NASA Astrophysics Data System (ADS)
Bieg, Bohdan; Chrzanowski, Janusz; Kravtsov, Yury A.; Orsitto, Francesco
Basic principles and recent findings of quasi-isotropic approximation (QIA) of a geometrical optics method are presented in a compact manner. QIA was developed in 1969 to describe electromagnetic waves in weakly anisotropic media. QIA represents the wave field as a power series in two small parameters, one of which is a traditional geometrical optics parameter, equal to wavelength ratio to plasma characteristic scale, and the other one is the largest component of anisotropy tensor. As a result, "" QIA ideally suits to tokamak polarimetry/interferometry systems in submillimeter range, where plasma manifests properties of weakly anisotropic medium.
NASA Astrophysics Data System (ADS)
Kravtsov, Yu. A.; Bieg, B.; Bliokh, K. Yu.; Hirsch, M.
2008-03-01
Three different theoretical approaches are presented: quasi-isotropic approximation (QIA), Stokes vector formalism and complex polarization angle method, which allow describing polarization of electromagnetic waves in weakly anisotropic plasma. QIA stems directly from the Maxwell equations under assumption of weak anisotropy and has a form of coupled differential equations for the transverse components of the electromagnetic wave field. Being applied to high frequency (microwave or IR) electromagnetic waves in magnetized plasma, QIA describes combined action of Faraday and Cotton-Mouton phenomena. QIA takes into account curvature and torsion of the ray, describes normal modes conversion in the inhomogeneous plasma and allows specifying area of applicability of the method. In distinction to QIA, Stokes vector formalism (SVF) deals with quantities, quadratic in a wave field. It is shown (and this is the main result of the paper) that equation for Stokes vector evolution can be derived directly from QIA. This evidences deep unity of two seemingly different approaches. In fact QIA suggests somewhat more information than SVF; in particular, it describes the phases of both transverse components of the electromagnetic field, whereas SVF operates only with the phase difference. At last, the coupled equations of the quasi-isotropic approximation can be reduced to a single equation for complex polarization angle (CPA), which describes both the shape and orientation of the polarization ellipse. In turn, equation for CPA allows obtaining equations for traditional parameters of polarization ellipse, which in fact are equivalent to the equation for Stokes vector evolution. It is pointed out that every method under discussion has its own advantages plasma polarimetry.
Tensile behavior of a quasi-isotropic carbon-carbon composite
Namiki, Fumiharu; Chou, T.W.
1998-01-01
This paper reports an analytical and experimental investigation of the tensile behavior of a two-dimensional woven carbon-carbon composite with a quasi-isotropic stacking sequence. Specimens in their as-received condition were already saturated with transverse cracks due to thermal stress induced during fabrication. Further cracking under load was not detected. The completely unloaded specimen showed that the slight residual strain tended to increase with applied tensile strain. Damage progression as seen on specimen edges and elastic moduli degradation were recorded. The predicted Young`s moduli were in good agreement with the measured data.
Thermo-oxidative degradation assessment in quasi-isotropic carbon fiber/epoxy composites
NASA Astrophysics Data System (ADS)
Daily, Connor; Barnard, Dan J.; Jones, Roger W.; McClelland, John F.; Bowler, Nicola
2015-03-01
Components made from polymer matrix composites (PMCs) are finding increasing use in armored vehicles for the purpose of weight savings and fuel efficiency. Often times, these PMC components are installed next to engines, or in other high-temperature environments within the vehicle. The present work investigates the change in surface chemistry and its correlation with changes in the interlaminar shear strength (ILSS) due to accelerated thermo-oxidative aging of a quasi-isotropic carbon fiber reinforced epoxy laminate. Samples are aged isothermally at various temperatures whose selection is guided by degradation steps revealed by thermo-gravimetric analysis. Fourier transform infrared (FTIR) photoacoustic spectroscopy is utilized to identify the chemical changes due to aging, and compression-test results reveal a non-linear decrease in ILSS with increasing aging temperature. A correlation between the FTIR and ILSS data sets suggests that nondestructive FTIR techniques may be used for assessing ILSS of PMCs.
High-Q/V Monolithic Diamond Microdisks Fabricated with Quasi-isotropic Etching.
Khanaliloo, Behzad; Mitchell, Matthew; Hryciw, Aaron C; Barclay, Paul E
2015-08-12
Optical microcavities enhance light-matter interactions and are essential for many experiments in solid state quantum optics, optomechanics, and nonlinear optics. Single crystal diamond microcavities are particularly sought after for applications involving diamond quantum emitters, such as nitrogen vacancy centers, and for experiments that benefit from diamond's excellent optical and mechanical properties. Light-matter coupling rates in experiments involving microcavities typically scale with Q/V, where Q and V are the microcavity quality-factor and mode-volume, respectively. Here we demonstrate that microdisk whispering gallery mode cavities with high Q/V can be fabricated directly from bulk single crystal diamond. By using a quasi-isotropic oxygen plasma to etch along diamond crystal planes and undercut passivated diamond structures, we create monolithic diamond microdisks. Fiber taper based measurements show that these devices support TE- and TM-like optical modes with Q > 1.1 × 10(5) and V < 11(λ/n) (3) at a wavelength of 1.5 μm. PMID:26134379
Leckey, Cara A C; Rogge, Matthew D; Raymond Parker, F
2014-01-01
Three-dimensional (3D) elastic wave simulations can be used to investigate and optimize nondestructive evaluation (NDE) and structural health monitoring (SHM) ultrasonic damage detection techniques for aerospace materials. 3D anisotropic elastodynamic finite integration technique (EFIT) has been implemented for ultrasonic waves in carbon fiber reinforced polymer (CFRP) composite laminates. This paper describes 3D EFIT simulations of guided wave propagation in undamaged and damaged anisotropic and quasi-isotropic composite plates. Comparisons are made between simulations of guided waves in undamaged anisotropic composite plates and both experimental laser Doppler vibrometer (LDV) wavefield data and dispersion curves. Time domain and wavenumber domain comparisons are described. Wave interaction with complex geometry delamination damage is then simulated to investigate how simulation tools incorporating realistic damage geometries can aid in the understanding of wave interaction with CFRP damage. In order to move beyond simplistic assumptions of damage geometry, volumetric delamination data acquired via X-ray microfocus computed tomography is directly incorporated into the simulation. Simulated guided wave interaction with the complex geometry delamination is compared to experimental LDV time domain data and 3D wave interaction with the volumetric damage is discussed.
Construction of a polarization insensitive lens from a quasi-isotropic metamaterial slab
NASA Astrophysics Data System (ADS)
Luo, Hailu; Ren, Zhongzhou; Shu, Weixing; Li, Fei
2007-02-01
We propose to employ the quasi-isotropic metamaterial (QIMM) slab to construct a polarization insensitive lens, in which both E - and H -polarized waves exhibit the same refocusing effect. For shallow incident angles, the QIMM slab will provide some degree of refocusing in the same manner as an isotropic negative index material. The refocusing effect allows us to introduce the ideas of paraxial beam focusing and phase compensation by the QIMM slab. On the basis of angular spectrum representation, a formalism describing paraxial beams propagating through a QIMM slab is presented. Because of the negative phase velocity in the QIMM slab, the inverse Gouy phase shift and the negative Rayleigh length of paraxial Gaussian beam are proposed. We find that the phase difference caused by the Gouy phase shift in vacuum can be compensated by that caused by the inverse Gouy phase shift in the QIMM slab. If certain matching conditions are satisfied, the intensity and phase distributions at object plane can be completely reconstructed at image plane. Our simulation results show that the superlensing effect with subwavelength image resolution could be achieved in the form of a QIMM slab.
Experimental data on single-bolt joints in quasi isotropic graphite/polyimide laminates
NASA Technical Reports Server (NTRS)
Wichorek, G. R.
1982-01-01
Sixteen ply, quasi-isotropic laminates of Celanese Celion 6000/PMR-15 and Celion 6000/LARC-160 with a fiber orientation of (0/45/90/-45) sub 2S were evaluated. Tensile and open hole specimens were tested at room temperature to establish laminate tensile strength and net tensile strength at an unloaded bolt hole. Double lap joint specimens with a single 4.83-mm (0.19 in.) diameter bolt torqued to 1.7 N-m (15 lbf-in.) were tested in tension at temperatures of 116 K (-250F), 297 K (75F), and 589 K (600F). The joint ratios of w/d (specimen width to hole diameter) and e/d (edge distance to hole diameter) were varied from 4 to 6 and from 2 to 4, respectively. The effect of joint geometry and temperature on failure mode and joint stresses are shown. Joint stresses calculated at maximum load for each joint geometry and test temperature are reported. Joint strength in net tension, bearing, and shear out at 116 K (-250F), 297 K (75F), and 589 K (600F) are given for the Celion 6000/PMR-15 and Celion 6000/LARC-160 laminates.
Durability-Based Design Criteria for a Quasi-Isotropic Carbon-Fiber Automotive Composite
Corum, J.M.
2002-04-17
This report provides recommended durability-based design properties and criteria for a quasi-isotropic carbon-fiber composite for possible automotive structural applications. The composite, which was made by a rapid molding process suitable for high-volume automotive applications, consisted of continuous Thornel T300 fibers (6K tow) in a Baydur 420 IMR urethane matrix. The reinforcement was in the form of four {+-}45{sup o} stitch-bonded mats in the following layup: [0/90{sup o}/{+-}45{sup o}]{sub S}. This material is the second in a progression of three candidate thermoset composites to be characterized and modeled as part of an Oak Ridge National Laboratory project entitled Durability of Carbon-Fiber Composites. The overall goal of the project, which is sponsored by the U.S. Department of Energy's Office of Advanced Automotive Technologies and is closely coordinated with the industry Automotive Composites Consortium, is to develop durability-driven design data and criteria to assure the long-term integrity of carbon-fiber-based composite systems for large automotive structural components. This document is in two parts. Part I provides the design criteria, and Part 2 provides the underlying experimental data and models. The durability issues addressed include the effects on deformation, strength, and stiffness of cyclic and sustained loads, operating temperature, automotive fluid environments, and low-energy impacts (e.g., tool drops and kickups of roadway debris). Guidance is provided for design analysis, time-dependent allowable stresses, rules for cyclic loadings, and damage tolerance design guidance, including the effects of holes. Chapter 6 provides a brief summary of the design criteria.
NASA Astrophysics Data System (ADS)
Zeng, Chunmei; Yu, Xia; Guo, Peiji
2014-08-01
A regularization stiffness coefficient method was verified further to optimize lay-up sequences of quasi-isotropic laminates for carbon fiber reinforced polymer (CFRP) composite mirrors. Firstly, the deformation due to gravity of 1G and temperature difference of 20-100°C and the modal were analyzed by finite element method (FEM). Secondly, the influence of angle error of ply stacking on quasi-isotropic of bending stiffness was evaluated. Finally, an active support system of 49 actuators in circular arrangement is designed for a 500mm CFRP mirror, and its goal is to deform the spherical CFRP mirror to a parabolic. Therefore, the response functions of the actuators were gotten, and the surface form errors and stresses were calculated and analyzed. The results show that the CFRP mirrors designed by the method have a better symmetrical bending deformation under gravity and thermal load and a higher fundamental frequency, and the larger n the better symmetry (for π/n quasi-isotropic laminates); the method reduces the sensitivity to misalignment of ply orientation for symmetric bending, and the mirror's maximum von Mises stress and maximum shear stress are less compared to those laminates not optimized in lay-up sequence.
NASA Astrophysics Data System (ADS)
Mall, S.; Coleman, J. M.
1998-12-01
This study investigated the effects of embedding piezoelectric lead zirconate-titanate (PZT) sensors on the tensile strength and fatigue behavior of a quasi-isotropic graphite/epoxy laminate as well as the embedded sensor's voltage degradation under these loading conditions. For this, AS4/3501-6 laminates were fabricated with a 0964-1726/7/6/010/img1 lay-up where PZT was inserted into a cut-out area in the two middle 0964-1726/7/6/010/img2 plies. Monotonic tensile tests showed that both the average ultimate strength and Young's modulus of the tested laminate with or without PZT were within 4% of each other. The fatigue lives with and without PZT were very close to each other as well. Overall, the sequence of damage in this study agreed with previous investigations of the damage mechanisms for 0964-1726/7/6/010/img1 quasi-isotropic laminates. The ranges of modulus reduction in both cases, with and without PZT, were within 5 to 15% of each other during fatigue loading. Delamination growths in both cases during most of the fatigue life were also very comparable to each other. Further, this study showed that the embedded PZT would maintain a steady voltage output indefinitely when mechanically cycled within its operational strain limit. It thus appears that the embedment of PZTs in a cut-out area of 0964-1726/7/6/010/img2 plies of quasi-isotropic graphite/epoxy laminates would not affect their monotonic tensile and fatigue behavior.
NASA Technical Reports Server (NTRS)
Kelkar, A. D.
1984-01-01
In thin composite laminates, the first level of visible damage occurs in the back face and is called back face spalling. A plate-membrane coupling model, and a finite element model to analyze the large deformation behavior of eight-ply quasi-isotropic circular composite plates under impact type point loads are developed. The back face spalling phenomenon in thin composite plates is explained by using the plate-membrane coupling model and the finite element model in conjunction with the fracture mechanics principles. The experimental results verifying these models are presented. Several conclusions concerning the deformation behavior are reached and discussed in detail.
NASA Technical Reports Server (NTRS)
Sohi, M. M.; Hahn, H. T.; Williams, J. G.
1986-01-01
Compressive failure mechanisms in quasi-isotropic graphite/epoxy laminates were characterized for both unnotched and notched specimens and also following damage by impact. Two types of fibers (Thornel 300 and 700) and four resin systems (Narmco 5208, American Cyanamid BP907, and Union Carbide 4901/MDA and 4901/mPDA) were studied. For all material combinations, failure of unnotched specimens was initiated by kinking of fibers in the 0-degree plies. A major difference was observed, however, in the mode of failure propagation after the 0-degree ply failure. The strength of quasi-isotropic laminates in general increased with increasing resin tensile modulus. The laminates made with Thornel 700 fibers exhibited slightly lower compressive strengths than did the laminates made with Thornel 300 fibers. The notch sensitivity as measured by the hole strength was lowest for the BP907 resin and highest for the 5208 resin. For the materials studied, however, the type of fiber had no effect on the notch sensitivity.
Thermomechanical fatigue behavior of a quasi-isotropic SCS-6/Ti-15-3 metal matrix composite
Hart, K.A.; Mall, S. . Dept. of Aeronautics and Astronautics)
1995-01-01
As the speed of new aerospace vehicles pushes the supersonic and hypersonic envelopes, aerodynamic heating and structural strength and weight are becoming even greater design factors. Here, the response of a quasi-isotropic laminate of metal matrix composite, SCS-6/Ti-15-3 in a thermomechanical fatigue (TMF) environment was investigated. To achieve this, three sets of fatigue tests were conducted: (1) in-phase TMF (IP-TMF), (2) out-of-phase TMF (OP-TMF), and (3) isothermal fatigue (IF). The fatigue response was dependent on the test condition and the maximum stress level during cycling. The IF, IP-TMF, and OP-TMF conditions yielded shortest fatigue life at higher, intermediate and lower stress levels, respectively. Examination of the failure mode through the variation of strain or modulus during cycling, and post-mortem microscopic evaluation revealed that it was dependent on the fatigue condition and applied stress level. Higher stresses, mostly with IP-TMF and IF conditions, produced a primarily fiber dominated failure. Lower stresses, mostly with the OP-TMF condition, produced a matrix dominated failure. Also, an empirical model based on the observed damage mechanisms was developed to represent the fatigue lives for the three conditions examined here.
NASA Technical Reports Server (NTRS)
Illg, W.
1986-01-01
A partial-bonding interlaminar toughening concept was evaluated for resistance to impact and for behavior of a loaded hole. Perforated Mylar sheets were interleaved between all 24 plies of a graphite/epoxy quasi-isotropic lay-up. Specimens were impacted by aluminum spheres while under tensile or compressive loads. Impact-failure thresholds and residual strengths were obtained. Loaded-hole specimens were tested in three configurations that were critical in bearing, shear, or tension. Partial bonding reduced the tensile and compressive strengths of undamaged specimens by about one-third. For impact, partial bonding did not change the threshold for impact failure under tensile preload. However, under compressive preload, partial bonding caused serious degradation of impact resistance. Partial bonding reduced the maximum load-carrying capacity of all three types of loaded-hole specimens. Overall, partial bonding degraded both impact resistance and bearing strength of holes.
Hergt, Steven; Schaefer, Gerhard
2008-05-15
The Kerr metric outside the ergosphere is transformed into Arnowitt-Deser-Misner coordinates up to the orders 1/r{sup 4} and a{sup 2}, respectively, in radial coordinate r and reduced angular momentum variable a, starting from the Kerr solution in quasi-isotropic as well as harmonic coordinates. The distributional source terms for the approximate solution are calculated. To leading order in linear momenta, higher-order-in-spin interaction Hamiltonians for black hole binaries are derived.
An approximate solution for interlaminar stresses in laminated composites: Applied mechanics program
NASA Technical Reports Server (NTRS)
Rose, Cheryl A.; Herakovich, Carl T.
1992-01-01
An approximate solution for interlaminar stresses in finite width, laminated composites subjected to uniform extensional, and bending loads is presented. The solution is based upon the principle of minimum complementary energy and an assumed, statically admissible stress state, derived by considering local material mismatch effects and global equilibrium requirements. The stresses in each layer are approximated by polynomial functions of the thickness coordinate, multiplied by combinations of exponential functions of the in-plane coordinate, expressed in terms of fourteen unknown decay parameters. Imposing the stationary condition of the laminate complementary energy with respect to the unknown variables yields a system of fourteen non-linear algebraic equations for the parameters. Newton's method is implemented to solve this system. Once the parameters are known, the stresses can be easily determined at any point in the laminate. Results are presented for through-thickness and interlaminar stress distributions for angle-ply, cross-ply (symmetric and unsymmetric laminates), and quasi-isotropic laminates subjected to uniform extension and bending. It is shown that the solution compares well with existing finite element solutions and represents an improved approximate solution for interlaminar stresses, primarily at interfaces where global equilibrium is satisfied by the in-plane stresses, but large local mismatch in properties requires the presence of interlaminar stresses.
Delamination growth analysis in quasi-isotropic laminates under loads simulating low-velocity impact
NASA Technical Reports Server (NTRS)
Shivakumar, K. N.; Elber, W.
1984-01-01
A geometrically nonlinear finite-element analysis was developed to calculate the strain energy released by delamination plates during impact loading. Only the first mode of deformation, which is equivalent to static deflection, was treated. Both the impact loading and delamination in the plate were assumed to be axisymmetric. The strain energy release rate in peeling, G sub I, and shear sliding, G sub II, modes were calculated using the fracture mechanics crack closure technique. Energy release rates for various delamination sizes and locations and for various plate configurations and materials were compared. The analysis indicated that shear sliding (G sub II) was the primary mode of delamination growth. The analysis also indicated that the midplane (maximum transverse shear stress plane) delamination was more critical and would grow before any other delamination of the same size near the midplane region. The delamination growth rate was higher (neutrally stable) for a low toughness (brittle) matrix and slower (stable) for high toughness matrix. The energy release rate in the peeling mode, G sub I, for a near-surface delamination can be as high as 0.5G sub II and can contribute significantly to the delamination growth.
The minimum energy decay rate in quasi-isotropic grid turbulence
NASA Astrophysics Data System (ADS)
Davidson, P. A.
2011-08-01
We consider high Reynolds number, freely-decaying, isotropic turbulence in which the large scales evolve in a self-similar manner when normalized by the integral scales, u and ℓ. As it is well known, a range of possible behaviors may be observed depending on the form of the longitudinal velocity correlation at large separation, uf∞=u 2f(r →∞). We consider the cases u2f∞=cmr-m,2≤m ≤6, whose spectral counterpart is E(k →0)~cmkm -1 for m <6, with or without a lnk correction, and E(k →0)~I k4 for m =6. (I is Loitsyansky's integral.) It has long been known that the cmm=constant during the decay. This, in turn, sets the energy decay rate as u2~t-(1-p)2m /(m+2), where p is the power-law exponent for the normalized dissipation rate, εℓ/εℓu3u3~t-p, observed empirically to be a small positive number in grid turbulence. We systematically explore the properties of these different classes of turbulence and arrive at the following conclusions. (i) The invariance of cm is a direct consequence of linear momentum conservation for m ≤4, and angular momentum conservation for m =5. (ii) The classical spectra of Saffman, E(k →0)~c3k2, and Batchelor, E(k →0)~Ik4, are robust in the sense that they emerge from a broad class of initial conditions. In particular, it is necessary only that <ωi ω'j >∞ ≤O(r-8) at t =0. The non-classical spectra (m =2,4,5), on the other hand, require very specific initial conditions in order to be realized, of the form <ωiω'j>∞=O(r-(m +2)). (Note the equality rather than the inequality here.) This makes the non-classical spectra less likely to be observed in practice. (iii) The case of m =2, which is usually associated with the u2~t-1 decay law, is pathological in a number of respects. For example, its spectral tensor diverges as k →0, and the long-range correlations
Elastic properties and fracture strength of quasi-isotropic graphite/epoxy composites
NASA Technical Reports Server (NTRS)
Sullivan, T. L.
1977-01-01
The layups of the studied laminates are (0, + or - 60) sub s, (0, + or - 45, 90) sub s, (0, + or - 30, + or - 60, 90) sub s (0, + or - 22 1/2, + or - 45, + or - 67 1/2, 90) sub s. The properties determined were tensile modulus, Poisson's ratio, bending stiffness, fracture strength and fracture strain. Measured properties and properties predicted using laminate theory were found to be in reasonable agreement. Reasons for data scatter were determined.
Hsu, T.L.
1998-09-01
This study primarily investigated the electro-mechanical fatigue behavior of the embedded piezoelectric actuators in graphite/epoxy laminate with a lay-up of 0/ {+-} 45 / 90s. A secondary focus was the investigation of the mechanical fatigue effects of the 0 / 0 / {+-} 45 / 0 / 0 / 90s laminate with embedded PZT under tensile loading. All the fatigue tests were conducted with a triangular loading waveform which had a frequency of 10 Hz and with R = 0.1. In the electro-mechanical testing, the embedded actuator was excited by a {minus}10 V to {minus}100 V or a 10 V to 100 V voltage input, which resulted in either in-phase or out-of-phase electrically induced strain waveform with respect to the mechanical loading or strain. It was found that the embedded PZTs performed very well during the out-of-phase electro-mechanical and low stress fatigue conditions when the applied strain was within the operating range of PZT. Beyond the upper strain limit, the voltage output of the PZT was primarily influenced by the mechanical fatigue loading. Results from the high stress fatigue tests showed that the embedded piezoelectric actuators did not have significant effect on the tensile strength of the laminates.
Rasin, A.
1994-04-01
We discuss the idea of approximate flavor symmetries. Relations between approximate flavor symmetries and natural flavor conservation and democracy models is explored. Implications for neutrino physics are also discussed.
NASA Astrophysics Data System (ADS)
Niiniluoto, Ilkka
2014-03-01
Approximation of laws is an important theme in the philosophy of science. If we can make sense of the idea that two scientific laws are "close" to each other, then we can also analyze such methodological notions as approximate explanation of laws, approximate reduction of theories, approximate empirical success of theories, and approximate truth of laws. Proposals for measuring the distance between quantitative scientific laws were given in Niiniluoto (1982, 1987). In this paper, these definitions are reconsidered as a response to the interesting critical remarks by Liu (1999).
Sparse pseudospectral approximation method
NASA Astrophysics Data System (ADS)
Constantine, Paul G.; Eldred, Michael S.; Phipps, Eric T.
2012-07-01
Multivariate global polynomial approximations - such as polynomial chaos or stochastic collocation methods - are now in widespread use for sensitivity analysis and uncertainty quantification. The pseudospectral variety of these methods uses a numerical integration rule to approximate the Fourier-type coefficients of a truncated expansion in orthogonal polynomials. For problems in more than two or three dimensions, a sparse grid numerical integration rule offers accuracy with a smaller node set compared to tensor product approximation. However, when using a sparse rule to approximately integrate these coefficients, one often finds unacceptable errors in the coefficients associated with higher degree polynomials. By reexamining Smolyak's algorithm and exploiting the connections between interpolation and projection in tensor product spaces, we construct a sparse pseudospectral approximation method that accurately reproduces the coefficients of basis functions that naturally correspond to the sparse grid integration rule. The compelling numerical results show that this is the proper way to use sparse grid integration rules for pseudospectral approximation.
Approximations for photoelectron scattering
NASA Astrophysics Data System (ADS)
Fritzsche, V.
1989-04-01
The errors of several approximations in the theoretical approach of photoelectron scattering are systematically studied, in tungsten, for electron energies ranging from 10 to 1000 eV. The large inaccuracies of the plane-wave approximation (PWA) are substantially reduced by means of effective scattering amplitudes in the modified small-scattering-centre approximation (MSSCA). The reduced angular momentum expansion (RAME) is so accurate that it allows reliable calculations of multiple-scattering contributions for all the energies considered.
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.
Approximate kernel competitive learning.
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.
Approximate kernel competitive learning.
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. PMID:25528318
ERIC Educational Resources Information Center
Wolff, Hans
This paper deals with a stochastic process for the approximation of the root of a regression equation. This process was first suggested by Robbins and Monro. The main result here is a necessary and sufficient condition on the iteration coefficients for convergence of the process (convergence with probability one and convergence in the quadratic…
NASA Astrophysics Data System (ADS)
Huang, Siendong
2009-11-01
The nonlocality of quantum states on a bipartite system \\mathcal {A+B} is tested by comparing probabilistic outcomes of two local observables of different subsystems. For a fixed observable A of the subsystem \\mathcal {A,} its optimal approximate double A' of the other system \\mathcal {B} is defined such that the probabilistic outcomes of A' are almost similar to those of the fixed observable A. The case of σ-finite standard von Neumann algebras is considered and the optimal approximate double A' of an observable A is explicitly determined. The connection between optimal approximate doubles and quantum correlations is explained. Inspired by quantum states with perfect correlation, like Einstein-Podolsky-Rosen states and Bohm states, the nonlocality power of an observable A for general quantum states is defined as the similarity that the outcomes of A look like the properties of the subsystem \\mathcal {B} corresponding to A'. As an application of optimal approximate doubles, maximal Bell correlation of a pure entangled state on \\mathcal {B}(\\mathbb {C}^{2})\\otimes \\mathcal {B}(\\mathbb {C}^{2}) is found explicitly.
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…
Naus, Dan J; Corum, James; Klett, Lynn B; Davenport, Mike; Battiste, Rick; Simpson, Jr., William A
2006-04-01
This report provides recommended durability-based design properties and criteria for a quais-isotropic carbon-fiber thermoplastic composite for possible automotive structural applications. The composite consisted of a PolyPhenylene Sulfide (PPS) thermoplastic matrix (Fortron's PPS - Ticona 0214B1 powder) reinforced with 16 plies of carbon-fiber unidirectional tape, [0?/90?/+45?/-45?]2S. The carbon fiber was Hexcel AS-4C and was present in a fiber volume of 53% (60%, by weight). The overall goal of the project, which is sponsored by the U.S. Department of Energy's Office of Freedom Car and Vehicle Technologies and is closely coordinated with the Advanced Composites Consortium, is to develop durability-driven design data and criteria to assure the long-term integrity of carbon-fiber-based composite systems for automotive structural applications. This document is in two parts. Part 1 provides design data and correlations, while Part 2 provides the underlying experimental data and models. The durability issues addressed include the effects of short-time, cyclic, and sustained loadings; temperature; fluid environments; and low-energy impacts (e.g., tool drops and kickups of roadway debris) on deformation, strength, and stiffness. Guidance for design analysis, time-independent and time-dependent allowable stresses, rules for cyclic loadings, and damage-tolerance design guidance are provided.
Optimizing the Zeldovich approximation
NASA Technical Reports Server (NTRS)
Melott, Adrian L.; Pellman, Todd F.; Shandarin, Sergei F.
1994-01-01
We have recently learned that the Zeldovich approximation can be successfully used for a far wider range of gravitational instability scenarios than formerly proposed; we study here how to extend this range. In previous work (Coles, Melott and Shandarin 1993, hereafter CMS) we studied the accuracy of several analytic approximations to gravitational clustering in the mildly nonlinear regime. We found that what we called the 'truncated Zeldovich approximation' (TZA) was better than any other (except in one case the ordinary Zeldovich approximation) over a wide range from linear to mildly nonlinear (sigma approximately 3) regimes. TZA was specified by setting Fourier amplitudes equal to zero for all wavenumbers greater than k(sub nl), where k(sub nl) marks the transition to the nonlinear regime. Here, we study the cross correlation of generalized TZA with a group of n-body simulations for three shapes of window function: sharp k-truncation (as in CMS), a tophat in coordinate space, or a Gaussian. We also study the variation in the crosscorrelation as a function of initial truncation scale within each type. We find that k-truncation, which was so much better than other things tried in CMS, is the worst of these three window shapes. We find that a Gaussian window e(exp(-k(exp 2)/2k(exp 2, sub G))) applied to the initial Fourier amplitudes is the best choice. It produces a greatly improved crosscorrelation in those cases which most needed improvement, e.g. those with more small-scale power in the initial conditions. The optimum choice of kG for the Gaussian window is (a somewhat spectrum-dependent) 1 to 1.5 times k(sub nl). Although all three windows produce similar power spectra and density distribution functions after application of the Zeldovich approximation, the agreement of the phases of the Fourier components with the n-body simulation is better for the Gaussian window. We therefore ascribe the success of the best-choice Gaussian window to its superior treatment
Chalasani, P.; Saias, I.; Jha, S.
1996-04-08
As increasingly large volumes of sophisticated options (called derivative securities) are traded in world financial markets, determining a fair price for these options has become an important and difficult computational problem. Many valuation codes use the binomial pricing model, in which the stock price is driven by a random walk. In this model, the value of an n-period option on a stock is the expected time-discounted value of the future cash flow on an n-period stock price path. Path-dependent options are particularly difficult to value since the future cash flow depends on the entire stock price path rather than on just the final stock price. Currently such options are approximately priced by Monte carlo methods with error bounds that hold only with high probability and which are reduced by increasing the number of simulation runs. In this paper the authors show that pricing an arbitrary path-dependent option is {number_sign}-P hard. They show that certain types f path-dependent options can be valued exactly in polynomial time. Asian options are path-dependent options that are particularly hard to price, and for these they design deterministic polynomial-time approximate algorithms. They show that the value of a perpetual American put option (which can be computed in constant time) is in many cases a good approximation to the value of an otherwise identical n-period American put option. In contrast to Monte Carlo methods, the algorithms have guaranteed error bounds that are polynormally small (and in some cases exponentially small) in the maturity n. For the error analysis they derive large-deviation results for random walks that may be of independent interest.
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.
Hierarchical Approximate Bayesian Computation
Turner, Brandon M.; Van Zandt, Trisha
2013-01-01
Approximate Bayesian computation (ABC) is a powerful technique for estimating the posterior distribution of a model’s parameters. It is especially important when the model to be fit has no explicit likelihood function, which happens for computational (or simulation-based) models such as those that are popular in cognitive neuroscience and other areas in psychology. However, ABC is usually applied only to models with few parameters. Extending ABC to hierarchical models has been difficult because high-dimensional hierarchical models add computational complexity that conventional ABC cannot accommodate. In this paper we summarize some current approaches for performing hierarchical ABC and introduce a new algorithm called Gibbs ABC. This new algorithm incorporates well-known Bayesian techniques to improve the accuracy and efficiency of the ABC approach for estimation of hierarchical models. We then use the Gibbs ABC algorithm to estimate the parameters of two models of signal detection, one with and one without a tractable likelihood function. PMID:24297436
Approximate Bayesian multibody tracking.
Lanz, Oswald
2006-09-01
Visual tracking of multiple targets is a challenging problem, especially when efficiency is an issue. Occlusions, if not properly handled, are a major source of failure. Solutions supporting principled occlusion reasoning have been proposed but are yet unpractical for online applications. This paper presents a new solution which effectively manages the trade-off between reliable modeling and computational efficiency. The Hybrid Joint-Separable (HJS) filter is derived from a joint Bayesian formulation of the problem, and shown to be efficient while optimal in terms of compact belief representation. Computational efficiency is achieved by employing a Markov random field approximation to joint dynamics and an incremental algorithm for posterior update with an appearance likelihood that implements a physically-based model of the occlusion process. A particle filter implementation is proposed which achieves accurate tracking during partial occlusions, while in cases of complete occlusion, tracking hypotheses are bound to estimated occlusion volumes. Experiments show that the proposed algorithm is efficient, robust, and able to resolve long-term occlusions between targets with identical appearance. PMID:16929730
NASA Astrophysics Data System (ADS)
Lubkin, Elihu
2002-04-01
In 1993,(E. & T. Lubkin, Int.J.Theor.Phys. 32), 993 (1993) we gave exact mean trace
Approximation by hinge functions
Faber, V.
1997-05-01
Breiman has defined {open_quotes}hinge functions{close_quotes} for use as basis functions in least squares approximations to data. A hinge function is the max (or min) function of two linear functions. In this paper, the author assumes the existence of smooth function f(x) and a set of samples of the form (x, f(x)) drawn from a probability distribution {rho}(x). The author hopes to find the best fitting hinge function h(x) in the least squares sense. There are two problems with this plan. First, Breiman has suggested an algorithm to perform this fit. The author shows that this algorithm is not robust and also shows how to create examples on which the algorithm diverges. Second, if the author tries to use the data to minimize the fit in the usual discrete least squares sense, the functional that must be minimized is continuous in the variables, but has a derivative which jumps at the data. This paper takes a different approach. This approach is an example of a method that the author has developed called {open_quotes}Monte Carlo Regression{close_quotes}. (A paper on the general theory is in preparation.) The author shall show that since the function f is continuous, the analytic form of the least squares equation is continuously differentiable. A local minimum is solved for by using Newton`s method, where the entries of the Hessian are estimated directly from the data by Monte Carlo. The algorithm has the desirable properties that it is quadratically convergent from any starting guess sufficiently close to a solution and that each iteration requires only a linear system solve.
Phenomenological applications of rational approximants
NASA Astrophysics Data System (ADS)
Gonzàlez-Solís, Sergi; Masjuan, Pere
2016-08-01
We illustrate the powerfulness of Padé approximants (PAs) as a summation method and explore one of their extensions, the so-called quadratic approximant (QAs), to access both space- and (low-energy) time-like (TL) regions. As an introductory and pedagogical exercise, the function 1 zln(1 + z) is approximated by both kind of approximants. Then, PAs are applied to predict pseudoscalar meson Dalitz decays and to extract Vub from the semileptonic B → πℓνℓ decays. Finally, the π vector form factor in the TL region is explored using QAs.
Approximating Functions with Exponential Functions
ERIC Educational Resources Information Center
Gordon, Sheldon P.
2005-01-01
The possibility of approximating a function with a linear combination of exponential functions of the form e[superscript x], e[superscript 2x], ... is considered as a parallel development to the notion of Taylor polynomials which approximate a function with a linear combination of power function terms. The sinusoidal functions sin "x" and cos "x"…
Approximate circuits for increased reliability
Hamlet, Jason R.; Mayo, Jackson R.
2015-12-22
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 reference 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.
Approximate circuits for increased reliability
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 reference 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.
Mathematical algorithms for approximate reasoning
NASA Technical Reports Server (NTRS)
Murphy, John H.; Chay, Seung C.; Downs, Mary M.
1988-01-01
Most state of the art expert system environments contain a single and often ad hoc strategy for approximate reasoning. Some environments provide facilities to program the approximate reasoning algorithms. However, the next generation of expert systems should have an environment which contain a choice of several mathematical algorithms for approximate reasoning. To meet the need for validatable and verifiable coding, the expert system environment must no longer depend upon ad hoc reasoning techniques but instead must include mathematically rigorous techniques for approximate reasoning. Popular approximate reasoning techniques are reviewed, including: certainty factors, belief measures, Bayesian probabilities, fuzzy logic, and Shafer-Dempster techniques for reasoning. A group of mathematically rigorous algorithms for approximate reasoning are focused on that could form the basis of a next generation expert system environment. These algorithms are based upon the axioms of set theory and probability theory. To separate these algorithms for approximate reasoning various conditions of mutual exclusivity and independence are imposed upon the assertions. Approximate reasoning algorithms presented include: reasoning with statistically independent assertions, reasoning with mutually exclusive assertions, reasoning with assertions that exhibit minimum overlay within the state space, reasoning with assertions that exhibit maximum overlay within the state space (i.e. fuzzy logic), pessimistic reasoning (i.e. worst case analysis), optimistic reasoning (i.e. best case analysis), and reasoning with assertions with absolutely no knowledge of the possible dependency among the assertions. A robust environment for expert system construction should include the two modes of inference: modus ponens and modus tollens. Modus ponens inference is based upon reasoning towards the conclusion in a statement of logical implication, whereas modus tollens inference is based upon reasoning away
Approximating random quantum optimization problems
NASA Astrophysics Data System (ADS)
Hsu, B.; Laumann, C. R.; Läuchli, A. M.; Moessner, R.; Sondhi, S. L.
2013-06-01
We report a cluster of results regarding the difficulty of finding approximate ground states to typical instances of the quantum satisfiability problem k-body quantum satisfiability (k-QSAT) on large random graphs. As an approximation strategy, we optimize the solution space over “classical” product states, which in turn introduces a novel autonomous classical optimization problem, PSAT, over a space of continuous degrees of freedom rather than discrete bits. Our central results are (i) the derivation of a set of bounds and approximations in various limits of the problem, several of which we believe may be amenable to a rigorous treatment; (ii) a demonstration that an approximation based on a greedy algorithm borrowed from the study of frustrated magnetism performs well over a wide range in parameter space, and its performance reflects the structure of the solution space of random k-QSAT. Simulated annealing exhibits metastability in similar “hard” regions of parameter space; and (iii) a generalization of belief propagation algorithms introduced for classical problems to the case of continuous spins. This yields both approximate solutions, as well as insights into the free energy “landscape” of the approximation problem, including a so-called dynamical transition near the satisfiability threshold. Taken together, these results allow us to elucidate the phase diagram of random k-QSAT in a two-dimensional energy-density-clause-density space.
Wavelet Sparse Approximate Inverse Preconditioners
NASA Technical Reports Server (NTRS)
Chan, Tony F.; Tang, W.-P.; Wan, W. L.
1996-01-01
There is an increasing interest in using sparse approximate inverses as preconditioners for Krylov subspace iterative methods. Recent studies of Grote and Huckle and Chow and Saad also show that sparse approximate inverse preconditioner can be effective for a variety of matrices, e.g. Harwell-Boeing collections. Nonetheless a drawback is that it requires rapid decay of the inverse entries so that sparse approximate inverse is possible. However, for the class of matrices that, come from elliptic PDE problems, this assumption may not necessarily hold. Our main idea is to look for a basis, other than the standard one, such that a sparse representation of the inverse is feasible. A crucial observation is that the kind of matrices we are interested in typically have a piecewise smooth inverse. We exploit this fact, by applying wavelet techniques to construct a better sparse approximate inverse in the wavelet basis. We shall justify theoretically and numerically that our approach is effective for matrices with smooth inverse. We emphasize that in this paper we have only presented the idea of wavelet approximate inverses and demonstrated its potential but have not yet developed a highly refined and efficient algorithm.
Adaptive approximation models in optimization
Voronin, A.N.
1995-05-01
The paper proposes a method for optimization of functions of several variables that substantially reduces the number of objective function evaluations compared to traditional methods. The method is based on the property of iterative refinement of approximation models of the optimand function in approximation domains that contract to the extremum point. It does not require subjective specification of the starting point, step length, or other parameters of the search procedure. The method is designed for efficient optimization of unimodal functions of several (not more than 10-15) variables and can be applied to find the global extremum of polymodal functions and also for optimization of scalarized forms of vector objective functions.
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…
Error Bounds for Interpolative Approximations.
ERIC Educational Resources Information Center
Gal-Ezer, J.; Zwas, G.
1990-01-01
Elementary error estimation in the approximation of functions by polynomials as a computational assignment, error-bounding functions and error bounds, and the choice of interpolation points are discussed. Precalculus and computer instruction are used on some of the calculations. (KR)
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.
Testing the frozen flow approximation
NASA Technical Reports Server (NTRS)
Lucchin, Francesco; Matarrese, Sabino; Melott, Adrian L.; Moscardini, Lauro
1993-01-01
We investigate the accuracy of the frozen-flow approximation (FFA), recently proposed by Matarrese, et al. (1992), for following the nonlinear evolution of cosmological density fluctuations under gravitational instability. We compare a number of statistics between results of the FFA and n-body simulations, including those used by Melott, Pellman & Shandarin (1993) to test the Zel'dovich approximation. The FFA performs reasonably well in a statistical sense, e.g. in reproducing the counts-in-cell distribution, at small scales, but it does poorly in the crosscorrelation with n-body which means it is generally not moving mass to the right place, especially in models with high small-scale power.
Approximate line shapes for hydrogen
NASA Technical Reports Server (NTRS)
Sutton, K.
1978-01-01
Two independent methods are presented for calculating radiative transport within hydrogen lines. In Method 1, a simple equation is proposed for calculating the line shape. In Method 2, the line shape is assumed to be a dispersion profile and an equation is presented for calculating the half half-width. The results obtained for the line shapes and curves of growth by the two approximate methods are compared with similar results using the detailed line shapes by Vidal et al.
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.
Computer Experiments for Function Approximations
Chang, A; Izmailov, I; Rizzo, S; Wynter, S; Alexandrov, O; Tong, C
2007-10-15
This research project falls in the domain of response surface methodology, which seeks cost-effective ways to accurately fit an approximate function to experimental data. Modeling and computer simulation are essential tools in modern science and engineering. A computer simulation can be viewed as a function that receives input from a given parameter space and produces an output. Running the simulation repeatedly amounts to an equivalent number of function evaluations, and for complex models, such function evaluations can be very time-consuming. It is then of paramount importance to intelligently choose a relatively small set of sample points in the parameter space at which to evaluate the given function, and then use this information to construct a surrogate function that is close to the original function and takes little time to evaluate. This study was divided into two parts. The first part consisted of comparing four sampling methods and two function approximation methods in terms of efficiency and accuracy for simple test functions. The sampling methods used were Monte Carlo, Quasi-Random LP{sub {tau}}, Maximin Latin Hypercubes, and Orthogonal-Array-Based Latin Hypercubes. The function approximation methods utilized were Multivariate Adaptive Regression Splines (MARS) and Support Vector Machines (SVM). The second part of the study concerned adaptive sampling methods with a focus on creating useful sets of sample points specifically for monotonic functions, functions with a single minimum and functions with a bounded first derivative.
Ultrafast approximation for phylogenetic bootstrap.
Minh, Bui Quang; Nguyen, Minh Anh Thi; von Haeseler, Arndt
2013-05-01
Nonparametric bootstrap has been a widely used tool in phylogenetic analysis to assess the clade support of phylogenetic trees. However, with the rapidly growing amount of data, this task remains a computational bottleneck. Recently, approximation methods such as the RAxML rapid bootstrap (RBS) and the Shimodaira-Hasegawa-like approximate likelihood ratio test have been introduced to speed up the bootstrap. Here, we suggest an ultrafast bootstrap approximation approach (UFBoot) to compute the support of phylogenetic groups in maximum likelihood (ML) based trees. To achieve this, we combine the resampling estimated log-likelihood method with a simple but effective collection scheme of candidate trees. We also propose a stopping rule that assesses the convergence of branch support values to automatically determine when to stop collecting candidate trees. UFBoot achieves a median speed up of 3.1 (range: 0.66-33.3) to 10.2 (range: 1.32-41.4) compared with RAxML RBS for real DNA and amino acid alignments, respectively. Moreover, our extensive simulations show that UFBoot is robust against moderate model violations and the support values obtained appear to be relatively unbiased compared with the conservative standard bootstrap. This provides a more direct interpretation of the bootstrap support. We offer an efficient and easy-to-use software (available at http://www.cibiv.at/software/iqtree) to perform the UFBoot analysis with ML tree inference.
Approximate Counting of Graphical Realizations
2015-01-01
In 1999 Kannan, Tetali and Vempala proposed a MCMC method to uniformly sample all possible realizations of a given graphical degree sequence and conjectured its rapidly mixing nature. Recently their conjecture was proved affirmative for regular graphs (by Cooper, Dyer and Greenhill, 2007), for regular directed graphs (by Greenhill, 2011) and for half-regular bipartite graphs (by Miklós, Erdős and Soukup, 2013). Several heuristics on counting the number of possible realizations exist (via sampling processes), and while they work well in practice, so far no approximation guarantees exist for such an approach. This paper is the first to develop a method for counting realizations with provable approximation guarantee. In fact, we solve a slightly more general problem; besides the graphical degree sequence a small set of forbidden edges is also given. We show that for the general problem (which contains the Greenhill problem and the Miklós, Erdős and Soukup problem as special cases) the derived MCMC process is rapidly mixing. Further, we show that this new problem is self-reducible therefore it provides a fully polynomial randomized approximation scheme (a.k.a. FPRAS) for counting of all realizations. PMID:26161994
Approximately Independent Features of Languages
NASA Astrophysics Data System (ADS)
Holman, Eric W.
To facilitate the testing of models for the evolution of languages, the present paper offers a set of linguistic features that are approximately independent of each other. To find these features, the adjusted Rand index (R‧) is used to estimate the degree of pairwise relationship among 130 linguistic features in a large published database. Many of the R‧ values prove to be near zero, as predicted for independent features, and a subset of 47 features is found with an average R‧ of -0.0001. These 47 features are recommended for use in statistical tests that require independent units of analysis.
The structural physical approximation conjecture
NASA Astrophysics Data System (ADS)
Shultz, Fred
2016-01-01
It was conjectured that the structural physical approximation (SPA) of an optimal entanglement witness is separable (or equivalently, that the SPA of an optimal positive map is entanglement breaking). This conjecture was disproved, first for indecomposable maps and more recently for decomposable maps. The arguments in both cases are sketched along with important related results. This review includes background material on topics including entanglement witnesses, optimality, duality of cones, decomposability, and the statement and motivation for the SPA conjecture so that it should be accessible for a broad audience.
Generalized Gradient Approximation Made Simple
Perdew, J.P.; Burke, K.; Ernzerhof, M.
1996-10-01
Generalized gradient approximations (GGA{close_quote}s) for the exchange-correlation energy improve upon the local spin density (LSD) description of atoms, molecules, and solids. We present a simple derivation of a simple GGA, in which all parameters (other than those in LSD) are fundamental constants. Only general features of the detailed construction underlying the Perdew-Wang 1991 (PW91) GGA are invoked. Improvements over PW91 include an accurate description of the linear response of the uniform electron gas, correct behavior under uniform scaling, and a smoother potential. {copyright} {ital 1996 The American Physical Society.}
Quantum tunneling beyond semiclassical approximation
NASA Astrophysics Data System (ADS)
Banerjee, Rabin; Ranjan Majhi, Bibhas
2008-06-01
Hawking radiation as tunneling by Hamilton-Jacobi method beyond semiclassical approximation is analysed. We compute all quantum corrections in the single particle action revealing that these are proportional to the usual semiclassical contribution. We show that a simple choice of the proportionality constants reproduces the one loop back reaction effect in the spacetime, found by conformal field theory methods, which modifies the Hawking temperature of the black hole. Using the law of black hole mechanics we give the corrections to the Bekenstein-Hawking area law following from the modified Hawking temperature. Some examples are explicitly worked out.
Fermion tunneling beyond semiclassical approximation
NASA Astrophysics Data System (ADS)
Majhi, Bibhas Ranjan
2009-02-01
Applying the Hamilton-Jacobi method beyond the semiclassical approximation prescribed in R. Banerjee and B. R. Majhi, J. High Energy Phys.JHEPFG1029-8479 06 (2008) 09510.1088/1126-6708/2008/06/095 for the scalar particle, Hawking radiation as tunneling of the Dirac particle through an event horizon is analyzed. We show that, as before, all quantum corrections in the single particle action are proportional to the usual semiclassical contribution. We also compute the modifications to the Hawking temperature and Bekenstein-Hawking entropy for the Schwarzschild black hole. Finally, the coefficient of the logarithmic correction to entropy is shown to be related with the trace anomaly.
Plasma Physics Approximations in Ares
Managan, R. A.
2015-01-08
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} (ζ ),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.
Wavelet Approximation in Data Assimilation
NASA Technical Reports Server (NTRS)
Tangborn, Andrew; Atlas, Robert (Technical Monitor)
2002-01-01
Estimation of the state of the atmosphere with the Kalman filter remains a distant goal because of high computational cost of evolving the error covariance for both linear and nonlinear systems. Wavelet approximation is presented here as a possible solution that efficiently compresses both global and local covariance information. We demonstrate the compression characteristics on the the error correlation field from a global two-dimensional chemical constituent assimilation, and implement an adaptive wavelet approximation scheme on the assimilation of the one-dimensional Burger's equation. In the former problem, we show that 99%, of the error correlation can be represented by just 3% of the wavelet coefficients, with good representation of localized features. In the Burger's equation assimilation, the discrete linearized equations (tangent linear model) and analysis covariance are projected onto a wavelet basis and truncated to just 6%, of the coefficients. A nearly optimal forecast is achieved and we show that errors due to truncation of the dynamics are no greater than the errors due to covariance truncation.
Approximating metal-insulator transitions
NASA Astrophysics Data System (ADS)
Danieli, Carlo; Rayanov, Kristian; Pavlov, Boris; Martin, Gaven; Flach, Sergej
2015-12-01
We consider quantum wave propagation in one-dimensional quasiperiodic lattices. We propose an iterative construction of quasiperiodic potentials from sequences of potentials with increasing spatial period. At each finite iteration step, the eigenstates reflect the properties of the limiting quasiperiodic potential properties up to a controlled maximum system size. We then observe approximate Metal-Insulator Transitions (MIT) at the finite iteration steps. We also report evidence on mobility edges, which are at variance to the celebrated Aubry-André model. The dynamics near the MIT shows a critical slowing down of the ballistic group velocity in the metallic phase, similar to the divergence of the localization length in the insulating phase.
New generalized gradient approximation functionals
NASA Astrophysics Data System (ADS)
Boese, A. Daniel; Doltsinis, Nikos L.; Handy, Nicholas C.; Sprik, Michiel
2000-01-01
New generalized gradient approximation (GGA) functionals are reported, using the expansion form of A. D. Becke, J. Chem. Phys. 107, 8554 (1997), with 15 linear parameters. Our original such GGA functional, called HCTH, was determined through a least squares refinement to data of 93 systems. Here, the data are extended to 120 systems and 147 systems, introducing electron and proton affinities, and weakly bound dimers to give the new functionals HCTH/120 and HCTH/147. HCTH/120 has already been shown to give high quality predictions for weakly bound systems. The functionals are applied in a comparative study of the addition reaction of water to formaldehyde and sulfur trioxide, respectively. Furthermore, the performance of the HCTH/120 functional in Car-Parrinello molecular dynamics simulations of liquid water is encouraging.
Interplay of approximate planning strategies.
Huys, Quentin J M; Lally, Níall; Faulkner, Paul; Eshel, Neir; Seifritz, Erich; Gershman, Samuel J; Dayan, Peter; Roiser, Jonathan P
2015-03-10
Humans routinely formulate plans in domains so complex that even the most powerful computers are taxed. To do so, they seem to avail themselves of many strategies and heuristics that efficiently simplify, approximate, and hierarchically decompose hard tasks into simpler subtasks. Theoretical and cognitive research has revealed several such strategies; however, little is known about their establishment, interaction, and efficiency. Here, we use model-based behavioral analysis to provide a detailed examination of the performance of human subjects in a moderately deep planning task. We find that subjects exploit the structure of the domain to establish subgoals in a way that achieves a nearly maximal reduction in the cost of computing values of choices, but then combine partial searches with greedy local steps to solve subtasks, and maladaptively prune the decision trees of subtasks in a reflexive manner upon encountering salient losses. Subjects come idiosyncratically to favor particular sequences of actions to achieve subgoals, creating novel complex actions or "options." PMID:25675480
Indexing the approximate number system.
Inglis, Matthew; Gilmore, Camilla
2014-01-01
Much recent research attention has focused on understanding individual differences in the approximate number system, a cognitive system believed to underlie human mathematical competence. To date researchers have used four main indices of ANS acuity, and have typically assumed that they measure similar properties. Here we report a study which questions this assumption. We demonstrate that the numerical ratio effect has poor test-retest reliability and that it does not relate to either Weber fractions or accuracy on nonsymbolic comparison tasks. Furthermore, we show that Weber fractions follow a strongly skewed distribution and that they have lower test-retest reliability than a simple accuracy measure. We conclude by arguing that in the future researchers interested in indexing individual differences in ANS acuity should use accuracy figures, not Weber fractions or numerical ratio effects. PMID:24361686
IONIS: Approximate atomic photoionization intensities
NASA Astrophysics Data System (ADS)
Heinäsmäki, Sami
2012-02-01
A program to compute relative atomic photoionization cross sections is presented. The code applies the output of the multiconfiguration Dirac-Fock method for atoms in the single active electron scheme, by computing the overlap of the bound electron states in the initial and final states. The contribution from the single-particle ionization matrix elements is assumed to be the same for each final state. This method gives rather accurate relative ionization probabilities provided the single-electron ionization matrix elements do not depend strongly on energy in the region considered. The method is especially suited for open shell atoms where electronic correlation in the ionic states is large. Program summaryProgram title: IONIS Catalogue identifier: AEKK_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEKK_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 1149 No. of bytes in distributed program, including test data, etc.: 12 877 Distribution format: tar.gz Programming language: Fortran 95 Computer: Workstations Operating system: GNU/Linux, Unix Classification: 2.2, 2.5 Nature of problem: Photoionization intensities for atoms. Solution method: The code applies the output of the multiconfiguration Dirac-Fock codes Grasp92 [1] or Grasp2K [2], to compute approximate photoionization intensities. The intensity is computed within the one-electron transition approximation and by assuming that the sum of the single-particle ionization probabilities is the same for all final ionic states. Restrictions: The program gives nonzero intensities for those transitions where only one electron is removed from the initial configuration(s). Shake-type many-electron transitions are not computed. The ionized shell must be closed in the initial state. Running time: Few seconds for a
NASA Astrophysics Data System (ADS)
Di Carlo, A.; Carbonell Garcia, A.
2012-07-01
The frequency response solution (SOL 111) of MSC Nastran versions prior to 2012 only allows the output of element stress components and element forces and does not allow the calculation of composite failure indices or Von-Mises stress for metallic parts. The analysis of a sandwich panel comprises several strength verifications, such as the check of facesheet and core failure as well as the check of facesheet and core local stability (shear crimping, wrinkling). In static analysis (SOL 101), MSC Nastran provides failure index output which can be used to generate fringe plots of Margins of Safety (MoS) in any post- processing tool. The other verifications (core strength and local stability) must be performed using different tools. For the dynamic analysis of sandwich panels, an analysis technique based on element forces and on failure envelope at laminate level has been developed and implemented in a Fortran program (SineMOS) which allows evaluating facesheet and core failure as well as local stability, taking into account modulus and phase information of the element forces. SineMOS is able to produce files containing information used to generate plots of minimum Margin of Safety in Patran for each failure mode. This paper shows the various steps of the analysis process, starting from the building of the failure envelope for the CFRP facesheet laminate. Finally some validation example is shown, comparing SineMOS results with results based on the application of static displacements to the nodes of the model.
Multidimensional stochastic approximation Monte Carlo.
Zablotskiy, Sergey V; Ivanov, Victor A; Paul, Wolfgang
2016-06-01
Stochastic Approximation Monte Carlo (SAMC) has been established as a mathematically founded powerful flat-histogram Monte Carlo method, used to determine the density of states, g(E), of a model system. We show here how it can be generalized for the determination of multidimensional probability distributions (or equivalently densities of states) of macroscopic or mesoscopic variables defined on the space of microstates of a statistical mechanical system. This establishes this method as a systematic way for coarse graining a model system, or, in other words, for performing a renormalization group step on a model. We discuss the formulation of the Kadanoff block spin transformation and the coarse-graining procedure for polymer models in this language. We also apply it to a standard case in the literature of two-dimensional densities of states, where two competing energetic effects are present g(E_{1},E_{2}). We show when and why care has to be exercised when obtaining the microcanonical density of states g(E_{1}+E_{2}) from g(E_{1},E_{2}). PMID:27415383
Interplay of approximate planning strategies
Huys, Quentin J. M.; Lally, Níall; Faulkner, Paul; Eshel, Neir; Seifritz, Erich; Gershman, Samuel J.; Dayan, Peter; Roiser, Jonathan P.
2015-01-01
Humans routinely formulate plans in domains so complex that even the most powerful computers are taxed. To do so, they seem to avail themselves of many strategies and heuristics that efficiently simplify, approximate, and hierarchically decompose hard tasks into simpler subtasks. Theoretical and cognitive research has revealed several such strategies; however, little is known about their establishment, interaction, and efficiency. Here, we use model-based behavioral analysis to provide a detailed examination of the performance of human subjects in a moderately deep planning task. We find that subjects exploit the structure of the domain to establish subgoals in a way that achieves a nearly maximal reduction in the cost of computing values of choices, but then combine partial searches with greedy local steps to solve subtasks, and maladaptively prune the decision trees of subtasks in a reflexive manner upon encountering salient losses. Subjects come idiosyncratically to favor particular sequences of actions to achieve subgoals, creating novel complex actions or “options.” PMID:25675480
Multidimensional stochastic approximation Monte Carlo
NASA Astrophysics Data System (ADS)
Zablotskiy, Sergey V.; Ivanov, Victor A.; Paul, Wolfgang
2016-06-01
Stochastic Approximation Monte Carlo (SAMC) has been established as a mathematically founded powerful flat-histogram Monte Carlo method, used to determine the density of states, g (E ) , of a model system. We show here how it can be generalized for the determination of multidimensional probability distributions (or equivalently densities of states) of macroscopic or mesoscopic variables defined on the space of microstates of a statistical mechanical system. This establishes this method as a systematic way for coarse graining a model system, or, in other words, for performing a renormalization group step on a model. We discuss the formulation of the Kadanoff block spin transformation and the coarse-graining procedure for polymer models in this language. We also apply it to a standard case in the literature of two-dimensional densities of states, where two competing energetic effects are present g (E1,E2) . We show when and why care has to be exercised when obtaining the microcanonical density of states g (E1+E2) from g (E1,E2) .
Semiclassics beyond the diagonal approximation
NASA Astrophysics Data System (ADS)
Turek, Marko
2004-05-01
The statistical properties of the energy spectrum of classically chaotic closed quantum systems are the central subject of this thesis. It has been conjectured by O.Bohigas, M.-J.Giannoni and C.Schmit that the spectral statistics of chaotic systems is universal and can be described by random-matrix theory. This conjecture has been confirmed in many experiments and numerical studies but a formal proof is still lacking. In this thesis we present a semiclassical evaluation of the spectral form factor which goes beyond M.V.Berry's diagonal approximation. To this end we extend a method developed by M.Sieber and K.Richter for a specific system: the motion of a particle on a two-dimensional surface of constant negative curvature. In particular we prove that these semiclassical methods reproduce the random-matrix theory predictions for the next to leading order correction also for a much wider class of systems, namely non-uniformly hyperbolic systems with f>2 degrees of freedom. We achieve this result by extending the configuration-space approach of M.Sieber and K.Richter to a canonically invariant phase-space approach.
Randomized approximate nearest neighbors algorithm.
Jones, Peter Wilcox; Osipov, Andrei; Rokhlin, Vladimir
2011-09-20
We present a randomized algorithm for the approximate nearest neighbor problem in d-dimensional Euclidean space. Given N points {x(j)} in R(d), the algorithm attempts to find k nearest neighbors for each of x(j), where k is a user-specified integer parameter. The algorithm is iterative, and its running time requirements are proportional to T·N·(d·(log d) + k·(d + log k)·(log N)) + N·k(2)·(d + log k), with T the number of iterations performed. The memory requirements of the procedure are of the order N·(d + k). A by-product of the scheme is a data structure, permitting a rapid search for the k nearest neighbors among {x(j)} for an arbitrary point x ∈ R(d). The cost of each such query is proportional to T·(d·(log d) + log(N/k)·k·(d + log k)), and the memory requirements for the requisite data structure are of the order N·(d + k) + T·(d + N). The algorithm utilizes random rotations and a basic divide-and-conquer scheme, followed by a local graph search. We analyze the scheme's behavior for certain types of distributions of {x(j)} and illustrate its performance via several numerical examples.
Saddlepoint distribution function approximations in biostatistical inference.
Kolassa, J E
2003-01-01
Applications of saddlepoint approximations to distribution functions are reviewed. Calculations are provided for marginal distributions and conditional distributions. These approximations are applied to problems of testing and generating confidence intervals, particularly in canonical exponential families.
An approximation technique for jet impingement flow
Najafi, Mahmoud; Fincher, Donald; Rahni, Taeibi; Javadi, KH.; Massah, H.
2015-03-10
The analytical approximate solution of a non-linear jet impingement flow model will be demonstrated. We will show that this is an improvement over the series approximation obtained via the Adomian decomposition method, which is itself, a powerful method for analysing non-linear differential equations. The results of these approximations will be compared to the Runge-Kutta approximation in order to demonstrate their validity.
A unified approach to the Darwin approximation
Krause, Todd B.; Apte, A.; Morrison, P. J.
2007-10-15
There are two basic approaches to the Darwin approximation. The first involves solving the Maxwell equations in Coulomb gauge and then approximating the vector potential to remove retardation effects. The second approach approximates the Coulomb gauge equations themselves, then solves these exactly for the vector potential. There is no a priori reason that these should result in the same approximation. Here, the equivalence of these two approaches is investigated and a unified framework is provided in which to view the Darwin approximation. Darwin's original treatment is variational in nature, but subsequent applications of his ideas in the context of Vlasov's theory are not. We present here action principles for the Darwin approximation in the Vlasov context, and this serves as a consistency check on the use of the approximation in this setting.
Approximate Analysis of Semiconductor Laser Arrays
NASA Technical Reports Server (NTRS)
Marshall, William K.; Katz, Joseph
1987-01-01
Simplified equation yields useful information on gains and output patterns. Theoretical method based on approximate waveguide equation enables prediction of lateral modes of gain-guided planar array of parallel semiconductor lasers. Equation for entire array solved directly using piecewise approximation of index of refraction by simple functions without customary approximation based on coupled waveguid modes of individual lasers. Improved results yield better understanding of laser-array modes and help in development of well-behaved high-power semiconductor laser arrays.
Bent approximations to synchrotron radiation optics
Heald, S.
1981-01-01
Ideal optical elements can be approximated by bending flats or cylinders. This paper considers the applications of these approximate optics to synchrotron radiation. Analytic and raytracing studies are used to compare their optical performance with the corresponding ideal elements. It is found that for many applications the performance is adequate, with the additional advantages of lower cost and greater flexibility. Particular emphasis is placed on obtaining the practical limitations on the use of the approximate elements in typical beamline configurations. Also considered are the possibilities for approximating very long length mirrors using segmented mirrors.
Novel bivariate moment-closure approximations.
Krishnarajah, Isthrinayagy; Marion, Glenn; Gibson, Gavin
2007-08-01
Nonlinear stochastic models are typically intractable to analytic solutions and hence, moment-closure schemes are used to provide approximations to these models. Existing closure approximations are often unable to describe transient aspects caused by extinction behaviour in a stochastic process. Recent work has tackled this problem in the univariate case. In this study, we address this problem by introducing novel bivariate moment-closure methods based on mixture distributions. Novel closure approximations are developed, based on the beta-binomial, zero-modified distributions and the log-Normal, designed to capture the behaviour of the stochastic SIS model with varying population size, around the threshold between persistence and extinction of disease. The idea of conditional dependence between variables of interest underlies these mixture approximations. In the first approximation, we assume that the distribution of infectives (I) conditional on population size (N) is governed by the beta-binomial and for the second form, we assume that I is governed by zero-modified beta-binomial distribution where in either case N follows a log-Normal distribution. We analyse the impact of coupling and inter-dependency between population variables on the behaviour of the approximations developed. Thus, the approximations are applied in two situations in the case of the SIS model where: (1) the death rate is independent of disease status; and (2) the death rate is disease-dependent. Comparison with simulation shows that these mixture approximations are able to predict disease extinction behaviour and describe transient aspects of the process.
Quirks of Stirling's Approximation
ERIC Educational Resources Information Center
Macrae, Roderick M.; Allgeier, Benjamin M.
2013-01-01
Stirling's approximation 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 approximation leads to incorrect conclusions. In this article, the problem is first illustrated using a familiar "toy…
Spline approximations for nonlinear hereditary control systems
NASA Technical Reports Server (NTRS)
Daniel, P. L.
1982-01-01
A sline-based approximation scheme is discussed for optimal control problems governed by nonlinear nonautonomous delay differential equations. The approximating 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.
Diagonal Pade approximations for initial value problems
Reusch, M.F.; Ratzan, L.; Pomphrey, N.; Park, W.
1987-06-01
Diagonal Pade approximations to the time evolution operator for initial value problems are applied in a novel way to the numerical solution of these problems by explicitly factoring the polynomials of the approximation. A remarkable gain over conventional methods in efficiency and accuracy of solution is obtained. 20 refs., 3 figs., 1 tab.
An approximate model for pulsar navigation simulation
NASA Astrophysics Data System (ADS)
Jovanovic, Ilija; Enright, John
2016-02-01
This paper presents an approximate model for the simulation of pulsar aided navigation systems. High fidelity simulations of these systems are computationally intensive and impractical for simulating periods of a day or more. Simulation of yearlong missions is done by abstracting navigation errors as periodic Gaussian noise injections. This paper presents an intermediary approximate model to simulate position errors for periods of several weeks, useful for building more accurate Gaussian error models. This is done by abstracting photon detection and binning, replacing it with a simple deterministic process. The approximate model enables faster computation of error injection models, allowing the error model to be inexpensively updated throughout a simulation. Testing of the approximate model revealed an optimistic performance prediction for non-millisecond pulsars with more accurate predictions for pulsars in the millisecond spectrum. This performance gap was attributed to noise which is not present in the approximate model but can be predicted and added to improve accuracy.
Approximate error conjugation gradient minimization methods
Kallman, Jeffrey S
2013-05-21
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 approximate error using the subset of rays, and calculating a minimum in a conjugate gradient direction based on the approximate 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 approximate error using the subset of rays, and logic for calculating a minimum in a conjugate gradient direction based on the approximate error. In other embodiments, computer program products, methods, and systems are described capable of using approximate error in constrained conjugate gradient minimization problems.
Approximate Shortest Path Queries Using Voronoi Duals
NASA Astrophysics Data System (ADS)
Honiden, Shinichi; Houle, Michael E.; Sommer, Christian; Wolff, Martin
We propose an approximation method to answer point-to-point shortest path queries in undirected edge-weighted graphs, based on random sampling and Voronoi duals. We compute a simplification of the graph by selecting nodes independently at random with probability p. Edges are generated as the Voronoi dual of the original graph, using the selected nodes as Voronoi sites. This overlay graph allows for fast computation of approximate shortest paths for general, undirected graphs. The time-quality tradeoff decision can be made at query time. We provide bounds on the approximation ratio of the path lengths as well as experimental results. The theoretical worst-case approximation ratio is bounded by a logarithmic factor. Experiments show that our approximation method based on Voronoi duals has extremely fast preprocessing time and efficiently computes reasonably short paths.
Frankenstein's glue: transition functions for approximate solutions
NASA Astrophysics Data System (ADS)
Yunes, Nicolás
2007-09-01
Approximations are commonly employed to find approximate solutions to the Einstein equations. These solutions, however, are usually only valid in some specific spacetime region. A global solution can be constructed by gluing approximate solutions together, but this procedure is difficult because discontinuities can arise, leading to large violations of the Einstein equations. In this paper, we provide an attempt to formalize this gluing scheme by studying transition functions that join approximate analytic solutions together. In particular, we propose certain sufficient conditions on these functions and prove that these conditions guarantee that the joined solution still satisfies the Einstein equations analytically to the same order as the approximate ones. An example is also provided for a binary system of non-spinning black holes, where the approximate solutions are taken to be given by a post-Newtonian expansion and a perturbed Schwarzschild solution. For this specific case, we show that if the transition functions satisfy the proposed conditions, then the joined solution does not contain any violations to the Einstein equations larger than those already inherent in the approximations. We further show that if these functions violate the proposed conditions, then the matter content of the spacetime is modified by the introduction of a matter shell, whose stress energy tensor depends on derivatives of these functions.
Approximate knowledge compilation: The first order case
Val, A. del
1996-12-31
Knowledge compilation procedures make a knowledge base more explicit so as make inference with respect to the compiled knowledge base tractable or at least more efficient. Most work to date in this area has been restricted to the propositional case, despite the importance of first order theories for expressing knowledge concisely. Focusing on (LUB) approximate compilation, our contribution is twofold: (1) We present a new ground algorithm for approximate compilation which can produce exponential savings with respect to the previously known algorithm. (2) We show that both ground algorithms can be lifted to the first order case preserving their correctness for approximate compilation.
Approximate Bruechner orbitals in electron propagator calculations
Ortiz, J.V.
1999-12-01
Orbitals and ground-state correlation amplitudes from the so-called Brueckner doubles approximation of coupled-cluster theory provide a useful reference state for electron propagator calculations. An operator manifold with hold, particle, two-hole-one-particle and two-particle-one-hole components is chosen. The resulting approximation, third-order algebraic diagrammatic construction [2ph-TDA, ADC (3)] and 3+ methods. The enhanced versatility of this approximation is demonstrated through calculations on valence ionization energies, core ionization energies, electron detachment energies of anions, and on a molecule with partial biradical character, ozone.
Alternative approximation concepts for space frame synthesis
NASA Technical Reports Server (NTRS)
Lust, R. V.; Schmit, L. A.
1985-01-01
A method for space frame synthesis based on the application of a full gamut of approximation concepts is presented. It is found that with the thoughtful selection of design space, objective function approximation, constraint approximation and mathematical programming problem formulation options it is possible to obtain near minimum mass designs for a significant class of space frame structural systems while requiring fewer than 10 structural analyses. Example problems are presented which demonstrate the effectiveness of the method for frame structures subjected to multiple static loading conditions with limits on structural stiffness and strength.
APPROXIMATING LIGHT RAYS IN THE SCHWARZSCHILD FIELD
Semerák, O.
2015-02-10
A short formula is suggested that approximates photon trajectories in the Schwarzschild field better than other simple prescriptions from the literature. We compare it with various ''low-order competitors'', namely, with those following from exact formulas for small M, with one of the results based on pseudo-Newtonian potentials, with a suitably adjusted hyperbola, and with the effective and often employed approximation by Beloborodov. Our main concern is the shape of the photon trajectories at finite radii, yet asymptotic behavior is also discussed, important for lensing. An example is attached indicating that the newly suggested approximation is usable—and very accurate—for practically solving the ray-deflection exercise.
Adiabatic approximation for the density matrix
NASA Astrophysics Data System (ADS)
Band, Yehuda B.
1992-05-01
An adiabatic approximation for the Liouville density-matrix equation which includes decay terms is developed. The adiabatic approximation employs the eigenvectors of the non-normal Liouville operator. The approximation is valid when there exists a complete set of eigenvectors of the non-normal Liouville operator (i.e., the eigenvectors span the density-matrix space), the time rate of change of the Liouville operator is small, and an auxiliary matrix is nonsingular. Numerical examples are presented involving efficient population transfer in a molecule by stimulated Raman scattering, with the intermediate level of the molecule decaying on a time scale that is fast compared with the pulse durations of the pump and Stokes fields. The adiabatic density-matrix approximation can be simply used to determine the density matrix for atomic or molecular systems interacting with cw electromagnetic fields when spontaneous emission or other decay mechanisms prevail.
Approximation concepts for efficient structural synthesis
NASA Technical Reports Server (NTRS)
Schmit, L. A., Jr.; Miura, H.
1976-01-01
It is shown that efficient structural synthesis capabilities can be created by using approximation 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 approximation concepts are described along with the technical foundations of the ACCESS 1 computer program, which implements several approximation 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 approximation 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.
Approximate probability distributions of the master equation
NASA Astrophysics Data System (ADS)
Thomas, Philipp; Grima, Ramon
2015-07-01
Master equations are common descriptions of mesoscopic systems. Analytical solutions to these equations can rarely be obtained. We here derive an analytical approximation of the time-dependent probability distribution of the master equation using orthogonal polynomials. The solution is given in two alternative formulations: a series with continuous and a series with discrete support, both of which can be systematically truncated. While both approximations satisfy the system size expansion of the master equation, the continuous distribution approximations become increasingly negative and tend to oscillations with increasing truncation order. In contrast, the discrete approximations rapidly converge to the underlying non-Gaussian distributions. The theory is shown to lead to particularly simple analytical expressions for the probability distributions of molecule numbers in metabolic reactions and gene expression systems.
Linear Approximation SAR Azimuth Processing Study
NASA Technical Reports Server (NTRS)
Lindquist, R. B.; Masnaghetti, R. K.; Belland, E.; Hance, H. V.; Weis, W. G.
1979-01-01
A segmented linear approximation 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 approximation 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 approximation SAR azimuth processor meeting requirements similar to those of the SEASAT-A SAR was developed. The study resulted in a design with approximately 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.
A Survey of Techniques for Approximate Computing
Mittal, Sparsh
2016-03-18
Approximate computing trades off computation quality with the effort expended and as rising performance demands confront with plateauing resource budgets, approximate computing has become, not merely attractive, but even imperative. Here, we present a survey of techniques for approximate computing (AC). We discuss strategies for finding approximable 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
Polynomial approximation of functions in Sobolev spaces
NASA Technical Reports Server (NTRS)
Dupont, T.; Scott, R.
1980-01-01
Constructive proofs and several generalizations of approximation results of J. H. Bramble and S. R. Hilbert are presented. Using an averaged Taylor series, we represent a function as a polynomial plus a remainder. The remainder can be manipulated in many ways to give different types of bounds. Approximation of functions in fractional order Sobolev spaces is treated as well as the usual integer order spaces and several nonstandard Sobolev-like spaces.
Introduction to the Maxwell Garnett approximation: tutorial.
Markel, Vadim A
2016-07-01
This tutorial is devoted to the Maxwell Garnett approximation and related theories. Topics covered in this first, introductory part of the tutorial include the Lorentz local field correction, the Clausius-Mossotti relation and its role in the modern numerical technique known as the discrete dipole approximation, the Maxwell Garnett mixing formula for isotropic and anisotropic media, multicomponent mixtures and the Bruggeman equation, the concept of smooth field, and Wiener and Bergman-Milton bounds. PMID:27409680
The Actinide Transition Revisited by Gutzwiller Approximation
NASA Astrophysics Data System (ADS)
Xu, Wenhu; Lanata, Nicola; Yao, Yongxin; Kotliar, Gabriel
2015-03-01
We revisit the problem of the actinide transition using the Gutzwiller approximation (GA) in combination with the local density approximation (LDA). In particular, we compute the equilibrium volumes of the actinide series and reproduce the abrupt change of density found experimentally near plutonium as a function of the atomic number. We discuss how this behavior relates with the electron correlations in the 5 f states, the lattice structure, and the spin-orbit interaction. Our results are in good agreement with the experiments.
Polynomial approximation of functions in Sobolev spaces
Dupont, T.; Scott, R.
1980-04-01
Constructive proofs and several generalizations of approximation results of J. H. Bramble and S. R. Hilbert are presented. Using an averaged Taylor series, we represent a function as a polynomical plus a remainder. The remainder can be manipulated in many ways to give different types of bounds. Approximation of functions in fractional order Sobolev spaces is treated as well as the usual integer order spaces and several nonstandard Sobolev-like spaces.
Computing functions by approximating the input
NASA Astrophysics Data System (ADS)
Goldberg, Mayer
2012-12-01
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 approximate. In this work, we take the opposite approach: we show how to compute the values of some transcendental functions by approximating the input to these functions, and obtaining exact answers for their output. Our approach assumes only the most rudimentary knowledge of algebra and trigonometry, and makes no use of calculus.
Approximate Solutions Of Equations Of Steady Diffusion
NASA Technical Reports Server (NTRS)
Edmonds, Larry D.
1992-01-01
Rigorous analysis yields reliable criteria for "best-fit" functions. Improved "curve-fitting" method yields approximate solutions to differential equations of steady-state diffusion. Method applies to problems in which rates of diffusion depend linearly or nonlinearly on concentrations of diffusants, approximate solutions analytic or numerical, and boundary conditions of Dirichlet type, of Neumann type, or mixture of both types. Applied to equations for diffusion of charge carriers in semiconductors in which mobilities and lifetimes of charge carriers depend on concentrations.
An improved proximity force approximation for electrostatics
Fosco, Cesar D.; Lombardo, Fernando C.; Mazzitelli, Francisco D.
2012-08-15
A quite straightforward approximation for the electrostatic interaction between two perfectly conducting surfaces suggests itself when the distance between them is much smaller than the characteristic lengths associated with their shapes. Indeed, in the so called 'proximity force approximation' the electrostatic force is evaluated by first dividing each surface into a set of small flat patches, and then adding up the forces due two opposite pairs, the contributions of which are approximated as due to pairs of parallel planes. This approximation has been widely and successfully applied in different contexts, ranging from nuclear physics to Casimir effect calculations. We present here an improvement on this approximation, based on a derivative expansion for the electrostatic energy contained between the surfaces. The results obtained could be useful for discussing the geometric dependence of the electrostatic force, and also as a convenient benchmark for numerical analyses of the tip-sample electrostatic interaction in atomic force microscopes. - Highlights: Black-Right-Pointing-Pointer The proximity force approximation (PFA) has been widely used in different areas. Black-Right-Pointing-Pointer The PFA can be improved using a derivative expansion in the shape of the surfaces. Black-Right-Pointing-Pointer We use the improved PFA to compute electrostatic forces between conductors. Black-Right-Pointing-Pointer The results can be used as an analytic benchmark for numerical calculations in AFM. Black-Right-Pointing-Pointer Insight is provided for people who use the PFA to compute nuclear and Casimir forces.
Approximation methods in gravitational-radiation theory
NASA Technical Reports Server (NTRS)
Will, C. M.
1986-01-01
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, approximation methods play a crucial role. Recent developments are summarized in two areas in which approximations 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 approximation gives accurate estimates of the complex frequencies of the modes.
The Cell Cycle Switch Computes Approximate Majority
NASA Astrophysics Data System (ADS)
Cardelli, Luca; Csikász-Nagy, Attila
2012-09-01
Both computational and biological systems have to make decisions about switching from one state to another. The `Approximate Majority' computational algorithm provides the asymptotically fastest way to reach a common decision by all members of a population between two possible outcomes, where the decision approximately matches the initial relative majority. The network that regulates the mitotic entry of the cell-cycle in eukaryotes also makes a decision before it induces early mitotic processes. Here we show that the switch from inactive to active forms of the mitosis promoting Cyclin Dependent Kinases is driven by a system that is related to both the structure and the dynamics of the Approximate Majority computation. We investigate the behavior of these two switches by deterministic, stochastic and probabilistic methods and show that the steady states and temporal dynamics of the two systems are similar and they are exchangeable as components of oscillatory networks.
Fast wavelet based sparse approximate inverse preconditioner
Wan, W.L.
1996-12-31
Incomplete LU factorization is a robust preconditioner for both general and PDE problems but unfortunately not easy to parallelize. Recent study of Huckle and Grote and Chow and Saad showed that sparse approximate inverse could be a potential alternative while readily parallelizable. However, for special class of matrix A that comes from elliptic PDE problems, their preconditioners are not optimal in the sense that independent of mesh size. A reason may be that no good sparse approximate inverse exists for the dense inverse matrix. Our observation is that for this kind of matrices, its inverse entries typically have piecewise smooth changes. We can take advantage of this fact and use wavelet compression techniques to construct a better sparse approximate inverse preconditioner. We shall show numerically that our approach is effective for this kind of matrices.
Exponential Approximations Using Fourier Series Partial Sums
NASA Technical Reports Server (NTRS)
Banerjee, Nana S.; Geer, James F.
1997-01-01
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 approximate 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 approximated to within O(N(sup -M-2), and the associated jump of the k(sup th) derivative of f is approximated 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 approximations to f. Each of these new approximations 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 approximations, 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.
Congruence Approximations for Entrophy Endowed Hyperbolic Systems
NASA Technical Reports Server (NTRS)
Barth, Timothy J.; Saini, Subhash (Technical Monitor)
1998-01-01
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 approximation. 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 approximation.
Approximate formulas for moderately small eikonal amplitudes
NASA Astrophysics Data System (ADS)
Kisselev, A. V.
2016-08-01
We consider the eikonal approximation for moderately small scattering amplitudes. To find numerical estimates of these approximations, we derive formulas that contain no Bessel functions and consequently no rapidly oscillating integrands. To obtain these formulas, we study improper integrals of the first kind containing products of the Bessel functions J0(z). We generalize the expression with four functions J0(z) and also find expressions for the integrals with the product of five and six Bessel functions. We generalize a known formula for the improper integral with two functions Jυ (az) to the case with noninteger υ and complex a.
ANALOG QUANTUM NEURON FOR FUNCTIONS APPROXIMATION
A. EZHOV; A. KHROMOV; G. BERMAN
2001-05-01
We describe a system able to perform universal stochastic approximations of continuous multivariable functions in both neuron-like and quantum manner. The implementation of this model in the form of multi-barrier multiple-silt system has been earlier proposed. For the simplified waveguide variant of this model it is proved, that the system can approximate any continuous function of many variables. This theorem is also applied to the 2-input quantum neural model analogical to the schemes developed for quantum control.
Approximate controllability of nonlinear impulsive differential systems
NASA Astrophysics Data System (ADS)
Sakthivel, R.; Mahmudov, N. I.; Kim, J. H.
2007-08-01
Many practical systems in physical and biological sciences have impulsive dynamical be- haviours during the evolution process which can be modeled by impulsive differential equations. This paper studies the approximate controllability issue for nonlinear impulsive differential and neutral functional differential equations in Hilbert spaces. Based on the semigroup theory and fixed point approach, sufficient conditions for approximate controllability of impulsive differential and neutral functional differential equations are established. Finally, two examples are presented to illustrate the utility of the proposed result. The results improve some recent results.
Median Approximations for Genomes Modeled as Matrices.
Zanetti, Joao Paulo Pereira; Biller, Priscila; Meidanis, Joao
2016-04-01
The genome median problem is an important problem in phylogenetic reconstruction under rearrangement models. It can be stated as follows: Given three genomes, find a fourth that minimizes the sum of the pairwise rearrangement distances between it and the three input genomes. In this paper, we model genomes as matrices and study the matrix median problem using the rank distance. It is known that, for any metric distance, at least one of the corners is a [Formula: see text]-approximation of the median. Our results allow us to compute up to three additional matrix median candidates, all of them with approximation ratios at least as good as the best corner, when the input matrices come from genomes. We also show a class of instances where our candidates are optimal. From the application point of view, it is usually more interesting to locate medians farther from the corners, and therefore, these new candidates are potentially more useful. In addition to the approximation algorithm, we suggest a heuristic to get a genome from an arbitrary square matrix. This is useful to translate the results of our median approximation algorithm back to genomes, and it has good results in our tests. To assess the relevance of our approach in the biological context, we ran simulated evolution tests and compared our solutions to those of an exact DCJ median solver. The results show that our method is capable of producing very good candidates. PMID:27072561
Approximation of virus structure by icosahedral tilings.
Salthouse, D G; Indelicato, G; Cermelli, P; Keef, T; Twarock, R
2015-07-01
Viruses are remarkable examples of order at the nanoscale, exhibiting protein containers that in the vast majority of cases are organized with icosahedral symmetry. Janner used lattice theory to provide blueprints for the organization of material in viruses. An alternative approach is provided here in terms of icosahedral tilings, motivated by the fact that icosahedral symmetry is non-crystallographic in three dimensions. In particular, a numerical procedure is developed to approximate the capsid of icosahedral viruses by icosahedral tiles via projection of high-dimensional tiles based on the cut-and-project scheme for the construction of three-dimensional quasicrystals. The goodness of fit of our approximation is assessed using techniques related to the theory of polygonal approximation of curves. The approach is applied to a number of viral capsids and it is shown that detailed features of the capsid surface can indeed be satisfactorily described by icosahedral tilings. This work complements previous studies in which the geometry of the capsid is described by point sets generated as orbits of extensions of the icosahedral group, as such point sets are by construction related to the vertex sets of icosahedral tilings. The approximations of virus geometry derived here can serve as coarse-grained models of viral capsids as a basis for the study of virus assembly and structural transitions of viral capsids, and also provide a new perspective on the design of protein containers for nanotechnology applications. PMID:26131897
Generalized string models and their semiclassical approximation
NASA Astrophysics Data System (ADS)
Elizalde, E.
1984-04-01
We construct an extensive family of Bose string models, all of them classically equivalent to the Nambu and Eguchi models. The new models involve an arbitrary analytical function f(u), with f(0)=0, and are based on the Brink-Di Vecchia-Howe and Polyakov string action. The semiclassical approximation of the models is worked out in detail.
Progressive Image Coding by Hierarchical Linear Approximation.
ERIC Educational Resources Information Center
Wu, Xiaolin; Fang, Yonggang
1994-01-01
Proposes a scheme of hierarchical piecewise linear approximation as an adaptive image pyramid. A progressive image coder comes naturally from the proposed image pyramid. The new pyramid is semantically more powerful than regular tessellation but syntactically simpler than free segmentation. This compromise between adaptability and complexity…
Alternative approximation concepts for space frame synthesis
NASA Technical Reports Server (NTRS)
Lust, R. V.; Schmit, L. A.
1985-01-01
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 approximation 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 approximate 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 approximate 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 approximation concepts options, the results could be compared with previously published work.
Kravchuk functions for the finite oscillator approximation
NASA Technical Reports Server (NTRS)
Atakishiyev, Natig M.; Wolf, Kurt Bernardo
1995-01-01
Kravchuk orthogonal functions - Kravchuk polynomials multiplied by the square root of the weight function - simplify the inversion algorithm for the analysis of discrete, finite signals in harmonic oscillator components. They can be regarded as the best approximation set. As the number of sampling points increases, the Kravchuk expansion becomes the standard oscillator expansion.
Approximation algorithms for planning and control
NASA Technical Reports Server (NTRS)
Boddy, Mark; Dean, Thomas
1989-01-01
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 approximate those that would result from having unlimited time for computation, where the degree of the approximation depends on the amount of time it actually has. There exist approximation 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 approximation 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.
Parameter Choices for Approximation by Harmonic Splines
NASA Astrophysics Data System (ADS)
Gutting, Martin
2016-04-01
The approximation by harmonic trial functions allows the construction of the solution of boundary value problems in geoscience, e.g., in terms of harmonic splines. Due to their localizing properties regional modeling or the improvement of a global model in a part of the Earth's surface is possible with splines. Fast multipole methods have been developed for some cases of the occurring kernels to obtain a fast matrix-vector multiplication. The main idea of the fast multipole algorithm consists of a hierarchical decomposition of the computational domain into cubes and a kernel approximation for the more distant points. This reduces the numerical effort of the matrix-vector multiplication from quadratic to linear in reference to the number of points for a prescribed accuracy of the kernel approximation. The application of the fast multipole method to spline approximation which also allows the treatment of noisy data requires the choice of a smoothing parameter. We investigate different methods to (ideally automatically) choose this parameter with and without prior knowledge of the noise level. Thereby, the performance of these methods is considered for different types of noise in a large simulation study. Applications to gravitational field modeling are presented as well as the extension to boundary value problems where the boundary is the known surface of the Earth itself.
Approximation and compression with sparse orthonormal transforms.
Sezer, Osman Gokhan; Guleryuz, Onur G; Altunbasak, Yucel
2015-08-01
We propose a new transform design method that targets the generation of compression-optimized transforms for next-generation multimedia applications. The fundamental idea behind transform compression is to exploit regularity within signals such that redundancy is minimized subject to a fidelity cost. Multimedia signals, in particular images and video, are well known to contain a diverse set of localized structures, leading to many different types of regularity and to nonstationary signal statistics. The proposed method designs sparse orthonormal transforms (SOTs) that automatically exploit regularity over different signal structures and provides an adaptation method that determines the best representation over localized regions. Unlike earlier work that is motivated by linear approximation constructs and model-based designs that are limited to specific types of signal regularity, our work uses general nonlinear approximation ideas and a data-driven setup to significantly broaden its reach. We show that our SOT designs provide a safe and principled extension of the Karhunen-Loeve transform (KLT) by reducing to the KLT on Gaussian processes and by automatically exploiting non-Gaussian statistics to significantly improve over the KLT on more general processes. We provide an algebraic optimization framework that generates optimized designs for any desired transform structure (multiresolution, block, lapped, and so on) with significantly better n -term approximation performance. For each structure, we propose a new prototype codec and test over a database of images. Simulation results show consistent increase in compression and approximation performance compared with conventional methods. PMID:25823033
Fostering Formal Commutativity Knowledge with Approximate Arithmetic.
Hansen, Sonja Maria; Haider, Hilde; Eichler, Alexandra; Godau, Claudia; Frensch, Peter A; Gaschler, Robert
2015-01-01
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 approximate 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. Approximate 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 approximation 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 approximation. 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
Can Distributional Approximations Give Exact Answers?
ERIC Educational Resources Information Center
Griffiths, Martin
2013-01-01
Some mathematical activities and investigations for the classroom or the lecture theatre can appear rather contrived. This cannot, however, be levelled at the idea given here, since it is based on a perfectly sensible question concerning distributional approximations that was posed by an undergraduate student. Out of this simple question, and…
Achievements and Problems in Diophantine Approximation Theory
NASA Astrophysics Data System (ADS)
Sprindzhuk, V. G.
1980-08-01
ContentsIntroduction I. Metrical theory of approximation on manifolds § 1. The basic problem § 2. Brief survey of results § 3. The principal conjecture II. Metrical theory of transcendental numbers § 1. Mahler's classification of numbers § 2. Metrical characterization of numbers with a given type of approximation § 3. Further problems III. Approximation of algebraic numbers by rationals § 1. Simultaneous approximations § 2. The inclusion of p-adic metrics § 3. Effective improvements of Liouville's inequality IV. Estimates of linear forms in logarithms of algebraic numbers § 1. The basic method § 2. Survey of results § 3. Estimates in the p-adic metric V. Diophantine equations § 1. Ternary exponential equations § 2. The Thue and Thue-Mahler equations § 3. Equations of hyperelliptic type § 4. Algebraic-exponential equations VI. The arithmetic structure of polynomials and the class number § 1. The greatest prime divisor of a polynomial in one variable § 2. The greatest prime divisor of a polynomial in two variables § 3. Square-free divisors of polynomials and the class number § 4. The general problem of the size of the class number Conclusion References
Quickly Approximating the Distance Between Two Objects
NASA Technical Reports Server (NTRS)
Hammen, David
2009-01-01
A method of quickly approximating 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.
Median Approximations for Genomes Modeled as Matrices.
Zanetti, Joao Paulo Pereira; Biller, Priscila; Meidanis, Joao
2016-04-01
The genome median problem is an important problem in phylogenetic reconstruction under rearrangement models. It can be stated as follows: Given three genomes, find a fourth that minimizes the sum of the pairwise rearrangement distances between it and the three input genomes. In this paper, we model genomes as matrices and study the matrix median problem using the rank distance. It is known that, for any metric distance, at least one of the corners is a [Formula: see text]-approximation of the median. Our results allow us to compute up to three additional matrix median candidates, all of them with approximation ratios at least as good as the best corner, when the input matrices come from genomes. We also show a class of instances where our candidates are optimal. From the application point of view, it is usually more interesting to locate medians farther from the corners, and therefore, these new candidates are potentially more useful. In addition to the approximation algorithm, we suggest a heuristic to get a genome from an arbitrary square matrix. This is useful to translate the results of our median approximation algorithm back to genomes, and it has good results in our tests. To assess the relevance of our approach in the biological context, we ran simulated evolution tests and compared our solutions to those of an exact DCJ median solver. The results show that our method is capable of producing very good candidates.
Fostering Formal Commutativity Knowledge with Approximate Arithmetic
Hansen, Sonja Maria; Haider, Hilde; Eichler, Alexandra; Godau, Claudia; Frensch, Peter A.; Gaschler, Robert
2015-01-01
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 approximate 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. Approximate 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 approximation 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 approximation. 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
Counting independent sets using the Bethe approximation
Chertkov, Michael; Chandrasekaran, V; Gamarmik, D; Shah, D; Sin, J
2009-01-01
The authors consider the problem of counting the number of independent sets or the partition function of a hard-core model in a graph. The problem in general is computationally hard (P hard). They study the quality of the approximation provided by the Bethe free energy. Belief propagation (BP) is a message-passing algorithm can be used to compute fixed points of the Bethe approximation; however, BP is not always guarantee to converge. As the first result, they propose a simple message-passing algorithm that converges to a BP fixed pont for any grapy. They find that their algorithm converges within a multiplicative error 1 + {var_epsilon} of a fixed point in {Omicron}(n{sup 2}E{sup -4} log{sup 3}(nE{sup -1})) iterations for any bounded degree graph of n nodes. In a nutshell, the algorithm can be thought of as a modification of BP with 'time-varying' message-passing. Next, they analyze the resulting error to the number of independent sets provided by such a fixed point of the Bethe approximation. Using the recently developed loop calculus approach by Vhertkov and Chernyak, they establish that for any bounded graph with large enough girth, the error is {Omicron}(n{sup -{gamma}}) for some {gamma} > 0. As an application, they find that for random 3-regular graph, Bethe approximation of log-partition function (log of the number of independent sets) is within o(1) of corret log-partition - this is quite surprising as previous physics-based predictions were expecting an error of o(n). In sum, their results provide a systematic way to find Bethe fixed points for any graph quickly and allow for estimating error in Bethe approximation using novel combinatorial techniques.
Analysing organic transistors based on interface approximation
Akiyama, Yuto; Mori, Takehiko
2014-01-15
Temperature-dependent characteristics of organic transistors are analysed thoroughly using interface approximation. In contrast to amorphous silicon transistors, it is characteristic of organic transistors that the accumulation layer is concentrated on the first monolayer, and it is appropriate to consider interface charge rather than band bending. On the basis of this model, observed characteristics of hexamethylenetetrathiafulvalene (HMTTF) and dibenzotetrathiafulvalene (DBTTF) transistors with various surface treatments are analysed, and the trap distribution is extracted. In turn, starting from a simple exponential distribution, we can reproduce the temperature-dependent transistor characteristics as well as the gate voltage dependence of the activation energy, so we can investigate various aspects of organic transistors self-consistently under the interface approximation. Small deviation from such an ideal transistor operation is discussed assuming the presence of an energetically discrete trap level, which leads to a hump in the transfer characteristics. The contact resistance is estimated by measuring the transfer characteristics up to the linear region.
Approximate inverse preconditioners for general sparse matrices
Chow, E.; Saad, Y.
1994-12-31
Preconditioned Krylov subspace methods are often very efficient in solving sparse linear matrices that arise from the discretization of elliptic partial differential equations. However, for general sparse indifinite matrices, the usual ILU preconditioners fail, often because of the fact that the resulting factors L and U give rise to unstable forward and backward sweeps. In such cases, alternative preconditioners based on approximate inverses may be attractive. We are currently developing a number of such preconditioners based on iterating on each column to get the approximate inverse. For this approach to be efficient, the iteration must be done in sparse mode, i.e., we must use sparse-matrix by sparse-vector type operatoins. We will discuss a few options and compare their performance on standard problems from the Harwell-Boeing collection.
Private Medical Record Linkage with Approximate Matching
Durham, Elizabeth; Xue, Yuan; Kantarcioglu, Murat; Malin, Bradley
2010-01-01
Federal regulations require patient data to be shared for reuse in a de-identified manner. However, disparate providers often share data on overlapping populations, such that a patient’s record may be duplicated or fragmented in the de-identified repository. To perform unbiased statistical analysis in a de-identified setting, it is crucial to integrate records that correspond to the same patient. Private record linkage techniques have been developed, but most methods are based on encryption and preclude the ability to determine similarity, decreasing the accuracy of record linkage. The goal of this research is to integrate a private string comparison method that uses Bloom filters to provide an approximate match, with a medical record linkage algorithm. We evaluate the approach with 100,000 patients’ identifiers and demographics from the Vanderbilt University Medical Center. We demonstrate that the private approximation method achieves sensitivity that is, on average, 3% higher than previous methods. PMID:21346965
Approximated solutions to Born-Infeld dynamics
NASA Astrophysics Data System (ADS)
Ferraro, Rafael; Nigro, Mauro
2016-02-01
The Born-Infeld equation in the plane is usefully captured in complex language. The general exact solution can be written as a combination of holomorphic and anti-holomorphic functions. However, this solution only expresses the potential in an implicit way. We rework the formulation to obtain the complex potential in an explicit way, by means of a perturbative procedure. We take care of the secular behavior common to this kind of approach, by resorting to a symmetry the equation has at the considered order of approximation. We apply the method to build approximated solutions to Born-Infeld electrodynamics. We solve for BI electromagnetic waves traveling in opposite directions. We study the propagation at interfaces, with the aim of searching for effects susceptible to experimental detection. In particular, we show that a reflected wave is produced when a wave is incident on a semi-space containing a magnetostatic field.
Laplace approximation in measurement error models.
Battauz, Michela
2011-05-01
Likelihood analysis for regression models with measurement errors in explanatory variables typically involves integrals that do not have a closed-form solution. In this case, numerical methods such as Gaussian quadrature are generally employed. However, when the dimension of the integral is large, these methods become computationally demanding or even unfeasible. This paper proposes the use of the Laplace approximation to deal with measurement error problems when the likelihood function involves high-dimensional integrals. The cases considered are generalized linear models with multiple covariates measured with error and generalized linear mixed models with measurement error in the covariates. The asymptotic order of the approximation and the asymptotic properties of the Laplace-based estimator for these models are derived. The method is illustrated using simulations and real-data analysis.
Planetary ephemerides approximation for radar astronomy
NASA Technical Reports Server (NTRS)
Sadr, R.; Shahshahani, M.
1991-01-01
The planetary ephemerides approximation for radar astronomy is discussed, and, in particular, the effect of this approximation on the performance of the programmable local oscillator (PLO) used in Goldstone Solar System Radar is presented. Four different approaches are considered and it is shown that the Gram polynomials outperform the commonly used technique based on Chebyshev polynomials. These methods are used to analyze the mean square, the phase error, and the frequency tracking error in the presence of the worst case Doppler shift that one may encounter within the solar system. It is shown that in the worst case the phase error is under one degree and the frequency tracking error less than one hertz when the frequency to the PLO is updated every millisecond.
Some approximation concepts for structural synthesis
NASA Technical Reports Server (NTRS)
Schmit, L. A., Jr.; Farshi, B.
1974-01-01
An efficient automated minimum weight design procedure is presented which is applicable to sizing structural systems that can be idealized by truss, shear panel, and constant strain triangles. Static stress and displacement constraints under alternative loading conditions are considered. The optimization algorithm is an adaptation of the method of inscribed hyperspheres and high efficiency is achieved by using several approximation concepts including temporary deletion of noncritical constraints, design variable linking, and Taylor series expansions for response variables in terms of design variables. Optimum designs for several planar and space truss examples problems are presented. The results reported support the contention that the innovative use of approximation concepts in structural synthesis can produce significant improvements in efficiency.
Some approximation concepts for structural synthesis.
NASA Technical Reports Server (NTRS)
Schmit, L. A., Jr.; Farshi, B.
1973-01-01
An efficient automated minimum weight design procedure is presented which is applicable to sizing structural systems that can be idealized by truss, shear panel, and constant strain triangles. Static stress and displacement constraints under alternative loading conditions are considered. The optimization algorithm is an adaptation of the method of inscribed hyperspheres and high efficiency is achieved by using several approximation concepts including temporary deletion of noncritical constraints, design variable linking, and Taylor series expansions for response variables in terms of design variables. Optimum designs for several planar and space truss example problems are presented. The results reported support the contention that the innovative use of approximation concepts in structural synthesis can produce significant improvements in efficiency.
Approximate Solutions in Planted 3-SAT
NASA Astrophysics Data System (ADS)
Hsu, Benjamin; Laumann, Christopher; Moessner, Roderich; Sondhi, Shivaji
2013-03-01
In many computational settings, there exists many instances where finding a solution requires a computing time that grows exponentially in the number of variables. Concrete examples occur in combinatorial optimization problems and cryptography in computer science or glassy systems in physics. However, while exact solutions are often known to require exponential time, a related and important question is the running time required to find approximate solutions. Treating this problem as a problem in statistical physics at finite temperature, we examine the computational running time in finding approximate solutions in 3-satisfiability for randomly generated 3-SAT instances which are guaranteed to have a solution. Analytic predictions are corroborated by numerical evidence using stochastic local search algorithms. A first order transition is found in the running time of these algorithms.
Approximate gauge symemtry of composite vector bosons
Suzuki, Mahiko
2010-06-01
It can be shown in a solvable field theory model that the couplings of the composite vector mesons made of a fermion pair approach the gauge couplings in the limit of strong binding. Although this phenomenon may appear accidental and special to the vector bosons made of a fermion pair, we extend it to the case of bosons being constituents and find that the same phenomenon occurs in more an intriguing way. The functional formalism not only facilitates computation but also provides us with a better insight into the generating mechanism of approximate gauge symmetry, in particular, how the strong binding and global current conservation conspire to generate such an approximate symmetry. Remarks are made on its possible relevance or irrelevance to electroweak and higher symmetries.
Signal recovery by best feasible approximation.
Combettes, P L
1993-01-01
The objective of set theoretical signal recovery is to find a feasible signal in the form of a point in the intersection of S of sets modeling the information available about the problem. For problems in which the true signal is known to lie near a reference signal r, the solution should not be any feasible point but one which best approximates r, i.e., a projection of r onto S. Such a solution cannot be obtained by the feasibility algorithms currently in use, e.g., the method of projections onto convex sets (POCS) and its offsprings. Methods for projecting a point onto the intersection of closed and convex sets in a Hilbert space are introduced and applied to signal recovery by best feasible approximation of a reference signal. These algorithms are closely related to the above projection methods, to which they add little computational complexity.
Weizsacker-Williams approximation in quantum chromodynamics
NASA Astrophysics Data System (ADS)
Kovchegov, Yuri V.
The Weizsacker-Williams approximation for a large nucleus in quantum chromodynamics is developed. The non-Abelian Wieizsacker Williams field for a large ultrarelativistic nucleus is constructed. This field is an exact solution of the classical Yang-Mills equations of motion in light cone gauge. The connection is made to the McLerran- Venugopalan model of a large nucleus, and the color charge density for a nucleus in this model is found. The density of states distribution, as a function of color charge density, is proved to be Gaussian. We construct the Feynman diagrams in the light cone gauge which correspond to the classical Weizsacker Williams field. Analyzing these diagrams we obtain a limitation on using the quasi-classical approximation for nuclear collisions.
Flow past a porous approximate spherical shell
NASA Astrophysics Data System (ADS)
Srinivasacharya, D.
2007-07-01
In this paper, the creeping flow of an incompressible viscous liquid past a porous approximate spherical shell is considered. The flow in the free fluid region outside the shell and in the cavity region of the shell is governed by the Navier Stokes equation. The flow within the porous annulus region of the shell is governed by Darcy’s Law. The boundary conditions used at the interface are continuity of the normal velocity, continuity of the pressure and Beavers and Joseph slip condition. An exact solution for the problem is obtained. An expression for the drag on the porous approximate spherical shell is obtained. The drag experienced by the shell is evaluated numerically for several values of the parameters governing the flow.
Numerical and approximate solutions for plume rise
NASA Astrophysics Data System (ADS)
Krishnamurthy, Ramesh; Gordon Hall, J.
Numerical and approximate analytical solutions are compared for turbulent plume rise in a crosswind. The numerical solutions were calculated using the plume rise model of Hoult, Fay and Forney (1969, J. Air Pollut. Control Ass.19, 585-590), over a wide range of pertinent parameters. Some wind shear and elevated inversion effects are included. The numerical solutions are seen to agree with the approximate solutions over a fairly wide range of the parameters. For the conditions considered in the study, wind shear effects are seen to be quite small. A limited study was made of the penetration of elevated inversions by plumes. The results indicate the adequacy of a simple criterion proposed by Briggs (1969, AEC Critical Review Series, USAEC Division of Technical Information extension, Oak Ridge, Tennesse).
Second derivatives for approximate spin projection methods
Thompson, Lee M.; Hratchian, Hrant P.
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 second derivatives for the Yamaguchi approximate projection scheme, building on recent work that has yielded an efficient implementation of the analytical first derivatives.
Rounded Approximate Step Functions For Interpolation
NASA Technical Reports Server (NTRS)
Nunes, Arthur C., Jr.
1993-01-01
Rounded approximate step functions of form x(Sup m)/(x(Sup n) + 1) and 1/(x(Sup n) + 1) useful in interpolating between local steep slopes or abrupt changes in tabulated data varying more smoothly elsewhere. Used instead of polynomial curve fits. Interpolation formulas based on these functions implemented quickly and easily on computers. Used in real-time control computations to interpolate between tabulated data governing control responses.
Solving Math Problems Approximately: A Developmental Perspective
Ganor-Stern, Dana
2016-01-01
Although solving arithmetic problems approximately 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 approximately, 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 approximated 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 approximate 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
Microscopic justification of the equal filling approximation
Perez-Martin, Sara; Robledo, L. M.
2008-07-15
The equal filling approximation, a procedure widely used in mean-field calculations to treat the dynamics of odd nuclei in a time-reversal invariant way, is justified as the consequence of a variational principle over an average energy functional. The ideas of statistical quantum mechanics are employed in the justification. As an illustration of the method, the ground and lowest-lying states of some octupole deformed radium isotopes are computed.
Approximation methods in relativistic eigenvalue perturbation theory
NASA Astrophysics Data System (ADS)
Noble, Jonathan Howard
In this dissertation, three questions, concerning approximation methods for the eigenvalues of quantum mechanical systems, are investigated: (i) What is a pseudo--Hermitian Hamiltonian, and how can its eigenvalues be approximated 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 approximated 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 approximated 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.
Approximation methods for stochastic petri nets
NASA Technical Reports Server (NTRS)
Jungnitz, Hauke Joerg
1992-01-01
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, approximation 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 approximation 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 approximation 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 approximation converges quickly and shows reasonable accuracy in most cases. The convergence of Marie's is slower, but the accuracy is generally better. Delay
Capacitor-Chain Successive-Approximation ADC
NASA Technical Reports Server (NTRS)
Cunningham, Thomas
2003-01-01
A proposed successive-approximation analog-to-digital converter (ADC) would contain a capacitively terminated chain of identical capacitor cells. Like a conventional successive-approximation 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, approximately 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 approximately 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.
Approximating spheroid inductive responses using spheres
Smith, J. Torquil; Morrison, H. Frank
2003-12-12
The response of high permeability ({mu}{sub r} {ge} 50) conductive spheroids of moderate aspect ratios (0.25 to 4) to excitation by uniform magnetic fields in the axial or transverse directions is approximated by the response of spheres of appropriate diameters, of the same conductivity and permeability, with magnitude rescaled based on the differing volumes, D.C. magnetizations, and high frequency limit responses of the spheres and modeled spheroids.
Analytic approximation to randomly oriented spheroid extinction
NASA Astrophysics Data System (ADS)
Evans, B. T. N.; Fournier, G. R.
1993-12-01
The estimation of electromagnetic extinction through dust or other nonspherical atmospheric aerosols and hydrosols is an essential first step in the evaluation of the performance of all electro-optic systems. Investigations were conducted to reduce the computational burden in calculating the extinction from nonspherical particles. An analytic semi-empirical approximation to the extinction efficiency Q(sub ext) for randomly oriented spheroids, based on an extension of the anomalous diffraction formula, is given and compared with the extended boundary condition or T-matrix method. This will allow for better and more general modeling of obscurants. Using this formula, Q(sub ext) can be evaluated over 10,000 times faster than with previous methods. This approximation has been verified for complex refractive indices m=n-ik, where n ranges from one to infinity and k from zero to infinity, and aspect ratios of 0.2 to 5. It is believed that the approximation is uniformly valid over all size parameters and aspect ratios. It has the correct Rayleigh, refractive index, and large particle asymptotic behaviors. The accuracy and limitations of this formula are extensively discussed.
Waveform feature extraction based on tauberian approximation.
De Figueiredo, R J; Hu, C L
1982-02-01
A technique is presented for feature extraction of a waveform y based on its Tauberian approximation, that is, on the approximation of y by a linear combination of appropriately delayed versions of a single basis function x, i.e., y(t) = ¿M i = 1 aix(t - ¿i), where the coefficients ai and the delays ¿i are adjustable parameters. Considerations in the choice or design of the basis function x are given. The parameters ai and ¿i, i=1, . . . , M, are retrieved by application of a suitably adapted version of Prony's method to the Fourier transform of the above approximation of y. A subset of the parameters ai and ¿i, i = 1, . . . , M, is used to construct the feature vector, the value of which can be used in a classification algorithm. Application of this technique to the classification of wide bandwidth radar return signatures is presented. Computer simulations proved successful and are also discussed.
Using Approximations to Accelerate Engineering Design Optimization
NASA Technical Reports Server (NTRS)
Torczon, Virginia; Trosset, Michael W.
1998-01-01
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 approximations 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 approximation 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 approximation. Our goal is to further improve the effectiveness of our general approach without sacrificing any of its rigor.
An Origami Approximation to the Cosmic Web
NASA Astrophysics Data System (ADS)
Neyrinck, Mark C.
2016-10-01
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 approximation, 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 approximation 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-approximation models giving velocity fields for the first time.
CMB-lensing beyond the Born approximation
NASA Astrophysics Data System (ADS)
Marozzi, Giovanni; Fanizza, Giuseppe; Di Dio, Enea; 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 lesssim 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-Gaussian 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.
A coastal ocean model with subgrid approximation
NASA Astrophysics Data System (ADS)
Walters, Roy A.
2016-06-01
A wide variety of coastal ocean models exist, each having attributes that reflect specific application areas. The model presented here is based on finite element methods with unstructured grids containing triangular and quadrilateral elements. The model optimizes robustness, accuracy, and efficiency by using semi-implicit methods in time in order to remove the most restrictive stability constraints, by using a semi-Lagrangian advection approximation to remove Courant number constraints, and by solving a wave equation at the discrete level for enhanced efficiency. An added feature is the approximation of the effects of subgrid objects. Here, the Reynolds-averaged Navier-Stokes equations and the incompressibility constraint are volume averaged over one or more computational cells. This procedure gives rise to new terms which must be approximated as a closure problem. A study of tidal power generation is presented as an example of this method. A problem that arises is specifying appropriate thrust and power coefficients for the volume averaged velocity when they are usually referenced to free stream velocity. A new contribution here is the evaluation of three approaches to this problem: an iteration procedure and two mapping formulations. All three sets of results for thrust (form drag) and power are in reasonable agreement.
Compression of strings with approximate repeats.
Allison, L; Edgoose, T; Dix, T I
1998-01-01
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 approximate 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 approximation algorithm is also given. The model is further extended to include approximate reverse complementary repeats when analyzing DNA strings. Tests include the recovery of parameter estimates from known sources and applications to real DNA strings.
Green-Ampt approximations: A comprehensive analysis
NASA Astrophysics Data System (ADS)
Ali, Shakir; Islam, Adlul; Mishra, P. K.; Sikka, Alok K.
2016-04-01
Green-Ampt (GA) model and its modifications are widely used for simulating infiltration process. Several explicit approximate solutions to the implicit GA model have been developed with varying degree of accuracy. In this study, performance of nine explicit approximations 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 approximate GA models for solving variety of hydrological problems.
Generalized Quasilinear Approximation: Application to Zonal Jets.
Marston, J B; Chini, G P; Tobias, S M
2016-05-27
Quasilinear theory is often utilized to approximate the dynamics of fluids exhibiting significant interactions between mean flows and eddies. We present a generalization of quasilinear theory to include dynamic mode interactions on the large scales. This generalized quasilinear (GQL) approximation is achieved by separating the state variables into large and small zonal scales via a spectral filter rather than by a decomposition into a formal mean and fluctuations. Nonlinear interactions involving only small zonal scales are then removed. The approximation is conservative and allows for scattering of energy between small-scale modes via the large scale (through nonlocal spectral interactions). We evaluate GQL for the paradigmatic problems of the driving of large-scale jets on a spherical surface and on the beta plane and show that it is accurate even for a small number of large-scale modes. As GQL is formally linear in the small zonal scales, it allows for the closure of the system and can be utilized in direct statistical simulation schemes that have proved an attractive alternative to direct numerical simulation for many geophysical and astrophysical problems. PMID:27284660
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.
Odic, Darko; Lisboa, Juan Valle; Eisinger, Robert; Olivera, Magdalena Gonzalez; Maiche, Alejandro; Halberda, Justin
2016-01-01
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 approximate 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 approximate 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 approximate number discrimination task, an approximate 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 approximate number precision and extend those findings by showing that math abilities also correlate with approximate time precision. But, contrary to approximate number and time sharing common representations, we find that each of these dimensions uniquely correlates with formal math: approximate 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 approximate number and approximate time and further clarify the relationship between quantity representations and mathematics.
Odic, Darko; Lisboa, Juan Valle; Eisinger, Robert; Olivera, Magdalena Gonzalez; Maiche, Alejandro; Halberda, Justin
2016-01-01
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 approximate 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 approximate 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 approximate number discrimination task, an approximate 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 approximate number precision and extend those findings by showing that math abilities also correlate with approximate time precision. But, contrary to approximate number and time sharing common representations, we find that each of these dimensions uniquely correlates with formal math: approximate 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 approximate number and approximate time and further clarify the relationship between quantity representations and mathematics. PMID:26587963
Approximate protein structural alignment in polynomial time.
Kolodny, Rachel; Linial, Nathan
2004-08-17
Alignment of protein structures is a fundamental task in computational molecular biology. Good structural alignments can help detect distant evolutionary relationships that are hard or impossible to discern from protein sequences alone. Here, we study the structural alignment problem as a family of optimization problems and develop an approximate polynomial-time algorithm to solve them. For a commonly used scoring function, the algorithm runs in O(n(10)/epsilon(6)) time, for globular protein of length n, and it detects alignments that score within an additive error of epsilon from all optima. Thus, we prove that this task is computationally feasible, although the method that we introduce is too slow to be a useful everyday tool. We argue that such approximate solutions are, in fact, of greater interest than exact ones because of the noisy nature of experimentally determined protein coordinates. The measurement of similarity between a pair of protein structures used by our algorithm involves the Euclidean distance between the structures (appropriately rigidly transformed). We show that an alternative approach, which relies on internal distance matrices, must incorporate sophisticated geometric ingredients if it is to guarantee optimality and run in polynomial time. We use these observations to visualize the scoring function for several real instances of the problem. Our investigations yield insights on the computational complexity of protein alignment under various scoring functions. These insights can be used in the design of scoring functions for which the optimum can be approximated efficiently and perhaps in the development of efficient algorithms for the multiple structural alignment problem. PMID:15304646
Strong washout approximation to resonant leptogenesis
Garbrecht, Björn; Gautier, Florian; Klaric, Juraj E-mail: florian.gautier@tum.de
2014-09-01
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 approximated 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 approximation 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 the 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 approximation 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.
Photoelectron spectroscopy and the dipole approximation
Hemmers, O.; Hansen, D.L.; Wang, H.
1997-04-01
Photoelectron spectroscopy is a powerful technique because it directly probes, via the measurement of photoelectron kinetic energies, orbital and band structure in valence and core levels in a wide variety of samples. The technique becomes even more powerful when it is performed in an angle-resolved mode, where photoelectrons are distinguished not only by their kinetic energy, but by their direction of emission as well. Determining the probability of electron ejection as a function of angle probes the different quantum-mechanical channels available to a photoemission process, because it is sensitive to phase differences among the channels. As a result, angle-resolved photoemission has been used successfully for many years to provide stringent tests of the understanding of basic physical processes underlying gas-phase and solid-state interactions with radiation. One mainstay in the application of angle-resolved photoelectron spectroscopy is the well-known electric-dipole approximation for photon interactions. In this simplification, all higher-order terms, such as those due to electric-quadrupole and magnetic-dipole interactions, are neglected. As the photon energy increases, however, effects beyond the dipole approximation become important. To best determine the range of validity of the dipole approximation, photoemission measurements on a simple atomic system, neon, where extra-atomic effects cannot play a role, were performed at BL 8.0. The measurements show that deviations from {open_quotes}dipole{close_quotes} expectations in angle-resolved valence photoemission are observable for photon energies down to at least 0.25 keV, and are quite significant at energies around 1 keV. From these results, it is clear that non-dipole angular-distribution effects may need to be considered in any application of angle-resolved photoelectron spectroscopy that uses x-ray photons of energies as low as a few hundred eV.
Virial expansion coefficients in the harmonic approximation.
Armstrong, J R; Zinner, N T; Fedorov, D V; Jensen, A S
2012-08-01
The virial expansion method is applied within a harmonic approximation to an interacting N-body system of identical fermions. We compute the canonical partition functions for two and three particles to get the two lowest orders in the expansion. The energy spectrum is carefully interpolated to reproduce ground-state properties at low temperature and the noninteracting high-temperature limit of constant virial coefficients. This resembles the smearing of shell effects in finite systems with increasing temperature. Numerical results are discussed for the second and third virial coefficients as functions of dimension, temperature, interaction, and transition temperature between low- and high-energy limits. PMID:23005730
Partially coherent contrast-transfer-function approximation.
Nesterets, Yakov I; Gureyev, Timur E
2016-04-01
The contrast-transfer-function (CTF) approximation, widely used in various phase-contrast imaging techniques, is revisited. CTF validity conditions are extended to a wide class of strongly absorbing and refracting objects, as well as to nonuniform partially coherent incident illumination. Partially coherent free-space propagators, describing amplitude and phase in-line contrast, are introduced and their properties are investigated. The present results are relevant to the design of imaging experiments with partially coherent sources, as well as to the analysis and interpretation of the corresponding images. PMID:27140752
Structural design utilizing updated, approximate sensitivity derivatives
NASA Technical Reports Server (NTRS)
Scotti, Stephen J.
1993-01-01
A method to improve the computational efficiency of structural optimization algorithms is investigated. In this method, the calculations of 'exact' sensitivity derivatives of constraint functions are performed only at selected iterations during the optimization process. The sensitivity derivatives utilized within other iterations are approximate derivatives which are calculated using an inexpensive derivative update formula. Optimization results are presented for an analytic optimization problem (i.e., one having simple polynomial expressions for the objective and constraint functions) and for two structural optimization problems. The structural optimization results indicate that up to a factor of three improvement in computation time is possible when using the updated sensitivity derivatives.
Relativistic Random Phase Approximation At Finite Temperature
Niu, Y. F.; Paar, N.; Vretenar, D.; Meng, J.
2009-08-26
The fully self-consistent finite temperature relativistic random phase approximation (FTRRPA) has been established in the single-nucleon basis of the temperature dependent Dirac-Hartree model (FTDH) based on effective Lagrangian with density dependent meson-nucleon couplings. Illustrative calculations in the FTRRPA framework show the evolution of multipole responses of {sup 132}Sn with temperature. With increased temperature, in both monopole and dipole strength distributions additional transitions appear in the low energy region due to the new opened particle-particle and hole-hole transition channels.
Shear viscosity in the postquasistatic approximation
Peralta, C.; Rosales, L.; Rodriguez-Mueller, B.; Barreto, W.
2010-05-15
We apply the postquasistatic approximation, an iterative method for the evolution of self-gravitating spheres of matter, to study the evolution of anisotropic nonadiabatic radiating and dissipative distributions in general relativity. Dissipation is described by viscosity and free-streaming radiation, assuming an equation of state to model anisotropy induced by the shear viscosity. We match the interior solution, in noncomoving coordinates, with the Vaidya exterior solution. Two simple models are presented, based on the Schwarzschild and Tolman VI solutions, in the nonadiabatic and adiabatic limit. In both cases, the eventual collapse or expansion of the distribution is mainly controlled by the anisotropy induced by the viscosity.
Approximations of nonlinear systems having outputs
NASA Technical Reports Server (NTRS)
Hunt, L. R.; Su, R.
1985-01-01
For a nonlinear system with output derivative x = f(x) and y = h(x), two types of linearizations about a point x(0) in state space are considered. One is the usual Taylor series approximation, and the other is defined by linearizing the appropriate Lie derivatives of the output with respect to f about x(0). The latter is called the obvservation model and appears to be quite natural for observation. It is noted that there is a coordinate system in which these two kinds of linearizations agree. In this coordinate system, a technique to construct an observer is introduced.
Pseudoscalar transition form factors from rational approximants
NASA Astrophysics Data System (ADS)
Masjuan, Pere
2014-06-01
The π0, η, and η' transition form factors in the space-like region are analyzed at low and intermediate energies in a model-independent way through the use of rational approximants. Slope and curvature parameters as well as their values at infinity are extracted from experimental data. These results are suited for constraining hadronic models such as the ones used for the hadronic light-by-light scattering part of the anomalous magnetic moment of the muon, and for the mixing parameters of the η - η' system.
Investigating Material Approximations in Spacecraft Radiation Analysis
NASA Technical Reports Server (NTRS)
Walker, Steven A.; Slaba, Tony C.; Clowdsley, Martha S.; Blattnig, Steve R.
2011-01-01
During the design process, the configuration of space vehicles and habitats changes frequently and the merits of design changes must be evaluated. Methods for rapidly assessing astronaut exposure are therefore required. Typically, approximations are made to simplify the geometry and speed up the evaluation of each design. In this work, the error associated with two common approximations used to simplify space radiation vehicle analyses, scaling into equivalent materials and material reordering, are investigated. Over thirty materials commonly found in spacesuits, vehicles, and human bodies are considered. Each material is placed in a material group (aluminum, polyethylene, or tissue), and the error associated with scaling and reordering was quantified for each material. Of the scaling methods investigated, range scaling is shown to be the superior method, especially for shields less than 30 g/cm2 exposed to a solar particle event. More complicated, realistic slabs are examined to quantify the separate and combined effects of using equivalent materials and reordering. The error associated with material reordering is shown to be at least comparable to, if not greater than, the error associated with range scaling. In general, scaling and reordering errors were found to grow with the difference between the average nuclear charge of the actual material and average nuclear charge of the equivalent material. Based on this result, a different set of equivalent materials (titanium, aluminum, and tissue) are substituted for the commonly used aluminum, polyethylene, and tissue. The realistic cases are scaled and reordered using the new equivalent materials, and the reduced error is shown.
Analytic approximate radiation effects due to Bremsstrahlung
Ben-Zvi I.
2012-02-01
The purpose of this note is to provide analytic approximate expressions that can provide quick estimates of the various effects of the Bremsstrahlung radiation produced relatively low energy electrons, such as the dumping of the beam into the beam stop at the ERL or field emission in superconducting cavities. The purpose of this work is not to replace a dependable calculation or, better yet, a measurement under real conditions, but to provide a quick but approximate estimate for guidance purposes only. These effects include dose to personnel, ozone generation in the air volume exposed to the radiation, hydrogen generation in the beam dump water cooling system and radiation damage to near-by magnets. These expressions can be used for other purposes, but one should note that the electron beam energy range is limited. In these calculations the good range is from about 0.5 MeV to 10 MeV. To help in the application of this note, calculations are presented as a worked out example for the beam dump of the R&D Energy Recovery Linac.
Variational extensions of the mean spherical approximation
NASA Astrophysics Data System (ADS)
Blum, L.; Ubriaco, M.
2000-04-01
In a previous work we have proposed a method to study complex systems with objects of arbitrary size. For certain specific forms of the atomic and molecular interactions, surprisingly simple and accurate theories (The Variational Mean Spherical Scaling Approximation, VMSSA) [(Velazquez, Blum J. Chem. Phys. 110 (1990) 10 931; Blum, Velazquez, J. Quantum Chem. (Theochem), in press)] can be obtained. The basic idea is that if the interactions can be expressed in a rapidly converging sum of (complex) exponentials, then the Ornstein-Zernike equation (OZ) has an analytical solution. This analytical solution is used to construct a robust interpolation scheme, the variation mean spherical scaling approximation (VMSSA). The Helmholtz excess free energy Δ A=Δ E- TΔ S is then written as a function of a scaling matrix Γ. Both the excess energy Δ E( Γ) and the excess entropy Δ S( Γ) will be functionals of Γ. In previous work of this series the form of this functional was found for the two- (Blum, Herrera, Mol. Phys. 96 (1999) 821) and three-exponential closures of the OZ equation (Blum, J. Stat. Phys., submitted for publication). In this paper we extend this to M Yukawas, a complete basis set: We obtain a solution for the one-component case and give a closed-form expression for the MSA excess entropy, which is also the VMSSA entropy.
Spectrally Invariant Approximation within Atmospheric Radiative Transfer
NASA Technical Reports Server (NTRS)
Marshak, A.; Knyazikhin, Y.; Chiu, J. C.; Wiscombe, W. J.
2011-01-01
Certain algebraic combinations of single scattering albedo and solar radiation reflected from, or transmitted through, vegetation canopies do not vary with wavelength. These spectrally invariant relationships are the consequence of wavelength independence of the extinction coefficient and scattering phase function in vegetation. In general, this wavelength independence does not hold in the atmosphere, but in cloud-dominated atmospheres the total extinction and total scattering phase function vary only weakly with wavelength. This paper identifies the atmospheric conditions under which the spectrally invariant approximation can accurately describe the extinction and scattering properties of cloudy atmospheres. The validity of the assumptions and the accuracy of the approximation are tested with 1D radiative transfer calculations using publicly available radiative transfer models: Discrete Ordinate Radiative Transfer (DISORT) and Santa Barbara DISORT Atmospheric Radiative Transfer (SBDART). It is shown for cloudy atmospheres with cloud optical depth above 3, and for spectral intervals that exclude strong water vapor absorption, that the spectrally invariant relationships found in vegetation canopy radiative transfer are valid to better than 5%. The physics behind this phenomenon, its mathematical basis, and possible applications to remote sensing and climate are discussed.
Approximation of Failure Probability Using Conditional Sampling
NASA Technical Reports Server (NTRS)
Giesy. Daniel P.; Crespo, Luis G.; Kenney, Sean P.
2008-01-01
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 approximation 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 approximating 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.
Function approximation using adaptive and overlapping intervals
Patil, R.B.
1995-05-01
A problem common to many disciplines is to approximate a function given only the values of the function at various points in input variable space. A method is proposed for approximating a function of several to one variable. The model takes the form of weighted averaging of overlapping basis functions defined over intervals. The number of such basis functions and their parameters (widths and centers) are automatically determined using given training data and a learning algorithm. The proposed algorithm can be seen as placing a nonuniform multidimensional grid in the input domain with overlapping cells. The non-uniformity and overlap of the cells is achieved by a learning algorithm to optimize a given objective function. This approach is motivated by the fuzzy modeling approach and a learning algorithms used for clustering and classification in pattern recognition. The basics of why and how the approach works are given. Few examples of nonlinear regression and classification are modeled. The relationship between the proposed technique, radial basis neural networks, kernel regression, probabilistic neural networks, and fuzzy modeling is explained. Finally advantages and disadvantages are discussed.
On some applications of diophantine approximations
Chudnovsky, G. V.
1984-01-01
Siegel's results [Siegel, C. L. (1929) Abh. Preuss. Akad. Wiss. Phys.-Math. Kl. 1] on the transcendence and algebraic independence of values of E-functions are refined to obtain the best possible bound for the measures of irrationality and linear independence of values of arbitrary E-functions at rational points. Our results show that values of E-functions at rational points have measures of diophantine approximations typical to “almost all” numbers. In particular, any such number has the “2 + ε” exponent of irrationality: ǀΘ - p/qǀ > ǀqǀ-2-ε for relatively prime rational integers p,q, with q ≥ q0 (Θ, ε). These results answer some problems posed by Lang. The methods used here are based on the introduction of graded Padé approximations to systems of functions satisfying linear differential equations with rational function coefficients. The constructions and proofs of this paper were used in the functional (nonarithmetic case) in a previous paper [Chudnovsky, D. V. & Chudnovsky, G. V. (1983) Proc. Natl. Acad. Sci. USA 80, 5158-5162]. PMID:16593441
Chiral Magnetic Effect in Hydrodynamic Approximation
NASA Astrophysics Data System (ADS)
Zakharov, Valentin I.
We review derivations of the chiral magnetic effect (ChME) in hydrodynamic approximation. The reader is assumed to be familiar with the basics of the effect. The main challenge now is to account for the strong interactions between the constituents of the fluid. The main result is that the ChME is not renormalized: in the hydrodynamic approximation it remains the same as for non-interacting chiral fermions moving in an external magnetic field. The key ingredients in the proof are general laws of thermodynamics and the Adler-Bardeen theorem for the chiral anomaly in external electromagnetic fields. The chiral magnetic effect in hydrodynamics represents a macroscopic manifestation of a quantum phenomenon (chiral anomaly). Moreover, one can argue that the current induced by the magnetic field is dissipation free and talk about a kind of "chiral superconductivity". More precise description is a quantum ballistic transport along magnetic field taking place in equilibrium and in absence of a driving force. The basic limitation is the exact chiral limit while temperature—excitingly enough—does not seemingly matter. What is still lacking, is a detailed quantum microscopic picture for the ChME in hydrodynamics. Probably, the chiral currents propagate through lower-dimensional defects, like vortices in superfluid. In case of superfluid, the prediction for the chiral magnetic effect remains unmodified although the emerging dynamical picture differs from the standard one.
NASA Astrophysics Data System (ADS)
Sultan, Cornel
2010-10-01
The design of vector second-order linear systems for accurate proportional damping approximation is addressed. For this purpose an error system is defined using the difference between the generalized coordinates of the non-proportionally damped system and its proportionally damped approximation in modal space. The accuracy of the approximation is characterized using the energy gain of the error system and the design problem is formulated as selecting parameters of the non-proportionally damped system to ensure that this gain is sufficiently small. An efficient algorithm that combines linear matrix inequalities and simultaneous perturbation stochastic approximation is developed to solve the problem and examples of its application to tensegrity structures design are presented.
Approximate flavor symmetries in the lepton sector
Rasin, A. ); Silva, J.P. )
1994-01-01
Approximate 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 MSW solution to the solar neutrino problem is always easily fit. Further, the [nu][sub [mu]-][nu][sub [tau
Generic sequential sampling for metamodel approximations
Turner, C. J.; Campbell, M. I.
2003-01-01
Metamodels approximate complex multivariate data sets from simulations and experiments. These data sets often are not based on an explicitly defined function. The resulting metamodel represents a complex system's behavior for subsequent analysis or optimization. Often an exhaustive data search to obtain the data for the metalnodel is impossible, so an intelligent sampling strategy is necessary. While inultiple approaches have been advocated, the majority of these approaches were developed in support of a particular class of metamodel, known as a Kriging. A more generic, cotninonsense approach to this problem allows sequential sampling techniques to be applied to other types of metamodeis. This research compares recent search techniques for Kriging inetamodels with a generic, inulti-criteria approach combined with a new type of B-spline metamodel. This B-spline metamodel is competitive with prior results obtained with a Kriging metamodel. Furthermore, the results of this research highlight several important features necessary for these techniques to be extended to more complex domains.
PROX: Approximated Summarization of Data Provenance
Ainy, Eleanor; Bourhis, Pierre; Davidson, Susan B.; Deutch, Daniel; Milo, Tova
2016-01-01
Many modern applications involve collecting large amounts of data from multiple sources, and then aggregating and manipulating it in intricate ways. The complexity of such applications, combined with the size of the collected data, makes it difficult to understand the application logic and how information was derived. Data provenance has been proven helpful in this respect in different contexts; however, maintaining and presenting the full and exact provenance may be infeasible, due to its size and complex structure. For that reason, we introduce the notion of approximated summarized provenance, where we seek a compact representation of the provenance at the possible cost of information loss. Based on this notion, we have developed PROX, a system for the management, presentation and use of data provenance for complex applications. We propose to demonstrate PROX in the context of a movies rating crowd-sourcing system, letting participants view provenance summarization and use it to gain insights on the application and its underlying data. PMID:27570843
Animal models and integrated nested Laplace approximations.
Holand, Anna Marie; Steinsland, Ingelin; Martino, Sara; Jensen, Henrik
2013-08-07
Animal models are generalized linear mixed models used in evolutionary biology and animal breeding to identify the genetic part of traits. Integrated Nested Laplace Approximation (INLA) is a methodology for making fast, nonsampling-based Bayesian inference for hierarchical Gaussian Markov models. In this article, we demonstrate that the INLA methodology can be used for many versions of Bayesian animal models. We analyze animal models for both synthetic case studies and house sparrow (Passer domesticus) population case studies with Gaussian, binomial, and Poisson likelihoods using INLA. Inference results are compared with results using Markov Chain Monte Carlo methods. For model choice we use difference in deviance information criteria (DIC). We suggest and show how to evaluate differences in DIC by comparing them with sampling results from simulation studies. We also introduce an R package, AnimalINLA, for easy and fast inference for Bayesian Animal models using INLA.
Exact and Approximate Probabilistic Symbolic Execution
NASA Technical Reports Server (NTRS)
Luckow, Kasper; Pasareanu, Corina S.; Dwyer, Matthew B.; Filieri, Antonio; Visser, Willem
2014-01-01
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 approximate 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.
Architecture-independent approximation of functions.
Ruiz De Angulo, V; Torras, C
2001-05-01
We show that minimizing the expected error of a feedforward network over a distribution of weights results in an approximation that tends to be independent of network size as the number of hidden units grows. This minimization can be easily performed, and the complexity of the resulting function implemented by the network is regulated by the variance of the weight distribution. For a fixed variance, there is a number of hidden units above which either the implemented function does not change or the change is slight and tends to zero as the size of the network grows. In sum, the control of the complexity depends on only the variance, not the architecture, provided it is large enough.
Approximate truncation robust computed tomography—ATRACT
NASA Astrophysics Data System (ADS)
Dennerlein, Frank; Maier, Andreas
2013-09-01
We present an approximate truncation robust algorithm to compute tomographic images (ATRACT). This algorithm targets at reconstructing volumetric images from cone-beam projections in scenarios where these projections are highly truncated in each dimension. It thus facilitates reconstructions of small subvolumes of interest, without involving prior knowledge about the object. Our method is readily applicable to medical C-arm imaging, where it may contribute to new clinical workflows together with a considerable reduction of x-ray dose. We give a detailed derivation of ATRACT that starts from the conventional Feldkamp filtered-backprojection algorithm and that involves, as one component, a novel original formula for the inversion of the two-dimensional Radon transform. Discretization and numerical implementation are discussed and reconstruction results from both, simulated projections and first clinical data sets are presented.
Optimal aeroassisted guidance using Loh's term approximations
NASA Technical Reports Server (NTRS)
Mceneaney, W. M.
1989-01-01
This paper presents three guidance algorithms for aerocapture and/or aeroassisted orbital transfer with plane change. All three algorithms are based on the approximate solution of an optimal control problem at each guidance update. The chief assumption is that Loh's term may be modeled as a function of the independent variable only. The first two algorithms maximize exit speed for fixed exit altitude, flight path angle and heading angle. The third minimizes, in one sense, the control effort for fixed exit altitude, flight path angle, heading angle and speed. Results are presented which indicate the near optimality of the solutions generated by the first two algorithms. Results are also presented which indicate the performance of the third algorithm in a simulation with unmodeled atmospheric density disturbances.
Fast Approximate Quadratic Programming for Graph Matching
Vogelstein, Joshua T.; Conroy, John M.; Lyzinski, Vince; Podrazik, Louis J.; Kratzer, Steven G.; Harley, Eric T.; Fishkind, Donniell E.; Vogelstein, R. Jacob; Priebe, Carey E.
2015-01-01
Quadratic assignment problems arise in a wide variety of domains, spanning operations research, graph theory, computer vision, and neuroscience, to name a few. The graph matching problem is a special case of the quadratic assignment problem, and graph matching is increasingly important as graph-valued data is becoming more prominent. With the aim of efficiently and accurately matching the large graphs common in big data, we present our graph matching algorithm, the Fast Approximate Quadratic assignment algorithm. We empirically demonstrate that our algorithm is faster and achieves a lower objective value on over 80% of the QAPLIB benchmark library, compared with the previous state-of-the-art. Applying our algorithm to our motivating example, matching C. elegans connectomes (brain-graphs), we find that it efficiently achieves performance. PMID:25886624
Collective pairing Hamiltonian in the GCM approximation
NASA Astrophysics Data System (ADS)
Góźdź, A.; Pomorski, K.; Brack, M.; Werner, E.
1985-08-01
Using the generator coordinate method and the gaussian overlap approximation we derived the collective Schrödinger-type equation starting from a microscopic single-particle plus pairing hamiltonian for one kind of particle. The BCS wave function was used as the generator function. The pairing energy-gap parameter Δ and the gauge transformation anglewere taken as the generator coordinates. Numerical results have been obtained for the full and the mean-field pairing hamiltonians and compared with the cranking estimates. A significant role played by the zero-point energy correction in the collective pairing potential is found. The ground-state energy dependence on the pairing strength agrees very well with the exact solution of the Richardson model for a set of equidistant doubly-degenerate single-particle levels.
Squashed entanglement and approximate private states
NASA Astrophysics Data System (ADS)
Wilde, Mark M.
2016-09-01
The squashed entanglement is a fundamental entanglement measure in quantum information theory, finding application as an upper bound on the distillable secret key or distillable entanglement of a quantum state or a quantum channel. This paper simplifies proofs that the squashed entanglement is an upper bound on distillable key for finite-dimensional quantum systems and solidifies such proofs for infinite-dimensional quantum systems. More specifically, this paper establishes that the logarithm of the dimension of the key system (call it log 2K ) in an ɛ -approximate private state is bounded from above by the squashed entanglement of that state plus a term that depends only ɛ and log 2K . Importantly, the extra term does not depend on the dimension of the shield systems of the private state. The result holds for the bipartite squashed entanglement, and an extension of this result is established for two different flavors of the multipartite squashed entanglement.
Improved effective vector boson approximation revisited
NASA Astrophysics Data System (ADS)
Bernreuther, Werner; Chen, Long
2016-03-01
We reexamine the improved effective vector boson approximation which is based on two-vector-boson luminosities Lpol for the computation of weak gauge-boson hard scattering subprocesses V1V2→W in high-energy hadron-hadron or e-e+ collisions. We calculate these luminosities for the nine combinations of the transverse and longitudinal polarizations of V1 and V2 in the unitary and axial gauge. For these two gauge choices the quality of this approach is investigated for the reactions e-e+→W-W+νeν¯ e and e-e+→t t ¯ νeν¯ e using appropriate phase-space cuts.
Improved approximations for control augmented structural synthesis
NASA Technical Reports Server (NTRS)
Thomas, H. L.; Schmit, L. A.
1990-01-01
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 approximations 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.
Comparing numerical and analytic approximate gravitational waveforms
NASA Astrophysics Data System (ADS)
Afshari, Nousha; Lovelace, Geoffrey; SXS Collaboration
2016-03-01
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 approximations.
Turbo Equalization Using Partial Gaussian Approximation
NASA Astrophysics Data System (ADS)
Zhang, Chuanzong; Wang, Zhongyong; Manchon, Carles Navarro; Sun, Peng; Guo, Qinghua; Fleury, Bernard Henri
2016-09-01
This paper deals with turbo-equalization for coded data transmission over intersymbol interference (ISI) channels. We propose a message-passing algorithm that uses the expectation-propagation rule to convert messages passed from the demodulator-decoder to the equalizer and computes messages returned by the equalizer by using a partial Gaussian approximation (PGA). Results from Monte Carlo simulations show that this approach leads to a significant performance improvement compared to state-of-the-art turbo-equalizers and allows for trading performance with complexity. We exploit the specific structure of the ISI channel model to significantly reduce the complexity of the PGA compared to that considered in the initial paper proposing the method.
Heat flow in the postquasistatic approximation
Rodriguez-Mueller, B.; Peralta, C.; Barreto, W.; Rosales, L.
2010-08-15
We apply the postquasistatic approximation to study the evolution of spherically symmetric fluid distributions undergoing dissipation in the form of radial heat flow. For a model that corresponds to an incompressible fluid departing from the static equilibrium, it is not possible to go far from the initial state after the emission of a small amount of energy. Initially collapsing distributions of matter are not permitted. Emission of energy can be considered as a mechanism to avoid the collapse. If the distribution collapses initially and emits one hundredth of the initial mass only the outermost layers evolve. For a model that corresponds to a highly compressed Fermi gas, only the outermost shell can evolve with a shorter hydrodynamic time scale.
Approximate Bayesian computation with functional statistics.
Soubeyrand, Samuel; Carpentier, Florence; Guiton, François; Klein, Etienne K
2013-03-26
Functional statistics are commonly used to characterize spatial patterns in general and spatial genetic structures in population genetics in particular. Such functional statistics also enable the estimation of parameters of spatially explicit (and genetic) models. Recently, Approximate Bayesian Computation (ABC) has been proposed to estimate model parameters from functional statistics. However, applying ABC with functional statistics may be cumbersome because of the high dimension of the set of statistics and the dependences among them. To tackle this difficulty, we propose an ABC procedure which relies on an optimized weighted distance between observed and simulated functional statistics. We applied this procedure to a simple step model, a spatial point process characterized by its pair correlation function and a pollen dispersal model characterized by genetic differentiation as a function of distance. These applications showed how the optimized weighted distance improved estimation accuracy. In the discussion, we consider the application of the proposed ABC procedure to functional statistics characterizing non-spatial processes.
Spline Approximation of Thin Shell Dynamics
NASA Technical Reports Server (NTRS)
delRosario, R. C. H.; Smith, R. C.
1996-01-01
A spline-based method for approximating 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.
An approximate CPHD filter for superpositional sensors
NASA Astrophysics Data System (ADS)
Mahler, Ronald; El-Fallah, Adel
2012-06-01
Most multitarget tracking algorithms, such as JPDA, MHT, and the PHD and CPHD filters, presume the following measurement model: (a) targets are point targets, (b) every target generates at most a single measurement, and (c) any measurement is generated by at most a single target. However, the most familiar sensors, such as surveillance and imaging radars, violate assumption (c). This is because they are actually superpositional-that is, any measurement is a sum of signals generated by all of the targets in the scene. At this conference in 2009, the first author derived exact formulas for PHD and CPHD filters that presume general superpositional measurement models. Unfortunately, these formulas are computationally intractable. In this paper, we modify and generalize a Gaussian approximation technique due to Thouin, Nannuru, and Coates to derive a computationally tractable superpositional-CPHD filter. Implementation requires sequential Monte Carlo (particle filter) techniques.
Approximation Preserving Reductions among Item Pricing Problems
NASA Astrophysics Data System (ADS)
Hamane, Ryoso; Itoh, Toshiya; Tomita, Kouhei
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 approximation preserving reductions among several item pricing problems and show that all of them have algorithms with good approximation ratio.
Robust Generalized Low Rank Approximations of Matrices.
Shi, Jiarong; Yang, Wei; Zheng, Xiuyun
2015-01-01
In recent years, the intrinsic low rank structure of some datasets has been extensively exploited to reduce dimensionality, remove noise and complete the missing entries. As a well-known technique for dimensionality reduction and data compression, Generalized Low Rank Approximations of Matrices (GLRAM) claims its superiority on computation time and compression ratio over the SVD. However, GLRAM is very sensitive to sparse large noise or outliers and its robust version does not have been explored or solved yet. To address this problem, this paper proposes a robust method for GLRAM, named Robust GLRAM (RGLRAM). We first formulate RGLRAM as an l1-norm optimization problem which minimizes the l1-norm of the approximation errors. Secondly, we apply the technique of Augmented Lagrange Multipliers (ALM) to solve this l1-norm minimization problem and derive a corresponding iterative scheme. Then the weak convergence of the proposed algorithm is discussed under mild conditions. Next, we investigate a special case of RGLRAM and extend RGLRAM to a general tensor case. Finally, the extensive experiments on synthetic data show that it is possible for RGLRAM to exactly recover both the low rank and the sparse components while it may be difficult for previous state-of-the-art algorithms. We also discuss three issues on RGLRAM: the sensitivity to initialization, the generalization ability and the relationship between the running time and the size/number of matrices. Moreover, the experimental results on images of faces with large corruptions illustrate that RGLRAM obtains the best denoising and compression performance than other methods. PMID:26367116
Robust Generalized Low Rank Approximations of Matrices
Shi, Jiarong; Yang, Wei; Zheng, Xiuyun
2015-01-01
In recent years, the intrinsic low rank structure of some datasets has been extensively exploited to reduce dimensionality, remove noise and complete the missing entries. As a well-known technique for dimensionality reduction and data compression, Generalized Low Rank Approximations of Matrices (GLRAM) claims its superiority on computation time and compression ratio over the SVD. However, GLRAM is very sensitive to sparse large noise or outliers and its robust version does not have been explored or solved yet. To address this problem, this paper proposes a robust method for GLRAM, named Robust GLRAM (RGLRAM). We first formulate RGLRAM as an l1-norm optimization problem which minimizes the l1-norm of the approximation errors. Secondly, we apply the technique of Augmented Lagrange Multipliers (ALM) to solve this l1-norm minimization problem and derive a corresponding iterative scheme. Then the weak convergence of the proposed algorithm is discussed under mild conditions. Next, we investigate a special case of RGLRAM and extend RGLRAM to a general tensor case. Finally, the extensive experiments on synthetic data show that it is possible for RGLRAM to exactly recover both the low rank and the sparse components while it may be difficult for previous state-of-the-art algorithms. We also discuss three issues on RGLRAM: the sensitivity to initialization, the generalization ability and the relationship between the running time and the size/number of matrices. Moreover, the experimental results on images of faces with large corruptions illustrate that RGLRAM obtains the best denoising and compression performance than other methods. PMID:26367116
Network histograms and universality of blockmodel approximation
Olhede, Sofia C.; Wolfe, Patrick J.
2014-01-01
In this paper we introduce the network histogram, a statistical summary of network interactions to be used as a tool for exploratory data analysis. A network histogram is obtained by fitting a stochastic blockmodel to a single observation of a network dataset. Blocks of edges play the role of histogram bins and community sizes that of histogram bandwidths or bin sizes. Just as standard histograms allow for varying bandwidths, different blockmodel estimates can all be considered valid representations of an underlying probability model, subject to bandwidth constraints. Here we provide methods for automatic bandwidth selection, by which the network histogram approximates the generating mechanism that gives rise to exchangeable random graphs. This makes the blockmodel a universal network representation for unlabeled graphs. With this insight, we discuss the interpretation of network communities in light of the fact that many different community assignments can all give an equally valid representation of such a network. To demonstrate the fidelity-versus-interpretability tradeoff inherent in considering different numbers and sizes of communities, we analyze two publicly available networks—political weblogs and student friendships—and discuss how to interpret the network histogram when additional information related to node and edge labeling is present. PMID:25275010
[Complex systems variability analysis using approximate entropy].
Cuestas, Eduardo
2010-01-01
Biological systems are highly complex systems, both spatially and temporally. They are rooted in an interdependent, redundant and pleiotropic interconnected dynamic network. The properties of a system are different from those of their parts, and they depend on the integrity of the whole. The systemic properties vanish when the system breaks down, while the properties of its components are maintained. The disease can be understood as a systemic functional alteration of the human body, which present with a varying severity, stability and durability. Biological systems are characterized by measurable complex rhythms, abnormal rhythms are associated with disease and may be involved in its pathogenesis, they are been termed "dynamic disease." Physicians have long time recognized that alterations of physiological rhythms are associated with disease. Measuring absolute values of clinical parameters yields highly significant, clinically useful information, however evaluating clinical parameters the variability provides additionally useful clinical information. The aim of this review was to study one of the most recent advances in the measurement and characterization of biological variability made possible by the development of mathematical models based on chaos theory and nonlinear dynamics, as approximate entropy, has provided us with greater ability to discern meaningful distinctions between biological signals from clinically distinct groups of patients.
A simple, approximate model of parachute inflation
Macha, J.M.
1992-01-01
A simple, approximate 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 fluid 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.
A simple, approximate model of parachute inflation
Macha, J.M.
1992-11-01
A simple, approximate 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 fluid 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.
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.
Adaptive approximation of higher order posterior statistics
Lee, Wonjung
2014-02-01
Filtering is an approach for incorporating observed data into time-evolving systems. Instead of a family of Dirac delta masses that is widely used in Monte Carlo methods, we here use the Wiener chaos expansion for the parametrization of the conditioned probability distribution to solve the nonlinear filtering problem. The Wiener chaos expansion is not the best method for uncertainty propagation without observations. Nevertheless, the projection of the system variables in a fixed polynomial basis spanning the probability space might be a competitive representation in the presence of relatively frequent observations because the Wiener chaos approach not only leads to an accurate and efficient prediction for short time uncertainty quantification, but it also allows to apply several data assimilation methods that can be used to yield a better approximate filtering solution. The aim of the present paper is to investigate this hypothesis. We answer in the affirmative for the (stochastic) Lorenz-63 system based on numerical simulations in which the uncertainty quantification method and the data assimilation method are adaptively selected by whether the dynamics is driven by Brownian motion and the near-Gaussianity of the measure to be updated, respectively.
The time-dependent Gutzwiller approximation
NASA Astrophysics Data System (ADS)
Fabrizio, Michele
2015-03-01
The time-dependent Gutzwiller Approximation (t-GA) is shown to be capable of tracking the off-equilibrium evolution both of coherent quasiparticles and of incoherent Hubbard bands. The method is used to demonstrate that the sharp dynamical crossover observed by time-dependent DMFT in the quench-dynamics of a half-filled Hubbard model can be identified within the t-GA as a genuine dynamical transition separating two distinct physical phases. This result, strictly variational for lattices of infinite coordination number, is intriguing as it actually questions the occurrence of thermalization. Next, we shall present how t-GA works in a multi-band model for V2O3 that displays a first-order Mott transition. We shall show that a physically accessible excitation pathway is able to collapse the Mott gap down and drive off-equilibrium the insulator into a metastable metal phase. Work supported by the European Union, Seventh Framework Programme, under the project GO FAST, Grant Agreement No. 280555.
[Complex systems variability analysis using approximate entropy].
Cuestas, Eduardo
2010-01-01
Biological systems are highly complex systems, both spatially and temporally. They are rooted in an interdependent, redundant and pleiotropic interconnected dynamic network. The properties of a system are different from those of their parts, and they depend on the integrity of the whole. The systemic properties vanish when the system breaks down, while the properties of its components are maintained. The disease can be understood as a systemic functional alteration of the human body, which present with a varying severity, stability and durability. Biological systems are characterized by measurable complex rhythms, abnormal rhythms are associated with disease and may be involved in its pathogenesis, they are been termed "dynamic disease." Physicians have long time recognized that alterations of physiological rhythms are associated with disease. Measuring absolute values of clinical parameters yields highly significant, clinically useful information, however evaluating clinical parameters the variability provides additionally useful clinical information. The aim of this review was to study one of the most recent advances in the measurement and characterization of biological variability made possible by the development of mathematical models based on chaos theory and nonlinear dynamics, as approximate entropy, has provided us with greater ability to discern meaningful distinctions between biological signals from clinically distinct groups of patients. PMID:21450141
Approximate algorithms for partitioning and assignment problems
NASA Technical Reports Server (NTRS)
Iqbal, M. A.
1986-01-01
The problem of optimally assigning the modules of a parallel/pipelined program over the processors of a multiple computer system under certain restrictions on the interconnection structure of the program as well as the multiple computer system was considered. For a variety of such programs it is possible to find linear time if a partition of the program exists in which the load on any processor is within a certain bound. This method, when combined with a binary search over a finite range, provides an approximate solution to the partitioning problem. The specific problems considered were: a chain structured parallel program over a chain-like computer system, multiple chain-like programs over a host-satellite system, and a tree structured parallel program over a host-satellite system. For a problem with m modules and n processors, the complexity of the algorithm is no worse than O(mnlog(W sub T/epsilon)), where W sub T is the cost of assigning all modules to one processor and epsilon the desired accuracy.
Approximate Model for Turbulent Stagnation Point Flow.
Dechant, Lawrence
2016-01-01
Here we derive an approximate 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 the 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.
Semiclassical approximation to supersymmetric quantum gravity
NASA Astrophysics Data System (ADS)
Kiefer, Claus; Lück, Tobias; Moniz, Paulo
2005-08-01
We develop a semiclassical approximation scheme for the constraint equations of supersymmetric canonical quantum gravity. This is achieved by a Born-Oppenheimer type of expansion, in analogy to the case of the usual Wheeler-DeWitt equation. The formalism is only consistent if the states at each order depend on the gravitino field. We recover at consecutive orders the Hamilton-Jacobi equation, the functional Schrödinger equation, and quantum gravitational correction terms to this Schrödinger equation. In particular, the following consequences are found: (i) the Hamilton-Jacobi equation and therefore the background spacetime must involve the gravitino, (ii) a (many-fingered) local time parameter has to be present on super Riem Σ (the space of all possible tetrad and gravitino fields), (iii) quantum supersymmetric gravitational corrections affect the evolution of the very early Universe. The physical meaning of these equations and results, in particular, the similarities to and differences from the pure bosonic case, are discussed.
Approximate theory for radial filtration/consolidation
Tiller, F.M.; Kirby, J.M.; Nguyen, H.L.
1996-10-01
Approximate solutions are developed for filtration and subsequent consolidation of compactible cakes on a cylindrical filter element. Darcy`s flow equation is coupled with equations for equilibrium stress under the conditions of plane strain and axial symmetry for radial flow inwards. The solutions are based on power function forms involving the relationships of the solidosity {epsilon}{sub s} (volume fraction of solids) and the permeability K to the solids effective stress p{sub s}. The solutions allow determination of the various parameters in the power functions and the ratio k{sub 0} of the lateral to radial effective stress (earth stress ratio). Measurements were made of liquid and effective pressures, flow rates, and cake thickness versus time. Experimental data are presented for a series of tests in a radial filtration cell with a central filter element. Slurries prepared from two materials (Microwate, which is mainly SrSO{sub 4}, and kaolin) were used in the experiments. Transient deposition of filter cakes was followed by static (i.e., no flow) conditions in the cake. The no-flow condition was accomplished by introducing bentonite which produced a nearly impermeable layer with negligible flow. Measurement of the pressure at the cake surface and the transmitted pressure on the central element permitted calculation of k{sub 0}.
Configuring Airspace Sectors with Approximate Dynamic Programming
NASA Technical Reports Server (NTRS)
Bloem, Michael; Gupta, Pramod
2010-01-01
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 approximate 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.
Magnetic reconnection under anisotropic magnetohydrodynamic approximation
Hirabayashi, K.; Hoshino, M.
2013-11-15
We study the formation of slow-mode shocks in collisionless magnetic reconnection by using one- and two-dimensional collisionless MHD codes based on the double adiabatic approximation and the Landau closure model. We bridge the gap between the Petschek-type MHD reconnection model accompanied by a pair of slow shocks and the observational evidence of the rare occasion of in-situ slow shock observations. Our results showed that once magnetic reconnection takes place, a firehose-sense (p{sub ∥}>p{sub ⊥}) pressure anisotropy arises in the downstream region, and the generated slow shocks are quite weak comparing with those in an isotropic MHD. In spite of the weakness of the shocks, however, the resultant reconnection rate is 10%–30% higher than that in an isotropic case. This result implies that the slow shock does not necessarily play an important role in the energy conversion in the reconnection system and is consistent with the satellite observation in the Earth's magnetosphere.
Rainbows: Mie computations and the Airy approximation.
Wang, R T; van de Hulst, H C
1991-01-01
Efficient and accurate computation of the scattered intensity pattern by the Mie formulas is now feasible for size parameters up to x = 50,000 at least, which in visual light means spherical drops with diameters up to 6 mm. We present a method for evaluating the Mie coefficients from the ratios between Riccati-Bessel and Neumann functions of successive order. We probe the applicability of the Airy approximation, which we generalize to rainbows of arbitrary p (number of internal reflections = p - 1), by comparing the Mie and Airy intensity patterns. Millimeter size water drops show a match in all details, including the position and intensity of the supernumerary maxima and the polarization. A fairly good match is still seen for drops of 0.1 mm. A small spread in sizes helps to smooth out irrelevant detail. The dark band between the rainbows is used to test more subtle features. We conclude that this band contains not only externally reflected light (p = 0) but also a sizable contribution f rom the p = 6 and p = 7 rainbows, which shift rapidly with wavelength. The higher the refractive index, the closer both theories agree on the first primary rainbow (p = 2) peak for drop diameters as small as 0.02 mm. This may be useful in supporting experimental work. PMID:20581954
Bond selective chemistry beyond the adiabatic approximation
Butler, L.J.
1993-12-01
One of the most important challenges in chemistry is to develop predictive ability for the branching between energetically allowed chemical reaction pathways. Such predictive capability, coupled with a fundamental understanding of the important molecular interactions, is essential to the development and utilization of new fuels and the design of efficient combustion processes. Existing transition state and exact quantum theories successfully predict the branching between available product channels for systems in which each reaction coordinate can be adequately described by different paths along a single adiabatic potential energy surface. In particular, unimolecular dissociation following thermal, infrared multiphoton, or overtone excitation in the ground state yields a branching between energetically allowed product channels which can be successfully predicted by the application of statistical theories, i.e. the weakest bond breaks. (The predictions are particularly good for competing reactions in which when there is no saddle point along the reaction coordinates, as in simple bond fission reactions.) The predicted lack of bond selectivity results from the assumption of rapid internal vibrational energy redistribution and the implicit use of a single adiabatic Born-Oppenheimer potential energy surface for the reaction. However, the adiabatic approximation is not valid for the reaction of a wide variety of energetic materials and organic fuels; coupling between the electronic states of the reacting species play a a key role in determining the selectivity of the chemical reactions induced. The work described below investigated the central role played by coupling between electronic states in polyatomic molecules in determining the selective branching between energetically allowed fragmentation pathways in two key systems.
Collisionless magnetic reconnection under anisotropic MHD approximation
NASA Astrophysics Data System (ADS)
Hirabayashi, Kota; Hoshino, Masahiro
We study the formation of slow-mode shocks in collisionless magnetic reconnection by using one- and two-dimensional collisionless magneto-hydro-dynamic (MHD) simulations based on the double adiabatic approximation, which is an important step to bridge the gap between the Petschek-type MHD reconnection model accompanied by a pair of slow shocks and the observational evidence of the rare occasion of in-situ slow shock observation. According to our results, a pair of slow shocks does form in the reconnection layer. The resultant shock waves, however, are quite weak compared with those in an isotropic MHD from the point of view of the plasma compression and the amount of the magnetic energy released across the shock. Once the slow shock forms, the downstream plasma are heated in highly anisotropic manner and a firehose-sense (P_{||}>P_{⊥}) pressure anisotropy arises. The maximum anisotropy is limited by the marginal firehose criterion, 1-(P_{||}-P_{⊥})/B(2) =0. In spite of the weakness of the shocks, the resultant reconnection rate is kept at the same level compared with that in the corresponding ordinary MHD simulations. It is also revealed that the sequential order of propagation of the slow shock and the rotational discontinuity, which appears when the guide field component exists, changes depending on the magnitude of the guide field. Especially, when no guide field exists, the rotational discontinuity degenerates with the contact discontinuity remaining at the position of the initial current sheet, while with the slow shock in the isotropic MHD. Our result implies that the slow shock does not necessarily play an important role in the energy conversion in the reconnection system and is consistent with the satellite observation in the Earth's magnetosphere.
Coronal Loops: Evolving Beyond the Isothermal Approximation
NASA Astrophysics Data System (ADS)
Schmelz, J. T.; Cirtain, J. W.; Allen, J. D.
2002-05-01
Are coronal loops isothermal? A controversy over this question has arisen recently because different investigators using different techniques have obtained very different answers. Analysis of SOHO-EIT and TRACE data using narrowband filter ratios to obtain temperature maps has produced several key publications that suggest that coronal loops may be isothermal. We have constructed a multi-thermal distribution for several pixels along a relatively isolated coronal loop on the southwest limb of the solar disk using spectral line data from SOHO-CDS taken on 1998 Apr 20. These distributions are clearly inconsistent with isothermal plasma along either the line of sight or the length of the loop, and suggested rather that the temperature increases from the footpoints to the loop top. We speculated originally that these differences could be attributed to pixel size -- CDS pixels are larger, and more `contaminating' material would be expected along the line of sight. To test this idea, we used CDS iron line ratios from our data set to mimic the isothermal results from the narrowband filter instruments. These ratios indicated that the temperature gradient along the loop was flat, despite the fact that a more complete analysis of the same data showed this result to be false! The CDS pixel size was not the cause of the discrepancy; rather, the problem lies with the isothermal approximation used in EIT and TRACE analysis. These results should serve as a strong warning to anyone using this simplistic method to obtain temperature. This warning is echoed on the EIT web page: ``Danger! Enter at your own risk!'' In other words, values for temperature may be found, but they may have nothing to do with physical reality. Solar physics research at the University of Memphis is supported by NASA grant NAG5-9783. This research was funded in part by the NASA/TRACE MODA grant for Montana State University.
Hydration thermodynamics beyond the linear response approximation.
Raineri, Fernando O
2016-10-19
The solvation energetics associated with the transformation of a solute molecule at infinite dilution in water from an initial state A to a final state B is reconsidered. The two solute states have different potentials energies of interaction, [Formula: see text] and [Formula: see text], with the solvent environment. Throughout the A [Formula: see text] B transformation of the solute, the solvation system is described by a Hamiltonian [Formula: see text] that changes linearly with the coupling parameter ξ. By focusing on the characterization of the probability density [Formula: see text] that the dimensionless perturbational solute-solvent interaction energy [Formula: see text] has numerical value y when the coupling parameter is ξ, we derive a hierarchy of differential equation relations between the ξ-dependent cumulant functions of various orders in the expansion of the appropriate cumulant generating function. On the basis of this theoretical framework we then introduce an inherently nonlinear solvation model for which we are able to find analytical results for both [Formula: see text] and for the solvation thermodynamic functions. The solvation model is based on the premise that there is an upper or a lower bound (depending on the nature of the interactions considered) to the amplitude of the fluctuations of Y in the solution system at equilibrium. The results reveal essential differences in behavior for the model when compared with the linear response approximation to solvation, particularly with regards to the probability density [Formula: see text]. The analytical expressions for the solvation properties show, however, that the linear response behavior is recovered from the new model when the room for the thermal fluctuations in Y is not restricted by the existence of a nearby bound. We compare the predictions of the model with the results from molecular dynamics computer simulations for aqueous solvation, in which either (1) the solute
Visual nesting impacts approximate number system estimation.
Chesney, Dana L; Gelman, Rochel
2012-08-01
The approximate number system (ANS) allows people to quickly but inaccurately enumerate large sets without counting. One popular account of the ANS is known as the accumulator model. This model posits that the ANS acts analogously to a graduated cylinder to which one "cup" is added for each item in the set, with set numerosity read from the "height" of the cylinder. Under this model, one would predict that if all the to-be-enumerated items were not collected into the accumulator, either the sets would be underestimated, or the misses would need to be corrected by a subsequent process, leading to longer reaction times. In this experiment, we tested whether such miss effects occur. Fifty participants judged numerosities of briefly presented sets of circles. In some conditions, circles were arranged such that some were inside others. This circle nesting was expected to increase the miss rate, since previous research had indicated that items in nested configurations cannot be preattentively individuated in parallel. Logically, items in a set that cannot be simultaneously individuated cannot be simultaneously added to an accumulator. Participants' response times were longer and their estimations were lower for sets whose configurations yielded greater levels of nesting. The level of nesting in a display influenced estimation independently of the total number of items present. This indicates that miss effects, predicted by the accumulator model, are indeed seen in ANS estimation. We speculate that ANS biases might, in turn, influence cognition and behavior, perhaps by influencing which kinds of sets are spontaneously counted. PMID:22810562
Rapid approximate inversion of airborne TEM
NASA Astrophysics Data System (ADS)
Fullagar, Peter K.; Pears, Glenn A.; Reid, James E.; Schaa, Ralf
2015-11-01
Rapid interpretation of large airborne transient electromagnetic (ATEM) datasets is highly desirable for timely decision-making in exploration. Full solution 3D inversion of entire airborne electromagnetic (AEM) surveys is often still not feasible on current day PCs. Therefore, two algorithms to perform rapid approximate 3D interpretation of AEM have been developed. The loss of rigour may be of little consequence if the objective of the AEM survey is regional reconnaissance. Data coverage is often quasi-2D rather than truly 3D in such cases, belying the need for `exact' 3D inversion. Incorporation of geological constraints reduces the non-uniqueness of 3D AEM inversion. Integrated interpretation can be achieved most readily when inversion is applied to a geological model, attributed with lithology as well as conductivity. Geological models also offer several practical advantages over pure property models during inversion. In particular, they permit adjustment of geological boundaries. In addition, optimal conductivities can be determined for homogeneous units. Both algorithms described here can operate on geological models; however, they can also perform `unconstrained' inversion if the geological context is unknown. VPem1D performs 1D inversion at each ATEM data location above a 3D model. Interpretation of cover thickness is a natural application; this is illustrated via application to Spectrem data from central Australia. VPem3D performs 3D inversion on time-integrated (resistive limit) data. Conversion to resistive limits delivers a massive increase in speed since the TEM inverse problem reduces to a quasi-magnetic problem. The time evolution of the decay is lost during the conversion, but the information can be largely recovered by constructing a starting model from conductivity depth images (CDIs) or 1D inversions combined with geological constraints if available. The efficacy of the approach is demonstrated on Spectrem data from Brazil. Both separately and in
Cophylogeny reconstruction via an approximate Bayesian computation.
Baudet, C; Donati, B; Sinaimeri, B; Crescenzi, P; Gautier, C; Matias, C; Sagot, M-F
2015-05-01
Despite an increasingly vast literature on cophylogenetic reconstructions for studying host-parasite associations, understanding the common evolutionary history of such systems remains a problem that is far from being solved. Most algorithms for host-parasite reconciliation use an event-based model, where the events include in general (a subset of) cospeciation, duplication, loss, and host switch. All known parsimonious event-based methods then assign a cost to each type of event in order to find a reconstruction of minimum cost. The main problem with this approach is that the cost of the events strongly influences the reconciliation obtained. Some earlier approaches attempt to avoid this problem by finding a Pareto set of solutions and hence by considering event costs under some minimization constraints. To deal with this problem, we developed an algorithm, called Coala, for estimating the frequency of the events based on an approximate Bayesian computation approach. The benefits of this method are 2-fold: (i) it provides more confidence in the set of costs to be used in a reconciliation, and (ii) it allows estimation of the frequency of the events in cases where the data set consists of trees with a large number of taxa. We evaluate our method on simulated and on biological data sets. We show that in both cases, for the same pair of host and parasite trees, different sets of frequencies for the events lead to equally probable solutions. Moreover, often these solutions differ greatly in terms of the number of inferred events. It appears crucial to take this into account before attempting any further biological interpretation of such reconciliations. More generally, we also show that the set of frequencies can vary widely depending on the input host and parasite trees. Indiscriminately applying a standard vector of costs may thus not be a good strategy. PMID:25540454
Hydration thermodynamics beyond the linear response approximation.
Raineri, Fernando O
2016-10-19
The solvation energetics associated with the transformation of a solute molecule at infinite dilution in water from an initial state A to a final state B is reconsidered. The two solute states have different potentials energies of interaction, [Formula: see text] and [Formula: see text], with the solvent environment. Throughout the A [Formula: see text] B transformation of the solute, the solvation system is described by a Hamiltonian [Formula: see text] that changes linearly with the coupling parameter ξ. By focusing on the characterization of the probability density [Formula: see text] that the dimensionless perturbational solute-solvent interaction energy [Formula: see text] has numerical value y when the coupling parameter is ξ, we derive a hierarchy of differential equation relations between the ξ-dependent cumulant functions of various orders in the expansion of the appropriate cumulant generating function. On the basis of this theoretical framework we then introduce an inherently nonlinear solvation model for which we are able to find analytical results for both [Formula: see text] and for the solvation thermodynamic functions. The solvation model is based on the premise that there is an upper or a lower bound (depending on the nature of the interactions considered) to the amplitude of the fluctuations of Y in the solution system at equilibrium. The results reveal essential differences in behavior for the model when compared with the linear response approximation to solvation, particularly with regards to the probability density [Formula: see text]. The analytical expressions for the solvation properties show, however, that the linear response behavior is recovered from the new model when the room for the thermal fluctuations in Y is not restricted by the existence of a nearby bound. We compare the predictions of the model with the results from molecular dynamics computer simulations for aqueous solvation, in which either (1) the solute
A comparison of approximate interval estimators for the Bernoulli parameter
NASA Technical Reports Server (NTRS)
Leemis, Lawrence; Trivedi, Kishor S.
1993-01-01
The goal of this paper is to compare the accuracy of two approximate confidence interval estimators for the Bernoulli parameter p. The approximate confidence intervals are based on the normal and Poisson approximations to the binomial distribution. Charts are given to indicate which approximation is appropriate for certain sample sizes and point estimators.
Hydration thermodynamics beyond the linear response approximation
NASA Astrophysics Data System (ADS)
Raineri, Fernando O.
2016-10-01
The solvation energetics associated with the transformation of a solute molecule at infinite dilution in water from an initial state A to a final state B is reconsidered. The two solute states have different potentials energies of interaction, {{\\Psi}\\text{A}} and {{\\Psi}\\text{B}} , with the solvent environment. Throughout the A \\to B transformation of the solute, the solvation system is described by a Hamiltonian H≤ft(ξ \\right) that changes linearly with the coupling parameter ξ. By focusing on the characterization of the probability density {{\\wp}ξ}≤ft( y\\right) that the dimensionless perturbational solute-solvent interaction energy Y=β ≤ft({{\\Psi}\\text{B}}-{{\\Psi}\\text{A}}\\right) has numerical value y when the coupling parameter is ξ, we derive a hierarchy of differential equation relations between the ξ-dependent cumulant functions of various orders in the expansion of the appropriate cumulant generating function. On the basis of this theoretical framework we then introduce an inherently nonlinear solvation model for which we are able to find analytical results for both {{\\wp}ξ} ≤ft( y\\right) and for the solvation thermodynamic functions. The solvation model is based on the premise that there is an upper or a lower bound (depending on the nature of the interactions considered) to the amplitude of the fluctuations of Y in the solution system at equilibrium. The results reveal essential differences in behavior for the model when compared with the linear response approximation to solvation, particularly with regards to the probability density {{\\wp}ξ} ≤ft( y\\right) . The analytical expressions for the solvation properties show, however, that the linear response behavior is recovered from the new model when the room for the thermal fluctuations in Y is not restricted by the existence of a nearby bound. We compare the predictions of the model with the results from molecular dynamics computer simulations for aqueous solvation, in
Approximate nearest neighbors via dictionary learning
NASA Astrophysics Data System (ADS)
Cherian, Anoop; Morellas, Vassilios; Papanikolopoulos, Nikolaos
2011-06-01
Approximate Nearest Neighbors (ANN) in high dimensional vector spaces is a fundamental, yet challenging problem in many areas of computer science, including computer vision, data mining and robotics. In this work, we investigate this problem from the perspective of compressive sensing, especially the dictionary learning aspect. High dimensional feature vectors are seldom seen to be sparse in the feature domain; examples include, but not limited to Scale Invariant Feature Transform (SIFT) descriptors, Histogram Of Gradients, Shape Contexts, etc. Compressive sensing advocates that if a given vector has a dense support in a feature space, then there should exist an alternative high dimensional subspace where the features are sparse. This idea is leveraged by dictionary learning techniques through learning an overcomplete projection from the feature space so that the vectors are sparse in the new space. The learned dictionary aids in refining the search for the nearest neighbors to a query feature vector into the most likely subspace combination indexed by its non-zero active basis elements. Since the size of the dictionary is generally very large, distinct feature vectors are most likely to have distinct non-zero basis. Utilizing this observation, we propose a novel representation of the feature vectors as tuples of non-zero dictionary indices, which then reduces the ANN search problem into hashing the tuples to an index table; thereby dramatically improving the speed of the search. A drawback of this naive approach is that it is very sensitive to feature perturbations. This can be due to two possibilities: (i) the feature vectors are corrupted by noise, (ii) the true data vectors undergo perturbations themselves. Existing dictionary learning methods address the first possibility. In this work we investigate the second possibility and approach it from a robust optimization perspective. This boils down to the problem of learning a dictionary robust to feature
Differential equation based method for accurate approximations in optimization
NASA Technical Reports Server (NTRS)
Pritchard, Jocelyn I.; Adelman, Howard M.
1990-01-01
A method to efficiently and accurately approximate 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 approximations, hence, the method is denoted the Differential Equation Based (DEB) method. Approximations were developed for vibration frequencies, mode shapes and static displacements. The DEB approximation method was applied to a cantilever beam and results compared with the commonly-used linear Taylor series approximations 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 approximation. The method is applicable to simultaneous perturbation of several design variables. Also, the approximations may be used to calculate other system response quantities. For example, the approximations for displacements are used to approximate bending stresses.
Difference equation state approximations for nonlinear hereditary control problems
NASA Technical Reports Server (NTRS)
Rosen, I. G.
1984-01-01
Discrete approximation schemes for the solution of nonlinear hereditary control problems are constructed. The methods involve approximation by a sequence of optimal control problems in which the original infinite dimensional state equation has been approximated by a finite dimensional discrete difference equation. Convergence of the state approximations is argued using linear semigroup theory and is then used to demonstrate that solutions to the approximating optimal control problems in some sense approximate solutions to the original control problem. Two schemes, one based upon piecewise constant approximation, and the other involving spline functions are discussed. Numerical results are presented, analyzed and used to compare the schemes to other available approximation methods for the solution of hereditary control problems. Previously announced in STAR as N83-33589
Difference equation state approximations for nonlinear hereditary control problems
NASA Technical Reports Server (NTRS)
Rosen, I. G.
1982-01-01
Discrete approximation schemes for the solution of nonlinear hereditary control problems are constructed. The methods involve approximation by a sequence of optimal control problems in which the original infinite dimensional state equation has been approximated by a finite dimensional discrete difference equation. Convergence of the state approximations is argued using linear semigroup theory and is then used to demonstrate that solutions to the approximating optimal control problems in some sense approximate solutions to the original control problem. Two schemes, one based upon piecewise constant approximation, and the other involving spline functions are discussed. Numerical results are presented, analyzed and used to compare the schemes to other available approximation methods for the solution of hereditary control problems.
NASA Astrophysics Data System (ADS)
Peng, Degao; Yang, Yang; Zhang, Peng; Yang, Weitao
2014-12-01
In this article, we develop systematically second random phase approximations (RPA) and Tamm-Dancoff approximations (TDA) of particle-hole and particle-particle channels for calculating molecular excitation energies. The second particle-hole RPA/TDA can capture double excitations missed by the particle-hole RPA/TDA and time-dependent density-functional theory (TDDFT), while the second particle-particle RPA/TDA recovers non-highest-occupied-molecular-orbital excitations missed by the particle-particle RPA/TDA. With proper orbital restrictions, these restricted second RPAs and TDAs have a formal scaling of only O(N4). The restricted versions of second RPAs and TDAs are tested with various small molecules to show some positive results. Data suggest that the restricted second particle-hole TDA (r2ph-TDA) has the best overall performance with a correlation coefficient similar to TDDFT, but with a larger negative bias. The negative bias of the r2ph-TDA may be induced by the unaccounted ground state correlation energy to be investigated further. Overall, the r2ph-TDA is recommended to study systems with both single and some low-lying double excitations with a moderate accuracy. Some expressions on excited state property evaluations, such as < hat{S}2rangle are also developed and tested.
Peng, Degao; Yang, Yang; Zhang, Peng; Yang, Weitao
2014-12-07
In this article, we develop systematically second random phase approximations (RPA) and Tamm-Dancoff approximations (TDA) of particle-hole and particle-particle channels for calculating molecular excitation energies. The second particle-hole RPA/TDA can capture double excitations missed by the particle-hole RPA/TDA and time-dependent density-functional theory (TDDFT), while the second particle-particle RPA/TDA recovers non-highest-occupied-molecular-orbital excitations missed by the particle-particle RPA/TDA. With proper orbital restrictions, these restricted second RPAs and TDAs have a formal scaling of only O(N{sup 4}). The restricted versions of second RPAs and TDAs are tested with various small molecules to show some positive results. Data suggest that the restricted second particle-hole TDA (r2ph-TDA) has the best overall performance with a correlation coefficient similar to TDDFT, but with a larger negative bias. The negative bias of the r2ph-TDA may be induced by the unaccounted ground state correlation energy to be investigated further. Overall, the r2ph-TDA is recommended to study systems with both single and some low-lying double excitations with a moderate accuracy. Some expressions on excited state property evaluations, such as 〈S{sup ^2}〉 are also developed and tested.
Pawlak Algebra and Approximate Structure on Fuzzy Lattice
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
Pawlak algebra and approximate structure on fuzzy lattice.
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.
Explicitly solvable complex Chebyshev approximation problems related to sine polynomials
NASA Technical Reports Server (NTRS)
Freund, Roland
1989-01-01
Explicitly solvable real Chebyshev approximation problems on the unit interval are typically characterized by simple error curves. A similar principle is presented for complex approximation problems with error curves induced by sine polynomials. As an application, some new explicit formulae for complex best approximations are derived.
Meta-Regression Approximations to Reduce Publication Selection Bias
ERIC Educational Resources Information Center
Stanley, T. D.; Doucouliagos, Hristos
2014-01-01
Publication selection bias is a serious challenge to the integrity of all empirical sciences. We derive meta-regression approximations to reduce this bias. Our approach employs Taylor polynomial approximations to the conditional mean of a truncated distribution. A quadratic approximation without a linear term, precision-effect estimate with…
Aspects of three field approximations: Darwin, frozen, EMPULSE
Boyd, J.K.; Lee, E.P.; Yu, S.S.
1985-05-25
The traditional approach used to study high energy beam propagation relies on the frozen field approximation. A minor modification of the frozen field approximation yields the set of equations applied to the analysis of the hose instability. These models are constrasted with the Darwin field approximation. A statement is made of the Darwin model equations relevant to the analysis of the hose instability.
Boundary control of parabolic systems - Finite-element approximation
NASA Technical Reports Server (NTRS)
Lasiecka, I.
1980-01-01
The finite element approximation of a Dirichlet type boundary control problem for parabolic systems is considered. An approach based on the direct approximation of an input-output semigroup formula is applied. Error estimates are derived for optimal state and optimal control, and it is noted that these estimates are actually optimal with respect to the approximation theoretic properties.
The Use of Approximations in a High School Chemistry Course
ERIC Educational Resources Information Center
Matsumoto, Paul S.; Tong, Gary; Lee, Stephanie; Kam, Bonita
2009-01-01
While approximations are used frequently in science, high school students may be unaware of the use of approximations in science, the motivation for their use, and the limitations of their use. In the article, we consider the use of approximations in a high school chemistry class as opportunities to increase student understanding of the use of…
NASA Technical Reports Server (NTRS)
Barth, Timothy J.; Larson, Mats G.
2000-01-01
We consider a posteriori error estimates for finite volume and finite element methods on arbitrary meshes subject to prescribed error functionals. Error estimates of this type are useful in a number of computational settings: (1) quantitative prediction of the numerical solution error, (2) adaptive meshing, and (3) load balancing of work on parallel computing architectures. Our analysis recasts the class of Godunov finite volumes schemes as a particular form of discontinuous Galerkin method utilizing broken space approximation obtained via reconstruction of cell-averaged data. In this general framework, weighted residual error bounds are readily obtained using duality arguments and Galerkin orthogonality. Additional consideration is given to issues such as nonlinearity, efficiency, and the relationship to other existing methods. Numerical examples are given throughout the talk to demonstrate the sharpness of the estimates and efficiency of the techniques. Additional information is contained in the original.
Mappings and accuracy for Chebyshev pseudo-spectral approximations
NASA Technical Reports Server (NTRS)
Bayliss, Alvin; Turkel, Eli
1992-01-01
The effect of mappings on the approximation, by Chebyshev collocation, of functions which exhibit localized regions of rapid variation is studied. A general strategy is introduced whereby mappings are adaptively constructed which map specified classes of rapidly varying functions into low order polynomials which can be accurately approximated by Chebyshev polynomial expansions. A particular family of mappings constructed in this way is tested on a variety of rapidly varying functions similar to those occurring in approximations. It is shown that the mapped function can be approximated much more accurately by Chebyshev polynomial approximations than in physical space or where mappings constructed from other strategies are employed.
Mappings and accuracy for Chebyshev pseudo-spectral approximations
NASA Technical Reports Server (NTRS)
Bayliss, Alvin; Turkel, Eli
1990-01-01
The effect of mappings on the approximation, by Chebyshev collocation, of functions which exhibit localized regions of rapid variation is studied. A general strategy is introduced whereby mappings are adaptively constructed which map specified classes of rapidly varying functions into low order polynomials which can be accurately approximated by Chebyshev polynomial expansions. A particular family of mappings constructed in this way is tested on a variety of rapidly varying functions similar to those occurring in approximations. It is shown that the mapped function can be approximated much more accurately by Chebyshev polynomial approximations than in physical space or where mappings constructed from other strategies are employed.
State space approximation for general fractional order dynamic systems
NASA Astrophysics Data System (ADS)
Liang, Shu; Peng, Cheng; Liao, Zeng; Wang, Yong
2014-10-01
Approximations for general fractional order dynamic systems are of much theoretical and practical interest. In this paper, a new approximate method for fractional order integrator is proposed. The poles of the approximate model are unrelated to the order of integrator. This feature shows benefits on extending the algorithm to the systems containing various fractional orders. Then a unified approximate method is derived for general fractional order linear or nonlinear dynamic systems via combining the proposed new method with the distributed frequency model approach. Numerical examples are given to show the wide applicability of our method and to illustrate the acceptable accuracy for approximations as well.
Discrete extrinsic curvatures and approximation of surfaces by polar polyhedra
NASA Astrophysics Data System (ADS)
Garanzha, V. A.
2010-01-01
Duality principle for approximation of geometrical objects (also known as Eu-doxus exhaustion method) was extended and perfected by Archimedes in his famous tractate “Measurement of circle”. The main idea of the approximation method by Archimedes is to construct a sequence of pairs of inscribed and circumscribed polygons (polyhedra) which approximate curvilinear convex body. This sequence allows to approximate length of curve, as well as area and volume of the bodies and to obtain error estimates for approximation. In this work it is shown that a sequence of pairs of locally polar polyhedra allows to construct piecewise-affine approximation to spherical Gauss map, to construct convergent point-wise approximations to mean and Gauss curvature, as well as to obtain natural discretizations of bending energies. The Suggested approach can be applied to nonconvex surfaces and in the case of multiple dimensions.
NASA Astrophysics Data System (ADS)
Luo, Hongjun; Kolb, Dietmar; Flad, Heinz-Jurgen; Hackbusch, Wolfgang; Koprucki, Thomas
2002-08-01
We have studied various aspects concerning the use of hyperbolic wavelets and adaptive approximation schemes for wavelet expansions of correlated wave functions. In order to analyze the consequences of reduced regularity of the wave function at the electron-electron cusp, we first considered a realistic exactly solvable many-particle model in one dimension. Convergence rates of wavelet expansions, with respect to L2 and H1 norms and the energy, were established for this model. We compare the performance of hyperbolic wavelets and their extensions through adaptive refinement in the cusp region, to a fully adaptive treatment based on the energy contribution of individual wavelets. Although hyperbolic wavelets show an inferior convergence behavior, they can be easily refined in the cusp region yielding an optimal convergence rate for the energy. Preliminary results for the helium atom are presented, which demonstrate the transferability of our observations to more realistic systems. We propose a contraction scheme for wavelets in the cusp region, which reduces the number of degrees of freedom and yields a favorable cost to benefit ratio for the evaluation of matrix elements.
Approximate solutions for certain bidomain problems in electrocardiography
NASA Astrophysics Data System (ADS)
Johnston, Peter R.
2008-10-01
The simulation of problems in electrocardiography using the bidomain model for cardiac tissue often creates issues with satisfaction of the boundary conditions required to obtain a solution. Recent studies have proposed approximate methods for solving such problems by satisfying the boundary conditions only approximately. This paper presents an analysis of their approximations using a similar method, but one which ensures that the boundary conditions are satisfied during the whole solution process. Also considered are additional functional forms, used in the approximate solutions, which are more appropriate to specific boundary conditions. The analysis shows that the approximations introduced by Patel and Roth [Phys. Rev. E 72, 051931 (2005)] generally give accurate results. However, there are certain situations where functional forms based on the geometry of the problem under consideration can give improved approximations. It is also demonstrated that the recent methods are equivalent to different approaches to solving the same problems introduced 20years earlier.
Multijet final states: exact results and the leading pole approximation
Ellis, R.K.; Owens, J.F.
1984-09-01
Exact results for the process gg ..-->.. ggg are compared with those obtained using the leading pole approximation. Regions of phase space where the approximation breaks down are discussed. A specific example relevant for background estimates to W boson production is presented. It is concluded that in this instance the leading pole approximation may underestimate the standard QCD background by more than a factor of two in certain kinematic regions of physical interest.
Beyond the Born approximation in one-dimensional profile reconstruction
NASA Astrophysics Data System (ADS)
Trantanella, Charles J.; Dudley, Donald G.; Nabulsi, Khalid A.
1995-07-01
A new method of one-dimensional profile reconstruction is presented. The method is based on an extension to the Born approximation and relates measurements of the scattered field to the Fourier transform of the slab profile. Since the Born and our new approximations are most valid at low frequency, we utilize superresolution to recover high-frequency information and then invert for the slab profile. Finally, we vary different parameters and examine the resulting reconstructions. approximation, profile reconstruction, superresolution.
Approximate analytical calculations of photon geodesics in the Schwarzschild metric
NASA Astrophysics Data System (ADS)
De Falco, Vittorio; Falanga, Maurizio; Stella, Luigi
2016-10-01
We develop a method for deriving approximate analytical formulae to integrate photon geodesics in a Schwarzschild spacetime. Based on this, we derive the approximate equations for light bending and propagation delay that have been introduced empirically. We then derive for the first time an approximate analytical equation for the solid angle. We discuss the accuracy and range of applicability of the new equations and present a few simple applications of them to known astrophysical problems.
Generalized Lorentzian approximations for the Voigt line shape.
Martin, P; Puerta, J
1981-01-15
The object of the work reported in this paper was to find a simple and easy to calculate approximation to the Voigt function using the Padé method. To do this we calculated the multipole approximation to the complex function as the error function or as the plasma dispersion function. This generalized Lorentzian approximation can be used instead of the exact function in experiments that do not require great accuracy. PMID:20309100
An approximation based global optimization strategy for structural synthesis
NASA Technical Reports Server (NTRS)
Sepulveda, A. E.; Schmit, L. A.
1991-01-01
A global optimization strategy for structural synthesis based on approximation concepts is presented. The methodology involves the solution of a sequence of highly accurate approximate problems using a global optimization algorithm. The global optimization algorithm implemented consists of a branch and bound strategy based on the interval evaluation of the objective function and constraint functions, combined with a local feasible directions algorithm. The approximate design optimization problems are constructed using first order approximations of selected intermediate response quantities in terms of intermediate design variables. Some numerical results for example problems are presented to illustrate the efficacy of the design procedure setforth.
How to Solve Schroedinger Problems by Approximating the Potential Function
Ledoux, Veerle; Van Daele, Marnix
2010-09-30
We give a survey over the efforts in the direction of solving the Schroedinger equation by using piecewise approximations of the potential function. Two types of approximating potentials have been considered in the literature, that is piecewise constant and piecewise linear functions. For polynomials of higher degree the approximating problem is not so easy to integrate analytically. This obstacle can be circumvented by using a perturbative approach to construct the solution of the approximating problem, leading to the so-called piecewise perturbation methods (PPM). We discuss the construction of a PPM in its most convenient form for applications and show that different PPM versions (CPM,LPM) are in fact equivalent.
Scattering from rough thin films: discrete-dipole-approximation simulations.
Parviainen, Hannu; Lumme, Kari
2008-01-01
We investigate the wave-optical light scattering properties of deformed thin circular films of constant thickness using the discrete-dipole approximation. Effects on the intensity distribution of the scattered light due to different statistical roughness models, model dependent roughness parameters, and uncorrelated, random, small-scale porosity of the inhomogeneous medium are studied. The suitability of the discrete-dipole approximation for rough-surface scattering problems is evaluated by considering thin films as computationally feasible rough-surface analogs. The effects due to small-scale inhomogeneity of the scattering medium are compared with the analytic approximation by Maxwell Garnett, and the results are found to agree with the approximation.
Tangent plane approximation and some of its generalizations
NASA Astrophysics Data System (ADS)
Voronovich, A. G.
2007-05-01
A review of the tangent plane approximation proposed by L.M. Brekhovskikh is presented. The advantage of the tangent plane approximation over methods based on the analysis of integral equations for surface sources is emphasized. A general formula is given for the scattering amplitude of scalar plane waves under an arbitrary boundary condition. The direct generalization of the tangent plane approximation is shown to yield approximations that include a correct description of the Bragg scattering and allow one to avoid the use of a two-scale model.
Sensitivity analysis and approximation methods for general eigenvalue problems
NASA Technical Reports Server (NTRS)
Murthy, D. V.; Haftka, R. T.
1986-01-01
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 approximate reanalysis methods. For the algebraic eigenvalue problem involving non-hermitian matrices, algorithms for sensitivity analysis and approximate 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 approximation methods for eigenvalues are proposed and evaluated. Linear and quadratic approximations are based directly on the Taylor series. Several approximation methods are developed based on the generalized Rayleigh quotient for the eigenvalue problem. Approximation methods based on trace theorem give high accuracy without needing any derivatives. Operation counts for the computation of the approximations are given. General recommendations are made for the selection of appropriate approximation 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 approximation is sought.
Approximating the physical inner product of loop quantum cosmology
NASA Astrophysics Data System (ADS)
Bahr, Benjamin; Thiemann, Thomas
2007-04-01
In this paper, we investigate the possibility of approximating the physical inner product of constrained quantum theories. In particular, we calculate the physical inner product of a simple cosmological model in two ways: firstly, we compute it analytically via a trick; secondly, we use the complexifier coherent states to approximate the physical inner product defined by the master constraint of the system. We find that the approximation is able to recover the analytic solution of the problem, which consolidates hopes that coherent states will help to approximate solutions of more complicated theories, like loop quantum gravity.
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.
Differential equation based method for accurate approximations in optimization
NASA Technical Reports Server (NTRS)
Pritchard, Jocelyn I.; Adelman, Howard M.
1990-01-01
This paper describes a method to efficiently and accurately approximate 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 approximations, hence, the method is denoted the Differential Equation Based (DEB) method. Approximations were developed for vibration frequencies, mode shapes and static displacements. The DEB approximation method was applied to a cantilever beam and results compared with the commonly-used linear Taylor series approximations 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 approximation. The method is applicable to simultaneous perturbation of several design variables. Also, the approximations may be used to calculate other system response quantities. For example, the approximations for displacement are used to approximate bending stresses.
Legendre-tau approximations for functional differential equations
NASA Technical Reports Server (NTRS)
Ito, K.; Teglas, R.
1986-01-01
The numerical approximation 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 approximate 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 approximations. Convergence is established, various numerical examples are presented, and comparison between the latter and cubic spline approximation is made.
Spatial Ability Explains the Male Advantage in Approximate Arithmetic
Wei, Wei; Chen, Chuansheng; Zhou, Xinlin
2016-01-01
Previous research has shown that females consistently outperform males in exact arithmetic, perhaps due to the former’s advantage in language processing. Much less is known about gender difference in approximate arithmetic. Given that approximate arithmetic is closely associated with visuospatial processing, which shows a male advantage we hypothesized that males would perform better than females in approximate arithmetic. In two experiments (496 children in Experiment 1 and 554 college students in Experiment 2), we found that males showed better performance in approximate arithmetic, which was accounted for by gender differences in spatial ability. PMID:27014124
Approximation functions for airblast environments from buried charges
Reichenbach, H.; Behrens, K.; Kuhl, A.L.
1993-11-01
In EMI report E 1/93, ``Airblast Environments from Buried HE-Charges,`` fit functions were used for the compact description of blastwave parameters. The coefficients of these functions were approximated by means of second order polynomials versus DOB. In most cases, the agreement with the measured data was satisfactory; to reduce remaining noticeable deviations, an approximation by polygons (i.e., piecewise-linear approximation) was used instead of polynomials. The present report describes the results of the polygon approximation and compares them to previous data. We conclude that the polygon representation leads to a better agreement with the measured data.
Impact of inflow transport approximation on light water reactor analysis
NASA Astrophysics Data System (ADS)
Choi, Sooyoung; Smith, Kord; Lee, Hyun Chul; Lee, Deokjung
2015-10-01
The impact of the inflow transport approximation on light water reactor analysis is investigated, and it is verified that the inflow transport approximation significantly improves the accuracy of the transport and transport/diffusion solutions. A methodology for an inflow transport approximation is implemented in order to generate an accurate transport cross section. The inflow transport approximation is compared to the conventional methods, which are the consistent-PN and the outflow transport approximations. The three transport approximations are implemented in the lattice physics code STREAM, and verification is performed for various verification problems in order to investigate their effects and accuracy. From the verification, it is noted that the consistent-PN and the outflow transport approximations cause significant error in calculating the eigenvalue and the power distribution. The inflow transport approximation shows very accurate and precise results for the verification problems. The inflow transport approximation shows significant improvements not only for the high leakage problem but also for practical large core problem analyses.
Various approximations made in augmented-plane-wave calculations
NASA Astrophysics Data System (ADS)
Bacalis, N. C.; Blathras, K.; Thomaides, P.; Papaconstantopoulos, D. A.
1985-10-01
The effects of various approximations used in performing augmented-plane-wave calculations were studied for elements of the fifth and sixth columns of the Periodic Table, namely V, Nb, Ta, Cr, Mo, and W. Two kinds of approximations have been checked: (i) variation of the number of k points used to iterate to self-consistency, and (ii) approximations for the treatment of the core states. In addition a comparison between relativistic and nonrelativistic calculations is made, and an approximate method of calculating the spin-orbit splitting is given.
The Role of Intuitive Approximation Skills for School Math Abilities
ERIC Educational Resources Information Center
Libertus, Melissa E.
2015-01-01
Research has shown that educated children and adults have access to two ways of representing numerical information: an approximate number system (ANS) that is present from birth and allows for quick approximations of numbers of objects encountered in one's environment, and an exact number system (ENS) that is acquired through experience and…
Finding the Best Quadratic Approximation of a Function
ERIC Educational Resources Information Center
Yang, Yajun; Gordon, Sheldon P.
2011-01-01
This article examines the question of finding the best quadratic function to approximate 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 approximations at various points in the interval under consideration, the other based on the fact…
The blind leading the blind: Mutual refinement of approximate theories
NASA Technical Reports Server (NTRS)
Kedar, Smadar T.; Bresina, John L.; Dent, C. Lisa
1991-01-01
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 approximate models which cause the failures. The approach focuses on using one approximate model to refine another.
Reply to Steele & Ferrer: Modeling Oscillation, Approximately or Exactly?
ERIC Educational Resources Information Center
Oud, Johan H. L.; Folmer, Henk
2011-01-01
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 approximate estimation procedure applied. This procedure is the latent version of the local linear approximation procedure…
General Entropic Approximations for Canonical Systems Described by Kinetic Equations
NASA Astrophysics Data System (ADS)
Pavan, V.
2011-02-01
In this paper we extend the general construction of entropic approximation for kinetic operators modelling canonical systems. More precisely, this paper aims at pursuing to thermalized systems the works of Levermore, Schneider and Junk on moments problems relying on entropy minimization in order to construct BGK approximations and moments based equations.
Approximation and modeling with ambient B-splines
NASA Astrophysics Data System (ADS)
Lehmann, N.; Maier, L.-B.; Odathuparambil, S.; Reif, U.
2016-06-01
We present a novel technique for solving approximation problems on manifolds in terms of standard tensor product B-splines. This method is easy to implement and provides optimal approximation order. Applications include the representation of smooth surfaces of arbitrary genus.
The use of neural networks for approximation of nuclear data
Korovin, Yu. A.; Maksimushkina, A. V.
2015-12-15
The article discusses the possibility of using neural networks for approximation or reconstruction of data such as the reaction cross sections. The quality of the approximation using fitting criteria is also evaluated. The activity of materials under irradiation is calculated from data obtained using neural networks.
Perturbation approximation for orbits in axially symmetric funnels
NASA Astrophysics Data System (ADS)
Nauenberg, Michael
2014-11-01
A perturbation method that can be traced back to Isaac Newton is applied to obtain approximate analytic solutions for objects sliding in axially symmetric funnels in near circular orbits. Some experimental observations are presented for balls rolling in inverted cones with different opening angles, and in a funnel with a hyperbolic surface that approximately simulates the gravitational force.
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.
A classical path approximation for diffractive surface scattering
NASA Astrophysics Data System (ADS)
Meyer, Hans-Dieter; Toennies, J. Peter
1984-12-01
The well-known classical path approximation is applied to a calculation of diffraction intensities in the scattering of atoms from a rigid crystal with a soft interaction potential. A general expression is derived for the diffraction intensities which can be applied to potentials with several higher-order terms in the Fourier series. For an uncorrugated Morse potential with a first-order exponential corrugation term an analytic solution is obtained which is compared with the infinite order suddent (IOS) approximation calculations for Ne/W(110) and He/LiF(100). Both approximations are very accurate for the weakly corrugated Ne/W system. For He/LiF the present approximation is more accurate than the sudden (IOS) approximation and has the added advantage of providing an analytic solution. Several improvements are suggested.
Oscillatory convection and limitations of the Boussinesq approximation
NASA Astrophysics Data System (ADS)
Wood, T. S.; Bushby, P. J.
2016-09-01
We determine the asymptotic conditions under which the Boussinesq approximation is valid for oscillatory convection in a rapidly rotating fluid. In the astrophysically relevant parameter regime of small Prandtl number, we show that the Boussinesq prediction for the onset of convection is valid only under much more restrictive conditions than those that are usually assumed. In the case of an ideal gas, we recover the Boussinesq results only if the ratio of the domain height to a typical scale height is much smaller than the Prandtl number. This requires an extremely shallow domain in the astrophysical parameter regime. Other commonly-used "sound-proof" approximations generally perform no better than the Boussinesq approximation. The exception is a particular implementation of the pseudo-incompressible approximation, which predicts the correct instability threshold beyond the range of validity of the Boussinesq approximation.
Properties of the Boltzmann equation in the classical approximation
Epelbaum, Thomas; Gelis, François; Tanji, Naoto; Wu, Bin
2014-12-30
We examine the Boltzmann equation with elastic point-like scalar interactions in two different versions of the the classical approximation. 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 approximations. This cutoff dependence in the classical approximations of the Boltzmann equation is closely related to the non-renormalizability of the classical statistical approximation 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 one has also access to the non-approximated result for comparison.
Approximating the Critical Domain Size of Integrodifference Equations.
Reimer, Jody R; Bonsall, Michael B; Maini, Philip K
2016-01-01
Integrodifference (IDE) models can be used to determine the critical domain size required for persistence of populations with distinct dispersal and growth phases. Using this modelling framework, we develop a novel spatially implicit approximation to the proportion of individuals lost to unfavourable habitat outside of a finite domain of favourable habitat, which consistently outperforms the most common approximations. We explore how results using this approximation compare to the existing IDE results on the critical domain size for populations in a single patch of good habitat, in a network of patches, in the presence of advection, and in structured populations. We find that the approximation consistently provides results which are in close agreement with those of an IDE model except in the face of strong advective forces, with the advantage of requiring fewer numerical approximations while providing insights into the significance of disperser retention in determining the critical domain size of an IDE. PMID:26721746
Sticky hard spheres beyond the Percus-Yevick approximation
NASA Astrophysics Data System (ADS)
Yuste, S. Bravo; Santos, A.
1993-12-01
The radial distribution function g(r) of a sticky-hard-sphere fluid is obtained by assuming a rational-function form for a function related to the Laplace transform of rg(r), compatible with the conditions of finite y(r)==g(r)ecphi(r)/kBT at c ontact point and finite isothermal compressibility. In a recent paper [S. Bravo Yuste and A. Santos, J. Stat. Phys. 72, 703 (1993)] we have shown that the simplest rational-function approximation, namely, the Padé approximant (2,3), leads to Baxter's exact solution of the Percus-Yevick equation. Here we consider the next approximation, i.e., the Padé approximant (3,4), and determine the two new parameters by imposing the values of y(r) at contact point and of the isothermal compressibility. Comparison with Monte Carlo simulation results shows a significant improvement over the Percus-Yevick approximation.
Properties of the Boltzmann equation in the classical approximation
Epelbaum, Thomas; Gelis, François; Tanji, Naoto; Wu, Bin
2014-12-30
We examine the Boltzmann equation with elastic point-like scalar interactions in two different versions of the the classical approximation. 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 approximations. This cutoff dependence in the classical approximations of the Boltzmann equation is closely related to the non-renormalizability of the classical statistical approximation 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-approximated result for comparison.« less
Approximate number word knowledge before the cardinal principle.
Gunderson, Elizabeth A; Spaepen, Elizabet; Levine, Susan C
2015-02-01
Approximate number word knowledge-understanding the relation between the count words and the approximate magnitudes of sets-is a critical piece of knowledge that predicts later math achievement. However, researchers disagree about when children first show evidence of approximate 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 approximate 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 approximate 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 approximate number word knowledge on the verbal production task even when only examining set sizes above 4. Across studies, children's age predicted approximate 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 approximation ability. Thus, children can develop approximate knowledge of number words up to 10 before learning the cardinal principle. Furthermore, approximate number word knowledge increases with age and might not be closely related to the development of exact number word knowledge. PMID:25462030
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.
Meromorphic approximants to complex Cauchy transforms with polar singularities
Baratchart, Laurent; Yattselev, Maxim L
2009-10-31
We study AAK-type meromorphic approximants to functions of the form F(z)={integral}(d{lambda}(t))/(z-t)+R(z), where R is a rational function and {lambda} is a complex measure with compact regular support included in (-1,1), whose argument has bounded variation on the support. The approximation is understood in the L{sup p}-norm of the unit circle, p{>=}2. We dwell on the fact that the denominators of such approximants satisfy certain non-Hermitian orthogonal relations with varying weights. They resemble the orthogonality relations that arise in the study of multipoint Pade approximants. However, the varying part of the weight implicitly depends on the orthogonal polynomials themselves, which constitutes the main novelty and the main difficulty of the undertaken analysis. We obtain that the counting measures of poles of the approximants converge to the Green equilibrium distribution on the support of {lambda} relative to the unit disc, that the approximants themselves converge in capacity to F, and that the poles of R attract at least as many poles of the approximants as their multiplicity and not much more. Bibliography: 35 titles.
A test of the adhesion approximation for gravitational clustering
NASA Technical Reports Server (NTRS)
Melott, Adrian L.; Shandarin, Sergei; Weinberg, David H.
1993-01-01
We quantitatively compare a particle implementation of the adhesion approximation to fully non-linear, numerical 'N-body' simulations. Our primary tool, cross-correlation of N-body simulations with the adhesion approximation, indicates good agreement, better than that found by the same test performed with the Zel-dovich approximation (hereafter ZA). However, the cross-correlation is not as good as that of the truncated Zel-dovich approximation (TZA), obtained by applying the Zel'dovich approximation after smoothing the initial density field with a Gaussian filter. We confirm that the adhesion approximation produces an excessively filamentary distribution. Relative to the N-body results, we also find that: (a) the power spectrum obtained from the adhesion approximation 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.
A test of the adhesion approximation for gravitational clustering
NASA Technical Reports Server (NTRS)
Melott, Adrian L.; Shandarin, Sergei F.; Weinberg, David H.
1994-01-01
We quantitatively compare a particle implementation of the adhesion approximation to fully nonlinear, numerical 'N-body' simulations. Our primary tool, cross-correlation of N-body simulations with the adhesion approximation, indicates good agreement, better than that found by the same test performed with the Zel'dovich approximation (hereafter ZA). However, the cross-correlation is not as good as that of the truncated Zel'dovich approximation (TZA), obtained by applying the Zel'dovich approximation after smoothing the initial density field with a Gaussian filter. We confirm that the adhesion approximation produces an excessively filamentary distribution. Relative to the N-body results, we also find that: (a) the power spectrum obtained from the adhesion approximation 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.
Pitch contour stylization using an optimal piecewise polynomial approximation
Ghosh, Prasanta Kumar; Narayanan, Shrikanth S.
2014-01-01
We propose a dynamic programming (DP) based piecewise polynomial approximation of discrete data such that the L2 norm of the approximation error is minimized. We apply this technique for the stylization of speech pitch contour. Objective evaluation verifies that the DP based technique indeed yields minimum mean square error (MSE) compared to other approximation methods. Subjective evaluation reveals that the quality of the synthesized speech using stylized pitch contour obtained by the DP method is almost identical to that of the original speech. PMID:24453471
Revisiting the envelope approximation: Gravitational waves from bubble collisions
NASA Astrophysics Data System (ADS)
Weir, David J.
2016-06-01
We study the envelope approximation and its applicability to first-order phase transitions in the early Universe. We demonstrate that the power laws seen in previous studies exist independently of the nucleation rate. We also compare the envelope approximation prediction to results from large-scale phase transition simulations. For phase transitions where the contribution to gravitational waves from scalar fields dominates over that from the coupled plasma of light particles, the envelope approximation is in agreement, giving a power spectrum of the same form and order of magnitude. In all other cases the form and amplitude of the gravitational wave power spectrum is markedly different and new techniques are required.
Analytic Approximate Solution for Falkner-Skan Equation
Marinca, Bogdan
2014-01-01
This paper deals with the Falkner-Skan nonlinear differential equation. An analytic approximate technique, namely, optimal homotopy asymptotic method (OHAM), is employed to propose a procedure to solve a boundary-layer problem. Our method does not depend upon small parameters and provides us with a convenient way to optimally control the convergence of the approximate solutions. The obtained results reveal that this procedure is very effective, simple, and accurate. A very good agreement was found between our approximate results and numerical solutions, which prove that OHAM is very efficient in practice, ensuring a very rapid convergence after only one iteration. PMID:24883417
Convergence of multipoint Pade approximants of piecewise analytic functions
Buslaev, Viktor I
2013-02-28
The behaviour as n{yields}{infinity} of multipoint Pade approximants to a function which is (piecewise) holomorphic on a union of finitely many continua is investigated. The convergence of multipoint Pade approximants is proved for a function which extends holomorphically from these continua to a union of domains whose boundaries have a certain symmetry property. An analogue of Stahl's theorem is established for two-point Pade approximants to a pair of functions, either of which is a multivalued analytic function with finitely many branch points. Bibliography: 11 titles.
Validity of the kink approximation to the tunneling action
NASA Astrophysics Data System (ADS)
Dutta, Koushik; Hector, Cecelie; Konstandin, Thomas; Vaudrevange, Pascal M.; Westphal, Alexander
2012-12-01
Coleman tunneling in a general scalar potential with two nondegenerate minima is known to have an approximation in terms of a piecewise linear triangular-shaped potential with sharp “kinks” at the place of the local minima. This approximate potential has a regime where the existence of the bounce solution needs the scalar field to “wait” for some amount of Euclidean time at one of the kinks. We discuss under which conditions a kink approximation of locally smooth “cap” regions provides a good estimate for the bounce action.
Non-perturbative QCD amplitudes in quenched and eikonal approximations
NASA Astrophysics Data System (ADS)
Fried, H. M.; Grandou, T.; Sheu, Y.-M.
2014-05-01
Even though approximated, strong coupling non-perturbative QCD amplitudes remain very difficult to obtain. In this article, in eikonal and quenched approximations at least, physical insights are presented that rely on the newly-discovered property of effective locality. The present article also provides a more rigorous mathematical basis for the crude approximations used in the previous derivation of the binding potential of quarks and nucleons. Furthermore, the techniques of Random Matrix calculus along with Meijer G-functions are applied to analyze the generic structure of fermionic amplitudes in QCD.
Baby Skyrme model, near-BPS approximations, and supersymmetric extensions
NASA Astrophysics Data System (ADS)
Bolognesi, S.; Zakrzewski, W.
2015-02-01
We study the baby Skyrme model as a theory that interpolates between two distinct BPS systems. For this, a near-BPS approximation can be used when there is a small deviation from each of the two BPS limits. We provide analytical explanation and numerical support for the validity of this approximation. We then study the set of all possible supersymmetric extensions of the baby Skyrme model with N =1 and the particular ones with extended N =2 supersymmetries and relate this to the above mentioned almost-BPS approximation.
Approximate polynomial preconditioning applied to biharmonic equations on vector supercomputers
NASA Technical Reports Server (NTRS)
Wong, Yau Shu; Jiang, Hong
1987-01-01
Applying a finite difference approximation to a biharmonic equation results in a very ill-conditioned system of equations. This paper examines the conjugate gradient method used in conjunction with the generalized and approximate polynomial preconditionings for solving such linear systems. An approximate polynomial preconditioning is introduced, and is shown to be more efficient than the generalized polynomial preconditionings. This new technique provides a simple but effective preconditioning polynomial, which is based on another coefficient matrix rather than the original matrix operator as commonly used.
Approximate Controllability of Fractional Neutral Stochastic System with Infinite Delay
NASA Astrophysics Data System (ADS)
Sakthivel, R.; Ganesh, R.; Suganya, S.
2012-12-01
The concept of controllability plays an important role in analysis and design of linear and nonlinear control systems. Further, fractional differential equations have wide applications in engineering and science. In this paper, the approximate controllability of neutral stochastic fractional integro-differential equation with infinite delay in a Hilbert space is studied. By using Krasnoselskii's fixed point theorem with stochastic analysis theory, we derive a new set of sufficient conditions for the approximate controllability of nonlinear fractional stochastic system under the assumption that the corresponding linear system is approximately controllable. Finally, an example is provided to illustrate the obtained theory.
On the Beebe-Linderberg two-electron integral approximation
NASA Astrophysics Data System (ADS)
Røeggen, I.; Wisløff-Nilssen, E.
1986-12-01
The Beebe-Linderberg two-electron integral approximation, which is generated by a Cholesky decomposition of the two-electron integral matrix ([μν|λσ]), is slightly modified. On the basis of test calculations, two key questions concerning this approximation are discussed: The numerical rank of the two-electron integral matrix and the relationship between the integral threshold and electronic properties. The numerical results presented in this work suggest that the modified Beebe-Linderberg approximation might be considered as an alternative to effective core potential methods.
New approximation for the effective energy of nonlinear conducting composites
NASA Astrophysics Data System (ADS)
Gibiansky, Leonid; Torquato, Salvatore
1998-07-01
Approximations for the effective energy and, thus, effective conductivity of nonlinear, isotropic conducting dispersions are developed. This is accomplished by using the Ponte Castaneda variational principles [Philos. Trans. R. Soc. London Ser. A 340, 1321 (1992)] and the Torquato approximation [J. Appl. Phys. 58, 3790 (1985)] of the effective conductivity of corresponding linear composites. The results are obtained for dispersions with superconducting or insulating inclusions, and, more generally, for phases with a power-law energy. It is shown that the new approximations lie within the best available rigorous upper and lower bounds on the effective energy.
Communication: Improved pair approximations in local coupled-cluster methods
Schwilk, Max; Werner, Hans-Joachim; Usvyat, Denis
2015-03-28
In local coupled cluster treatments the electron pairs can be classified according to the magnitude of their energy contributions or distances into strong, close, weak, and distant pairs. Different approximations are introduced for the latter three classes. In this communication, an improved simplified treatment of close and weak pairs is proposed, which is based on long-range cancellations of individually slowly decaying contributions in the amplitude equations. Benchmark calculations for correlation, reaction, and activation energies demonstrate that these approximations work extremely well, while pair approximations based on local second-order Møller-Plesset theory can lead to errors that are 1-2 orders of magnitude larger.
Quadrupole Collective Inertia in Nuclear Fission: Cranking Approximation
Baran, A.; Sheikh, J. A.; Dobaczewski, J.; Nazarewicz, Witold
2011-01-01
Collective mass tensor derived from the cranking approximation to the adiabatic time-dependent Hartree-Fock-Bogoliubov (ATDHFB) approach is compared with that obtained in the Gaussian Overlap Approximation (GOA) to the generator coordinate method. Illustrative calculations are carried out for one-dimensional quadrupole fission pathways in ^{256}Fm. It is shown that the collective mass exhibits strong variations with the quadrupole collective coordinate. These variations are related to the changes in the intrinsic shell structure. The differences between collective inertia obtained in cranking and perturbative cranking approximations to ATDHFB, and within GOA, are discussed.
Best approximation of Gaussian neural networks with nodes uniformly spaced.
Mulero-Martinez, J I
2008-02-01
This paper is aimed at exposing the reader to certain aspects in the design of the best approximants with Gaussian radial basis functions (RBFs). The class of functions to which this approach applies consists of those compactly supported in frequency. The approximative properties of uniqueness and existence are restricted to this class. Functions which are smooth enough can be expanded in Gaussian series converging uniformly to the objective function. The uniqueness of these series is demonstrated by the context of the orthonormal basis in a Hilbert space. Furthermore, the best approximation to a given band-limited function from a truncated Gaussian series is analyzed by an energy-based argument. This analysis not only gives a theoretical proof concerned with the existence of best approximations but addresses the problems of architectural selection. Specifically, guidance for selecting the variance and the oversampling parameters is provided for practitioners. PMID:18269959
Real-time creased approximate subdivision surfaces with displacements.
Kovacs, Denis; Mitchell, Jason; Drone, Shanon; Zorin, Denis
2010-01-01
We present an extension of Loop and Schaefer's approximation of Catmull-Clark surfaces (ACC) for surfaces with creases and corners. We discuss the integration of ACC into Valve's Source game engine and analyze performance of our implementation.
Integral approximations to classical diffusion and smoothed particle hydrodynamics
Du, Qiang; Lehoucq, R. B.; Tartakovsky, A. M.
2014-12-31
The contribution of the paper is the approximation of a classical diffusion operator by an integral equation with a volume constraint. A particular focus is on classical diffusion problems associated with Neumann boundary conditions. By exploiting this approximation, we can also approximate other quantities such as the flux out of a domain. Our analysis of the model equation on the continuum level is closely related to the recent work on nonlocal diffusion and peridynamic mechanics. In particular, we elucidate the role of a volumetric constraint as an approximation to a classical Neumann boundary condition in the presence of physical boundary. The volume-constrained integral equation then provides the basis for accurate and robust discretization methods. As a result, an immediate application is to the understanding and improvement of the Smoothed Particle Hydrodynamics (SPH) method.
Integral approximations to classical diffusion and smoothed particle hydrodynamics
Du, Qiang; Lehoucq, R. B.; Tartakovsky, A. M.
2014-12-31
The contribution of the paper is the approximation of a classical diffusion operator by an integral equation with a volume constraint. A particular focus is on classical diffusion problems associated with Neumann boundary conditions. By exploiting this approximation, we can also approximate other quantities such as the flux out of a domain. Our analysis of the model equation on the continuum level is closely related to the recent work on nonlocal diffusion and peridynamic mechanics. In particular, we elucidate the role of a volumetric constraint as an approximation to a classical Neumann boundary condition in the presence of physical boundary.more » The volume-constrained integral equation then provides the basis for accurate and robust discretization methods. As a result, an immediate application is to the understanding and improvement of the Smoothed Particle Hydrodynamics (SPH) method.« less
Integral approximations to classical diffusion and smoothed particle hydrodynamics
Du, Q.; Lehoucq, Richard B.; Tartakovsky, Alexandre M.
2015-04-01
The contribution of the paper is the approximation of a classical diffusion operator by an integral equation with a volume constraint. A particular focus is on classical diffusion problems associated with Neumann boundary conditions. By exploiting this approximation, we can also approximate other quantities such as the flux out of a domain. Our analysis of the model equation on the continuum level is closely related to the recent work on nonlocal diffusion and peridynamic mechanics. In particular, we elucidate the role of a volumetric constraint as an approximation to a classical Neumann boundary condition in the presence of physical boundary. The volume-constrained integral equation then provides the basis for accurate and robust discretization methods. An immediate application is to the understanding and improvement of the Smoothed Particle Hydrodynamics (SPH) method.
Approximate Riemann solvers for the Godunov SPH (GSPH)
NASA Astrophysics Data System (ADS)
Puri, Kunal; Ramachandran, Prabhu
2014-08-01
The Godunov Smoothed Particle Hydrodynamics (GSPH) method is coupled with non-iterative, approximate Riemann solvers for solutions to the compressible Euler equations. The use of approximate solvers avoids the expensive solution of the non-linear Riemann problem for every interacting particle pair, as required by GSPH. In addition, we establish an equivalence between the dissipative terms of GSPH and the signal based SPH artificial viscosity, under the restriction of a class of approximate Riemann solvers. This equivalence is used to explain the anomalous “wall heating” experienced by GSPH and we provide some suggestions to overcome it. Numerical tests in one and two dimensions are used to validate the proposed Riemann solvers. A general SPH pairing instability is observed for two-dimensional problems when using unequal mass particles. In general, Ducowicz Roe's and HLLC approximate Riemann solvers are found to be suitable replacements for the iterative Riemann solver in the original GSPH scheme.
4. BUILDING 422, WEST SIDE, FROM APPROXIMATELY 25 FEET SOUTHWEST ...
4. BUILDING 422, WEST SIDE, FROM APPROXIMATELY 25 FEET SOUTHWEST OF SOUTHWEST CORNER, LOOKING NORTHEAST. - Oakland Naval Supply Center, Aeronautical Materials Storehouses, Between E & G Streets, between Fourth & Sixth Streets, Oakland, Alameda County, CA
6. BUILDING 522, INTERIOR, STOREROOM, FROM APPROXIMATELY TWOTHIRDS OF DISTANCE ...
6. BUILDING 522, INTERIOR, STOREROOM, FROM APPROXIMATELY TWO-THIRDS OF DISTANCE FROM EAST END, LOOKING WEST. - Oakland Naval Supply Center, Aeronautical Materials Storehouses, Between E & G Streets, between Fourth & Sixth Streets, Oakland, Alameda County, CA
5. BUILDING 522, INTERIOR, STOREROOM, FROM APPROXIMATELY 50 FEET SOUTHEAST ...
5. BUILDING 522, INTERIOR, STOREROOM, FROM APPROXIMATELY 50 FEET SOUTHEAST OF NORTHWEST CORNER, LOOKING EAST. - Oakland Naval Supply Center, Aeronautical Materials Storehouses, Between E & G Streets, between Fourth & Sixth Streets, Oakland, Alameda County, CA
1. WEST AND SOUTH SIDES, FROM APPROXIMATELY 75 FEET SOUTHWEST ...
1. WEST AND SOUTH SIDES, FROM APPROXIMATELY 75 FEET SOUTHWEST OF BUILDING, LOOKING EAST-NORTHEAST. - Oakland Naval Supply Center, Heating Plant, North of B Street & West of Third Street, Oakland, Alameda County, CA
1. WEST AND SOUTH SIDES, FROM APPROXIMATELY 25 FEET SOUTH ...
1. WEST AND SOUTH SIDES, FROM APPROXIMATELY 25 FEET SOUTH OF SOUTHEASTERN CORNER OF BUILDING 441-B, LOOKING NORTHEAST. - Oakland Naval Supply Center, Heating Plant, On Northwest Corner of K Street & Fifth Street, Oakland, Alameda County, CA
Perspective view looking from the northeast, from approximately the same ...
Perspective view looking from the northeast, from approximately the same vantage point as in MD-1109-K-12 - National Park Seminary, Japanese Bungalow, 2801 Linden Lane, Silver Spring, Montgomery County, MD
15. Looking north from east bank of ditch, approximately halfway ...
15. Looking north from east bank of ditch, approximately 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
Low-complexity approximations to maximum likelihood MPSK modulation classification
NASA Technical Reports Server (NTRS)
Hamkins, Jon
2004-01-01
We present a new approximation to the maximum likelihood classifier to discriminate between M-ary and M'-ary phase-shift-keying transmitted on an additive white Gaussian noise (AWGN) channel and received noncoherentl, partially coherently, or coherently.
Non-ideal boson system in the Gaussian approximation
Tommasini, P.R.; de Toledo Piza, A.F.
1997-01-01
We investigate ground-state and thermal properties of a system of non-relativistic bosons interacting through repulsive, two-body interactions in a self-consistent Gaussian mean-field approximation which consists in writing the variationally determined density operator as the most general Gaussian functional of the quantized field operators. Finite temperature results are obtained in a grand canonical framework. Contact is made with the results of Lee, Yang, and Huang in terms of particular truncations of the Gaussian approximation. The full Gaussian approximation supports a free phase or a thermodynamically unstable phase when contact forces and a standard renormalization scheme are used. When applied to a Hamiltonian with zero range forces interpreted as an effective theory with a high momentum cutoff, the full Gaussian approximation generates a quasi-particle spectrum having an energy gap, in conflict with perturbation theory results. {copyright} 1997 Academic Press, Inc.
Numerical Stability and Convergence of Approximate Methods for Conservation Laws
NASA Astrophysics Data System (ADS)
Galkin, V. A.
We present the new approach to background of approximate methods convergence based on functional solutions theory for conservation laws. The applications to physical kinetics, gas and fluid dynamics are considered.
Vacancy-rearrangement theory in the first Magnus approximation
Becker, R.L.
1984-01-01
In the present paper we employ the first Magnus approximation (M1A), a unitarized Born approximation, in semiclassical collision theory. We have found previously that the M1A gives a substantial improvement over the first Born approximation (B1A) and can give a good approximation to a full coupled channels calculation of the mean L-shell vacancy probability per electron, p/sub L/, when the L-vacancies are accompanied by a K-shell vacancy (p/sub L/ is obtained experimentally from measurements of K/sub ..cap alpha../-satellite intensities). For sufficiently strong projectile-electron interactions (sufficiently large Z/sub p/ or small v) the M1A ceases to reproduce the coupled channels results, but it is accurate over a much wider range of Z/sub p/ and v than the B1A. 27 references.
A coefficient average approximation towards Gutzwiller wavefunction formalism
NASA Astrophysics Data System (ADS)
Liu, Jun; Yao, Yongxin; Wang, Cai-Zhuang; Ho, Kai-Ming
2015-06-01
Gutzwiller wavefunction is a physically well-motivated trial wavefunction for describing correlated electron systems. In this work, a new approximation is introduced to facilitate the evaluation of the expectation value of any operator within the Gutzwiller wavefunction formalism. The basic idea is to make use of a specially designed average over Gutzwiller wavefunction coefficients expanded in the many-body Fock space to approximate the ratio of expectation values between a Gutzwiller wavefunction and its underlying noninteracting wavefunction. To check with the standard Gutzwiller approximation (GA), we test its performance on single band systems and find quite interesting properties. On finite systems, we noticed that it gives superior performance over GA, while on infinite systems it asymptotically approaches GA. Analytic analysis together with numerical tests are provided to support this claimed asymptotical behavior. Finally, possible improvements on the approximation and its generalization towards multiband systems are illustrated and discussed.
A coefficient average approximation towards Gutzwiller wavefunction formalism.
Liu, Jun; Yao, Yongxin; Wang, Cai-Zhuang; Ho, Kai-Ming
2015-06-24
Gutzwiller wavefunction is a physically well-motivated trial wavefunction for describing correlated electron systems. In this work, a new approximation is introduced to facilitate the evaluation of the expectation value of any operator within the Gutzwiller wavefunction formalism. The basic idea is to make use of a specially designed average over Gutzwiller wavefunction coefficients expanded in the many-body Fock space to approximate the ratio of expectation values between a Gutzwiller wavefunction and its underlying noninteracting wavefunction. To check with the standard Gutzwiller approximation (GA), we test its performance on single band systems and find quite interesting properties. On finite systems, we noticed that it gives superior performance over GA, while on infinite systems it asymptotically approaches GA. Analytic analysis together with numerical tests are provided to support this claimed asymptotical behavior. Finally, possible improvements on the approximation and its generalization towards multiband systems are illustrated and discussed.
86. SITE INSTRUMENTATION: VIEW OF COMMUNICATIONS WIRING APPROXIMATELY THREE MILES ...
86. SITE INSTRUMENTATION: VIEW OF COMMUNICATIONS WIRING APPROXIMATELY THREE MILES NORTH OF GROUND ZERO, LOOKING NORTH - White Sands Missile Range, Trinity Site, Vicinity of Routes 13 & 20, White Sands, Dona Ana County, NM
Approximating the ground state of gapped quantum spin systems
Michalakis, Spyridon; Hamza, Eman; Nachtergaele, Bruno; Sims, Robert
2009-01-01
We consider quantum spin systems defined on finite sets V equipped with a metric. In typical examples, V is a large, but finite subset of Z{sup d}. For finite range Hamiltonians with uniformly bounded interaction terms and a unique, gapped ground state, we demonstrate a locality property of the corresponding ground state projector. In such systems, this ground state projector can be approximated by the product of observables with quantifiable supports. In fact, given any subset {chi} {contained_in} V the ground state projector can be approximated by the product of two projections, one supported on {chi} and one supported on {chi}{sup c}, and a bounded observable supported on a boundary region in such a way that as the boundary region increases, the approximation becomes better. Such an approximation was useful in proving an area law in one dimension, and this result corresponds to a multi-dimensional analogue.
An approximation for homogeneous freezing temperature of water droplets
NASA Astrophysics Data System (ADS)
O, K.-T.; Wood, R.
2015-11-01
In this work, based on the well-known formulae of classical nucleation theory (CNT), the temperature TNc = 1 at which the mean number of critical embryos inside a droplet is unity is derived and proposed as a new approximation for homogeneous freezing temperature of water droplets. Without consideration of time dependence and stochastic nature of the ice nucleation process, the approximation TNc = 1 is able to reproduce the dependence of homogeneous freezing temperature on drop size and water activity of aqueous drops observed in a wide range of experimental studies. We use the TNc = 1 approximation to argue that the distribution of homogeneous freezing temperatures observed in the experiments may largely be explained by the spread in the size distribution of droplets used in the particular experiment. It thus appears that this approximation is useful for predicting homogeneous freezing temperatures of water droplets in the atmosphere.
Generalized eikonal approximation for strong-field ionization
NASA Astrophysics Data System (ADS)
Cajiao Vélez, F.; Krajewska, K.; Kamiński, J. Z.
2015-05-01
We develop the eikonal perturbation theory to describe the strong-field ionization by finite laser pulses. This approach in the first order with respect to the binding potential (the so-called generalized eikonal approximation) avoids a singularity at the potential center. Thus, in contrast to the ordinary eikonal approximation, it allows one to treat rescattering phenomena in terms of quantum trajectories. We demonstrate how the first Born approximation and its domain of validity follow from eikonal perturbation theory. Using this approach, we study the coherent interference patterns in photoelectron energy spectra and their modifications induced by the interaction of photoelectrons with the atomic potential. Along with these first results, we discuss the prospects of using the generalized eikonal approximation to study strong-field ionization from multicentered atomic systems and to study other strong-field phenomena.
Approximate supernova remnant dynamics with cosmic ray production
NASA Technical Reports Server (NTRS)
Voelk, H. J.; Drury, L. O.; Dorfi, E. A.
1985-01-01
Supernova explosions are the most violent and energetic events in the galaxy and have long been considered probably sources of Cosmic Rays. Recent shock acceleration models treating the Cosmic Rays (CR's) as test particles nb a prescribed Supernova Remnant (SNR) evolution, indeed indicate an approximate power law momentum distribution f sub source (p) approximation p(-a) for the particles ultimately injected into the Interstellar Medium (ISM). This spectrum extends almost to the momentum p = 1 million GeV/c, where the break in the observed spectrum occurs. The calculated power law index approximately less than 4.2 agrees with that inferred for the galactic CR sources. The absolute CR intensity can however not be well determined in such a test particle approximation.
Model reduction using new optimal Routh approximant technique
NASA Technical Reports Server (NTRS)
Hwang, Chyi; Guo, Tong-Yi; Sheih, Leang-San
1992-01-01
An optimal Routh approximant of a single-input single-output dynamic system is a reduced-order transfer function of which the denominator is obtained by the Routh approximation method while the numerator is determined by minimizing a time-response integral-squared-error (ISE) criterion. In this paper, a new elegant approach is presented for obtaining the optimal Routh approximants for linear time-invariant continuous-time systems. The approach is based on the Routh canonical expansion, which is a finite-term orthogonal series of rational basis functions, and minimization of the ISE criterion. A procedure for combining the above approach with the bilinear transformation is also presented in order to obtain the optimal bilinear Routh approximants of linear time-invariant discrete-time systems. The proposed technique is simple in formulation and is amenable to practical implementation.
Interpolation function for approximating knee joint behavior in human gait
NASA Astrophysics Data System (ADS)
Toth-Taşcǎu, Mirela; Pater, Flavius; Stoia, Dan Ioan
2013-10-01
Starting from the importance of analyzing the kinematic data of the lower limb in gait movement, especially the angular variation of the knee joint, the paper propose an approximation function that can be used for processing the correlation among a multitude of knee cycles. The approximation of the raw knee data was done by Lagrange polynomial interpolation on a signal acquired using Zebris Gait Analysis System. The signal used in approximation belongs to a typical subject extracted from a lot of ten investigated subjects, but the function domain of definition belongs to the entire group. The study of the knee joint kinematics plays an important role in understanding the kinematics of the gait, this articulation having the largest range of motion in whole joints, in gait. The study does not propose to find an approximation function for the adduction-abduction movement of the knee, this being considered a residual movement comparing to the flexion-extension.
6. NORTH SIDE, FROM APPROXIMATELY 25 FEET SOUTHEAST OF SOUTHWEST ...
6. NORTH SIDE, FROM APPROXIMATELY 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
Second post-Newtonian approximation of Einstein-aether theory
Xie Yi; Huang Tianyi
2008-06-15
In this paper, second post-Newtonian approximation of Einstein-aether theory is obtained by Chandrasekhar's approach. Five parametrized post-Newtonian parameters in first post-Newtonian approximation are presented after a time transformation and they are identical with previous works, in which {gamma}=1, {beta}=1, and two preferred-frame parameters remain. Meanwhile, in second post-Newtonian approximation, a parameter, which represents third order nonlinearity for gravity, is zero--the same as in general relativity. For an application for future deep space laser ranging missions, we reduce the metric coefficients for light propagation in a case of N point masses as a simplified model of the Solar System. The resulting light deflection angle in second post-Newtonian approximation poses another constraint on the Einstein-aether theory.
Quantum instanton approximation for thermal rate constants of chemical reactions
NASA Astrophysics Data System (ADS)
Miller, William H.; Zhao, Yi; Ceotto, Michele; Yang, Sandy
2003-07-01
A quantum mechanical theory for chemical reaction rates is presented which is modeled after the [semiclassical (SC)] instanton approximation. It incorporates the desirable aspects of the instanton picture, which involves only properties of the (SC approximation to the) Boltzmann operator, but corrects its quantitative deficiencies by replacing the SC approximation for the Boltzmann operator by the quantum Boltzmann operator, exp(-βĤ). Since a calculation of the quantum Boltzmann operator is feasible for quite complex molecular systems (by Monte Carlo path integral methods), having an accurate rate theory that involves only the Boltzmann operator could be quite useful. The application of this quantum instanton approximation to several one- and two-dimensional model problems illustrates its potential; e.g., it is able to describe thermal rate constants accurately (˜10-20% error) from high to low temperatures deep in the tunneling regime, and applies equally well to asymmetric and symmetric potentials.
2. NORTHWEST SIDE, OBLIQUE VIEW, FROM APPROXIMATELY 10 FEET SOUTH ...
2. NORTHWEST SIDE, OBLIQUE VIEW, FROM APPROXIMATELY 10 FEET SOUTH OF SOUTH CORNER, LOOKING EAST-NORTHEAST. - Oakland Naval Supply Center, Administrative Offices, On Seventh Street East of Maritime Street, Oakland, Alameda County, CA
Approximate inverse preconditioning of iterative methods for nonsymmetric linear systems
Benzi, M.; Tuma, M.
1996-12-31
A method for computing an incomplete factorization of the inverse of a nonsymmetric matrix A is presented. The resulting factorized sparse approximate inverse is used as a preconditioner in the iterative solution of Ax = b by Krylov subspace methods.
Importance Sampling Approach for the Nonstationary Approximation Error Method
NASA Astrophysics Data System (ADS)
Huttunen, J. M. J.; Lehikoinen, A.; Hämäläinen, J.; Kaipio, J. P.
2010-09-01
The approximation error approach has been earlier proposed to handle modelling, numerical and computational errors in inverse problems. The idea of the approach is to include the errors to the forward model and compute the approximate statistics of the errors using Monte Carlo sampling. This can be a computationally tedious task but the key property of the approach is that the approximate statistics can be calculated off-line before measurement process takes place. In nonstationary problems, however, information is accumulated over time, and the initial uncertainties may turn out to have been exaggerated. In this paper, we propose an importance weighing algorithm with which the approximation error statistics can be updated during the accumulation of measurement information. As a computational example, we study an estimation problem that is related to a convection-diffusion problem in which the velocity field is not accurately specified.
Kullback-Leibler divergence and the Pareto-Exponential approximation.
Weinberg, G V
2016-01-01
Recent radar research interests in the Pareto distribution as a model for X-band maritime surveillance radar clutter returns have resulted in analysis of the asymptotic behaviour of this clutter model. In particular, it is of interest to understand when the Pareto distribution is well approximated by an Exponential distribution. The justification for this is that under the latter clutter model assumption, simpler radar detection schemes can be applied. An information theory approach is introduced to investigate the Pareto-Exponential approximation. By analysing the Kullback-Leibler divergence between the two distributions it is possible to not only assess when the approximation is valid, but to determine, for a given Pareto model, the optimal Exponential approximation.
VIEW INLAND (MAUKA) FROM BEACH ROAD. NOTE THE APPROXIMATE 46' ...
VIEW INLAND (MAUKA) FROM BEACH ROAD. NOTE THE APPROXIMATE 46' DISTANCE BETWEEN RESIDENCES 26 AND 28 WORCHESTER AVENUE. VIEW FACING NORTHEAST. - Hickam Field, Fort Kamehameha Historic Housing, Along Worchester Avenue & Hope Street, Honolulu, Honolulu County, HI
Scattering of electromagnetic wave by dielectric cylinder in eikonal approximation
NASA Astrophysics Data System (ADS)
Syshchenko, V. V.
2016-07-01
The scattering of the plane electromagnetic wave on a spatially extended, fiber lake target is considered. The formula for the scattering cross section is obtained using the approximation analogous to eikonal one in quantum mechanics.
Hamilton's Principle and Approximate Solutions to Problems in Classical Mechanics
ERIC Educational Resources Information Center
Schlitt, D. W.
1977-01-01
Shows how to use the Ritz method for obtaining approximate solutions to problems expressed in variational form directly from the variational equation. Application of this method to classical mechanics is given. (MLH)
Approximating the Helium Wavefunction in Positronium-Helium Scattering
NASA Technical Reports Server (NTRS)
DiRienzi, Joseph; Drachman, Richard J.
2003-01-01
In the Kohn variational treatment of the positronium- hydrogen scattering problem the scattering wave function is approximated 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 approximation to d e t e e the zero-energy scattering length for the Ps-He system, using a suite of approximate target functions. The results we obtain will be compared with each other and with corresponding values found by other approximation techniques.
Contextual classification of multispectral image data: Approximate algorithm
NASA Technical Reports Server (NTRS)
Tilton, J. C. (Principal Investigator)
1980-01-01
An approximation to a classification algorithm incorporating spatial context information in a general, statistical manner is presented which is computationally less intensive. Classifications that are nearly as accurate are produced.
Approximate penetration factors for nuclear reactions of astrophysical interest
NASA Technical Reports Server (NTRS)
Humblet, J.; Fowler, W. A.; Zimmerman, B. A.
1987-01-01
The ranges of validity of approximations of P(l), the penetration factor which appears in the parameterization of nuclear-reaction cross sections at low energies and is employed in the extrapolation of laboratory data to even lower energies of astrophysical interest, are investigated analytically. Consideration is given to the WKB approximation, P(l) at the energy of the total barrier, approximations derived from the asymptotic expansion of G(l) for large eta, approximations for small values of the parameter x, applications of P(l) to nuclear reactions, and the dependence of P(l) on channel radius. Numerical results are presented in tables and graphs, and parameter ranges where the danger of serious errors is high are identified.
Berkel, M. van; Hogeweij, G. M. D.; Tamura, N.; Ida, K.; Zwart, H. J.; Inagaki, S.; Baar, M. R. de
2014-11-15
In this paper, a number of new explicit approximations are introduced to estimate the perturbative diffusivity (χ), convectivity (V), and damping (τ) in a cylindrical geometry. For this purpose, the harmonic components of heat waves induced by localized deposition of modulated power are used. The approximations are based upon the heat equation in a semi-infinite cylindrical domain. The approximations are based upon continued fractions, asymptotic expansions, and multiple harmonics. The relative error for the different derived approximations is presented for different values of frequency, transport coefficients, and dimensionless radius. Moreover, it is shown how combinations of different explicit formulas can yield good approximations over a wide parameter space for different cases, such as no convection and damping, only damping, and both convection and damping. This paper is the second part (Part II) of a series of three papers. In Part I, the semi-infinite slab approximations have been treated. In Part III, cylindrical approximations are treated for heat waves traveling towards the center of the plasma.
Quark matter in the Hartree-Fock approximation
Grassi, F.
1987-07-01
An equation of state is computed for quark matter interacting through a phenomenological potential in the Hartree-Fock approximation. It is shown that for color-independent confining potentials, it can be approximated by the Hartree result and leads to a first order mass phase transition. For color-dependent confining potentials, a phase transition from a Fermi sphere to a Fermi shell is possible.
Interior, building 810, view to west from approximately midhangar. Area ...
Interior, building 810, view to west from approximately mid-hangar. Area of photo encompasses approximately 1/4 of the interior space, with the KC-10 tanker aircraft and the figures beneath it giving an idea of scale, 90mm lens plus electronic flash fill lightening. - Travis Air Force Base, B-36 Hangar, Between Woodskill Avenue & Ellis, adjacent to Taxiway V & W, Fairfield, Solano County, CA
Problems with the quenched approximation in the chiral limit
Sharpe, S.R.
1992-01-01
In the quenched approximation, loops of the light singlet meson (the [eta][prime]) give rise to a type of chiral logarithm absent in full QCD. These logarithms are singular in the chiral limit, throwing doubt upon the utility of the quenched approximation. In previous work, I summed a class of diagrams, leading to non-analytic power dependencies such as [l angle][anti [psi
Optimal feedback control infinite dimensional parabolic evolution systems: Approximation techniques
NASA Technical Reports Server (NTRS)
Banks, H. T.; Wang, C.
1989-01-01
A general approximation framework is discussed for computation of optimal feedback controls in linear quadratic regular problems for nonautonomous parabolic distributed parameter systems. This is done in the context of a theoretical framework using general evolution systems in infinite dimensional Hilbert spaces. Conditions are discussed for preservation under approximation of stabilizability and detectability hypotheses on the infinite dimensional system. The special case of periodic systems is also treated.
Approximate quantitative relationships for rotating magnetic field current drive
NASA Astrophysics Data System (ADS)
Hugrass, W. N.; Ohnishi, M.
1999-08-01
A simplified model for the rotating magnetic field (RMF) current drive in an infinitely long cylindrical plasma is used to obtain approximate relationships between the fluid flow velocities, collisionality and degree of nonlinearity. These approximate relationships provide simple quantitative estimates for the basic conditions required for the RMF current drive technique to be applied successfully. In particular, the condition required for the motion of the ion fluid not to be flux-preserving, is evaluated quantitatively for the first time.
On Nash-Equilibria of Approximation-Stable Games
NASA Astrophysics Data System (ADS)
Awasthi, Pranjal; Balcan, Maria-Florina; Blum, Avrim; Sheffet, Or; Vempala, Santosh
One reason for wanting to compute an (approximate) Nash equilibrium of a game is to predict how players will play. However, if the game has multiple equilibria that are far apart, or ɛ-equilibria that are far in variation distance from the true Nash equilibrium strategies, then this prediction may not be possible even in principle. Motivated by this consideration, in this paper we define the notion of games that are approximation stable, meaning that all ɛ-approximate equilibria are contained inside a small ball of radius Δ around a true equilibrium, and investigate a number of their properties. Many natural small games such as matching pennies and rock-paper-scissors are indeed approximation stable. We show furthermore there exist 2-player n-by-n approximation-stable games in which the Nash equilibrium and all approximate equilibria have support Ω(log n). On the other hand, we show all (ɛ,Δ) approximation-stable games must have an ɛ-equilibrium of support O(Δ^{2-o(1)}/ɛ2{log n}), yielding an immediate n^{O(Δ^{2-o(1)}/ɛ^2log n)}-time algorithm, improving over the bound of [11] for games satisfying this condition. We in addition give a polynomial-time algorithm for the case that Δ and ɛ are sufficiently close together. We also consider an inverse property, namely that all non-approximate equilibria are far from some true equilibrium, and give an efficient algorithm for games satisfying that condition.
Robustness of controllers designed using Galerkin type approximations
NASA Technical Reports Server (NTRS)
Morris, K. A.
1990-01-01
One of the difficulties in designing controllers for infinite-dimensional systems arises from attempting to calculate a state for the system. It is shown that Galerkin type approximations can be used to design controllers which will perform as designed when implemented on the original infinite-dimensional system. No assumptions, other than those typically employed in numerical analysis, are made on the approximating scheme.
Low rank approximation in G 0 W 0 calculations
NASA Astrophysics Data System (ADS)
Shao, MeiYue; Lin, Lin; Yang, Chao; Liu, Fang; Da Jornada, Felipe H.; Deslippe, Jack; Louie, Steven G.
2016-08-01
The single particle energies obtained in a Kohn--Sham density functional theory (DFT) calculation are generally known to be poor approximations to electron excitation energies that are measured in transport, tunneling and spectroscopic experiments such as photo-emission spectroscopy. The correction to these energies can be obtained from the poles of a single particle Green's function derived from a many-body perturbation theory. From a computational perspective, the accuracy and efficiency of such an approach depends on how a self energy term that properly accounts for dynamic screening of electrons is approximated. The $G_0W_0$ approximation is a widely used technique in which the self energy is expressed as the convolution of a non-interacting Green's function ($G_0$) and a screened Coulomb interaction ($W_0$) in the frequency domain. The computational cost associated with such a convolution is high due to the high complexity of evaluating $W_0$ at multiple frequencies. In this paper, we discuss how the cost of $G_0W_0$ calculation can be reduced by constructing a low rank approximation to the frequency dependent part of $W_0$. In particular, we examine the effect of such a low rank approximation on the accuracy of the $G_0W_0$ approximation. We also discuss how the numerical convolution of $G_0$ and $W_0$ can be evaluated efficiently and accurately by using a contour deformation technique with an appropriate choice of the contour.
Simultaneous Approximation to Real and p-adic Numbers
NASA Astrophysics Data System (ADS)
Zelo, Dmitrij
2009-02-01
We study the problem of simultaneous approximation to a fixed family of real and p-adic numbers by roots of integer polynomials of restricted type. The method that we use for this purpose was developed by H. Davenport and W.M. Schmidt in their study of approximation to real numbers by algebraic integers. This method based on Mahler's Duality requires to study the dual problem of approximation to successive powers of these numbers by rational numbers with the same denominators. Dirichlet's Box Principle provides estimates for such approximations but one can do better. In this thesis we establish constraints on how much better one can do when dealing with the numbers and their squares. We also construct examples showing that at least in some instances these constraints are optimal. Going back to the original problem, we obtain estimates for simultaneous approximation to real and p-adic numbers by roots of integer polynomials of degree 3 or 4 with fixed coefficients in degree at least 3. In the case of a single real number (and no p-adic numbers), we extend work of D. Roy by showing that the square of the golden ratio is the optimal exponent of approximation by algebraic numbers of degree 4 with bounded denominator and trace.
Computing gap free Pareto front approximations with stochastic search algorithms.
Schütze, Oliver; Laumanns, Marco; Tantar, Emilia; Coello, Carlos A Coello; Talbi, El-Ghazali
2010-01-01
Recently, a convergence proof of stochastic search algorithms toward finite size Pareto set approximations of continuous multi-objective optimization problems has been given. The focus was on obtaining a finite approximation that captures the entire solution set in some suitable sense, which was defined by the concept of epsilon-dominance. Though bounds on the quality of the limit approximation-which are entirely determined by the archiving strategy and the value of epsilon-have been obtained, the strategies do not guarantee to obtain a gap free approximation of the Pareto front. That is, such approximations A can reveal gaps in the sense that points f in the Pareto front can exist such that the distance of f to any image point F(a), a epsilon A, is "large." Since such gap free approximations are desirable in certain applications, and the related archiving strategies can be advantageous when memetic strategies are included in the search process, we are aiming in this work for such methods. We present two novel strategies that accomplish this task in the probabilistic sense and under mild assumptions on the stochastic search algorithm. In addition to the convergence proofs, we give some numerical results to visualize the behavior of the different archiving strategies. Finally, we demonstrate the potential for a possible hybridization of a given stochastic search algorithm with a particular local search strategy-multi-objective continuation methods-by showing that the concept of epsilon-dominance can be integrated into this approach in a suitable way.
Validity of the Aluminum Equivalent Approximation in Space Radiation Shielding
NASA Technical Reports Server (NTRS)
Badavi, Francis F.; Adams, Daniel O.; Wilson, John W.
2009-01-01
The origin of the aluminum equivalent shield approximation in space radiation analysis can be traced back to its roots in the early years of the NASA space programs (Mercury, Gemini and Apollo) wherein the primary radiobiological concern was the intense sources of ionizing radiation causing short term effects which was thought to jeopardize the safety of the crew and hence the mission. Herein, it is shown that the aluminum equivalent shield approximation, although reasonably well suited for that time period and to the application for which it was developed, is of questionable usefulness to the radiobiological concerns of routine space operations of the 21 st century which will include long stays onboard the International Space Station (ISS) and perhaps the moon. This is especially true for a risk based protection system, as appears imminent for deep space exploration where the long-term effects of Galactic Cosmic Ray (GCR) exposure is of primary concern. The present analysis demonstrates that sufficiently large errors in the interior particle environment of a spacecraft result from the use of the aluminum equivalent approximation, and such approximations should be avoided in future astronaut risk estimates. In this study, the aluminum equivalent approximation is evaluated as a means for estimating the particle environment within a spacecraft structure induced by the GCR radiation field. For comparison, the two extremes of the GCR environment, the 1977 solar minimum and the 2001 solar maximum, are considered. These environments are coupled to the Langley Research Center (LaRC) deterministic ionized particle transport code High charge (Z) and Energy TRaNsport (HZETRN), which propagates the GCR spectra for elements with charges (Z) in the range I <= Z <= 28 (H -- Ni) and secondary neutrons through selected target materials. The coupling of the GCR extremes to HZETRN allows for the examination of the induced environment within the interior' of an idealized spacecraft
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
Liu, Jian; Miller, William H.
2006-09-06
The thermal Gaussian approximation (TGA) recently developed by Mandelshtam et al has been demonstrated to be a practical way for approximating the Boltzmann operator exp(-{beta}H) for multidimensional systems. In this paper the TGA is combined with semiclassical (SC) initial value representations (IVRs) for thermal time correlation functions. Specifically, it is used with the linearized SC-IVR (LSC-IVR, equivalent to the classical Wigner model), and the 'forward-backward semiclassical dynamics' (FBSD) approximation developed by Makri et al. Use of the TGA with both of these approximate SC-IVRs allows the oscillatory part of the IVR to be integrated out explicitly, providing an extremely simple result that is readily applicable to large molecular systems. Calculation of the force-force autocorrelation for a strongly anharmonic oscillator demonstrates its accuracy, and of the velocity autocorrelation function (and thus the diffusion coefficient) of liquid neon demonstrates its applicability.
Comparison of the Radiative Two-Flux and Diffusion Approximations
NASA Technical Reports Server (NTRS)
Spuckler, Charles M.
2006-01-01
Approximate solutions are sometimes used to determine the heat transfer and temperatures in a semitransparent material in which conduction and thermal radiation are acting. A comparison of the Milne-Eddington two-flux approximation and the diffusion approximation for combined conduction and radiation heat transfer in a ceramic material was preformed to determine the accuracy of the diffusion solution. A plane gray semitransparent layer without a substrate and a non-gray semitransparent plane layer on an opaque substrate were considered. For the plane gray layer the material is semitransparent for all wavelengths and the scattering and absorption coefficients do not vary with wavelength. For the non-gray plane layer the material is semitransparent with constant absorption and scattering coefficients up to a specified wavelength. At higher wavelengths the non-gray plane layer is assumed to be opaque. The layers are heated on one side and cooled on the other by diffuse radiation and convection. The scattering and absorption coefficients were varied. The error in the diffusion approximation compared to the Milne-Eddington two flux approximation was obtained as a function of scattering coefficient and absorption coefficient. The percent difference in interface temperatures and heat flux through the layer obtained using the Milne-Eddington two-flux and diffusion approximations are presented as a function of scattering coefficient and absorption coefficient. The largest errors occur for high scattering and low absorption except for the back surface temperature of the plane gray layer where the error is also larger at low scattering and low absorption. It is shown that the accuracy of the diffusion approximation can be improved for some scattering and absorption conditions if a reflectance obtained from a Kubelka-Munk type two flux theory is used instead of a reflection obtained from the Fresnel equation. The Kubelka-Munk reflectance accounts for surface reflection and
Berkel, M. van; Zwart, H. J.; Tamura, N.; Ida, K.; Hogeweij, G. M. D.; Inagaki, S.; Baar, M. R. de
2014-11-15
In this paper, a number of new approximations are introduced to estimate the perturbative diffusivity (χ), convectivity (V), and damping (τ) in cylindrical geometry. For this purpose, the harmonic components of heat waves induced by localized deposition of modulated power are used. The approximations are based on semi-infinite slab approximations of the heat equation. The main result is the approximation of χ under the influence of V and τ based on the phase of two harmonics making the estimate less sensitive to calibration errors. To understand why the slab approximations can estimate χ well in cylindrical geometry, the relationships between heat transport models in slab and cylindrical geometry are studied. In addition, the relationship between amplitude and phase with respect to their derivatives, used to estimate χ, is discussed. The results are presented in terms of the relative error for the different derived approximations for different values of frequency, transport coefficients, and dimensionless radius. The approximations show a significant region in which χ, V, and τ can be estimated well, but also regions in which the error is large. Also, it is shown that some compensation is necessary to estimate V and τ in a cylindrical geometry. On the other hand, errors resulting from the simplified assumptions are also discussed showing that estimating realistic values for V and τ based on infinite domains will be difficult in practice. This paper is the first part (Part I) of a series of three papers. In Part II and Part III, cylindrical approximations based directly on semi-infinite cylindrical domain (outward propagating heat pulses) and inward propagating heat pulses in a cylindrical domain, respectively, will be treated.
The Born approximation in time-distance helioseismology
NASA Astrophysics Data System (ADS)
Birch, A. C.; Kosovichev, A. G.
2001-01-01
Time-distance helioseismology, which measures the time for acoustic waves to travel between points on the solar surface, has been used to study small-scale three-dimensional features in the sun, for example active regions, as well as large-scale features, for example meridional flow, that are not accessible by standard global helioseismology. The interpretation of travel times has typically been done in the ray approximation. The interaction of acoustic waves with features smaller than their wavelength, such as in active regions or in the tachocline, is not expected to be well represented by ray theory. In order to develop a wave interpretation of time-distance data we employ the first Born approximation, which takes into account finite-wavelength effects and allows a single scattering between the source and receiver of the acoustic wave. We show that in the case of spherically symmetric perturbations the Born approximation can be easily related to normal mode perturbation theory. The Born approximation agrees with ray theory when applied to large scale perturbations, and performs better than ray theory when applied to perturbations with small spatial scale. We show, via an example OLA inversion of artificial data, that reasonable averaging kernels can be built from Born approximation kernels.
Detection of approximal caries with a new laser fluorescence device.
Lussi, A; Hack, A; Hug, I; Heckenberger, H; Megert, B; Stich, H
2006-01-01
The laser device DIAGNOdent developed for the detection of occlusal caries has limited value on approximal surfaces. The aim of this study was to develop and to test a new laser fluorescence (LF) device for the detection of approximal caries. Light with a wavelength of 655 nm was transported to the approximal surface using two different sapphire fibre tips. Seventy-five teeth were selected from a pool of extracted permanent human molars, frozen at -20 degrees C until use. Before being measured, they were defrosted, cleaned and calculus was removed with a scaler. The molars were set in blocks simulating the contact area of adults. Bitewing radiographs were obtained using Kodak Insight films. After two independent assessments with the new LF device, the teeth were histologically prepared, and assessed for caries extension. Using the laser, specificity values for D1 threshold (outer half of enamel), D2 threshold (inner half of enamel), D3 threshold (dentine) ranged between 0.81 and 0.93, sensitivity between 0.84 and 0.92 with no difference between the two tips. Bitewing radiography showed an inferior performance compared to LF (p<0.05). Intraex aminer reproducibility was high (kappa>.74). The new LF system might be a useful additional tool in detecting approximal caries. Because of its good reproducibility, it could be used to monitor caries regression or progression on approximal surfaces.
Dissociation between exact and approximate addition in developmental dyslexia.
Yang, Xiujie; Meng, Xiangzhi
2016-09-01
Previous research has suggested that number sense and language are involved in number representation and calculation, in which number sense supports approximate arithmetic, and language permits exact enumeration and calculation. Meanwhile, individuals with dyslexia have a core deficit in phonological processing. Based on these findings, we thus hypothesized that children with dyslexia may exhibit exact calculation impairment while doing mental arithmetic. The reaction time and accuracy while doing exact and approximate addition with symbolic Arabic digits and non-symbolic visual arrays of dots were compared between typically developing children and children with dyslexia. Reaction time analyses did not reveal any differences across two groups of children, the accuracies, interestingly, revealed a distinction of approximation and exact addition across two groups of children. Specifically, two groups of children had no differences in approximation. Children with dyslexia, however, had significantly lower accuracy in exact addition in both symbolic and non-symbolic tasks than that of typically developing children. Moreover, linguistic performances were selectively associated with exact calculation across individuals. These results suggested that children with dyslexia have a mental arithmetic deficit specifically in the realm of exact calculation, while their approximation ability is relatively intact. PMID:27310366
A consistent collinear triad approximation for operational wave models
NASA Astrophysics Data System (ADS)
Salmon, J. E.; Smit, P. B.; Janssen, T. T.; Holthuijsen, L. H.
2016-08-01
In shallow water, the spectral evolution associated with energy transfers due to three-wave (or triad) interactions is important for the prediction of nearshore wave propagation and wave-driven dynamics. The numerical evaluation of these nonlinear interactions involves the evaluation of a weighted convolution integral in both frequency and directional space for each frequency-direction component in the wave field. For reasons of efficiency, operational wave models often rely on a so-called collinear approximation that assumes that energy is only exchanged between wave components travelling in the same direction (collinear propagation) to eliminate the directional convolution. In this work, we show that the collinear approximation as presently implemented in operational models is inconsistent. This causes energy transfers to become unbounded in the limit of unidirectional waves (narrow aperture), and results in the underestimation of energy transfers in short-crested wave conditions. We propose a modification to the collinear approximation to remove this inconsistency and to make it physically more realistic. Through comparison with laboratory observations and results from Monte Carlo simulations, we demonstrate that the proposed modified collinear model is consistent, remains bounded, smoothly converges to the unidirectional limit, and is numerically more robust. Our results show that the modifications proposed here result in a consistent collinear approximation, which remains bounded and can provide an efficient approximation to model nonlinear triad effects in operational wave models.
Efficient solution of parabolic equations by Krylov approximation methods
NASA Technical Reports Server (NTRS)
Gallopoulos, E.; Saad, Y.
1990-01-01
Numerical techniques for solving parabolic equations by the method of lines is addressed. The main motivation for the proposed approach is the possibility of exploiting a high degree of parallelism in a simple manner. The basic idea of the method is to approximate the action of the evolution operator on a given state vector by means of a projection process onto a Krylov subspace. Thus, the resulting approximation consists of applying an evolution operator of a very small dimension to a known vector which is, in turn, computed accurately by exploiting well-known rational approximations to the exponential. Because the rational approximation is only applied to a small matrix, the only operations required with the original large matrix are matrix-by-vector multiplications, and as a result the algorithm can easily be parallelized and vectorized. Some relevant approximation and stability issues are discussed. We present some numerical experiments with the method and compare its performance with a few explicit and implicit algorithms.
A quantum relaxation-time approximation for finite fermion systems
Reinhard, P.-G.; Suraud, E.
2015-03-15
We propose a relaxation time approximation for the description of the dynamics of strongly excited fermion systems. Our approach is based on time-dependent density functional theory at the level of the local density approximation. This mean-field picture is augmented by collisional correlations handled in relaxation time approximation which is inspired from the corresponding semi-classical picture. The method involves the estimate of microscopic relaxation rates/times which is presently taken from the well established semi-classical experience. The relaxation time approximation implies evaluation of the instantaneous equilibrium state towards which the dynamical state is progressively driven at the pace of the microscopic relaxation time. As test case, we consider Na clusters of various sizes excited either by a swift ion projectile or by a short and intense laser pulse, driven in various dynamical regimes ranging from linear to strongly non-linear reactions. We observe a strong effect of dissipation on sensitive observables such as net ionization and angular distributions of emitted electrons. The effect is especially large for moderate excitations where typical relaxation/dissipation time scales efficiently compete with ionization for dissipating the available excitation energy. Technical details on the actual procedure to implement a working recipe of such a quantum relaxation approximation are given in appendices for completeness.
Validity criterion for the Born approximation convergence in microscopy imaging.
Trattner, Sigal; Feigin, Micha; Greenspan, Hayit; Sochen, Nir
2009-05-01
The need for the reconstruction and quantification of visualized objects from light microscopy images requires an image formation model that adequately describes the interaction of light waves with biological matter. Differential interference contrast (DIC) microscopy, as well as light microscopy, uses the common model of the scalar Helmholtz equation. Its solution is frequently expressed via the Born approximation. A theoretical bound is known that limits the validity of such an approximation to very small objects. We present an analytic criterion for the validity region of the Born approximation. In contrast to the theoretical known bound, the suggested criterion considers the field at the lens, external to the object, that corresponds to microscopic imaging and extends the validity region of the approximation. An analytical proof of convergence is presented to support the derived criterion. The suggested criterion for the Born approximation validity region is described in the context of a DIC microscope, yet it is relevant for any light microscope with similar fundamental apparatus. PMID:19412231
Rational trigonometric approximations using Fourier series partial sums
NASA Technical Reports Server (NTRS)
Geer, James F.
1993-01-01
A class of approximations (S(sub N,M)) to a periodic function f which uses the ideas of Pade, or rational function, approximations based on the Fourier series representation of f, rather than on the Taylor series representation of f, is introduced and studied. Each approximation S(sub N,M) is the quotient of a trigonometric polynomial of degree N and a trigonometric polynomial of degree M. The coefficients in these polynomials are determined by requiring that an appropriate number of the Fourier coefficients of S(sub N,M) agree with those of f. Explicit expressions are derived for these coefficients in terms of the Fourier coefficients of f. It is proven that these 'Fourier-Pade' approximations converge point-wise to (f(x(exp +))+f(x(exp -)))/2 more rapidly (in some cases by a factor of 1/k(exp 2M)) than the Fourier series partial sums on which they are based. The approximations are illustrated by several examples and an application to the solution of an initial, boundary value problem for the simple heat equation is presented.
Optical approximation in the theory of geometric impedance
NASA Astrophysics Data System (ADS)
Stupakov, G.; Bane, K. L. F.; Zagorodnov, I.
2007-05-01
In this paper we introduce an optical approximation into the theory of impedance calculation, one valid in the limit of high frequencies. This approximation neglects diffraction effects in the radiation process, and is conceptually equivalent to the approximation of geometric optics in electromagnetic theory. Using this approximation, we derive equations for the longitudinal impedance for arbitrary offsets, with respect to a reference orbit, of source and test particles. With the help of the Panofsky-Wenzel theorem, we also obtain expressions for the transverse impedance (also for arbitrary offsets). We further simplify these expressions for the case of the small offsets that are typical for practical applications. Our final expressions for the impedance, in the general case, involve two-dimensional integrals over various cross sections of the transition. We further demonstrate, for several known axisymmetric examples, how our method is applied to the calculation of impedances. Finally, we discuss the accuracy of the optical approximation and its relation to the diffraction regime in the theory of impedance.
Massive neutrinos in cosmology: Analytic solutions and fluid approximation
Shoji, Masatoshi; Komatsu, Eiichiro
2010-06-15
We study the evolution of linear density fluctuations of free-streaming massive neutrinos at redshift of z<1000, with an explicit justification on the use of a fluid approximation. We solve the collisionless Boltzmann equation in an Einstein de-Sitter (EdS) universe, truncating the Boltzmann hierarchy at l{sub max}=1 and 2, and compare the resulting density contrast of neutrinos {delta}{sub {nu}}{sup fluid} with that of the exact solutions of the Boltzmann equation that we derive in this paper. Roughly speaking, the fluid approximation is accurate if neutrinos were already nonrelativistic when the neutrino density fluctuation of a given wave number entered the horizon. We find that the fluid approximation is accurate at subpercent levels for massive neutrinos with m{sub {nu}>}0.05 eV at the scale of k < or approx. 1.0h Mpc{sup -1} and redshift of z<100. This result validates the use of the fluid approximation, at least for the most massive species of neutrinos suggested by the neutrino oscillation experiments. We also find that the density contrast calculated from fluid equations (i.e., continuity and Euler equations) becomes a better approximation at a lower redshift, and the accuracy can be further improved by including an anisotropic stress term in the Euler equation. The anisotropic stress term effectively increases the pressure term by a factor of 9/5.
Approximating smooth functions using algebraic-trigonometric polynomials
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
Analytical approximations for x-ray cross sections III
Biggs, F; Lighthill, R
1988-08-01
This report updates our previous work that provided analytical approximations to cross sections for both photoelectric absorption of photons by atoms and incoherent scattering of photons by atoms. This representation is convenient for use in programmable calculators and in computer programs to evaluate these cross sections numerically. The results apply to atoms of atomic numbers between 1 and 100 and for photon energiesgreater than or equal to10 eV. The photoelectric cross sections are again approximated by four-term polynomials in reciprocal powers of the photon energy. There are now more fitting intervals, however, than were used previously. The incoherent-scattering cross sections are based on the Klein-Nishina relation, but use simpler approximate equations for efficient computer evaluation. We describe the averaging scheme for applying these atomic results to any composite material. The fitting coefficients are included in tables, and the cross sections are shown graphically. 100 graphs, 1 tab.
Integral approximants for functions of higher monodromic dimension
Baker, G.A. Jr.
1987-01-01
In addition to the description of multiform, locally analytic functions as covering a many sheeted version of the complex plane, Riemann also introduced the notion of considering them as describing a space whose ''monodromic'' dimension is the number of linearly independent coverings by the monogenic analytic function at each point of the complex plane. I suggest that this latter concept is natural for integral approximants (sub-class of Hermite-Pade approximants) and discuss results for both ''horizontal'' and ''diagonal'' sequences of approximants. Some theorems are now available in both cases and make clear the natural domain of convergence of the horizontal sequences is a disk centered on the origin and that of the diagonal sequences is a suitably cut complex-plane together with its identically cut pendant Riemann sheets. Some numerical examples have also been computed.
Axially symmetric dissipative fluids in the quasi-static approximation
NASA Astrophysics Data System (ADS)
Herrera, L.; di Prisco, A.; Ospino, J.; Carot, J.
2016-01-01
Using a framework based on the 1 + 3 formalism, we carry out a study on axially and reflection symmetric dissipative fluids, in the quasi-static regime. We first derive a set of invariantly defined “velocities”, which allow for an inambiguous definition of the quasi-static approximation. Next, we rewrite all the relevant equations in this approximation and extract all the possible, physically relevant, consequences ensuing the adoption of such an approximation. In particular, we show how the vorticity, the shear and the dissipative flux, may lead to situations where different kind of “velocities” change their sign within the fluid distribution with respect to their sign on the boundary surface. It is shown that states of gravitational radiation are not a priori incompatible with the quasi-static regime. However, any such state must last for an infinite period of time, thereby diminishing its physical relevance.
Optimal Slater-determinant approximation of fermionic wave functions
NASA Astrophysics Data System (ADS)
Zhang, J. M.; Mauser, Norbert J.
2016-09-01
We study the optimal Slater-determinant approximation of an N -fermion wave function analytically. That is, we seek the Slater-determinant (constructed out of N orthonormal single-particle orbitals) wave function having largest overlap with a given N -fermion wave function. Some simple lemmas have been established and their usefulness is demonstrated on some structured states, such as the Greenberger-Horne-Zeilinger state. In the simplest nontrivial case of three fermions in six orbitals, which the celebrated Borland-Dennis discovery is about, the optimal Slater approximation wave function is proven to be built out of the natural orbitals in an interesting way. We also show that the Hadamard inequality is useful for finding the optimal Slater approximation of some special target wave functions.
Kernel polynomial approximations for densities of states and spectral functions
Silver, R.N.; Voter, A.F.; Kress, J.D.; Roeder, H.
1996-03-01
Chebyshev polynomial approximations are an efficient and numerically stable way to calculate properties of the very large Hamiltonians important in computational condensed matter physics. The present paper derives an optimal kernal polynomial which enforces positivity of density of states and spectral estimates, achieves the best energy resolution, and preserves normalization. This kernel polynomial method (KPM) is demonstrated for electronic structure and dynamic magnetic susceptibility calculations. For tight binding Hamiltonians of Si, we show how to achieve high precision and rapid convergence of the cohesive energy and vacancy formation energy by careful attention to the order of approximation. For disordered XXZ-magnets, we show that the KPM provides a simpler and more reliable procedure for calculating spectral functions than Lanczos recursion methods. Polynomial approximations to Fermi projection operators are also proposed. 26 refs., 10 figs.
Burgers approximation for two-dimensional flow past an ellipse
NASA Technical Reports Server (NTRS)
Dorrepaal, J. M.
1982-01-01
A linearization of the Navier-Stokes equation due to Burgers in which vorticity is transported by the velocity field corresponding to continuous potential flow is examined. The governing equations are solved exactly for the two dimensional steady flow past an ellipse of arbitrary aspect ratio. The requirement of no slip along the surface of the ellipse results in an infinite algebraic system of linear equations for coefficients appearing in the solution. The system is truncated at a point which gives reliable results for Reynolds numbers R in the range 0 R 5. Predictions of the Burgers approximation regarding separation, drag and boundary layer behavior are investigated. In particular, Burgers linearization gives drag coefficients which are closer to observed experimental values than those obtained from Oseen's approximation. In the special case of flow past a circular cylinder, Burgers approximation predicts a boundary layer whose thickness is roughly proportional to R-1/2.
Two Timescale Approximation Applied to Gravitational Waves from Eccentric EMRIs
NASA Astrophysics Data System (ADS)
Moxon, Jordan; Flanagan, Eanna; Hinderer, Tanja; Pound, Adam
2016-03-01
Gravitational-wave driven inspirals of compact objects into massive black holes (Extreme Mass Ratio Inspirals - EMRIs) form an interesting, long-lived signal for future space-based gravitational wave detectors. Accurate signal predictions will be necessary to take full advantage of matched filtering techniques, motivating the development of a calculational technique for deriving the gravitational wave signal to good approximation throughout the inspiral. We report on recent work on developing the two-timescale technique with the goal of predicting waveforms from eccentric equatorial systems to subleading (post-adiabatic) order in the phase, building on recent work by Pound in the scalar case. The computation requires us to understand the dissipative component of the second-order self force. It also demands careful consideration of how the two timescale (near-zone) approximation should match with the post-Minkowski approximation of the gravitational waves at great distances.
Inertia and Compressibility Effects in the Boussinesq Approximation
NASA Astrophysics Data System (ADS)
Shirgaonkar, Anup; Lele, Sanjiva
2006-11-01
The Boussinesq approximation is typically applied to flows where buoyancy is the dominant driving force. To extend its applicability to flows with substantial inertial perturbations, we examine the flow equations using perturbation analysis about the hydrostatic state. The physical effects corresponding to stratification, compressibility, small initial entropy perturbations, and inertia are characterized in terms of nondimensional parameters derived from the analysis. A simple and computationally efficient extension to the traditional Boussinesq approximation is proposed to include the interaction of buoyancy and inertia. The role of fluid compressibility in stratified low Mach number flows is highlighted and distinguished from the flow compressibility which is caused by motion. A nondimensional parameter is derived to demarcate compressible and nearly-incompressible hydrostatic states. The significance of the extended Boussinesq approximation is illustrated with numerical solutions to model problems. Application to the problem of aircraft vortex wake-exhaust jet interaction is discussed.
Approximating smooth functions using algebraic-trigonometric polynomials
NASA Astrophysics Data System (ADS)
Sharapudinov, Idris I.
2011-01-01
The problem under consideration is that of approximating classes of smooth functions by algebraic-trigonometric polynomials of the form p_n(t)+\\tau_m(t), where p_n(t) is an algebraic polynomial of degree n and \\tau_m(t)=a_0+\\sum_{k=1}^ma_k\\cos k\\pi t+b_k\\sin k\\pi t is a trigonometric polynomial of order m. The precise order of approximation by such polynomials in the classes W^r_\\infty(M) and an upper bound for similar approximations in the class W^r_p(M) with \\frac43 are found. The proof of these estimates uses mixed series in Legendre polynomials which the author has introduced and investigated previously. Bibliography: 13 titles.
Algebraic approximations for transcendental equations with applications in nanophysics
NASA Astrophysics Data System (ADS)
Barsan, Victor
2015-09-01
Using algebraic approximations of trigonometric or hyperbolic functions, a class of transcendental equations can be transformed in tractable, algebraic equations. Studying transcendental equations this way gives the eigenvalues of Sturm-Liouville problems associated to wave equation, mainly to Schroedinger equation; these algebraic approximations provide approximate analytical expressions for the energy of electrons and phonons in quantum wells, quantum dots (QDs) and quantum wires, in the frame of one-particle models of such systems. The advantage of this approach, compared to the numerical calculations, is that the final result preserves the functional dependence on the physical parameters of the problem. The errors of this method, situated between some few percentages and ?, are carefully analysed. Several applications, for quantum wells, QDs and quantum wires, are presented.
Resonant-state-expansion Born approximation for waveguides with dispersion
NASA Astrophysics Data System (ADS)
Doost, M. B.
2016-02-01
The resonant-state-expansion (RSE) Born approximation, a rigorous perturbative method developed for electrodynamic and quantum mechanical open systems, is further developed to treat waveguides with a Sellmeier dispersion. For media that can be described by these types of dispersion over the relevant frequency range, such as optical glass, I show that the the perturbed RSE problem can be solved by diagonalizing a second-order eigenvalue problem. In the case of a single resonance at zero frequency, this is simplified to a generalized eigenvalue problem. Results are presented using analytically solvable planar waveguides and parameters of borosilicate BK7 glass, for a perturbation in the waveguide width. The efficiency of using either an exact dispersion over all frequencies or an approximate dispersion over a narrow frequency range is compared. I included a derivation of the RSE Born approximation for waveguides to make use of the resonances calculated by the RSE.
Dual methods and approximation concepts in structural synthesis
NASA Technical Reports Server (NTRS)
Fleury, C.; Schmit, L. A., Jr.
1980-01-01
Approximation concepts and dual method algorithms are combined to create a method for minimum weight design of structural systems. Approximation concepts convert the basic mathematical programming statement of the structural synthesis problem into a sequence of explicit primal problems of separable form. These problems are solved by constructing explicit dual functions, which are maximized subject to nonnegativity constraints on the dual variables. It is shown that the joining together of approximation concepts and dual methods can be viewed as a generalized optimality criteria approach. The dual method is successfully extended to deal with pure discrete and mixed continuous-discrete design variable problems. The power of the method presented is illustrated with numerical results for example problems, including a metallic swept wing and a thin delta wing with fiber composite skins.
Galerkin approximation for inverse problems for nonautonomous nonlinear distributed systems
NASA Technical Reports Server (NTRS)
Banks, H. T.; Reich, Simeon; Rosen, I. G.
1988-01-01
An abstract framework and convergence theory is developed for Galerkin approximation for inverse problems involving the identification of nonautonomous nonlinear distributed parameter systems. A set of relatively easily verified conditions is provided which are sufficient to guarantee the existence of optimal solutions and their approximation by a sequence of solutions to a sequence of approximating finite dimensional identification problems. The approach is based on the theory of monotone operators in Banach spaces and is applicable to a reasonably broad class of nonlinear distributed systems. Operator theoretic and variational techniques are used to establish a fundamental convergence result. An example involving evolution systems with dynamics described by nonstationary quasilinear elliptic operators along with some applications are presented and discussed.
Approximate Bisimulation-Based Reduction of Power System Dynamic Models
Stankovic, AM; Dukic, SD; Saric, AT
2015-05-01
In this paper we propose approximate bisimulation relations and functions for reduction of power system dynamic models in differential- algebraic (descriptor) form. The full-size dynamic model is obtained by linearization of the nonlinear transient stability model. We generalize theoretical results on approximate bisimulation relations and bisimulation functions, originally derived for a class of constrained linear systems, to linear systems in descriptor form. An algorithm for transient stability assessment is proposed and used to determine whether the power system is able to maintain the synchronism after a large disturbance. Two benchmark power systems are used to illustrate the proposed algorithm and to evaluate the applicability of approximate bisimulation relations and bisimulation functions for reduction of the power system dynamic models.
An approximate method for residual stress calculation infunctionally graded materials
Becker, T.L.
1999-06-02
Thermal residual stresses in functionally graded materials(FGMs) arise primarily from nonlinear spatial variations in the thermalexpansion coefficient, but can be significantly adjusted by variations inmodulus. Thermoelastic analysis of FGMs is complicated by such modulusgradients. A class of problems for which thermal stress solutions formaterials with constant modulus can be used as a basis for approximationsfor FGMs is discussed. The size of the error in this approximation due togradients in elastic modulus is investigated. Analytical and finiteelement solutions for the thermal stresses in various FGM geometries arecompared to results from this approximate method. In a geometry ofpractical interest, a right cylinder graded along the z-axis, the errorfor a Ni-Al2O3 FGM was found to be within 15 percent for all gradientsconsidered. The form of the approximation makes it easier to identifydesirable types of spatial nonlinearity in expansion coefficient andvariations in modulus: this would allow the manipulation of the locationof compressive stresses.
Efficient algorithm for approximating one-dimensional ground states
Aharonov, Dorit; Arad, Itai; Irani, Sandy
2010-07-15
The density-matrix renormalization-group method is very effective at finding ground states of one-dimensional (1D) quantum systems in practice, but it is a heuristic method, and there is no known proof for when it works. In this article we describe an efficient classical algorithm which provably finds a good approximation of the ground state of 1D systems under well-defined conditions. More precisely, our algorithm finds a matrix product state of bond dimension D whose energy approximates the minimal energy such states can achieve. The running time is exponential in D, and so the algorithm can be considered tractable even for D, which is logarithmic in the size of the chain. The result also implies trivially that the ground state of any local commuting Hamiltonian in 1D can be approximated efficiently; we improve this to an exact algorithm.
Sparse Multinomial Logistic Regression via Approximate Message Passing
NASA Astrophysics Data System (ADS)
Byrne, Evan; Schniter, Philip
2016-11-01
For the problem of multi-class linear classification and feature selection, we propose approximate message passing approaches to sparse multinomial logistic regression (MLR). First, we propose two algorithms based on the Hybrid Generalized Approximate Message Passing (HyGAMP) framework: one finds the maximum a posteriori (MAP) linear classifier and the other finds an approximation of the test-error-rate minimizing linear classifier. Then we design computationally simplified variants of these two algorithms. Next, we detail methods to tune the hyperparameters of their assumed statistical models using Stein's unbiased risk estimate (SURE) and expectation-maximization (EM), respectively. Finally, using both synthetic and real-world datasets, we demonstrate improved error-rate and runtime performance relative to existing state-of-the-art approaches to sparse MLR.
Theory of periodically specified problems: Complexity and approximability
Marathe, M.V.; Hunt, H.B. III; Stearns, R.E.; Rosenkrantz, D.J.
1997-12-05
We study the complexity and the efficient approximability of graph and satisfiability problems when specified using various kinds of periodic specifications studied. The general results obtained include the following: (1) We characterize the complexities of several basic generalized CNF satisfiability problems SAT(S) [Sc78], when instances are specified using various kinds of 1- and 2-dimensional periodic specifications. We outline how this characterization can be used to prove a number of new hardness results for the complexity classes DSPACE(n), NSPACE(n), DEXPTIME, NEXPTIME, EXPSPACE etc. These results can be used to prove in a unified way the hardness of a number of combinatorial problems when instances are specified succinctly using various succient specifications considered in the literature. As one corollary, we show that a number of basic NP-hard problems because EXPSPACE-hard when inputs are represented using 1-dimensional infinite periodic wide specifications. This answers a long standing open question posed by Orlin. (2) We outline a simple yet a general technique to devise approximation algorithms with provable worst case performance guarantees for a number of combinatorial problems specified periodically. Our efficient approximation algorithms and schemes are based on extensions of the ideas and represent the first non-trivial characterization of a class of problems having an {epsilon}-approximation (or PTAS) for periodically specified NEXPTIME-hard problems. Two of properties of our results are: (i) For the first time, efficient approximation algorithms and schemes have been developed for natural NEXPTIME-complete problems. (ii) Our results are the first polynomial time approximation algorithms with good performance guarantees for hard problems specified using various kinds of periodic specifications considered in this paper.
Solving the infeasible trust-region problem using approximations.
Renaud, John E.; Perez, Victor M.; Eldred, Michael Scott
2004-07-01
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 approximations for both cases is an alternative investigated by many researchers including the authors. When approximations 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 proven effective. The same approach has been used to manage the local move limits in sequential approximate 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 order 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 approximate optimization framework using homotopy ideas to relax the constraints. This algorithm called interior-point trust-region sequential approximate optimization (IPTRSAO) presents some similarities to the two normal-tangential steps algorithms. In this paper, a description of the similarities is presented and an
Quantum algorithm for an additive approximation of Ising partition functions
NASA Astrophysics Data System (ADS)
Matsuo, Akira; Fujii, Keisuke; Imoto, Nobuyuki
2014-08-01
We investigate quantum-computational complexity of calculating partition functions of Ising models. We construct a quantum algorithm for an additive approximation of Ising partition functions on square lattices. To this end, we utilize the overlap mapping developed by M. Van den Nest, W. Dür, and H. J. Briegel [Phys. Rev. Lett. 98, 117207 (2007), 10.1103/PhysRevLett.98.117207] and its interpretation through measurement-based quantum computation (MBQC). We specify an algorithmic domain, on which the proposed algorithm works, and an approximation scale, which determines the accuracy of the approximation. We show that the proposed algorithm performs a nontrivial task, which would be intractable on any classical computer, by showing that the problem that is solvable by the proposed quantum algorithm is BQP-complete. In the construction of the BQP-complete problem coupling strengths and magnetic fields take complex values. However, the Ising models that are of central interest in statistical physics and computer science consist of real coupling strengths and magnetic fields. Thus we extend the algorithmic domain of the proposed algorithm to such a real physical parameter region and calculate the approximation scale explicitly. We found that the overlap mapping and its MBQC interpretation improve the approximation scale exponentially compared to a straightforward constant-depth quantum algorithm. On the other hand, the proposed quantum algorithm also provides partial evidence that there exist no efficient classical algorithm for a multiplicative approximation of the Ising partition functions even on the square lattice. This result supports the observation that the proposed quantum algorithm also performs a nontrivial task in the physical parameter region.
Are there approximate relations among transverse momentum dependent distribution functions?
Harutyun AVAKIAN; Anatoli Efremov; Klaus Goeke; Andreas Metz; Peter Schweitzer; Tobias Teckentrup
2007-10-11
Certain {\\sl exact} relations among transverse momentum dependent parton distribution functions due to QCD equations of motion turn into {\\sl approximate} ones upon the neglect of pure twist-3 terms. On the basis of available data from HERMES we test the practical usefulness of one such ``Wandzura-Wilczek-type approximation'', namely of that connecting $h_{1L}^{\\perp(1)a}(x)$ to $h_L^a(x)$, and discuss how it can be further tested by future CLAS and COMPASS data.
Generating exact solutions to Einstein's equation using linearized approximations
NASA Astrophysics Data System (ADS)
Harte, Abraham I.; Vines, Justin
2016-10-01
We show that certain solutions to the linearized Einstein equation can—by the application of a particular type of linearized gauge transformation—be straightforwardly transformed into solutions of the exact Einstein equation. In cases with nontrivial matter content, the exact stress-energy tensor of the transformed metric has the same eigenvalues and eigenvectors as the linearized stress-energy tensor of the initial approximation. When our gauge exists, the tensorial structure of transformed metric perturbations identically eliminates all nonlinearities in Einstein's equation. As examples, we derive the exact Kerr and gravitational plane wave metrics from standard harmonic-gauge approximations.
A Poisson process approximation for generalized K-5 confidence regions
NASA Technical Reports Server (NTRS)
Arsham, H.; Miller, D. R.
1982-01-01
One-sided confidence regions for continuous cumulative distribution functions are constructed using empirical cumulative distribution functions and the generalized Kolmogorov-Smirnov distance. The band width of such regions becomes narrower in the right or left tail of the distribution. To avoid tedious computation of confidence levels and critical values, an approximation based on the Poisson process is introduced. This aproximation provides a conservative confidence region; moreover, the approximation error decreases monotonically to 0 as sample size increases. Critical values necessary for implementation are given. Applications are made to the areas of risk analysis, investment modeling, reliability assessment, and analysis of fault tolerant systems.
Compressibility Corrections to Closure Approximations for Turbulent Flow Simulations
Cloutman, L D
2003-02-01
We summarize some modifications to the usual closure approximations for statistical models of turbulence that are necessary for use with compressible fluids at all Mach numbers. We concentrate here on the gradient-flu approximation for the turbulent heat flux, on the buoyancy production of turbulence kinetic energy, and on a modification of the Smagorinsky model to include buoyancy. In all cases, there are pressure gradient terms that do not appear in the incompressible models and are usually omitted in compressible-flow models. Omission of these terms allows unphysical rates of entropy change.
Exponentially accurate approximations to piece-wise smooth periodic functions
NASA Technical Reports Server (NTRS)
Greer, James; Banerjee, Saheb
1995-01-01
A family of simple, periodic basis functions with 'built-in' discontinuities are introduced, and their properties are analyzed and discussed. Some of their potential usefulness is illustrated in conjunction with the Fourier series representations of functions with discontinuities. In particular, it is demonstrated how they can be used to construct a sequence of approximations which converges exponentially in the maximum norm to a piece-wise smooth function. The theory is illustrated with several examples and the results are discussed in the context of other sequences of functions which can be used to approximate discontinuous functions.
Nonlinear acoustic behavior at a caustic - An approximate analytical solution
NASA Technical Reports Server (NTRS)
Gill, P. M.; Seebass, A. R.
1975-01-01
The present paper discusses an approximate analytical solution to the nonlinear behavior of a discontinuous acoustic signal near a caustic. The Seebass transformation (1970) is refined to provide results which satisfy the governing equation to any prescribed accuracy, except across the shock wave produced by reflection of the simple wave at the caustic. The solution is approximate in the sense that the basic equation is satisfied wherever the solution is continuous but can satisfy only one of the two jump conditions at the reflected shock. The results give essential geometric features of the exact solution and provide a quantitative estimate of the strength of the so-called superboom.
Quark propagator in a truncation scheme beyond the rainbow approximation
NASA Astrophysics Data System (ADS)
Fu, Hui-Feng; Wang, Qing
2016-01-01
The quark propagator is studied under a truncation scheme beyond the rainbow approximation by dressing the quark-gluon vertex nonperturbatively. It is found that, in the chiral limit with dynamical symmetry breaking, the dynamical quark mass and the quark condensate are significantly enhanced due to the non-Abelian contribution arising from the three-gluon interaction compared to those under the rainbow approximation, and the critical strength of the dynamical chiral symmetry breaking is much lowered. The Abelian contribution is much smaller than the non-Abelian contribution. A technical issue on removing the ultraviolet divergences, including the overlapping divergences, is discussed.
An approximation concepts method for space frame synthesis
NASA Technical Reports Server (NTRS)
Mills-Curran, W. C.; Lust, R. V.; Schmit, L. A.
1982-01-01
A method is presented for the minimum mass design of three dimensional space frames constructed of thin walled rectangular cross-section members. Constraints on nodal displacements and rotations, material stress, local buckling, and cross sectional dimensions are included. A high quality separable approximate problem is formed in terms of the reciprocals of the four section properties of the frame element cross section, replacing all implicit functions with simplified explicit relations. The cross sectional dimensions are efficiently calculated without using multilevel techniques. Several test problems are solved, demonstrating that a series of approximate problem solutions converge rapidly to an optimal design.
Approximated maximum likelihood estimation in multifractal random walks
NASA Astrophysics Data System (ADS)
Løvsletten, O.; Rypdal, M.
2012-04-01
We present an approximated maximum likelihood method for the multifractal random walk processes of [E. Bacry , Phys. Rev. EPLEEE81539-375510.1103/PhysRevE.64.026103 64, 026103 (2001)]. The likelihood is computed using a Laplace approximation and a truncation in the dependency structure for the latent volatility. The procedure is implemented as a package in the r computer language. Its performance is tested on synthetic data and compared to an inference approach based on the generalized method of moments. The method is applied to estimate parameters for various financial stock indices.
Observation and Structure Determination of an Oxide Quasicrystal Approximant
NASA Astrophysics Data System (ADS)
Förster, S.; Trautmann, M.; Roy, S.; Adeagbo, W. A.; Zollner, E. M.; Hammer, R.; Schumann, F. O.; Meinel, K.; Nayak, S. K.; Mohseni, K.; Hergert, W.; Meyerheim, H. L.; Widdra, W.
2016-08-01
We report on the first observation of an approximant structure to the recently discovered two-dimensional oxide quasicrystal. Using scanning tunneling microscopy, low-energy electron diffraction, and surface x-ray diffraction in combination with ab initio calculations, the atomic structure and the bonding scheme are determined. The oxide approximant follows a 32 .4.3.4 Archimedean tiling. Ti atoms reside at the corners of each tiling element and are threefold coordinated to oxygen atoms. Ba atoms separate the TiO3 clusters, leading to a fundamental edge length of the tiling 6.7 Å.
Optimal approximation of harmonic growth clusters by orthogonal polynomials
Teodorescu, Razvan
2008-01-01
Interface dynamics in two-dimensional systems with a maximal number of conservation laws gives an accurate theoreticaI model for many physical processes, from the hydrodynamics of immiscible, viscous flows (zero surface-tension limit of Hele-Shaw flows), to the granular dynamics of hard spheres, and even diffusion-limited aggregation. Although a complete solution for the continuum case exists, efficient approximations of the boundary evolution are very useful due to their practical applications. In this article, the approximation scheme based on orthogonal polynomials with a deformed Gaussian kernel is discussed, as well as relations to potential theory.
On the approximation of crack shapes found during inservice inspection
Bhate, S.R.; Chawla, D.S.; Kushwaha, H.S.
1997-04-01
This paper addresses the characterization of axial internal flaw found during inservice inspection of a pipe. J-integral distribution for various flaw shapes is obtained using line spring finite, element method. The peak J-value and its distribution across the crack is found to be characteristic feature of each shape. The triangular shape yields peak J-value away from the center, the point of depth. The elliptic approximation results in large overestimate of J-value for unsymmetric flaws. Triangular approximation is recommended for such flaws so that further service can be obtained from the component.
Approximate analytic solutions for singular non-linear oscillators
NASA Technical Reports Server (NTRS)
Bota, K. B.; Mickens, R. E.
1984-01-01
Mickens (1981, 1984) has considered analytic techniques for obtaining approximate solutions to one-dimensional nonlinear oscillatory systems x(double-dot) + x = lambda f(x, x/dot/, lambda) where lambda is a small positive parameter and f is a nonlinear polynomial function of its arguments. However, in certain cases there is interest in the analysis of physical systems for which the nonlinear function f(x, x/dot/, lambda) is singular for finite values of x or x(dot). The present investigation is concerned with the use of existing approximate analytic schemes to obtain solutions to singular nonlinear oscillatory differential equations.
Decoupling approximation design using the peak to peak gain
NASA Astrophysics Data System (ADS)
Sultan, Cornel
2013-04-01
Linear system design for accurate decoupling approximation is examined using the peak to peak gain of the error system. The design problem consists in finding values of system parameters to ensure that this gain is small. For this purpose a computationally inexpensive upper bound on the peak to peak gain, namely the star norm, is minimized using a stochastic method. Examples of the methodology's application to tensegrity structures design are presented. Connections between the accuracy of the approximation, the damping matrix, and the natural frequencies of the system are examined, as well as decoupling in the context of open and closed loop control.
Observation and Structure Determination of an Oxide Quasicrystal Approximant.
Förster, S; Trautmann, M; Roy, S; Adeagbo, W A; Zollner, E M; Hammer, R; Schumann, F O; Meinel, K; Nayak, S K; Mohseni, K; Hergert, W; Meyerheim, H L; Widdra, W
2016-08-26
We report on the first observation of an approximant structure to the recently discovered two-dimensional oxide quasicrystal. Using scanning tunneling microscopy, low-energy electron diffraction, and surface x-ray diffraction in combination with ab initio calculations, the atomic structure and the bonding scheme are determined. The oxide approximant follows a 3^{2}.4.3.4 Archimedean tiling. Ti atoms reside at the corners of each tiling element and are threefold coordinated to oxygen atoms. Ba atoms separate the TiO_{3} clusters, leading to a fundamental edge length of the tiling 6.7 Å. PMID:27610863
An approximate method for calculating aircraft downwash on parachute trajectories
Strickland, J.H.
1989-01-01
An approximate method for calculating velocities induced by aircraft on parachute trajectories is presented herein. A simple system of quadrilateral vortex panels is used to model the aircraft wing and its wake. The purpose of this work is to provide a simple analytical tool which can be used to approximate the effect of aircraft-induced velocities on parachute performance. Performance issues such as turnover and wake recontact may be strongly influenced by velocities induced by the wake of the delivering aircraft, especially if the aircraft is maneuvering at the time of parachute deployment. 7 refs., 9 figs.
Superfluidity of heated Fermi systems in the static fluctuation approximation
Khamzin, A. A.; Nikitin, A. S.; Sitdikov, A. S.
2015-10-15
Superfluidity properties of heated finite Fermi systems are studied in the static fluctuation approximation, which is an original method. This method relies on a single and controlled approximation, which permits taking correctly into account quasiparticle correlations and thereby going beyond the independent-quasiparticle model. A closed self-consistent set of equations for calculating correlation functions at finite temperature is obtained for a finite Fermi system described by the Bardeen–Cooper–Schrieffer Hamiltonian. An equation for the energy gap is found with allowance for fluctuation effects. It is shown that the phase transition to the supefluid state is smeared upon the inclusion of fluctuations.
Intermediate boundary conditions for LOD, ADI and approximate factorization methods
NASA Technical Reports Server (NTRS)
Leveque, R. J.
1985-01-01
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 order accurate approximation 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 approximate factorization method is considered in slightly more generality.
Trigonometric Pade approximants for functions with regularly decreasing Fourier coefficients
Labych, Yuliya A; Starovoitov, Alexander P
2009-08-31
Sufficient conditions describing the regular decrease of the coefficients of a Fourier series f(x)=a{sub 0}/2 + {sigma} a{sub n} cos kx are found which ensure that the trigonometric Pade approximants {pi}{sup t}{sub n,m}(x;f) converge to the function f in the uniform norm at a rate which coincides asymptotically with the highest possible one. The results obtained are applied to problems dealing with finding sharp constants for rational approximations. Bibliography: 31 titles.
Approximation in control of flexible structures, theory and application
NASA Technical Reports Server (NTRS)
Gibson, J. S.
1983-01-01
The sense in which the feedback control law based on an approximate finite dimensional model of a continuous structure approximates a control law which is optimal for the distributed, or infinite dimensional, model of the structure is studied. From the analysis of the various control and stability issues associated with this basis question, useful information for designing finite dimensional compensators which produce near-optimal performance in infinite dimensional systems is gained. Some of the important predictions that can be made about large-order finite dimensional control laws, using the theory of infinite dimensional Riccati equations are indicated.
abcpmc: Approximate Bayesian Computation for Population Monte-Carlo code
NASA Astrophysics Data System (ADS)
Akeret, Joel
2015-04-01
abcpmc is a Python Approximate Bayesian Computing (ABC) Population Monte Carlo (PMC) implementation based on Sequential Monte Carlo (SMC) with Particle Filtering techniques. It is extendable with k-nearest neighbour (KNN) or optimal local covariance matrix (OLCM) pertubation kernels and has built-in support for massively parallelized sampling on a cluster using MPI.
1. VIEW ALONG APPROXIMATE CENTERLINE OF PROPOSED NEW BRIDGE. FLUME ...
1. VIEW ALONG APPROXIMATE CENTERLINE OF PROPOSED NEW BRIDGE. FLUME AT RIGHT CENTER IS FED FROM SMALL DIVERSION DAM ON MIDDLE FORK OF TULE RIVER, LOOKING SOUTH. (65mm lens) - Tule River Hydroelectric Complex, Tule River Bridge, Spanning North Fork of Middle Fork of Tule River, Springville, Tulare County, CA
Bose gases, Bose–Einstein condensation, and the Bogoliubov approximation
Seiringer, Robert
2014-07-15
We review recent progress towards a rigorous understanding of the Bogoliubov approximation for bosonic quantum many-body systems. We focus, in particular, on the excitation spectrum of a Bose gas in the mean-field (Hartree) limit. A list of open problems will be discussed at the end.
Abrupt Depletion Layer Approximation for the Metal Insulator Semiconductor Diode.
ERIC Educational Resources Information Center
Jones, Kenneth
1979-01-01
Determines the excess surface change carrier density, surface potential, and relative capacitance of a metal insulator semiconductor diode as a function of the gate voltage, using the precise questions and the equations derived with the abrupt depletion layer approximation. (Author/GA)
Towards numerical consistency and conservation for SPH approximations
NASA Astrophysics Data System (ADS)
Adams, Nikolaus; Hu, Xiangyu; Litvinov, Sergej
2014-11-01
Typical conservative Smoothed particle hydrodynamics (SPH) approximations introduce two errors: smoothing error is due to smoothing of the gradient by an integration associated with a kernel function; integration error due to approximating of the integration by summation over all particles within the kernel support. The integration error leads to violation of zero-order consistency, i.e., the inability to reproduce a constant field. We show that partition of unity is the condition under which the conservative SPH approximation achieves both consistency and convergence. The condition can be met by relaxing a particle distribution under a constant pressure field and invariant particle volume. The resulting particle distribution is very similar to those observed for liquid molecules. We further show that with two different typical kernel functions the SPH approximation satisfying the partition of unity property is able to achieve very high-order of the integration error for random particle locations. The background pressure used in a weakly compressible SPH simulation implies a self-relaxation mechanism, which explains that convergence with respect to increasing particle numbers could be obtained in SPH simulations, although not predicted by previous numerical analysis. Furthermore, by relating the integration error to the background pressure, we explain why the previously proposed transport-velocity formulation of SPH is able to achieve unprecedented accuracy and stability.
An approximation formula for a class of Markov reliability models
NASA Technical Reports Server (NTRS)
White, A. L.
1984-01-01
A way of considering a small but often used class of reliability model and approximating algebraically the systems reliability is shown. The models considered are appropriate for redundant reconfigurable digital control systems that operate for a short period of time without maintenance, and for such systems the method gives a formula in terms of component fault rates, system recovery rates, and system operating time.
8. BUILDING 522, INTERIOR, STOREROOM (NORTHERN PORTION), FROM APPROXIMATELY TWOTHIRDS ...
8. BUILDING 522, INTERIOR, STOREROOM (NORTHERN PORTION), FROM APPROXIMATELY TWO-THIRDS DISTANCE FROM EAST END, LOOKING EAST, WITH WINDOW-OPENING APPARATUS ALONG WALL TO LEFT, AND CONVEYOR APPARATUS IN FOREGROUND. - Oakland Naval Supply Center, Aeronautical Materials Storehouses, Between E & G Streets, between Fourth & Sixth Streets, Oakland, Alameda County, CA
11. BUILDING 522, INTERIOR, SOUTH MEZZANINE, FROM SOUTH WALL APPROXIMATELY ...
11. BUILDING 522, INTERIOR, SOUTH MEZZANINE, FROM SOUTH WALL APPROXIMATELY ONE-THIRD OF DISTANCE WEST FROM EAST END, LOOKING EAST-NORTHEAST. - Oakland Naval Supply Center, Aeronautical Materials Storehouses, Between E & G Streets, between Fourth & Sixth Streets, Oakland, Alameda County, CA
Violation of the condon approximation in semiconducting carbon nanotubes.
Duque, Juan G; Chen, Hang; Swan, Anna K; Shreve, Andrew P; Kilina, Svetlana; Tretiak, Sergei; Tu, Xiaomin; Zheng, Ming; Doorn, Stephen K
2011-06-28
The Condon approximation is widely applied in molecular and condensed matter spectroscopy and states that electronic transition dipoles are independent of nuclear positions. This approximation is related to the Franck-Condon principle, which in its simplest form holds that electronic transitions are instantaneous on the time scale of nuclear motion. The Condon approximation leads to a long-held assumption in Raman spectroscopy of carbon nanotubes: intensities arising from resonance with incident and scattered photons are equal. Direct testing of this assumption has not been possible due to the lack of homogeneous populations of specific carbon nanotube chiralities. Here, we present the first complete Raman excitation profiles (REPs) for the nanotube G band for 10 pure semiconducting chiralities. In contrast to expectations, a strong asymmetry is observed in the REPs for all chiralities, with the scattered resonance always appearing weaker than the incident resonance. The observed behavior results from violation of the Condon approximation and originates in changes in the electronic transition dipole due to nuclear motion (non-Condon effect), as confirmed by our quantum chemical calculations. The agreement of our calculations with the experimental REP asymmetries and observed trends in family dependence indicates the behavior is intrinsic. PMID:21612303
2. UPPER NOTTINGHAM MINE, WOODEN BOXES. BOXES ARE LOCATED APPROXIMATELY ...
2. UPPER NOTTINGHAM MINE, WOODEN BOXES. BOXES ARE LOCATED APPROXIMATELY 10 YARDS TO THE RIGHT AND DOWNSLOPE OF THE ADIT IN ID-31-F-1. CAMERA IS POINTED EAST-SOUTHEAST. - Florida Mountain Mining Sites, Upper Nottingham Mine, West face of Florida Mountain, head of Jacobs Gulch, Silver City, Owyhee County, ID
4. WEST PORTION OF SOUTH SIDE, OBLIQUE VIEW, FROM APPROXIMATELY ...
4. WEST PORTION OF SOUTH SIDE, OBLIQUE VIEW, FROM APPROXIMATELY 30 FEET WEST OF SOUTHWEST CORNER, LOOKING EAST, WITH WEST SIDE OF BUILDING 123 AT RIGHT BEYOND LIGHTING POLE. - Oakland Naval Supply Center, Pier Transit Shed, South of D Street between First & Second Streets, Oakland, Alameda County, CA
Revised Thomas-Fermi approximation for singular potentials
NASA Astrophysics Data System (ADS)
Dufty, James W.; Trickey, S. B.
2016-08-01
Approximations 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 approximation 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. Approximate expressions for the regularized potential are given to leading order in a nonlocality parameter, and the limiting behavior at high and low temperatures is described. The noninteracting part of the free energy in this approximation 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.
A Simple Geometric Approach to Approximating the Gini Coefficient
ERIC Educational Resources Information Center
Kasper, Hirschel; Golden, John
2008-01-01
The author shows how a quick approximation of the Lorenz curve's Gini coefficient can be calculated empirically using numerical data presented in cumulative income quintiles. When the technique here was used to estimate 621 income quintile/Gini coefficient observations from the Deninger and Squire/World Bank data set, this approach performed…
Breakdown of the few-level approximation in collective systems
Kiffner, M.; Evers, J.; Keitel, C. H.
2007-07-15
The validity of the few-level approximation in dipole-dipole interacting collective systems is discussed. As an example system, we study the archetype case of two dipole-dipole interacting atoms, each modeled by two complete sets of angular momentum multiplets. We establish the breakdown of the few-level approximation by first proving the intuitive result that the dipole-dipole induced energy shifts between collective two-atom states depend on the length of the vector connecting the atoms, but not on its orientation, if complete and degenerate multiplets are considered. A careful analysis of our findings reveals that the simplification of the atomic level scheme by artificially omitting Zeeman sublevels in a few-level approximation generally leads to incorrect predictions. We find that this breakdown can be traced back to the dipole-dipole coupling of transitions with orthogonal dipole moments. Our interpretation enables us to identify special geometries in which partial few-level approximations to two- or three-level systems are valid.
Laplace Approximation for Divisive Gaussian Processes for Nonstationary Regression.
Muñoz-González, Luis; Lázaro-Gredilla, Miguel; Figueiras-Vidal, Aníbal R
2016-03-01
The standard Gaussian Process regression (GP) is usually formulated under stationary hypotheses: The noise power is considered constant throughout the input space and the covariance of the prior distribution is typically modeled as depending only on the difference between input samples. These assumptions can be too restrictive and unrealistic for many real-world problems. Although nonstationarity can be achieved using specific covariance functions, they require a prior knowledge of the kind of nonstationarity, not available for most applications. In this paper we propose to use the Laplace approximation to make inference in a divisive GP model to perform nonstationary regression, including heteroscedastic noise cases. The log-concavity of the likelihood ensures a unimodal posterior and makes that the Laplace approximation converges to a unique maximum. The characteristics of the likelihood also allow to obtain accurate posterior approximations when compared to the Expectation Propagation (EP) approximations and the asymptotically exact posterior provided by a Markov Chain Monte Carlo implementation with Elliptical Slice Sampling (ESS), but at a reduced computational load with respect to both, EP and ESS.
Discrete dipole approximation in time domain through the Laplace transform.
Chaumet, Patrick C; Zhang, Ting; Rahmani, Adel; Gralak, Boris; Belkebir, Kamal
2013-12-01
We present a form of the discrete dipole approximation for electromagnetic scattering computations in time domain. We show that the introduction of complex frequencies, through the Laplace transform, significantly improves the computation time. We also show that the Laplace transform and its inverse can be combined to extract the field inside a scatterer at a real resonance frequency.
Animating Nested Taylor Polynomials to Approximate a Function
ERIC Educational Resources Information Center
Mazzone, Eric F.; Piper, Bruce R.
2010-01-01
The way that Taylor polynomials approximate functions can be demonstrated by moving the center point while keeping the degree fixed. These animations are particularly nice when the Taylor polynomials do not intersect and form a nested family. We prove a result that shows when this nesting occurs. The animations can be shown in class or…
Generalised quasilinear approximation of the helical magnetorotational instability
NASA Astrophysics Data System (ADS)
Child, Adam; Hollerbach, Rainer; Marston, Brad; Tobias, Steven
2016-06-01
> Motivated by recent advances in direct statistical simulation (DSS) of astrophysical phenomena such as out-of-equilibrium jets, we perform a direct numerical simulation (DNS) of the helical magnetorotational instability (HMRI) under the generalised quasilinear approximation (GQL). This approximation generalises the quasilinear approximation (QL) to include the self-consistent interaction of large-scale modes, interpolating between fully nonlinear DNS and QL DNS whilst still remaining formally linear in the small scales. In this paper we address whether GQL can more accurately describe low-order statistics of axisymmetric HMRI when compared with QL by performing DNS under various degrees of GQL approximation. We utilise various diagnostics, such as energy spectra in addition to first and second cumulants, for calculations performed for a range of Reynolds and Hartmann numbers (describing rotation and imposed magnetic field strength respectively). We find that GQL performs significantly better than QL in describing the statistics of the HMRI even when relatively few large-scale modes are kept in the formalism. We conclude that DSS based on GQL (GCE2) will be significantly more accurate than that based on QL (CE2).
Basis Function Approximation of Transonic Aerodynamic Influence Coefficient Matrix
NASA Technical Reports Server (NTRS)
Li, Wesley W.; Pak, Chan-gi
2011-01-01
A technique for approximating the modal aerodynamic influence coefficients matrices by using basis functions has been developed and validated. An application of the resulting approximated modal aerodynamic influence coefficients matrix for a flutter analysis in transonic speed regime has been demonstrated. This methodology can be applied to the unsteady subsonic, transonic, and supersonic aerodynamics. The method requires the unsteady aerodynamics in frequency-domain. The flutter solution can be found by the classic methods, such as rational function approximation, k, p-k, p, root-locus et cetera. The unsteady aeroelastic analysis for design optimization using unsteady transonic aerodynamic approximation is being demonstrated using the ZAERO flutter solver (ZONA Technology Incorporated, Scottsdale, Arizona). The technique presented has been shown to offer consistent flutter speed prediction on an aerostructures test wing 2 configuration with negligible loss in precision in transonic speed regime. These results may have practical significance in the analysis of aircraft aeroelastic calculation and could lead to a more efficient design optimization cycle.
Basis Function Approximation of Transonic Aerodynamic Influence Coefficient Matrix
NASA Technical Reports Server (NTRS)
Li, Wesley Waisang; Pak, Chan-Gi
2010-01-01
A technique for approximating the modal aerodynamic influence coefficients [AIC] matrices by using basis functions has been developed and validated. An application of the resulting approximated modal AIC matrix for a flutter analysis in transonic speed regime has been demonstrated. This methodology can be applied to the unsteady subsonic, transonic and supersonic aerodynamics. The method requires the unsteady aerodynamics in frequency-domain. The flutter solution can be found by the classic methods, such as rational function approximation, k, p-k, p, root-locus et cetera. The unsteady aeroelastic analysis for design optimization using unsteady transonic aerodynamic approximation is being demonstrated using the ZAERO(TradeMark) flutter solver (ZONA Technology Incorporated, Scottsdale, Arizona). The technique presented has been shown to offer consistent flutter speed prediction on an aerostructures test wing [ATW] 2 configuration with negligible loss in precision in transonic speed regime. These results may have practical significance in the analysis of aircraft aeroelastic calculation and could lead to a more efficient design optimization cycle
Cubic spline approximation techniques for parameter estimation in distributed systems
NASA Technical Reports Server (NTRS)
Banks, H. T.; Crowley, J. M.; Kunisch, K.
1983-01-01
Approximation schemes employing cubic splines in the context of a linear semigroup framework are developed for both parabolic and hyperbolic second-order partial differential equation parameter estimation problems. Convergence results are established for problems with linear and nonlinear systems, and a summary of numerical experiments with the techniques proposed is given.
A method of approximating range size of small mammals
Stickel, L.F.
1965-01-01
In summary, trap success trends appear to provide a useful approximation to range size of easily trapped small mammals such as Peromyscus. The scale of measurement can be adjusted as desired. Further explorations of the usefulness of the plan should be made and modifications possibly developed before adoption.
22. Photographic copy of 1889 linen drawing of reservoir. Approximately ...
22. Photographic copy of 1889 linen drawing of reservoir. Approximately two and a half feet by four feet. Delineator unknown, original currently located in the Sangre de Cristo Water Company files. - Two Mile Reservoir, Santa Fe River, intersection of Canyon & Cerro Gordo Roads, Santa Fe, Santa Fe County, NM
How Good Are Statistical Models at Approximating Complex Fitness Landscapes?
du Plessis, Louis; Leventhal, Gabriel E.; Bonhoeffer, Sebastian
2016-01-01
Fitness landscapes determine the course of adaptation by constraining and shaping evolutionary trajectories. Knowledge of the structure of a fitness landscape can thus predict evolutionary outcomes. Empirical fitness landscapes, however, have so far only offered limited insight into real-world questions, as the high dimensionality of sequence spaces makes it impossible to exhaustively measure the fitness of all variants of biologically meaningful sequences. We must therefore revert to statistical descriptions of fitness landscapes that are based on a sparse sample of fitness measurements. It remains unclear, however, how much data are required for such statistical descriptions to be useful. Here, we assess the ability of regression models accounting for single and pairwise mutations to correctly approximate a complex quasi-empirical fitness landscape. We compare approximations based on various sampling regimes of an RNA landscape and find that the sampling regime strongly influences the quality of the regression. On the one hand it is generally impossible to generate sufficient samples to achieve a good approximation of the complete fitness landscape, and on the other hand systematic sampling schemes can only provide a good description of the immediate neighborhood of a sequence of interest. Nevertheless, we obtain a remarkably good and unbiased fit to the local landscape when using sequences from a population that has evolved under strong selection. Thus, current statistical methods can provide a good approximation to the landscape of naturally evolving populations. PMID:27189564
Dentate Gyrus Circuitry Features Improve Performance of Sparse Approximation Algorithms
Petrantonakis, Panagiotis C.; Poirazi, Panayiota
2015-01-01
Memory-related activity in the Dentate Gyrus (DG) is characterized by sparsity. Memory representations are seen as activated neuronal populations of granule cells, the main encoding cells in DG, which are estimated to engage 2–4% of the total population. This sparsity is assumed to enhance the ability of DG to perform pattern separation, one of the most valuable contributions of DG during memory formation. In this work, we investigate how features of the DG such as its excitatory and inhibitory connectivity diagram can be used to develop theoretical algorithms performing Sparse Approximation, a widely used strategy in the Signal Processing field. Sparse approximation stands for the algorithmic identification of few components from a dictionary that approximate a certain signal. The ability of DG to achieve pattern separation by sparsifing its representations is exploited here to improve the performance of the state of the art sparse approximation algorithm “Iterative Soft Thresholding” (IST) by adding new algorithmic features inspired by the DG circuitry. Lateral inhibition of granule cells, either direct or indirect, via mossy cells, is shown to enhance the performance of the IST. Apart from revealing the potential of DG-inspired theoretical algorithms, this work presents new insights regarding the function of particular cell types in the pattern separation task of the DG. PMID:25635776
Approximating Attractors of Boolean Networks by Iterative CTL Model Checking.
Klarner, Hannes; Siebert, Heike
2015-01-01
This paper introduces the notion of approximating asynchronous attractors of Boolean networks by minimal trap spaces. We define three criteria for determining the quality of an approximation: "faithfulness" which requires that the oscillating variables of all attractors in a trap space correspond to their dimensions, "univocality" which requires that there is a unique attractor in each trap space, and "completeness" which requires that there are no attractors outside of a given set of trap spaces. Each is a reachability property for which we give equivalent model checking queries. Whereas faithfulness and univocality can be decided by model checking the corresponding subnetworks, the naive query for completeness must be evaluated on the full state space. Our main result is an alternative approach which is based on the iterative refinement of an initially poor approximation. The algorithm detects so-called autonomous sets in the interaction graph, variables that contain all their regulators, and considers their intersection and extension in order to perform model checking on the smallest possible state spaces. A benchmark, in which we apply the algorithm to 18 published Boolean networks, is given. In each case, the minimal trap spaces are faithful, univocal, and complete, which suggests that they are in general good approximations for the asymptotics of Boolean networks. PMID:26442247
The Inertial Property of Approximately Inertial Frames of Reference
ERIC Educational Resources Information Center
Chubykalo, Andrew E.; Espinoza, Augusto; Kosyakov, B. P.
2011-01-01
Is it possible to compare approximately inertial frames in the inertial property? If this is the case, the inertial property becomes a measurable quantity. We give a positive answer to this question, and discuss the general principle of design of devices for making the required measurements. This paper is intended for advanced undergraduate and…
Approximate Approaches to the One-Dimensional Finite Potential Well
ERIC Educational Resources Information Center
Singh, Shilpi; Pathak, Praveen; Singh, Vijay A.
2011-01-01
The one-dimensional finite well is a textbook problem. We propose approximate approaches to obtain the energy levels of the well. The finite well is also encountered in semiconductor heterostructures where the carrier mass inside the well (m[subscript i]) is taken to be distinct from mass outside (m[subscript o]). A relevant parameter is the mass…
Stochastic Approximation Methods for Latent Regression Item Response Models
ERIC Educational Resources Information Center
von Davier, Matthias; Sinharay, Sandip
2010-01-01
This article presents an application of a stochastic approximation expectation maximization (EM) algorithm using a Metropolis-Hastings (MH) sampler to estimate the parameters of an item response latent regression model. Latent regression item response models are extensions of item response theory (IRT) to a latent variable model with covariates…
Cold pasta phase in the extended Thomas-Fermi approximation
NASA Astrophysics Data System (ADS)
Avancini, S. S.; Bertolino, B. P.
2015-10-01
In this paper, we aim to obtain more accurate values for the transition density to the homogenous phase in the nuclear pasta that occurs in the inner crust of neutron stars. To that end, we use the nonlinear Walecka model at zero temperature and an approach based on the extended Thomas-Fermi (ETF) approximation.
7. WESTCENTRAL PORTION OF NORTH SIDE, FROM APPROXIMATELY 10 FEET ...
7. WEST-CENTRAL PORTION OF NORTH SIDE, FROM APPROXIMATELY 10 FEET EAST OF NORTHEAST CORNER OF LUMBER SHED ('NORTHWEST WING' OF BUILDING), LOOKING SOUTH, WITH LUMBER SHED ON RIGHT. - Oakland Naval Supply Center, Lumber Storage & Box Factory, East of Fifth Street, between H & I Streets, Oakland, Alameda County, CA
1. Photocopy of photograph, showing a view approximately northnorthwest of ...
1. Photocopy of photograph, showing a view approximately north-northwest of the sixteen original kilns. Photographer and date unknown, but believed to be ca. 1895. Courtesy of Felicia Nichols, Pocatello, Id. - Warren King Charcoal Kilns, 5 miles west of Idaho Highway 28, Targhee National Forest, Leadore, Lemhi County, ID
An application of artificial neural networks to experimental data approximation
NASA Technical Reports Server (NTRS)
Meade, Andrew J., Jr.
1993-01-01
As an initial step in the evaluation of networks, a feedforward architecture is trained to approximate experimental data by the backpropagation algorithm. Several drawbacks were detected and an alternative learning algorithm was then developed to partially address the drawbacks. This noniterative algorithm has a number of advantages over the backpropagation method and is easily implemented on existing hardware.
Adequacy of selected evapotranspiration approximations for hydrologic simulation
Sumner, D.M.
2006-01-01
Evapotranspiration (ET) approximations, usually based on computed potential ET (PET) and diverse PET-to-ET conceptualizations, are routinely used in hydrologic analyses. This study presents an approach to incorporate measured (actual) ET data, increasingly available using micrometeorological methods, to define the adequacy of ET approximations for hydrologic simulation. The approach is demonstrated at a site where eddy correlation-measured ET values were available. A baseline hydrologic model incorporating measured ET values was used to evaluate the sensitivity of simulated water levels, subsurface recharge, and surface runoff to error in four ET approximations. An annually invariant pattern of mean monthly vegetation coefficients was shown to be most effective, despite the substantial year-to-year variation in measured vegetation coefficients. The temporal variability of available water (precipitation minus ET) at the humid, subtropical site was largely controlled by the relatively high temporal variability of precipitation, benefiting the effectiveness of coarse ET approximations, a result that is likely to prevail at other humid sites.
Temperature dependence of electronic eigenenergies in the adiabatic harmonic approximation
NASA Astrophysics Data System (ADS)
Poncé, S.; Antonius, G.; Gillet, Y.; Boulanger, P.; Laflamme Janssen, J.; Marini, A.; Côté, M.; Gonze, X.
2014-12-01
The renormalization of electronic eigenenergies due to electron-phonon interactions (temperature dependence and zero-point motion effect) is important in many materials. We address it in the adiabatic harmonic approximation, based on first principles (e.g., density-functional theory), from different points of view: directly from atomic position fluctuations or, alternatively, from Janak's theorem generalized to the case where the Helmholtz free energy, including the vibrational entropy, is used. We prove their equivalence, based on the usual form of Janak's theorem and on the dynamical equation. We then also place the Allen-Heine-Cardona (AHC) theory of the renormalization in a first-principles context. The AHC theory relies on the rigid-ion approximation, and naturally leads to a self-energy (Fan) contribution and a Debye-Waller contribution. Such a splitting can also be done for the complete harmonic adiabatic expression, in which the rigid-ion approximation is not required. A numerical study within the density-functional perturbation theory framework allows us to compare the AHC theory with frozen-phonon calculations, with or without the rigid-ion approximation. For the two different numerical approaches without non-rigid-ion terms, the agreement is better than 7 μ eV in the case of diamond, which represent an agreement to five significant digits. The magnitude of the non-rigid-ion terms in this case is also presented, distinguishing specific phonon modes contributions to different electronic eigenenergies.
Single-mode approximation of parametric down-conversion
Li Yongmin; Mikami, Hideharu; Wang Haibo; Kobayashi, Takayoshi
2005-12-15
In this paper, we study the theory of spontaneous parametric down-conversion pumped by a short coherent pulse. It is shown that when single spatial mode filters and narrow band optical filters are used to filter the output state of parametric down-conversion, the postselected state of parametric down-conversion can be approximately described by a simple single-mode theory.
Hyperspherical Sparse Approximation Techniques for High-Dimensional Discontinuity Detection
Zhang, Guannan; Webster, Clayton G.; Gunzburger, Max; Burkardt, John
2016-08-04
This work proposes a hyperspherical sparse approximation framework for detecting jump discontinuities in functions in high-dimensional spaces. The need for a novel approach results from the theoretical and computational inefficiencies of well-known approaches, such as adaptive sparse grids, for discontinuity detection. Our approach constructs the hyperspherical coordinate representation of the discontinuity surface of a function. Then sparse approximations of the transformed function are built in the hyperspherical coordinate system, with values at each point estimated by solving a one-dimensional discontinuity detection problem. Due to the smoothness of the hypersurface, the new technique can identify jump discontinuities with significantly reduced computationalmore » cost, compared to existing methods. Several approaches are used to approximate the transformed discontinuity surface in the hyperspherical system, including adaptive sparse grid and radial basis function interpolation, discrete least squares projection, and compressed sensing approximation. Moreover, hierarchical acceleration techniques are also incorporated to further reduce the overall complexity. In conclusion, rigorous complexity analyses of the new methods are provided, as are several numerical examples that illustrate the effectiveness of our approach.« less
1. ONTARIO MINE IS LOCATED ALONG FAR RIGHT ROADWAY APPROXIMATELY ...
1. ONTARIO MINE IS LOCATED ALONG FAR RIGHT ROADWAY APPROXIMATELY 100 YARDS FROM CAMERA POSITION. TAILING PILE DOWN SLOPE AND WEST OF CAMERA POSITION IN ID-31-C-44. - Florida Mountain Mining Sites, Ontario Mine, Northwest side of Florida Mountain, Silver City, Owyhee County, ID
Calculating Resonance Positions and Widths Using the Siegert Approximation Method
ERIC Educational Resources Information Center
Rapedius, Kevin
2011-01-01
Here, we present complex resonance states (or Siegert states) that describe the tunnelling decay of a trapped quantum particle from an intuitive point of view that naturally leads to the easily applicable Siegert approximation method. This can be used for analytical and numerical calculations of complex resonances of both the linear and nonlinear…
Young Children "Solve for X" Using the Approximate Number System
ERIC Educational Resources Information Center
Kibbe, Melissa M.; Feigenson, Lisa
2015-01-01
The Approximate Number System (ANS) supports basic arithmetic computation in early childhood, but it is unclear whether the ANS also supports the more complex computations introduced later in formal education. "Solving for x" in addend-unknown problems is notoriously difficult for children, who often struggle with these types of problems…
The Relation between Idealisation and Approximation in Scientific Model Construction
ERIC Educational Resources Information Center
Portides, Demetris P.
2007-01-01
The notions of "idealisation" and "approximation" are strongly linked to the question of "how our theories represent the phenomena in their scope". Although there is no consensus amongst Philosophers on the nature of the process of idealisation and how it affects theoretical representation, at the level of science education much can be gained from…
Approximation algorithms for the fixed-topology phylogenetic number problem
Cryan, M.; Goldberg, L.A.; Phillips, C.A.
1997-04-01
In the {ell}-phylogeny problem, one wishes to construct an evolutionary tree for a set of species represented by characters, in which each state of each character induces no more than {ell} connected components. The authors consider the fixed-topology version of this problem for fixed-topologies of arbitrary degree. This version of the problem is known to be NP-complete for {ell} {ge} 3 even for degree-3 trees in which no state labels more than {ell} + 1 leaves (and therefore there is a trivial {ell} + 1 phylogeny). They give a 2-approximation algorithm for all {ell} {ge} 3 for arbitrary input topologies and they given an optimal approximation algorithm that constructs a 4-phylogeny when a 3-phylogeny exists. Dynamic programming techniques, which are typically used in fixed-topology problems, cannot be applied to {ell}-phylogeny problems. The 2-approximation algorithm is the first application of linear programming to approximation algorithms for phylogeny problems. They extend their results to a related problem in which characters are polymorphic.
7. BUILDING 432, INTERIOR, EASTERN STOREROOM, FROM APPROXIMATELY 50 FEET ...
7. BUILDING 432, INTERIOR, EASTERN STOREROOM, FROM APPROXIMATELY 50 FEET NORTHEAST OF SOUTHWEST CORNER, LOOKING WEST, WITH OTHER STOREROOMS THROUGH CONNECTING FIRE DOORS AT CENTER. - Oakland Naval Supply Center, Paint & Oil Storehouses, Between G & I Streets, between Fourth & Fifth Streets, Oakland, Alameda County, CA
3. VIEW OF ARVFS BUNKER TAKEN FROM APPROXIMATELY 150 FEET ...
3. VIEW OF ARVFS BUNKER TAKEN FROM APPROXIMATELY 150 FEET EAST OF BUNKER DOOR. CAMERA FACING WEST. VIEW SHOWS EARTH MOUND COVERING CONTROL BUNKER AND REMAINS OF CABLE CHASE. - Idaho National Engineering Laboratory, Advanced Reentry Vehicle Fusing System, Scoville, Butte County, ID
Nonlinear functional approximation with networks using adaptive neurons
NASA Technical Reports Server (NTRS)
Tawel, Raoul
1992-01-01
A novel mathematical framework for the rapid learning of nonlinear mappings and topological transformations is presented. It is based on allowing the neuron's parameters to adapt as a function of learning. This fully recurrent adaptive neuron model (ANM) has been successfully applied to complex nonlinear function approximation problems such as the highly degenerate inverse kinematics problem in robotics.
Application of Approximate Unsteady Aerodynamics for Flutter Analysis
NASA Technical Reports Server (NTRS)
Pak, Chan-gi; Li, Wesley W.
2010-01-01
A technique for approximating the modal aerodynamic influence coefficient (AIC) matrices by using basis functions has been developed. A process for using the resulting approximated modal AIC matrix in aeroelastic analysis has also been developed. The method requires the unsteady aerodynamics in frequency domain, and this methodology can be applied to the unsteady subsonic, transonic, and supersonic aerodynamics. The flutter solution can be found by the classic methods, such as rational function approximation, k, p-k, p, root locus et cetera. The unsteady aeroelastic analysis using unsteady subsonic aerodynamic approximation is demonstrated herein. The technique presented is shown to offer consistent flutter speed prediction on an aerostructures test wing (ATW) 2 and a hybrid wing body (HWB) type of vehicle configuration with negligible loss in precision. This method computes AICs that are functions of the changing parameters being studied and are generated within minutes of CPU time instead of hours. These results may have practical application in parametric flutter analyses as well as more efficient multidisciplinary design and optimization studies.
Construction of the approximant of complete diagram for rock deformation
NASA Astrophysics Data System (ADS)
Tsvetkov, A. B.; Pavlova, L. D.; Fryanov, V. N.
2016-10-01
The function for approximation of the diagram of rock deformation, intended for nonlinear model of stress-strain state of coal geomassive, is developed, which is realized in the package of problem-oriented software. The computing experiments and the analysis of outcomes of numerical modeling are performed, confirming the correspondence of the calculated results with the measured settlings of underworked seams.
Engine With Regression and Neural Network Approximators Designed
NASA Technical Reports Server (NTRS)
Patnaik, Surya N.; Hopkins, Dale A.
2001-01-01
At the NASA Glenn Research Center, the NASA engine performance program (NEPP, ref. 1) and the design optimization testbed COMETBOARDS (ref. 2) with regression and neural network analysis-approximators have been coupled to obtain a preliminary engine design methodology. The solution to a high-bypass-ratio subsonic waverotor-topped turbofan engine, which is shown in the preceding figure, was obtained by the simulation depicted in the following figure. This engine is made of 16 components mounted on two shafts with 21 flow stations. The engine is designed for a flight envelope with 47 operating points. The design optimization utilized both neural network and regression approximations, along with the cascade strategy (ref. 3). The cascade used three algorithms in sequence: the method of feasible directions, the sequence of unconstrained minimizations technique, and sequential quadratic programming. The normalized optimum thrusts obtained by the three methods are shown in the following figure: the cascade algorithm with regression approximation is represented by a triangle, a circle is shown for the neural network solution, and a solid line indicates original NEPP results. The solutions obtained from both approximate methods lie within one standard deviation of the benchmark solution for each operating point. The simulation improved the maximum thrust by 5 percent. The performance of the linear regression and neural network methods as alternate engine analyzers was found to be satisfactory for the analysis and operation optimization of air-breathing propulsion engines (ref. 4).
Renormalization of the frozen Gaussian approximation to the quantum propagator
Tatchen, Joerg; Pollak, Eli; Tao Guohua; Miller, William H.
2011-04-07
The frozen Gaussian approximation to the quantum propagator may be a viable method for obtaining 'on the fly' quantum dynamical information on systems with many degrees of freedom. However, it has two severe limitations, it rapidly loses normalization and one needs to know the Gaussian averaged potential, hence it is not a purely local theory in the force field. These limitations are in principle remedied by using the Herman-Kluk (HK) form for the semiclassical propagator. The HK propagator approximately conserves unitarity for relatively long times and depends only locally on the bare potential and its second derivatives. However, the HK propagator involves a much more expensive computation due to the need for evaluating the monodromy matrix elements. In this paper, we (a) derive a new formula for the normalization integral based on a prefactor free HK propagator which is amenable to 'on the fly' computations; (b) show that a frozen Gaussian version of the normalization integral is not readily computable 'on the fly'; (c) provide a new insight into how the HK prefactor leads to approximate unitarity; and (d) how one may construct a prefactor free approximation which combines the advantages of the frozen Gaussian and the HK propagators. The theoretical developments are backed by numerical examples on a Morse oscillator and a quartic double well potential.
Polynomial Approximation of Functions: Historical Perspective and New Tools
ERIC Educational Resources Information Center
Kidron, Ivy
2003-01-01
This paper examines the effect of applying symbolic computation and graphics to enhance students' ability to move from a visual interpretation of mathematical concepts to formal reasoning. The mathematics topics involved, Approximation and Interpolation, were taught according to their historical development, and the students tried to follow the…
How Good Are Statistical Models at Approximating Complex Fitness Landscapes?
du Plessis, Louis; Leventhal, Gabriel E; Bonhoeffer, Sebastian
2016-09-01
Fitness landscapes determine the course of adaptation by constraining and shaping evolutionary trajectories. Knowledge of the structure of a fitness landscape can thus predict evolutionary outcomes. Empirical fitness landscapes, however, have so far only offered limited insight into real-world questions, as the high dimensionality of sequence spaces makes it impossible to exhaustively measure the fitness of all variants of biologically meaningful sequences. We must therefore revert to statistical descriptions of fitness landscapes that are based on a sparse sample of fitness measurements. It remains unclear, however, how much data are required for such statistical descriptions to be useful. Here, we assess the ability of regression models accounting for single and pairwise mutations to correctly approximate a complex quasi-empirical fitness landscape. We compare approximations based on various sampling regimes of an RNA landscape and find that the sampling regime strongly influences the quality of the regression. On the one hand it is generally impossible to generate sufficient samples to achieve a good approximation of the complete fitness landscape, and on the other hand systematic sampling schemes can only provide a good description of the immediate neighborhood of a sequence of interest. Nevertheless, we obtain a remarkably good and unbiased fit to the local landscape when using sequences from a population that has evolved under strong selection. Thus, current statistical methods can provide a good approximation to the landscape of naturally evolving populations.
Analytic Approximations for the Extrapolation of Lattice Data
Masjuan, Pere
2010-12-22
We present analytic approximations of chiral SU(3) amplitudes for the extrapolation of lattice data to the physical masses and the determination of Next-to-Next-to-Leading-Order low-energy constants. Lattice data for the ratio F{sub K}/F{sub {pi}} is used to test the method.
Approximate furrow infiltration model for time-variable ponding depth
Technology Transfer Automated Retrieval System (TEKTRAN)
A methodology is proposed for estimating furrow infiltration under time-variable ponding depth conditions. The methodology approximates the solution to the two-dimensional Richards equation, and is a modification of a procedure that was originally proposed for computing infiltration under constant ...
Poisson process approximation for sequence repeats, and sequencing by hybridization.
Arratia, R; Martin, D; Reinert, G; Waterman, M S
1996-01-01
Sequencing by hybridization is a tool to determine a DNA sequence from the unordered list of all l-tuples contained in this sequence; typical numbers for l are l = 8, 10, 12. For theoretical purposes we assume that the multiset of all l-tuples is known. This multiset determines the DNA sequence uniquely if none of the so-called Ukkonen transformations are possible. These transformations require repeats of (l-1)-tuples in the sequence, with these repeats occurring in certain spatial patterns. We model DNA as an i.i.d. sequence. We first prove Poisson process approximations for the process of indicators of all leftmost long repeats allowing self-overlap and for the process of indicators of all left-most long repeats without self-overlap. Using the Chen-Stein method, we get bounds on the error of these approximations. As a corollary, we approximate the distribution of longest repeats. In the second step we analyze the spatial patterns of the repeats. Finally we combine these two steps to prove an approximation for the probability that a random sequence is uniquely recoverable from its list of l-tuples. For all our results we give some numerical examples including error bounds. PMID:8891959
Glytsis, E N; Harrigan, M E; Hirayama, K; Gaylord, T K
1998-01-01
Practical collimating diffractive cylindrical lenses of 2, 4, 8, and 16 discrete levels are analyzed with a sequential application of the two-region formulation of the rigorous electromagnetic boundary-element method (BEM). A Gaussian beam of TE or TM polarization is incident upon the finite-thickness lens. F/4, F/2, and F/1.4 lenses are analyzed and near-field electric-field patterns are presented. The near-field wave-front quality is quantified by its mean-square deviation from a planar wave front. This deviation is found to be less than 0.05 free-space wavelengths. The far-field intensity patterns are determined and compared with the ones predicted by the approximate Fraunhofer scalar diffraction analysis. The diffraction efficiencies determined with the rigorous BEM are found to be generally lower than those obtained with the scalar approximation. For comparison, the performance characteristics of the corresponding continuous Fresnel (continuous profile within a zone but discontinuous at zone boundaries) and continuous refractive lenses are determined by the use of both the BEM and the scalar approximation. The diffraction efficiency of the continuous Fresnel lens is found to be similar to that of the 16-level diffractive lens but less than that of the continuous refractive lens. It is shown that the validity of the scalar approximation deteriorates as the lens f-number decreases.
Evaluating the Accuracy of Hessian Approximations for Direct Dynamics Simulations.
Zhuang, Yu; Siebert, Matthew R; Hase, William L; Kay, Kenneth G; Ceotto, Michele
2013-01-01
Direct dynamics simulations are a very useful and general approach for studying the atomistic properties of complex chemical systems, since an electronic structure theory representation of a system's potential energy surface is possible without the need for fitting an analytic potential energy function. In this paper, recently introduced compact finite difference (CFD) schemes for approximating the Hessian [J. Chem. Phys.2010, 133, 074101] are tested by employing the monodromy matrix equations of motion. Several systems, including carbon dioxide and benzene, are simulated, using both analytic potential energy surfaces and on-the-fly direct dynamics. The results show, depending on the molecular system, that electronic structure theory Hessian direct dynamics can be accelerated up to 2 orders of magnitude. The CFD approximation is found to be robust enough to deal with chaotic motion, concomitant with floppy and stiff mode dynamics, Fermi resonances, and other kinds of molecular couplings. Finally, the CFD approximations allow parametrical tuning of different CFD parameters to attain the best possible accuracy for different molecular systems. Thus, a direct dynamics simulation requiring the Hessian at every integration step may be replaced with an approximate Hessian updating by tuning the appropriate accuracy. PMID:26589009
Least squares approximation of two-dimensional FIR digital filters
NASA Astrophysics Data System (ADS)
Alliney, S.; Sgallari, F.
1980-02-01
In this paper, a new method for the synthesis of two-dimensional FIR digital filters is presented. The method is based on a least-squares approximation of the ideal frequency response; an orthogonality property of certain functions, related to the frequency sampling design, improves the computational efficiency.
A simple analytic approximation for dusty Stroemgren spheres
NASA Technical Reports Server (NTRS)
Petrosian, V.; Silk, J.; Field, G. B.
1972-01-01
An analytic approximation is illustrated to Stromgren's solution for H II regions which permits explicit exhibition of the effects of internal dust on the ionization structure. Far infrared observations of H II regions are accounted for in terms of true absorption by internal dust of a significant fraction of the Lyman continuum photons.
A Comparison of Approximation Modeling Techniques: Polynomial Versus Interpolating Models
NASA Technical Reports Server (NTRS)
Giunta, Anthony A.; Watson, Layne T.
1998-01-01
Two methods of creating approximation models are compared through the calculation of the modeling accuracy on test problems involving one, five, and ten independent variables. Here, the test problems are representative of the modeling challenges typically encountered in realistic engineering optimization problems. The first approximation model is a quadratic polynomial created using the method of least squares. This type of polynomial model has seen considerable use in recent engineering optimization studies due to its computational simplicity and ease of use. However, quadratic polynomial models may be of limited accuracy when the response data to be modeled have multiple local extrema. The second approximation model employs an interpolation scheme known as kriging developed in the fields of spatial statistics and geostatistics. This class of interpolating model has the flexibility to model response data with multiple local extrema. However, this flexibility is obtained at an increase in computational expense and a decrease in ease of use. The intent of this study is to provide an initial exploration of the accuracy and modeling capabilities of these two approximation methods.
Universal approximation by radial basis function networks of Delsarte translates.
Arteaga, Cristian; Marrero, Isabel
2013-10-01
We prove that, under certain mild conditions on the kernel function (or activation function), the family of radial basis function neural networks obtained by replacing the usual translation with the Delsarte one, and taking the same smoothing factor in all kernel nodes, has the universal approximation property.
Meta-regression approximations to reduce publication selection bias.
Stanley, T D; Doucouliagos, Hristos
2014-03-01
Publication selection bias is a serious challenge to the integrity of all empirical sciences. We derive meta-regression approximations to reduce this bias. Our approach employs Taylor polynomial approximations to the conditional mean of a truncated distribution. A quadratic approximation without a linear term, precision-effect estimate with standard error (PEESE), is shown to have the smallest bias and mean squared error in most cases and to outperform conventional meta-analysis estimators, often by a great deal. Monte Carlo simulations also demonstrate how a new hybrid estimator that conditionally combines PEESE and the Egger regression intercept can provide a practical solution to publication selection bias. PEESE is easily expanded to accommodate systematic heterogeneity along with complex and differential publication selection bias that is related to moderator variables. By providing an intuitive reason for these approximations, we can also explain why the Egger regression works so well and when it does not. These meta-regression methods are applied to several policy-relevant areas of research including antidepressant effectiveness, the value of a statistical life, the minimum wage, and nicotine replacement therapy. PMID:26054026
Methods to approximate reliabilities in single-step genomic evaluation
Technology Transfer Automated Retrieval System (TEKTRAN)
Reliability of predictions from single-step genomic BLUP (ssGBLUP) can be calculated by inversion, but that is not feasible for large data sets. Two methods of approximating reliability were developed based on decomposition of a function of reliability into contributions from records, pedigrees, and...
New approximating results for data with errors in both variables
NASA Astrophysics Data System (ADS)
Bogdanova, N.; Todorov, S.
2015-05-01
We introduce new data from mineral water probe Lenovo Bulgaria, measured with errors in both variables. We apply our Orthonormal Polynomial Expansion Method (OPEM), based on Forsythe recurrence formula to describe the data in the new error corridor. The development of OPEM gives the approximating curves and their derivatives in optimal orthonormal and usual expansions including the errors in both variables with special criteria.
Prince William Forest Park American Beech , Approximately one mile ...
Prince William Forest Park American Beech , Approximately one mile from visitors center, south bank of the south fork of Quantico Creek, about 75 yards upstream from its confluence with Quantico Creek, Near Birch Bluff Trail, Triangle, Prince William County, VA
Approximate neutral point of a subsonic canard aircraft
NASA Technical Reports Server (NTRS)
Phillips, J. D.
1985-01-01
An approximate formula is derived for the position of the neutral point in canard aircraft. This formula accounts for the aerodynamic interference between the wing (rear wing) and the canard (forward wing). Topics covered include determination of the canard downwash derivative, determination of the canard and wing liftslopes including the aerodynamic interference, and calculation of the neutral point position.
Approximation of reliability of direct genomic breeding values
Technology Transfer Automated Retrieval System (TEKTRAN)
Two methods to efficiently approximate theoretical genomic reliabilities are presented. The first method is based on the direct inverse of the left hand side (LHS) of mixed model equations. It uses the genomic relationship matrix for a small subset of individuals with the highest genomic relationshi...
Approximating Attractors of Boolean Networks by Iterative CTL Model Checking.
Klarner, Hannes; Siebert, Heike
2015-01-01
This paper introduces the notion of approximating asynchronous attractors of Boolean networks by minimal trap spaces. We define three criteria for determining the quality of an approximation: "faithfulness" which requires that the oscillating variables of all attractors in a trap space correspond to their dimensions, "univocality" which requires that there is a unique attractor in each trap space, and "completeness" which requires that there are no attractors outside of a given set of trap spaces. Each is a reachability property for which we give equivalent model checking queries. Whereas faithfulness and univocality can be decided by model checking the corresponding subnetworks, the naive query for completeness must be evaluated on the full state space. Our main result is an alternative approach which is based on the iterative refinement of an initially poor approximation. The algorithm detects so-called autonomous sets in the interaction graph, variables that contain all their regulators, and considers their intersection and extension in order to perform model checking on the smallest possible state spaces. A benchmark, in which we apply the algorithm to 18 published Boolean networks, is given. In each case, the minimal trap spaces are faithful, univocal, and complete, which suggests that they are in general good approximations for the asymptotics of Boolean networks.
Symmetric approximations of the Navier-Stokes equations
Kobel'kov, G M
2002-08-31
A new method for the symmetric approximation of the non-stationary Navier-Stokes equations by a Cauchy-Kovalevskaya-type system is proposed. Properties of the modified problem are studied. In particular, the convergence as {epsilon}{yields}0 of the solutions of the modified problem to the solutions of the original problem on an infinite interval is established.
Symmetric approximations of the Navier-Stokes equations
NASA Astrophysics Data System (ADS)
Kobel'kov, G. M.
2002-08-01
A new method for the symmetric approximation of the non-stationary Navier-Stokes equations by a Cauchy-Kovalevskaya-type system is proposed. Properties of the modified problem are studied. In particular, the convergence as \\varepsilon\\to0 of the solutions of the modified problem to the solutions of the original problem on an infinite interval is established.
Benchmarking mean-field approximations to level densities
NASA Astrophysics Data System (ADS)
Alhassid, Y.; Bertsch, G. F.; Gilbreth, C. N.; Nakada, H.
2016-04-01
We assess the accuracy of finite-temperature mean-field theory using as a standard the Hamiltonian and model space of the shell model Monte Carlo calculations. Two examples are considered: the nucleus 162Dy, representing a heavy deformed nucleus, and 148Sm, representing a nearby heavy spherical nucleus with strong pairing correlations. The errors inherent in the finite-temperature Hartree-Fock and Hartree-Fock-Bogoliubov approximations are analyzed by comparing the entropies of the grand canonical and canonical ensembles, as well as the level density at the neutron resonance threshold, with shell model Monte Carlo calculations, which are accurate up to well-controlled statistical errors. The main weak points in the mean-field treatments are found to be: (i) the extraction of number-projected densities from the grand canonical ensembles, and (ii) the symmetry breaking by deformation or by the pairing condensate. In the absence of a pairing condensate, we confirm that the usual saddle-point approximation to extract the number-projected densities is not a significant source of error compared to other errors inherent to the mean-field theory. We also present an alternative formulation of the saddle-point approximation that makes direct use of an approximate particle-number projection and avoids computing the usual three-dimensional Jacobian of the saddle-point integration. We find that the pairing condensate is less amenable to approximate particle-number projection methods because of the explicit violation of particle-number conservation in the pairing condensate. Nevertheless, the Hartree-Fock-Bogoliubov theory is accurate to less than one unit of entropy for 148Sm at the neutron threshold energy, which is above the pairing phase transition. This result provides support for the commonly used "back-shift" approximation, treating pairing as only affecting the excitation energy scale. When the ground state is strongly deformed, the Hartree-Fock entropy is significantly
Brown, L. F.; Travis, B. J.
2004-01-01
Investigators (e.g., Seebauer 1994, Bogillo and Shkilev 1999) have used the condensation approximation (CA) successfully for determining broad nonuniform surface site-energy distributions (SEDs) from temperature-programmed desorption (TPD) spectra and for identifying constant pre-exponential factors from peak analysis. The CA assumes that at any temperature T, desorption occurs only at sites with a single desorption activation energy (E{sub cdn}). E{sub cdn} is of course a function of T. Further, the approximation assumes that during TPD all sites with desorption energy E{sub cdn} empty at T.
Ideal amino acid exchange forms for approximating substitution matrices.
Pokarowski, Piotr; Kloczkowski, Andrzej; Nowakowski, Szymon; Pokarowska, Maria; Jernigan, Robert L; Kolinski, Andrzej
2007-11-01
We have analyzed 29 published substitution matrices (SMs) and five statistical protein contact potentials (CPs) for comparison. We find that popular, 'classical' SMs obtained mainly from sequence alignments of globular proteins are mostly correlated by at least a value of 0.9. The BLOSUM62 is the central element of this group. A second group includes SMs derived from alignments of remote homologs or transmembrane proteins. These matrices correlate better with classical SMs (0.8) than among themselves (0.7). A third group consists of intermediate links between SMs and CPs - matrices and potentials that exhibit mutual correlations of at least 0.8. Next, we show that SMs can be approximated with a correlation of 0.9 by expressions c(0) + x(i)x(j) + y(i)y(j) + z(i)z(j), 1
Analyzing the errors of DFT approximations for compressed water systems
NASA Astrophysics Data System (ADS)
Alfè, D.; Bartók, A. P.; Csányi, G.; Gillan, M. J.
2014-07-01
We report an extensive study of the errors of density functional theory (DFT) approximations for compressed water systems. The approximations studied are based on the widely used PBE and BLYP exchange-correlation functionals, and we characterize their errors before and after correction for 1- and 2-body errors, the corrections being performed using the methods of Gaussian approximation potentials. The errors of the uncorrected and corrected approximations are investigated for two related types of water system: first, the compressed liquid at temperature 420 K and density 1.245 g/cm3 where the experimental pressure is 15 kilobars; second, thermal samples of compressed water clusters from the trimer to the 27-mer. For the liquid, we report four first-principles molecular dynamics simulations, two generated with the uncorrected PBE and BLYP approximations and a further two with their 1- and 2-body corrected counterparts. The errors of the simulations are characterized by comparing with experimental data for the pressure, with neutron-diffraction data for the three radial distribution functions, and with quantum Monte Carlo (QMC) benchmarks for the energies of sets of configurations of the liquid in periodic boundary conditions. The DFT errors of the configuration samples of compressed water clusters are computed using QMC benchmarks. We find that the 2-body and beyond-2-body errors in the liquid are closely related to similar errors exhibited by the clusters. For both the liquid and the clusters, beyond-2-body errors of DFT make a substantial contribution to the overall errors, so that correction for 1- and 2-body errors does not suffice to give a satisfactory description. For BLYP, a recent representation of 3-body energies due to Medders, Babin, and Paesani [J. Chem. Theory Comput. 9, 1103 (2013)] gives a reasonably good way of correcting for beyond-2-body errors, after which the remaining errors are typically 0.5 mEh ≃ 15 meV/monomer for the liquid and the
Analyzing the errors of DFT approximations for compressed water systems.
Alfè, D; Bartók, A P; Csányi, G; Gillan, M J
2014-07-01
We report an extensive study of the errors of density functional theory (DFT) approximations for compressed water systems. The approximations studied are based on the widely used PBE and BLYP exchange-correlation functionals, and we characterize their errors before and after correction for 1- and 2-body errors, the corrections being performed using the methods of Gaussian approximation potentials. The errors of the uncorrected and corrected approximations are investigated for two related types of water system: first, the compressed liquid at temperature 420 K and density 1.245 g/cm(3) where the experimental pressure is 15 kilobars; second, thermal samples of compressed water clusters from the trimer to the 27-mer. For the liquid, we report four first-principles molecular dynamics simulations, two generated with the uncorrected PBE and BLYP approximations and a further two with their 1- and 2-body corrected counterparts. The errors of the simulations are characterized by comparing with experimental data for the pressure, with neutron-diffraction data for the three radial distribution functions, and with quantum Monte Carlo (QMC) benchmarks for the energies of sets of configurations of the liquid in periodic boundary conditions. The DFT errors of the configuration samples of compressed water clusters are computed using QMC benchmarks. We find that the 2-body and beyond-2-body errors in the liquid are closely related to similar errors exhibited by the clusters. For both the liquid and the clusters, beyond-2-body errors of DFT make a substantial contribution to the overall errors, so that correction for 1- and 2-body errors does not suffice to give a satisfactory description. For BLYP, a recent representation of 3-body energies due to Medders, Babin, and Paesani [J. Chem. Theory Comput. 9, 1103 (2013)] gives a reasonably good way of correcting for beyond-2-body errors, after which the remaining errors are typically 0.5 mE(h) ≃ 15 meV/monomer for the liquid and the
NASA Astrophysics Data System (ADS)
Baran, V.; Palade, D. I.; Colonna, M.; Di Toro, M.; Croitoru, A.; Nicolin, A. I.
2015-05-01
Within schematic models based on the Tamm-Dancoff approximation and the random-phase approximation with separable interactions, we investigate the physical conditions that may determine the emergence of the pygmy dipole resonance in the E 1 response of atomic nuclei. By introducing a generalization of the Brown-Bolsterli schematic model with a density-dependent particle-hole residual interaction, we find that an additional mode will be affected by the interaction, whose energy centroid is closer to the distance between two major shells and therefore well below the giant dipole resonance (GDR). This state, together with the GDR, exhausts all the transition strength in the Tamm-Dancoff approximation and all the energy-weighted sum rule in the random-phase approximation. Thus, within our scheme, this mode, which could be associated with the pygmy dipole resonance, is of collective nature. By relating the coupling constants appearing in the separable interaction to the symmetry energy value at and below saturation density we explore the role of density dependence of the symmetry energy on the low-energy dipole response.
Polynomial approximations of a class of stochastic multiscale elasticity problems
NASA Astrophysics Data System (ADS)
Hoang, Viet Ha; Nguyen, Thanh Chung; Xia, Bingxing
2016-06-01
We consider a class of elasticity equations in {mathbb{R}^d} whose elastic moduli depend on n separated microscopic scales. The moduli are random and expressed as a linear expansion of a countable sequence of random variables which are independently and identically uniformly distributed in a compact interval. The multiscale Hellinger-Reissner mixed problem that allows for computing the stress directly and the multiscale mixed problem with a penalty term for nearly incompressible isotropic materials are considered. The stochastic problems are studied via deterministic problems that depend on a countable number of real parameters which represent the probabilistic law of the stochastic equations. We study the multiscale homogenized problems that contain all the macroscopic and microscopic information. The solutions of these multiscale homogenized problems are written as generalized polynomial chaos (gpc) expansions. We approximate these solutions by semidiscrete Galerkin approximating problems that project into the spaces of functions with only a finite number of N gpc modes. Assuming summability properties for the coefficients of the elastic moduli's expansion, we deduce bounds and summability properties for the solutions' gpc expansion coefficients. These bounds imply explicit rates of convergence in terms of N when the gpc modes used for the Galerkin approximation are chosen to correspond to the best N terms in the gpc expansion. For the mixed problem with a penalty term for nearly incompressible materials, we show that the rate of convergence for the best N term approximation is independent of the Lamé constants' ratio when it goes to {infty}. Correctors for the homogenization problem are deduced. From these we establish correctors for the solutions of the parametric multiscale problems in terms of the semidiscrete Galerkin approximations. For two-scale problems, an explicit homogenization error which is uniform with respect to the parameters is deduced. Together
Subsonic Aircraft With Regression and Neural-Network Approximators Designed
NASA Technical Reports Server (NTRS)
Patnaik, Surya N.; Hopkins, Dale A.
2004-01-01
At the NASA Glenn Research Center, NASA Langley Research Center's Flight Optimization System (FLOPS) and the design optimization testbed COMETBOARDS with regression and neural-network-analysis approximators have been coupled to obtain a preliminary aircraft design methodology. For a subsonic aircraft, the optimal design, that is the airframe-engine combination, is obtained by the simulation. The aircraft is powered by two high-bypass-ratio engines with a nominal thrust of about 35,000 lbf. It is to carry 150 passengers at a cruise speed of Mach 0.8 over a range of 3000 n mi and to operate on a 6000-ft runway. The aircraft design utilized a neural network and a regression-approximations-based analysis tool, along with a multioptimizer cascade algorithm that uses sequential linear programming, sequential quadratic programming, the method of feasible directions, and then sequential quadratic programming again. Optimal aircraft weight versus the number of design iterations is shown. The central processing unit (CPU) time to solution is given. It is shown that the regression-method-based analyzer exhibited a smoother convergence pattern than the FLOPS code. The optimum weight obtained by the approximation technique and the FLOPS code differed by 1.3 percent. Prediction by the approximation technique exhibited no error for the aircraft wing area and turbine entry temperature, whereas it was within 2 percent for most other parameters. Cascade strategy was required by FLOPS as well as the approximators. The regression method had a tendency to hug the data points, whereas the neural network exhibited a propensity to follow a mean path. The performance of the neural network and regression methods was considered adequate. It was at about the same level for small, standard, and large models with redundancy ratios (defined as the number of input-output pairs to the number of unknown coefficients) of 14, 28, and 57, respectively. In an SGI octane workstation (Silicon Graphics
Quantum Calisthenics: Gaussians, The Path Integral and Guided Numerical Approximations
Weinstein, Marvin; /SLAC
2009-02-12
It is apparent to anyone who thinks about it that, to a large degree, the basic concepts of Newtonian physics are quite intuitive, but quantum mechanics is not. My purpose in this talk is to introduce you to a new, much more intuitive way to understand how quantum mechanics works. I begin with an incredibly easy way to derive the time evolution of a Gaussian wave-packet for the case free and harmonic motion without any need to know the eigenstates of the Hamiltonian. This discussion is completely analytic and I will later use it to relate the solution for the behavior of the Gaussian packet to the Feynman path-integral and stationary phase approximation. It will be clear that using the information about the evolution of the Gaussian in this way goes far beyond what the stationary phase approximation tells us. Next, I introduce the concept of the bucket brigade approach to dealing with problems that cannot be handled totally analytically. This approach combines the intuition obtained in the initial discussion, as well as the intuition obtained from the path-integral, with simple numerical tools. My goal is to show that, for any specific process, there is a simple Hilbert space interpretation of the stationary phase approximation. I will then argue that, from the point of view of numerical approximations, the trajectory obtained from my generalization of the stationary phase approximation specifies that subspace of the full Hilbert space that is needed to compute the time evolution of the particular state under the full Hamiltonian. The prescription I will give is totally non-perturbative and we will see, by the grace of Maple animations computed for the case of the anharmonic oscillator Hamiltonian, that this approach allows surprisingly accurate computations to be performed with very little work. I think of this approach to the path-integral as defining what I call a guided numerical approximation scheme. After the discussion of the anharmonic oscillator I will
NASA Astrophysics Data System (ADS)
Pietracaprina, Francesca; Ros, Valentina; Scardicchio, Antonello
2016-02-01
In this paper we analyze the predictions of the forward approximation in some models which exhibit an Anderson (single-body) or many-body localized phase. This approximation, which consists of summing over the amplitudes of only the shortest paths in the locator expansion, is known to overestimate the critical value of the disorder which determines the onset of the localized phase. Nevertheless, the results provided by the approximation become more and more accurate as the local coordination (dimensionality) of the graph, defined by the hopping matrix, is made larger. In this sense, the forward approximation can be regarded as a mean-field theory for the Anderson transition in infinite dimensions. The sum can be efficiently computed using transfer matrix techniques, and the results are compared with the most precise exact diagonalization results available. For the Anderson problem, we find a critical value of the disorder which is 0.9 % off the most precise available numerical value already in 5 spatial dimensions, while for the many-body localized phase of the Heisenberg model with random fields the critical disorder hc=4.0 ±0.3 is strikingly close to the most recent results obtained by exact diagonalization. In both cases we obtain a critical exponent ν =1 . In the Anderson case, the latter does not show dependence on the dimensionality, as it is common within mean-field approximations. We discuss the relevance of the correlations between the shortest paths for both the single- and many-body problems, and comment on the connections of our results with the problem of directed polymers in random medium.
Counting statistics for genetic switches based on effective interaction approximation
NASA Astrophysics Data System (ADS)
Ohkubo, Jun
2012-09-01
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 order 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 order to count the number of transitions from inactive to active states in the gene. To avoid having the non-closed equations, an effective interaction approximation is employed. As a result, it is shown that the switching problem can be treated as a simple two-state model approximately, which immediately indicates that the switching obeys non-Poisson statistics.
Short-bearing approximation for full journal bearings
NASA Technical Reports Server (NTRS)
Ocvirk, F W
1952-01-01
A short-bearing approximation of pressure distribution in the oil film is presented which is an extension of the pressure-distribution function of Michell and Cardullo and includes end-leakage effects. Equations giving applied load, attitude angle, location and magnitude of peak film pressure, friction, and required oil flow rate as functions of the eccentricity ratio are also given. The capacity number, a basic non dimensional quantity resulting from this analysis is the product of the Sommerfeld number and the square of the length-diameter ratio. Curves determined by this analysis are compared with previously published experimental data and theoretical curves of Sommerfeld and Cameron and Wood. Conclusions reached indicate that this approximation is of practical value for analysis of short bearings.
Network meta-analysis with integrated nested Laplace approximations.
Sauter, Rafael; Held, Leonhard
2015-11-01
Analyzing the collected evidence of a systematic review in form of a network meta-analysis (NMA) enjoys increasing popularity and provides a valuable instrument for decision making. Bayesian inference of NMA models is often propagated, especially if correlated random effects for multiarm trials are included. The standard choice for Bayesian inference is Markov chain Monte Carlo (MCMC) sampling, which is computationally intensive. An alternative to MCMC sampling is the recently suggested approximate Bayesian method of integrated nested Laplace approximations (INLA) that dramatically saves computation time without any substantial loss in accuracy. We show how INLA apply to NMA models for summary level as well as trial-arm level data. Specifically, we outline the modeling of multiarm trials and inference for functional contrasts with INLA. We demonstrate how INLA facilitate the assessment of network inconsistency with node-splitting. Three applications illustrate the use of INLA for a NMA. PMID:26360927
Swiss-cheese models and the Dyer-Roeder approximation
NASA Astrophysics Data System (ADS)
Fleury, Pierre
2014-06-01
In view of interpreting the cosmological observations precisely, especially when they involve narrow light beams, it is crucial to understand how light propagates in our statistically homogeneous, clumpy, Universe. Among the various approaches to tackle this issue, Swiss-cheese models propose an inhomogeneous spacetime geometry which is an exact solution of Einstein's equation, while the Dyer-Roeder approximation deals with inhomogeneity in an effective way. In this article, we demonstrate that the distance-redshift relation of a certain class of Swiss-cheese models is the same as the one predicted by the Dyer-Roeder approach, at a well-controlled level of approximation. Both methods are therefore equivalent when applied to the interpretation of, e.g., supernova obervations. The proof relies on completely analytical arguments, and is illustrated by numerical results.
Quasielastic electron-deuteron scattering in the weak binding approximation
Ethier, Jacob J.; Doshi, Nidhi P.; Malace, Simona P.; Melnitchouk, Wally
2014-06-01
We perform a global analysis of all available electron-deuteron quasielastic scattering data using Q^2-dependent smearing functions that describe inclusive inelastic e-d scattering within the weak binding approximation. We study the dependence of the cross sections on the deuteron wave function and the off-shell extrapolation of the elastic electron-nucleon cross section, which show particular sensitivity at x >> 1. The excellent overall agreement with data over a large range of Q^2 and x suggest a limited need for effects beyond the impulse approximation, with the exception of the very high-x or very low-Q^2 regions, where short-distance effects in the deuteron become more relevant.
Network meta-analysis with integrated nested Laplace approximations.
Sauter, Rafael; Held, Leonhard
2015-11-01
Analyzing the collected evidence of a systematic review in form of a network meta-analysis (NMA) enjoys increasing popularity and provides a valuable instrument for decision making. Bayesian inference of NMA models is often propagated, especially if correlated random effects for multiarm trials are included. The standard choice for Bayesian inference is Markov chain Monte Carlo (MCMC) sampling, which is computationally intensive. An alternative to MCMC sampling is the recently suggested approximate Bayesian method of integrated nested Laplace approximations (INLA) that dramatically saves computation time without any substantial loss in accuracy. We show how INLA apply to NMA models for summary level as well as trial-arm level data. Specifically, we outline the modeling of multiarm trials and inference for functional contrasts with INLA. We demonstrate how INLA facilitate the assessment of network inconsistency with node-splitting. Three applications illustrate the use of INLA for a NMA.
Post-Newtonian approximation in Maxwell-like form
Kaplan, Jeffrey D.; Nichols, David A.; Thorne, Kip S.
2009-12-15
The equations of the linearized first post-Newtonian approximation to general relativity are often written in 'gravitoelectromagnetic' Maxwell-like form, since that facilitates physical intuition. Damour, Soffel, and Xu (DSX) (as a side issue in their complex but elegant papers on relativistic celestial mechanics) have expressed the first post-Newtonian approximation, including all nonlinearities, in Maxwell-like form. This paper summarizes that DSX Maxwell-like formalism (which is not easily extracted from their celestial mechanics papers), and then extends it to include the post-Newtonian (Landau-Lifshitz-based) gravitational momentum density, momentum flux (i.e. gravitational stress tensor), and law of momentum conservation in Maxwell-like form. The authors and their colleagues have found these Maxwell-like momentum tools useful for developing physical intuition into numerical-relativity simulations of compact binaries with spin.
Adaptive Nonlinear Signal Approximation Using Bacterial Foraging Strategy
NASA Astrophysics Data System (ADS)
Kumar, Naik Manoj; Rutuparna, Panda
Uniform approximation of signals has been an area of interest for researchers working in different disciplines of science and engineering. This paper presents an adaptive algorithm based on E. coli bacteria foraging strategy (EBFS) for uniform approximation of signals by linear combinations of shifted nonlinear basis functions. New class of nonlinear basis functions has been derived from a sigmoid function. The weight factor of the newly proposed nonlinear basis functions has been optimized by using the EBFS to minimize the mean square error. Different test signals are considered for validation of the present technique. Results are also compared with Genetic algorithm approach. The proposed technique could also be useful in fractional signal processing applications.
Approximating the Generalized Voronoi Diagram of Closely Spaced Objects
Edwards, John; Daniel, Eric; Pascucci, Valerio; Bajaj, Chandrajit
2016-01-01
We present an algorithm to compute an approximation of the generalized Voronoi diagram (GVD) on arbitrary collections of 2D or 3D geometric objects. In particular, we focus on datasets with closely spaced objects; GVD approximation is expensive and sometimes intractable on these datasets using previous algorithms. With our approach, the GVD can be computed using commodity hardware even on datasets with many, extremely tightly packed objects. Our approach is to subdivide the space with an octree that is represented with an adjacency structure. We then use a novel adaptive distance transform to compute the distance function on octree vertices. The computed distance field is sampled more densely in areas of close object spacing, enabling robust and parallelizable GVD surface generation. We demonstrate our method on a variety of data and show example applications of the GVD in 2D and 3D. PMID:27540272
Exponential approximation for one-component Yukawa plasma
Hlushak, Stepan
2014-11-28
A theory based on the exponential approximation of the liquid-state theory is applied to study properties of several models of one-component Yukawa plasma characterized by different values of the screening parameter z. The results of the new theory are compared to the results of a conventional theory, which is based on the first-order mean spherical approximation, and to the results of a Monte Carlo simulation. The new theory shows improvements in the predictions for the thermodynamic and structural properties of Yukawa plasmas with high and intermediate values of the screening parameter, z, and coupling parameter, Γ. For low values of z and Γ, the new theory is comparable in accuracy to the conventional theory, which in turn agrees well with the results of the Monte Carlo simulation.
Problems with the quenched approximation in the chiral limit
Sharpe, S.R.
1992-12-31
In the quenched approximation, loops of the light singlet meson (the {eta}{prime}) give rise to a type of chiral logarithm absent in full QCD. These logarithms are singular in the chiral limit, throwing doubt upon the utility of the quenched approximation. In previous work, I summed a class of diagrams, leading to non-analytic power dependencies such as {l_angle}{anti {psi}}{psi}{r_angle} {proportional_to} m{sub q}{sup {minus}{delta}}/(1+{delta}) I suggested, however, that these peculiar results could be redefined away. Here I give an alternative derivation of the results, based on the renormalization group, and argue that they cannot be redefined away. I discuss the evidence (or lack thereof) for such effects in numerical data.
Understanding multi-quantum NMR through secular approximation.
Srivastava, Deepansh; SubbaRao, R Venkata; Ramachandran, Ramesh
2013-05-14
With the development of technology and improved understanding of nuclear spin-spin interactions and their behavior in static/oscillating magnetic fields, NMR spectroscopy has emerged as a powerful tool for characterizing molecular structure in a wide range of systems of chemical, physical and biological relevance. Here in this article, we revisit the important connection between "Secular-Approximation" (a well-known fundamental concept) and NMR spectroscopy. Employing recent experimental results as the background, an alternate interpretation of the secular approximation is presented for describing and understanding the nuances of Multi-Quantum (MQ) NMR spectroscopy of quadrupolar nuclei. Since MQ NMR spectroscopy of quadrupolar nuclei forms the basis of the structural characterization of inorganic solids and clusters, we believe that the analytic theory presented herein would be beneficial both in the understanding and design of MQ NMR experiments. Additionally, the analytic results are corroborated with rigorous numerical simulations and could be employed in the quantitative interpretation of experimental results.
Complexity and efficient approximability of two dimensional periodically specified problems
Marathe, M.V.; Hunt, H.B. III; Stearns, R.E.
1996-09-01
The authors consider the two dimensional periodic specifications: a method to specify succinctly objects with highly regular repetitive structure. These specifications arise naturally when processing engineering designs including VLSI designs. These specifications can specify objects whose sizes are exponentially larger than the sizes of the specification themselves. Consequently solving a periodically specified problem by explicitly expanding the instance is prohibitively expensive in terms of computational resources. This leads one to investigate the complexity and efficient approximability of solving graph theoretic and combinatorial problems when instances are specified using two dimensional periodic specifications. They prove the following results: (1) several classical NP-hard optimization problems become NEXPTIME-hard, when instances are specified using two dimensional periodic specifications; (2) in contrast, several of these NEXPTIME-hard problems have polynomial time approximation algorithms with guaranteed worst case performance.
On the convergence of difference approximations to scalar conservation laws
NASA Technical Reports Server (NTRS)
Osher, S.; Tadmor, E.
1985-01-01
A unified treatment of explicit in time, two level, second order resolution, total variation diminishing, approximations 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 order accurate in general. Convergence for total variation diminishing-second order resolution schemes approximating convex or concave conservation laws is shown by enforcing a single discrete entropy inequality.
Dynamical observer for a flexible beam via finite element approximations
NASA Technical Reports Server (NTRS)
Manitius, Andre; Xia, Hong-Xing
1994-01-01
The purpose of this view-graph presentation is a computational investigation of the closed-loop output feedback control of a Euler-Bernoulli beam based on finite element approximation. The observer is part of the classical observer plus state feedback control, but it is finite-dimensional. In the theoretical work on the subject it is assumed (and sometimes proved) that increasing the number of finite elements will improve accuracy of the control. In applications, this may be difficult to achieve because of numerical problems. The main difficulty in computing the observer and simulating its work is the presence of high frequency eigenvalues in the finite-element model and poor numerical conditioning of some of the system matrices (e.g. poor observability properties) when the dimension of the approximating system increases. This work dealt with some of these difficulties.
Analysis and approximations for crossing two nearby spin resonances
Ranjbar, V. H.
2014-01-07
Solutions to the T-BMT spin equation have to date been confined to the single resonance crossing. However, in reality most cases of interest concern the overlapping of several resonances. To date there has been several serious studies of this problem; however, a good analytical solution or even approximation has eluded the community. We show that the T-BMT equation can be transformed into a Hill’s like equation. In this representation it can be shown that, while the single resonance crossing represents the solution to the Parabolic Cylinder equation, the overlapping case becomes a parametric type of resonance. We present possible approximations for both the non-accelerating case and accelerating case.
Assessment of Time Series Complexity Using Improved Approximate Entropy
NASA Astrophysics Data System (ADS)
Kong, De-Ren; Xie, Hong-Bo
2011-09-01
Approximate entropy (ApEn), a measure quantifying complexity and/or regularity, is believed to be an effective method of analyzing diverse settings. However, the similarity definition of vectors based on Heaviside function may cause some problems in the validity and accuracy of ApEn. To overcome the problems, an improved approximate entropy (iApEn) based on the sigmoid function is proposed. The performance of iApEn is tested on the independent identically distributed (IID) Gaussian noise, the MIX stochastic model, the Rossler map, the logistic map, and the high-dimensional Mackey—Glass oscillator. The results show that iApEn is superior to ApEn in several aspects, including better relative consistency, freedom of parameter selection, robust to noise, and more independence on record length when characterizing time series with different complexities.
Location constrained approximate message passing for compressed sensing MRI.
Sung, Kyunghyun; Daniel, Bruce L; Hargreaves, Brian A
2013-08-01
Iterative thresholding methods have been extensively studied as faster alternatives to convex optimization methods for solving large-sized problems in compressed sensing. A novel iterative thresholding method called LCAMP (Location Constrained Approximate Message Passing) is presented for reducing computational complexity and improving reconstruction accuracy when a nonzero location (or sparse support) constraint can be obtained from view shared images. LCAMP modifies the existing approximate message passing algorithm by replacing the thresholding stage with a location constraint, which avoids adjusting regularization parameters or thresholding levels. This work is first compared with other conventional reconstruction methods using random one-dimention signals and then applied to dynamic contrast-enhanced breast magnetic resonance imaging to demonstrate the excellent reconstruction accuracy (less than 2% absolute difference) and low computation time (5-10 s using Matlab) with highly undersampled three-dimentional data (244 × 128 × 48; overall reduction factor = 10). PMID:23042658