-1)) poles to achieve an error tolerance ɛ at temperature β-1 over an energy interval Δ. We apply minimax <span class="hlt">approximation</span> to reduce the number of poles by a factor of four and replace Δ with Δocc, the occupied energy interval. This is particularly beneficial when Δ ≫ Δocc, such as in electronic structure calculations that use a large basis set.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017BrJPh..47...72T','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017BrJPh..47...72T"><span>Reduced <span class="hlt">Order</span> Podolsky Model</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Thibes, Ronaldo</p> <p>2017-02-01</p> <p>We perform the canonical and path integral quantizations of a lower-<span class="hlt">order</span> derivatives model describing Podolsky's generalized electrodynamics. The physical content of the model shows an auxiliary massive vector field coupled to the usual electromagnetic field. The equivalence with Podolsky's original model is studied at classical and quantum levels. Concerning the dynamical time evolution, we obtain a theory with two first-class and two second-class constraints in phase space. We calculate explicitly the corresponding Dirac brackets involving both vector fields. We use the Senjanovic procedure to implement the second-class constraints and the Batalin-Fradkin-Vilkovisky path integral quantization scheme to deal with the symmetries generated by the first-class constraints. The physical interpretation of the results turns out to be simpler due to the reduced derivatives <span class="hlt">order</span> permeating the equations of motion, Dirac brackets and effective action.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017PhDT........77D','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017PhDT........77D"><span>Essays on variational <span class="hlt">approximation</span> techniques for stochastic optimization problems</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Deride Silva, Julio A.</p> <p></p> <p>This dissertation presents five essays on <span class="hlt">approximation</span> and modeling techniques, based on variational analysis, applied to stochastic optimization problems. It is divided into two parts, where the first is devoted to equilibrium problems and maxinf optimization, and the second corresponds to two essays in statistics and uncertainty modeling. Stochastic optimization lies at the core of this research as we were interested in relevant equilibrium applications that contain an uncertain component, and the design of a solution strategy. In addition, every stochastic optimization problem relies heavily on the underlying probability distribution that models the uncertainty. We studied these distributions, in particular, their design process and theoretical properties such as their convergence. Finally, the last aspect of stochastic optimization that we covered is the scenario creation problem, in which we described a procedure based on a probabilistic model to create scenarios for the applied problem of power estimation of renewable energies. In the first part, Equilibrium problems and maxinf optimization, we considered three Walrasian equilibrium problems: from economics, we studied a stochastic general equilibrium problem in a pure exchange economy, described in Chapter 3, and a stochastic general equilibrium with financial contracts, in Chapter 4; finally from engineering, we studied an infrastructure planning problem in Chapter 5. We stated these problems as belonging to the maxinf optimization class and, in each instance, we provided an <span class="hlt">approximation</span> scheme based on the notion of lopsided convergence and non-concave duality. This strategy is the foundation of the augmented Walrasian algorithm, whose convergence is guaranteed by lopsided convergence, that was implemented computationally, obtaining numerical results for relevant examples. The second part, Essays about statistics and uncertainty modeling, contains two essays covering a convergence problem for a sequence</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=5646186','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=5646186"><span>The checkpoint <span class="hlt">ordering</span> problem</span></a></p> <p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p> <p>Hungerländer, P.</p> <p>2017-01-01</p> <p>Abstract We suggest a new variant of a row layout problem: Find an <span class="hlt">ordering</span> of n departments with given lengths such that the total weighted sum of their distances to a given checkpoint is minimized. The Checkpoint <span class="hlt">Ordering</span> Problem (COP) is both of theoretical and practical interest. It has several applications and is conceptually related to some well-studied combinatorial optimization problems, namely the Single-Row Facility Layout Problem, the Linear <span class="hlt">Ordering</span> Problem and a variant of parallel machine scheduling. In this paper we study the complexity of the (COP) and its special cases. The general version of the (COP) with an arbitrary but fixed number of checkpoints is NP-hard in the weak sense. We propose both a dynamic programming algorithm and an integer linear programming approach for the (COP) . Our computational experiments indicate that the (COP) is hard to solve in practice. While the run time of the dynamic programming algorithm strongly depends on the length of the departments, the integer linear programming approach is able to solve instances with up to 25 departments to optimality. PMID:29170574</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4976340','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4976340"><span>Competing <span class="hlt">Orders</span> and Anomalies</span></a></p> <p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p> <p>Moon, Eun-Gook</p> <p>2016-01-01</p> <p>A conservation law is one of the most fundamental properties in nature, but a certain class of conservation “laws” could be spoiled by intrinsic quantum mechanical effects, so-called quantum anomalies. Profound properties of the anomalies have deepened our understanding in quantum many body systems. Here, we investigate quantum anomaly effects in quantum phase transitions between competing <span class="hlt">orders</span> and striking consequences of their presence. We explicitly calculate topological nature of anomalies of non-linear sigma models (NLSMs) with the Wess-Zumino-Witten (WZW) terms. The non-perturbative nature is directly related with the ’t Hooft anomaly matching condition: anomalies are conserved in renormalization group flow. By applying the matching condition, we show massless excitations are enforced by the anomalies in a whole phase diagram in sharp contrast to the case of the Landau-Ginzburg-Wilson theory which only has massive excitations in symmetric phases. Furthermore, we find non-perturbative criteria to characterize quantum phase transitions between competing <span class="hlt">orders</span>. For example, in 4D, we show the two competing <span class="hlt">order</span> parameter theories, CP(1) and the NLSM with WZW, describe different universality class. Physical realizations and experimental implication of the anomalies are also discussed. PMID:27499184</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/28368840','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/28368840"><span>Minimal-<span class="hlt">Approximation</span>-Based Distributed Consensus Tracking of a Class of Uncertain Nonlinear Multiagent Systems With Unknown Control Directions.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Choi, Yun Ho; Yoo, Sung Jin</p> <p>2017-03-28</p> <p>A minimal-<span class="hlt">approximation</span>-based distributed adaptive consensus tracking approach is presented for strict-feedback multiagent systems with unknown heterogeneous nonlinearities and control directions under a directed network. Existing <span class="hlt">approximation</span>-based consensus results for uncertain nonlinear multiagent systems in lower-triangular form have used multiple function <span class="hlt">approximators</span> in each local controller to <span class="hlt">approximate</span> unmatched nonlinearities of each follower. Thus, as the follower's <span class="hlt">order</span> increases, the number of the <span class="hlt">approximators</span> used in its local controller increases. However, the proposed approach employs only one function <span class="hlt">approximator</span> to construct the local controller of each follower regardless of the <span class="hlt">order</span> of the follower. The recursive design methodology using a new error transformation is derived for the proposed minimal-<span class="hlt">approximation</span>-based design. Furthermore, a bounding lemma on parameters of Nussbaum functions is presented to handle the unknown control direction problem in the minimal-<span class="hlt">approximation</span>-based distributed consensus tracking framework and the stability of the overall closed-loop system is rigorously analyzed in the Lyapunov sense.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016OcMod.107...21Z','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016OcMod.107...21Z"><span>A diffusion <span class="hlt">approximation</span> for ocean wave scatterings by randomly distributed ice floes</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Zhao, Xin; Shen, Hayley</p> <p>2016-11-01</p> <p>This study presents a continuum approach using a diffusion <span class="hlt">approximation</span> method to solve the scattering of ocean waves by randomly distributed ice floes. In <span class="hlt">order</span> to model both strong and weak scattering, the proposed method decomposes the wave action density function into two parts: the transmitted part and the scattered part. For a given wave direction, the transmitted part of the wave action density is defined as the part of wave action density in the same direction before the scattering; and the scattered part is a first <span class="hlt">order</span> Fourier series <span class="hlt">approximation</span> for the directional spreading caused by scattering. An additional <span class="hlt">approximation</span> is also adopted for simplification, in which the net directional redistribution of wave action by a single scatterer is assumed to be the reflected wave action of a normally incident wave into a semi-infinite ice cover. Other required input includes the mean shear modulus, diameter and thickness of ice floes, and the ice concentration. The directional spreading of wave energy from the diffusion <span class="hlt">approximation</span> is found to be in reasonable agreement with the previous solution using the Boltzmann equation. The diffusion model provides an alternative method to implement wave scattering into an operational wave model.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/26093569','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/26093569"><span>A consistent hierarchy of generalized kinetic equation <span class="hlt">approximations</span> to the master equation applied to surface catalysis.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Herschlag, Gregory J; Mitran, Sorin; Lin, Guang</p> <p>2015-06-21</p> <p>We develop a hierarchy of <span class="hlt">approximations</span> to the master equation for systems that exhibit translational invariance and finite-range spatial correlation. Each <span class="hlt">approximation</span> within the hierarchy is a set of ordinary differential equations that considers spatial correlations of varying lattice distance; the assumption is that the full system will have finite spatial correlations and thus the behavior of the models within the hierarchy will approach that of the full system. We provide evidence of this convergence in the context of one- and two-dimensional numerical examples. Lower levels within the hierarchy that consider shorter spatial correlations are shown to be up to three <span class="hlt">orders</span> of magnitude faster than traditional kinetic Monte Carlo methods (KMC) for one-dimensional systems, while predicting similar system dynamics and steady states as KMC methods. We then test the hierarchy on a two-dimensional model for the oxidation of CO on RuO2(110), showing that low-<span class="hlt">order</span> truncations of the hierarchy efficiently capture the essential system dynamics. By considering sequences of models in the hierarchy that account for longer spatial correlations, successive model predictions may be used to establish empirical <span class="hlt">approximation</span> of error estimates. The hierarchy may be thought of as a class of generalized phenomenological kinetic models since each element of the hierarchy <span class="hlt">approximates</span> the master equation and the lowest level in the hierarchy is identical to a simple existing phenomenological kinetic models.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/18516140','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/18516140"><span>Scattering from very rough layers under the geometric optics <span class="hlt">approximation</span>: further investigation.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Pinel, Nicolas; Bourlier, Christophe</p> <p>2008-06-01</p> <p>Scattering from very rough homogeneous layers is studied in the high-frequency limit (under the geometric optics <span class="hlt">approximation</span>) by taking the shadowing effect into account. To do so, the iterated Kirchhoff <span class="hlt">approximation</span>, recently developed by Pinel et al. [Waves Random Complex Media17, 283 (2007)] and reduced to the geometric optics <span class="hlt">approximation</span>, is used and investigated in more detail. The contributions from the higher <span class="hlt">orders</span> of scattering inside the rough layer are calculated under the iterated Kirchhoff <span class="hlt">approximation</span>. The method can be applied to rough layers of either very rough or perfectly flat lower interfaces, separating either lossless or lossy media. The results are compared with the PILE (propagation-inside-layer expansion) method, recently developed by Déchamps et al. [J. Opt. Soc. Am. A23, 359 (2006)], and accelerated by the forward-backward method with spectral acceleration. They highlight that there is very good agreement between the developed method and the reference numerical method for all scattering <span class="hlt">orders</span> and that the method can be applied to root-mean-square (RMS) heights at least down to 0.25lambda.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017PhRvD..95l3503P','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017PhRvD..95l3503P"><span>Validity of the Born <span class="hlt">approximation</span> for beyond Gaussian weak lensing observables</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Petri, Andrea; Haiman, Zoltán; May, Morgan</p> <p>2017-06-01</p> <p>Accurate forward modeling of weak lensing (WL) observables from cosmological parameters is necessary for upcoming galaxy surveys. Because WL probes structures in the nonlinear regime, analytical forward modeling is very challenging, if not impossible. Numerical simulations of WL features rely on ray tracing through the outputs of N -body simulations, which requires knowledge of the gravitational potential and accurate solvers for light ray trajectories. A less accurate procedure, based on the Born <span class="hlt">approximation</span>, only requires knowledge of the density field, and can be implemented more efficiently and at a lower computational cost. In this work, we use simulations to show that deviations of the Born-<span class="hlt">approximated</span> convergence power spectrum, skewness and kurtosis from their fully ray-traced counterparts are consistent with the smallest nontrivial O (Φ3) post-Born corrections (so-called geodesic and lens-lens terms). Our results imply a cancellation among the larger O (Φ4) (and higher <span class="hlt">order</span>) terms, consistent with previous analytic work. We also find that cosmological parameter bias induced by the Born-<span class="hlt">approximated</span> power spectrum is negligible even for a LSST-like survey, once galaxy shape noise is considered. When considering higher <span class="hlt">order</span> statistics such as the κ skewness and kurtosis, however, we find significant bias of up to 2.5 σ . Using the LensTools software suite, we show that the Born <span class="hlt">approximation</span> saves a factor of 4 in computing time with respect to the full ray tracing in reconstructing the convergence.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=20040030512&hterms=Kohn&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D20%26Ntt%3DKohn','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=20040030512&hterms=Kohn&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D20%26Ntt%3DKohn"><span><span class="hlt">Approximating</span> the Helium Wavefunction in Positronium-Helium Scattering</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>DiRienzi, Joseph; Drachman, Richard J.</p> <p>2003-01-01</p> <p>In the Kohn variational treatment of the positronium- hydrogen scattering problem the scattering wave function is <span class="hlt">approximated</span> by an expansion in some appropriate basis set, but the target and projectile wave functions are known exactly. In the positronium-helium case, however, a difficulty immediately arises in that the wave function of the helium target atom is not known exactly, and there are several ways to deal with the associated eigenvalue in formulating the variational scattering equations to be solved. In this work we will use the Kohn variational principle in the static exchange <span class="hlt">approximation</span> to d e t e e the zero-energy scattering length for the Ps-He system, using a suite of <span class="hlt">approximate</span> target functions. The results we obtain will be compared with each other and with corresponding values found by other <span class="hlt">approximation</span> techniques.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.loc.gov/pictures/collection/hh/item/ca2206.photos.182659p/','SCIGOV-HHH'); return false;" href="https://www.loc.gov/pictures/collection/hh/item/ca2206.photos.182659p/"><span>6. NORTH SIDE, FROM <span class="hlt">APPROXIMATELY</span> 25 FEET SOUTHEAST OF SOUTHWEST ...</span></a></p> <p><a target="_blank" href="http://www.loc.gov/pictures/collection/hh/">Library of Congress Historic Buildings Survey, Historic Engineering Record, Historic Landscapes Survey</a></p> <p></p> <p></p> <p>6. NORTH SIDE, FROM <span class="hlt">APPROXIMATELY</span> 25 FEET SOUTHEAST OF SOUTHWEST CORNER OF BUILDING 320, LOOKING SOUTH. - Oakland Naval Supply Center, Administration Building-Dental Annex-Dispensary, Between E & F Streets, East of Third Street, Oakland, Alameda County, CA</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=5837704','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=5837704"><span>Fundamentals and Recent Developments in <span class="hlt">Approximate</span> Bayesian Computation</span></a></p> <p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p> <p>Lintusaari, Jarno; Gutmann, Michael U.; Dutta, Ritabrata; Kaski, Samuel; Corander, Jukka</p> <p>2017-01-01</p> <p>Abstract Bayesian inference plays an important role in phylogenetics, evolutionary biology, and in many other branches of science. It provides a principled framework for dealing with uncertainty and quantifying how it changes in the light of new evidence. For many complex models and inference problems, however, only <span class="hlt">approximate</span> quantitative answers are obtainable. <span class="hlt">Approximate</span> Bayesian computation (ABC) refers to a family of algorithms for <span class="hlt">approximate</span> inference that makes a minimal set of assumptions by only requiring that sampling from a model is possible. We explain here the fundamentals of ABC, review the classical algorithms, and highlight recent developments. [ABC; <span class="hlt">approximate</span> Bayesian computation; Bayesian inference; likelihood-free inference; phylogenetics; simulator-based models; stochastic simulation models; tree-based models.] PMID:28175922</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2004OptCo.238..351G','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2004OptCo.238..351G"><span>General properties and analytical <span class="hlt">approximations</span> of photorefractive solitons</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Geisler, A.; Homann, F.; Schmidt, H.-J.</p> <p>2004-08-01</p> <p>We investigate general properties of spatial 1-dimensional bright photorefractive solitons and discuss various analytical <span class="hlt">approximations</span> for the soliton profile and the half width, both depending on an intensity parameter r. The case of dark solitons is also briefly addressed.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.loc.gov/pictures/collection/hh/item/ca1808.photos.180885p/','SCIGOV-HHH'); return false;" href="https://www.loc.gov/pictures/collection/hh/item/ca1808.photos.180885p/"><span>15. Looking north from east bank of ditch, <span class="hlt">approximately</span> halfway ...</span></a></p> <p><a target="_blank" href="http://www.loc.gov/pictures/collection/hh/">Library of Congress Historic Buildings Survey, Historic Engineering Record, Historic Landscapes Survey</a></p> <p></p> <p></p> <p>15. Looking north from east bank of ditch, <span class="hlt">approximately</span> halfway between cement pipe to north and burned irrigation pump station to south - Natomas Ditch System, Blue Ravine Segment, Juncture of Blue Ravine & Green Valley Roads, Folsom, Sacramento County, CA</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.loc.gov/pictures/collection/hh/item/co0042.photos.021480p/','SCIGOV-HHH'); return false;" href="https://www.loc.gov/pictures/collection/hh/item/co0042.photos.021480p/"><span>1. Rockwork <span class="hlt">approximately</span> 6 of a mile upstream from Keystone ...</span></a></p> <p><a target="_blank" href="http://www.loc.gov/pictures/collection/hh/">Library of Congress Historic Buildings Survey, Historic Engineering Record, Historic Landscapes Survey</a></p> <p></p> <p></p> <p>1. Rockwork <span class="hlt">approximately</span> 6 of a mile upstream from Keystone Bridge. View looking south from a distance of 50 feet. - Denver & Rio Grande Rockwork, East of South Platte, Waterton, Jefferson County, CO</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016PhRvB..94p5149T','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016PhRvB..94p5149T"><span><span class="hlt">Approximations</span> to the exact exchange potential: KLI versus semilocal</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Tran, Fabien; Blaha, Peter; Betzinger, Markus; Blügel, Stefan</p> <p>2016-10-01</p> <p>In the search for an accurate and computationally efficient <span class="hlt">approximation</span> to the exact exchange potential of Kohn-Sham density functional theory, we recently compared various semilocal exchange potentials to the exact one [F. Tran et al., Phys. Rev. B 91, 165121 (2015), 10.1103/PhysRevB.91.165121]. It was concluded that the Becke-Johnson (BJ) potential is a very good starting point, but requires the use of empirical parameters to obtain good agreement with the exact exchange potential. In this work, we extend the comparison by considering the Krieger-Li-Iafrate (KLI) <span class="hlt">approximation</span>, which is a beyond-semilocal <span class="hlt">approximation</span>. It is shown that overall the KLI- and BJ-based potentials are the most reliable <span class="hlt">approximations</span> to the exact exchange potential, however, sizable differences, especially for the antiferromagnetic transition-metal oxides, can be obtained.</p> </li> </ol> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li><a href="#" onclick='return showDiv("page_22");'>22</a></li> <li><a href="#" onclick='return showDiv("page_23");'>23</a></li> <li><a href="#" onclick='return showDiv("page_24");'>24</a></li> <li class="active"><span>25</span></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div><!-- col-sm-12 --> </div><!-- row --> </div><!-- page_25 --> <center> <div class="footer-extlink text-muted"><small>Some links on this page may take you to non-federal websites. 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