Global Monte Carlo Simulation with High Order Polynomial Expansions
William R. Martin; James Paul Holloway; Kaushik Banerjee; Jesse Cheatham; Jeremy Conlin
2007-12-13
The functional expansion technique (FET) was recently developed for Monte Carlo simulation. The basic idea of the FET is to expand a Monte Carlo tally in terms of a high order expansion, the coefficients of which can be estimated via the usual random walk process in a conventional Monte Carlo code. If the expansion basis is chosen carefully, the lowest order coefficient is simply the conventional histogram tally, corresponding to a flat mode. This research project studied the applicability of using the FET to estimate the fission source, from which fission sites can be sampled for the next generation. The idea is that individual fission sites contribute to expansion modes that may span the geometry being considered, possibly increasing the communication across a loosely coupled system and thereby improving convergence over the conventional fission bank approach used in most production Monte Carlo codes. The project examined a number of basis functions, including global Legendre polynomials as well as “local” piecewise polynomials such as finite element hat functions and higher order versions. The global FET showed an improvement in convergence over the conventional fission bank approach. The local FET methods showed some advantages versus global polynomials in handling geometries with discontinuous material properties. The conventional finite element hat functions had the disadvantage that the expansion coefficients could not be estimated directly but had to be obtained by solving a linear system whose matrix elements were estimated. An alternative fission matrix-based response matrix algorithm was formulated. Studies were made of two alternative applications of the FET, one based on the kernel density estimator and one based on Arnoldi’s method of minimized iterations. Preliminary results for both methods indicate improvements in fission source convergence. These developments indicate that the FET has promise for speeding up Monte Carlo fission source
Range Image Flow using High-Order Polynomial Expansion
2013-09-01
give a special thanks to Dr. Steve Hobbs for his help with the high-order tensor calculations. MATLAB ® is a registered...that using multiple spatial scales and past information improve the final flow estimation, as we would expect. Also, we will port the MATLAB R...taken column- wise and diagonalized, and f is the range image data, taken column-wise. The values of these weights for a Velodyne R© and Odetic lidar
Analysis of spacecraft disposal solutions from LPO to the Moon with high order polynomial expansions
NASA Astrophysics Data System (ADS)
Vetrisano, Massimo; Vasile, Massimiliano
2017-07-01
This paper presents the analysis of disposal trajectories from libration point orbits to the Moon under uncertainty. The paper proposes the use of polynomial chaos expansions to quantify the uncertainty in the final conditions given an uncertainty in initial conditions and disposal manoeuvre. The paper will compare the use of polynomial chaos expansions against high order Taylor expansions computed with point-wise integration of the partials of the dynamics, the use of the covariance matrix propagated using a unscented transformation and a standard Monte Carlo simulation. It will be shown that the use of the ellipsoid of uncertainty, that corresponds to the propagation of the covariance matrix with a first order Taylor expansions, is not adequate to correctly capture the dispersion of the trajectories that can intersect the Moon. Furthermore, it will be shown that polynomial chaos expansions better represent the distribution of the final states compared to Taylor expansions of equal order and are comparable to a full scale Monte Carlo simulations but at a fraction of the computational cost.
High-Order Polynomial Expansions (HOPE) for flux-vector splitting
NASA Technical Reports Server (NTRS)
Liou, Meng-Sing; Steffen, Chris J., Jr.
1991-01-01
The Van Leer flux splitting is known to produce excessive numerical dissipation for Navier-Stokes calculations. Researchers attempt to remedy this deficiency by introducing a higher order polynomial expansion (HOPE) for the mass flux. In addition to Van Leer's splitting, a term is introduced so that the mass diffusion error vanishes at M = 0. Several splittings for pressure are proposed and examined. The effectiveness of the HOPE scheme is illustrated for 1-D hypersonic conical viscous flow and 2-D supersonic shock-wave boundary layer interactions.
High-Order Polynomial Expansions (HOPE) for flux-vector splitting
NASA Technical Reports Server (NTRS)
Liou, Meng-Sing; Steffen, Chris J., Jr.
1991-01-01
The Van Leer flux splitting is known to produce excessive numerical dissipation for Navier-Stokes calculations. Researchers attempt to remedy this deficiency by introducing a higher order polynomial expansion (HOPE) for the mass flux. In addition to Van Leer's splitting, a term is introduced so that the mass diffusion error vanishes at M equals 0. Several splittings for pressure are proposed and examined. The effectiveness of the HOPE scheme is illustrated for 1-D hypersonic conical viscous flow and 2-D supersonic shock-wave boundary layer interactions. Also, the authors give the weakness of the scheme and suggest areas for further investigation.
Adapted polynomial chaos expansion for failure detection
Paffrath, M. Wever, U.
2007-09-10
In this paper, we consider two methods of computation of failure probabilities by adapted polynomial chaos expansions. The performance of the two methods is demonstrated by a predator-prey model and a chemical reaction problem.
Adapted polynomial chaos expansion for failure detection
NASA Astrophysics Data System (ADS)
Paffrath, M.; Wever, U.
2007-09-01
In this paper, we consider two methods of computation of failure probabilities by adapted polynomial chaos expansions. The performance of the two methods is demonstrated by a predator-prey model and a chemical reaction problem.
NASA Astrophysics Data System (ADS)
Liu, Y.; Zheng, L.; Pau, G. S. H.
2016-12-01
A careful assessment of the risk associated with geologic CO2 storage is critical to the deployment of large-scale storage projects. While numerical modeling is an indispensable tool for risk assessment, there has been increasing need in considering and addressing uncertainties in the numerical models. However, uncertainty analyses have been significantly hindered by the computational complexity of the model. As a remedy, reduced-order models (ROM), which serve as computationally efficient surrogates for high-fidelity models (HFM), have been employed. The ROM is constructed at the expense of an initial set of HFM simulations, and afterwards can be relied upon to predict the model output values at minimal cost. The ROM presented here is part of National Risk Assessment Program (NRAP) and intends to predict the water quality change in groundwater in response to hypothetical CO2 and brine leakage. The HFM based on which the ROM is derived is a multiphase flow and reactive transport model, with 3-D heterogeneous flow field and complex chemical reactions including aqueous complexation, mineral dissolution/precipitation, adsorption/desorption via surface complexation and cation exchange. Reduced-order modeling techniques based on polynomial basis expansion, such as polynomial chaos expansion (PCE), are widely used in the literature. However, the accuracy of such ROMs can be affected by the sparse structure of the coefficients of the expansion. Failing to identify vanishing polynomial coefficients introduces unnecessary sampling errors, the accumulation of which deteriorates the accuracy of the ROMs. To address this issue, we treat the PCE as a sparse Bayesian learning (SBL) problem, and the sparsity is obtained by detecting and including only the non-zero PCE coefficients one at a time by iteratively selecting the most contributing coefficients. The computational complexity due to predicting the entire 3-D concentration fields is further mitigated by a dimension
Need for higher order polynomial basis for polynomial nodal methods employed in LWR calculations
Taiwo, T.A.; Palmiotti, G.
1997-08-01
The paper evaluates the accuracy and efficiency of sixth order polynomial solutions and the use of one radial node per core assembly for pressurized water reactor (PWR) core power distributions and reactivities. The computer code VARIANT was modified to calculate sixth order polynomial solutions for a hot zero power benchmark problem in which a control assembly along a core axis is assumed to be out of the core. Results are presented for the VARIANT, DIF3D-NODAL, and DIF3D-finite difference codes. The VARIANT results indicate that second order expansion of the within-node source and linear representation of the node surface currents are adequate for this problem. The results also demonstrate the improvement in the VARIANT solution when the order of the polynomial expansion of the within-node flux is increased from fourth to sixth order. There is a substantial saving in computational time for using one radial node per assembly with the sixth order expansion compared to using four or more nodes per assembly and fourth order polynomial solutions. 11 refs., 1 tab.
Asymptotic expansions of Feynman integrals of exponentials with polynomial exponent
NASA Astrophysics Data System (ADS)
Kravtseva, A. K.; Smolyanov, O. G.; Shavgulidze, E. T.
2016-10-01
In the paper, an asymptotic expansion of path integrals of functionals having exponential form with polynomials in the exponent is constructed. The definition of the path integral in the sense of analytic continuation is considered.
Explicit energy expansion for general odd-degree polynomial potentials
NASA Astrophysics Data System (ADS)
Nanayakkara, Asiri; Mathanaranjan, Thilagarajah
2013-11-01
In this paper we derive an almost explicit analytic formula for asymptotic eigenenergy expansion of arbitrary odd-degree polynomial potentials of the form V (x) = (ix)2N+1 + β1x2N + β2x2N-1 + ··· + β2Nx, where β‧k are real or complex for 1 ⩽ k ⩽ 2N. The formula can be used to find semiclassical analytic expressions for eigenenergies up to any order, very efficiently. Each term of the expansion is given explicitly as a multinomial of the parameters β1,β2… and β2N of the potential. Unlike in the even-degree polynomial case, the highest-order term in the potential is pure imaginary and hence the system is non-Hermitian. Therefore all the integrations have been carried out along a contour enclosing two complex branch points, which lies within a wedge in the complex plane. With the help of some examples we demonstrate the accuracy of the method for both real and complex eigenspectra.
Optical homodyne tomography with polynomial series expansion
Benichi, Hugo; Furusawa, Akira
2011-09-15
We present and demonstrate a method for optical homodyne tomography based on the inverse Radon transform. Different from the usual filtered back-projection algorithm, this method uses an appropriate polynomial series to expand the Wigner function and the marginal distribution, and discretize Fourier space. We show that this technique solves most technical difficulties encountered with kernel deconvolution-based methods and reconstructs overall better and smoother Wigner functions. We also give estimators of the reconstruction errors for both methods and show improvement in noise handling properties and resilience to statistical errors.
Compressive sampling of polynomial chaos expansions: Convergence analysis and sampling strategies
Hampton, Jerrad; Doostan, Alireza
2015-01-01
Sampling orthogonal polynomial bases via Monte Carlo is of interest for uncertainty quantification of models with random inputs, using Polynomial Chaos (PC) expansions. It is known that bounding a probabilistic parameter, referred to as coherence, yields a bound on the number of samples necessary to identify coefficients in a sparse PC expansion via solution to an ℓ{sub 1}-minimization problem. Utilizing results for orthogonal polynomials, we bound the coherence parameter for polynomials of Hermite and Legendre type under their respective natural sampling distribution. In both polynomial bases we identify an importance sampling distribution which yields a bound with weaker dependence on the order of the approximation. For more general orthonormal bases, we propose the coherence-optimal sampling: a Markov Chain Monte Carlo sampling, which directly uses the basis functions under consideration to achieve a statistical optimality among all sampling schemes with identical support. We demonstrate these different sampling strategies numerically in both high-order and high-dimensional, manufactured PC expansions. In addition, the quality of each sampling method is compared in the identification of solutions to two differential equations, one with a high-dimensional random input and the other with a high-order PC expansion. In both cases, the coherence-optimal sampling scheme leads to similar or considerably improved accuracy.
Polynomial expansion nodal transport method in hexagonal geometry
Cho, Jin Young; Kim, Chang Hyo; Noh, Taewan
1997-12-01
Recently, the polynomial expansion nodal (PEN) method was developed as a new nodal diffusion scheme for hexagonal core analyses. Using the direct polynomial expansion for the node flux, the PEN method not only eliminates the complicated transverse integration procedures-especially in hexagonal geometry-which are frequently used in conventional nodal methods, but also provides a number of features such as the convenient energy group expendability and much enhanced accuracy with less computational effort. In this paper, we further develop the PEN method for the transport equation for the cases where the transport effects are important: highly heterogeneous, small (high-leakage), and fast reactors, etc. Here, we take the even-parity form of transport equation. The main reason is that the diffusion-like nature of the even-parity equation is adequate to establish the new nodal transport method (PEN-TR) using the earlier developed PEN method in diffusion theory.
Modelling Trends in Ordered Correspondence Analysis Using Orthogonal Polynomials.
Lombardo, Rosaria; Beh, Eric J; Kroonenberg, Pieter M
2016-06-01
The core of the paper consists of the treatment of two special decompositions for correspondence analysis of two-way ordered contingency tables: the bivariate moment decomposition and the hybrid decomposition, both using orthogonal polynomials rather than the commonly used singular vectors. To this end, we will detail and explain the basic characteristics of a particular set of orthogonal polynomials, called Emerson polynomials. It is shown that such polynomials, when used as bases for the row and/or column spaces, can enhance the interpretations via linear, quadratic and higher-order moments of the ordered categories. To aid such interpretations, we propose a new type of graphical display-the polynomial biplot.
From sequences to polynomials and back, via operator orderings
NASA Astrophysics Data System (ADS)
Amdeberhan, Tewodros; De Angelis, Valerio; Dixit, Atul; Moll, Victor H.; Vignat, Christophe
2013-12-01
Bender and Dunne ["Polynomials and operator orderings," J. Math. Phys. 29, 1727-1731 (1988)] showed that linear combinations of words qkpnqn-k, where p and q are subject to the relation qp - pq = ı, may be expressed as a polynomial in the symbol z = 1/2(qp+pq). Relations between such polynomials and linear combinations of the transformed coefficients are explored. In particular, examples yielding orthogonal polynomials are provided.
Zhang, Yan; Sahinidis, Nikolaos V.
2013-03-06
In this paper, surrogate models are iteratively built using polynomial chaos expansion (PCE) and detailed numerical simulations of a carbon sequestration system. Output variables from a numerical simulator are approximated as polynomial functions of uncertain parameters. Once generated, PCE representations can be used in place of the numerical simulator and often decrease simulation times by several orders of magnitude. However, PCE models are expensive to derive unless the number of terms in the expansion is moderate, which requires a relatively small number of uncertain variables and a low degree of expansion. To cope with this limitation, instead of using a classical full expansion at each step of an iterative PCE construction method, we introduce a mixed-integer programming (MIP) formulation to identify the best subset of basis terms in the expansion. This approach makes it possible to keep the number of terms small in the expansion. Monte Carlo (MC) simulation is then performed by substituting the values of the uncertain parameters into the closed-form polynomial functions. Based on the results of MC simulation, the uncertainties of injecting CO{sub 2} underground are quantified for a saline aquifer. Moreover, based on the PCE model, we formulate an optimization problem to determine the optimal CO{sub 2} injection rate so as to maximize the gas saturation (residual trapping) during injection, and thereby minimize the chance of leakage.
Enhancing sparsity of Hermite polynomial expansions by iterative rotations
Yang, Xiu; Lei, Huan; Baker, Nathan A.; Lin, Guang
2016-02-01
Compressive sensing has become a powerful addition to uncertainty quantification in recent years. This paper identifies new bases for random variables through linear mappings such that the representation of the quantity of interest is more sparse with new basis functions associated with the new random variables. This sparsity increases both the efficiency and accuracy of the compressive sensing-based uncertainty quantification method. Specifically, we consider rotation- based linear mappings which are determined iteratively for Hermite polynomial expansions. We demonstrate the effectiveness of the new method with applications in solving stochastic partial differential equations and high-dimensional (O(100)) problems.
Limitations of polynomial chaos expansions in the Bayesian solution of inverse problems
Lu, Fei; Morzfeld, Matthias; Tu, Xuemin; Chorin, Alexandre J.
2015-02-01
Polynomial chaos expansions are used to reduce the computational cost in the Bayesian solutions of inverse problems by creating a surrogate posterior that can be evaluated inexpensively. We show, by analysis and example, that when the data contain significant information beyond what is assumed in the prior, the surrogate posterior can be very different from the posterior, and the resulting estimates become inaccurate. One can improve the accuracy by adaptively increasing the order of the polynomial chaos, but the cost may increase too fast for this to be cost effective compared to Monte Carlo sampling without a surrogate posterior.
Uncertainty Quantification for Polynomial Systems via Bernstein Expansions
NASA Technical Reports Server (NTRS)
Crespo, Luis G.; Kenny, Sean P.; Giesy, Daniel P.
2012-01-01
This paper presents a unifying framework to uncertainty quantification for systems having polynomial response metrics that depend on both aleatory and epistemic uncertainties. The approach proposed, which is based on the Bernstein expansions of polynomials, enables bounding the range of moments and failure probabilities of response metrics as well as finding supersets of the extreme epistemic realizations where the limits of such ranges occur. These bounds and supersets, whose analytical structure renders them free of approximation error, can be made arbitrarily tight with additional computational effort. Furthermore, this framework enables determining the importance of particular uncertain parameters according to the extent to which they affect the first two moments of response metrics and failure probabilities. This analysis enables determining the parameters that should be considered uncertain as well as those that can be assumed to be constants without incurring significant error. The analytical nature of the approach eliminates the numerical error that characterizes the sampling-based techniques commonly used to propagate aleatory uncertainties as well as the possibility of under predicting the range of the statistic of interest that may result from searching for the best- and worstcase epistemic values via nonlinear optimization or sampling.
On an Ordering-Dependent Generalization of the Tutte Polynomial
NASA Astrophysics Data System (ADS)
Geloun, Joseph Ben; Caravelli, Francesco
2017-07-01
A generalization of the Tutte polynomial involved in the evaluation of the moments of the integrated geometric Brownian in the Itô formalism is discussed. The new combinatorial invariant depends on the order in which the sequence of contraction-deletions have been performed on the graph. Thus, this work provides a motivation for studying an order-dependent Tutte polynomial in the context of stochastic differential equations. We show that in the limit of the control parameters encoding the ordering going to zero, the multivariate Tutte-Fortuin-Kasteleyn polynomial is recovered.
On an Ordering-Dependent Generalization of the Tutte Polynomial
NASA Astrophysics Data System (ADS)
Geloun, Joseph Ben; Caravelli, Francesco
2017-09-01
A generalization of the Tutte polynomial involved in the evaluation of the moments of the integrated geometric Brownian in the Itô formalism is discussed. The new combinatorial invariant depends on the order in which the sequence of contraction-deletions have been performed on the graph. Thus, this work provides a motivation for studying an order-dependent Tutte polynomial in the context of stochastic differential equations. We show that in the limit of the control parameters encoding the ordering going to zero, the multivariate Tutte-Fortuin-Kasteleyn polynomial is recovered.
Huberts, W; Donders, W P; Delhaas, T; van de Vosse, F N
2014-12-01
Patient-specific modeling requires model personalization, which can be achieved in an efficient manner by parameter fixing and parameter prioritization. An efficient variance-based method is using generalized polynomial chaos expansion (gPCE), but it has not been applied in the context of model personalization, nor has it ever been compared with standard variance-based methods for models with many parameters. In this work, we apply the gPCE method to a previously reported pulse wave propagation model and compare the conclusions for model personalization with that of a reference analysis performed with Saltelli's efficient Monte Carlo method. We furthermore differentiate two approaches for obtaining the expansion coefficients: one based on spectral projection (gPCE-P) and one based on least squares regression (gPCE-R). It was found that in general the gPCE yields similar conclusions as the reference analysis but at much lower cost, as long as the polynomial metamodel does not contain unnecessary high order terms. Furthermore, the gPCE-R approach generally yielded better results than gPCE-P. The weak performance of the gPCE-P can be attributed to the assessment of the expansion coefficients using the Smolyak algorithm, which might be hampered by the high number of model parameters and/or by possible non-smoothness in the output space.
C. ALLEN
2000-08-01
We consider halo formation in continuous beams oscillating at natural modes by inspecting particle trajectories. Trajectory equations containing field nonlinearities are derived from a weighted polynomial expansion. We then use perturbational techniques to further analyze particle motion.
Cross gramian approximation with Laguerre polynomials for model order reduction
NASA Astrophysics Data System (ADS)
Perev, Kamen
2015-11-01
This paper considers the problem of model order reduction by approximate balanced truncation with Laguerre polynomials approximation of the system cross gramian. The cross gramian contains information both for the reachability of the system as well as for its observability. The main property of the cross gramian for a square symmetric stable linear system is that its square is equal to the product of the reachability and observability gramians and therefore, the absolute values of its eigenvalues are equal to the Hankel singular values. This is the reason to use the cross gramian for computing balancing transformations for model reduction. Laguerre polynomial series representations are used to approximate the cross gramian of the system at infinity. The orthogonal polynomials of Laguerre possess good convergence properties and allow to reduce the computational complexity of the model reduction problem. Numerical experiments are performed confirming the effectiveness of the proposed method.
NASA Astrophysics Data System (ADS)
Konakli, Katerina; Sudret, Bruno
2016-09-01
The growing need for uncertainty analysis of complex computational models has led to an expanding use of meta-models across engineering and sciences. The efficiency of meta-modeling techniques relies on their ability to provide statistically-equivalent analytical representations based on relatively few evaluations of the original model. Polynomial chaos expansions (PCE) have proven a powerful tool for developing meta-models in a wide range of applications; the key idea thereof is to expand the model response onto a basis made of multivariate polynomials obtained as tensor products of appropriate univariate polynomials. The classical PCE approach nevertheless faces the "curse of dimensionality", namely the exponential increase of the basis size with increasing input dimension. To address this limitation, the sparse PCE technique has been proposed, in which the expansion is carried out on only a few relevant basis terms that are automatically selected by a suitable algorithm. An alternative for developing meta-models with polynomial functions in high-dimensional problems is offered by the newly emerged low-rank approximations (LRA) approach. By exploiting the tensor-product structure of the multivariate basis, LRA can provide polynomial representations in highly compressed formats. Through extensive numerical investigations, we herein first shed light on issues relating to the construction of canonical LRA with a particular greedy algorithm involving a sequential updating of the polynomial coefficients along separate dimensions. Specifically, we examine the selection of optimal rank, stopping criteria in the updating of the polynomial coefficients and error estimation. In the sequel, we confront canonical LRA to sparse PCE in structural-mechanics and heat-conduction applications based on finite-element solutions. Canonical LRA exhibit smaller errors than sparse PCE in cases when the number of available model evaluations is small with respect to the input dimension, a
Konakli, Katerina Sudret, Bruno
2016-09-15
The growing need for uncertainty analysis of complex computational models has led to an expanding use of meta-models across engineering and sciences. The efficiency of meta-modeling techniques relies on their ability to provide statistically-equivalent analytical representations based on relatively few evaluations of the original model. Polynomial chaos expansions (PCE) have proven a powerful tool for developing meta-models in a wide range of applications; the key idea thereof is to expand the model response onto a basis made of multivariate polynomials obtained as tensor products of appropriate univariate polynomials. The classical PCE approach nevertheless faces the “curse of dimensionality”, namely the exponential increase of the basis size with increasing input dimension. To address this limitation, the sparse PCE technique has been proposed, in which the expansion is carried out on only a few relevant basis terms that are automatically selected by a suitable algorithm. An alternative for developing meta-models with polynomial functions in high-dimensional problems is offered by the newly emerged low-rank approximations (LRA) approach. By exploiting the tensor–product structure of the multivariate basis, LRA can provide polynomial representations in highly compressed formats. Through extensive numerical investigations, we herein first shed light on issues relating to the construction of canonical LRA with a particular greedy algorithm involving a sequential updating of the polynomial coefficients along separate dimensions. Specifically, we examine the selection of optimal rank, stopping criteria in the updating of the polynomial coefficients and error estimation. In the sequel, we confront canonical LRA to sparse PCE in structural-mechanics and heat-conduction applications based on finite-element solutions. Canonical LRA exhibit smaller errors than sparse PCE in cases when the number of available model evaluations is small with respect to the input
First-Order Polynomial Heisenberg Algebras and Coherent States
NASA Astrophysics Data System (ADS)
Castillo-Celeita, M.; Fernández C, D. J.
2016-03-01
The polynomial Heisenberg algebras (PHA) are deformations of the Heisenberg- Weyl algebra characterizing the underlying symmetry of the supersymmetric partners of the Harmonic oscillator. When looking for the simplest system ruled by PHA, however, we end up with the harmonic oscillator. In this paper we are going to realize the first-order PHA through the harmonic oscillator. The associated coherent states will be also constructed, which turn out to be the well known even and odd coherent states.
NASA Technical Reports Server (NTRS)
Apostol, Tom M. (Editor)
1991-01-01
In this 'Project Mathematics! series, sponsored by California Institute for Technology (CalTech), the mathematical concept of polynomials in rectangular coordinate (x, y) systems are explored. sing film footage of real life applications and computer animation sequences, the history of, the application of, and the different linear coordinate systems for quadratic, cubic, intersecting, and higher degree of polynomials are discussed.
NASA Astrophysics Data System (ADS)
Jacquelin, E.; Adhikari, S.; Sinou, J.-J.; Friswell, M. I.
2015-11-01
Polynomial chaos solution for the frequency response of linear non-proportionally damped dynamic systems has been considered. It has been observed that for lightly damped systems the convergence of the solution can be very poor in the vicinity of the deterministic resonance frequencies. To address this, Aitken's transformation and its generalizations are suggested. The proposed approach is successfully applied to the sequences defined by the first two moments of the responses, and this process significantly accelerates the polynomial chaos convergence. In particular, a 2-dof system with respectively 1 and 2 parameter uncertainties has been studied. The first two moments of the frequency response were calculated by Monte Carlo simulation, polynomial chaos expansion and Aitken's transformation of the polynomial chaos expansion. Whereas 200 polynomials are required to have a good agreement with Monte Carlo results around the deterministic eigenfrequencies, less than 50 polynomials transformed by the Aitken's method are enough. This latter result is improved if a generalization of Aitken's method (recursive Aitken's transformation, Shank's transformation) is applied. With the proposed convergence acceleration, polynomial chaos may be reconsidered as an efficient method to estimate the first two moments of a random dynamic response.
NASA Astrophysics Data System (ADS)
Bazargan, Hamid; Christie, Mike; Elsheikh, Ahmed H.; Ahmadi, Mohammad
2015-12-01
Markov Chain Monte Carlo (MCMC) methods are often used to probe the posterior probability distribution in inverse problems. This allows for computation of estimates of uncertain system responses conditioned on given observational data by means of approximate integration. However, MCMC methods suffer from the computational complexities in the case of expensive models as in the case of subsurface flow models. Hence, it is of great interest to develop alterative efficient methods utilizing emulators, that are cheap to evaluate, in order to replace the full physics simulator. In the current work, we develop a technique based on sparse response surfaces to represent the flow response within a subsurface reservoir and thus enable efficient exploration of the posterior probability density function and the conditional expectations given the data. Polynomial Chaos Expansion (PCE) is a powerful tool to quantify uncertainty in dynamical systems when there is probabilistic uncertainty in the system parameters. In the context of subsurface flow model, it has been shown to be more accurate and efficient compared with traditional experimental design (ED). PCEs have a significant advantage over other response surfaces as the convergence to the true probability distribution when the order of the PCE is increased can be proved for the random variables with finite variances. However, the major drawback of PCE is related to the curse of dimensionality as the number of terms to be estimated grows drastically with the number of the input random variables. This renders the computational cost of classical PCE schemes unaffordable for reservoir simulation purposes when the deterministic finite element model is expensive to evaluate. To address this issue, we propose the reduced-terms polynomial chaos representation which uses an impact factor to only retain the most relevant terms of the PCE decomposition. Accordingly, the reduced-terms polynomial chaos proxy can be used as the pseudo
Lüchow, Arne; Sturm, Alexander; Schulte, Christoph; Haghighi Mood, Kaveh
2015-02-28
Jastrow correlation factors play an important role in quantum Monte Carlo calculations. Together with an orbital based antisymmetric function, they allow the construction of highly accurate correlation wave functions. In this paper, a generic expansion of the Jastrow correlation function in terms of polynomials that satisfy both the electron exchange symmetry constraint and the cusp conditions is presented. In particular, an expansion of the three-body electron-electron-nucleus contribution in terms of cuspless homogeneous symmetric polynomials is proposed. The polynomials can be expressed in fairly arbitrary scaling function allowing a generic implementation of the Jastrow factor. It is demonstrated with a few examples that the new Jastrow factor achieves 85%–90% of the total correlation energy in a variational quantum Monte Carlo calculation and more than 90% of the diffusion Monte Carlo correlation energy.
NASA Astrophysics Data System (ADS)
Hatano, Naomichi; Feinberg, Joshua
2016-12-01
We study Chebyshev-polynomial expansion of the inverse localization length of Hermitian and non-Hermitian random chains as a function of energy. For Hermitian models, the expansion produces this energy-dependent function numerically in one run of the algorithm. This is in strong contrast to the standard transfer-matrix method, which produces the inverse localization length for a fixed energy in each run. For non-Hermitian models, as in the transfer-matrix method, our algorithm computes the inverse localization length for a fixed (complex) energy. We also find a formula of the Chebyshev-polynomial expansion of the density of states of non-Hermitian models. As explained in detail, our algorithm for non-Hermitian models may be the only available efficient algorithm for finding the density of states of models with interactions.
Jakeman, John D.; Narayan, Akil; Zhou, Tao
2017-06-22
We propose an algorithm for recovering sparse orthogonal polynomial expansions via collocation. A standard sampling approach for recovering sparse polynomials uses Monte Carlo sampling, from the density of orthogonality, which results in poor function recovery when the polynomial degree is high. Our proposed approach aims to mitigate this limitation by sampling with respect to the weighted equilibrium measure of the parametric domain and subsequently solves a preconditionedmore » $$\\ell^1$$-minimization problem, where the weights of the diagonal preconditioning matrix are given by evaluations of the Christoffel function. Our algorithm can be applied to a wide class of orthogonal polynomial families on bounded and unbounded domains, including all classical families. We present theoretical analysis to motivate the algorithm and numerical results that show our method is superior to standard Monte Carlo methods in many situations of interest. In conclusion, numerical examples are also provided to demonstrate that our proposed algorithm leads to comparable or improved accuracy even when compared with Legendre- and Hermite-specific algorithms.« less
NASA Astrophysics Data System (ADS)
Ishizuka, Hiroaki; Udagawa, Masafumi; Motome, Yukitoshi
2013-12-01
We present the benchmark of the polynomial expansion Monte Carlo method to a Kondo lattice model with classical localized spins on a geometrically frustrated lattice. The method enables us to reduce the calculation amount by using the Chebyshev polynomial expansion of the density of states compared to a conventional Monte Carlo technique based on the exact diagonalization of the fermion Hamiltonian matrix. Further reduction is brought about by a real-space truncation of the vector-matrix operations. We apply the method to the model with spin-ice type Ising spins on a three-dimensional pyrochlore lattice and carefully examine the convergence in terms of the order of polynomials and the truncation distance. We find that, in a wide range of electron density at a relatively weak Kondo coupling compared to the noninteracting bandwidth, the results by the polynomial expansion method show good convergence to those by the conventional method within reasonable numbers of polynomials. This enables us to study the systems up to 4×83=2048 sites, while the previous study by the conventional method was limited to 4×43=256 sites. On the other hand, the real-space truncation is not helpful in reducing the calculation amount for the system sizes that we reached, as the sufficient convergence is obtained when most of the sites are involved within the truncation distance. The necessary truncation distance, however, appears not to show significant system size dependence, suggesting that the truncation method becomes efficient for larger system sizes.
Nonmonotonic Recursive Polynomial Expansions for Linear Scaling Calculation of the Density Matrix.
Rubensson, Emanuel H
2011-05-10
As it stands, density matrix purification is a powerful tool for linear scaling electronic structure calculations. The convergence is rapid and depends only weakly on the band gap. However, as will be shown in this letter, there is room for improvements. The key is to allow for nonmonotonicity in the recursive polynomial expansion. On the basis of this idea, new purification schemes are proposed that require only half the number of matrix-matrix multiplications compared to previous schemes. The speedup is essentially independent of the location of the chemical potential and increases with decreasing band gap.
Fast and accurate sensitivity analysis of IMPT treatment plans using Polynomial Chaos Expansion
NASA Astrophysics Data System (ADS)
Perkó, Zoltán; van der Voort, Sebastian R.; van de Water, Steven; Hartman, Charlotte M. H.; Hoogeman, Mischa; Lathouwers, Danny
2016-06-01
The highly conformal planned dose distribution achievable in intensity modulated proton therapy (IMPT) can severely be compromised by uncertainties in patient setup and proton range. While several robust optimization approaches have been presented to address this issue, appropriate methods to accurately estimate the robustness of treatment plans are still lacking. To fill this gap we present Polynomial Chaos Expansion (PCE) techniques which are easily applicable and create a meta-model of the dose engine by approximating the dose in every voxel with multidimensional polynomials. This Polynomial Chaos (PC) model can be built in an automated fashion relatively cheaply and subsequently it can be used to perform comprehensive robustness analysis. We adapted PC to provide among others the expected dose, the dose variance, accurate probability distribution of dose-volume histogram (DVH) metrics (e.g. minimum tumor or maximum organ dose), exact bandwidths of DVHs, and to separate the effects of random and systematic errors. We present the outcome of our verification experiments based on 6 head-and-neck (HN) patients, and exemplify the usefulness of PCE by comparing a robust and a non-robust treatment plan for a selected HN case. The results suggest that PCE is highly valuable for both research and clinical applications.
Fast and accurate sensitivity analysis of IMPT treatment plans using Polynomial Chaos Expansion.
Perkó, Zoltán; van der Voort, Sebastian R; van de Water, Steven; Hartman, Charlotte M H; Hoogeman, Mischa; Lathouwers, Danny
2016-06-21
The highly conformal planned dose distribution achievable in intensity modulated proton therapy (IMPT) can severely be compromised by uncertainties in patient setup and proton range. While several robust optimization approaches have been presented to address this issue, appropriate methods to accurately estimate the robustness of treatment plans are still lacking. To fill this gap we present Polynomial Chaos Expansion (PCE) techniques which are easily applicable and create a meta-model of the dose engine by approximating the dose in every voxel with multidimensional polynomials. This Polynomial Chaos (PC) model can be built in an automated fashion relatively cheaply and subsequently it can be used to perform comprehensive robustness analysis. We adapted PC to provide among others the expected dose, the dose variance, accurate probability distribution of dose-volume histogram (DVH) metrics (e.g. minimum tumor or maximum organ dose), exact bandwidths of DVHs, and to separate the effects of random and systematic errors. We present the outcome of our verification experiments based on 6 head-and-neck (HN) patients, and exemplify the usefulness of PCE by comparing a robust and a non-robust treatment plan for a selected HN case. The results suggest that PCE is highly valuable for both research and clinical applications.
Chen, Yi; Jakeman, John; Gittelson, Claude; Xiu, Dongbin
2015-01-08
In this paper we present a localized polynomial chaos expansion for partial differential equations (PDE) with random inputs. In particular, we focus on time independent linear stochastic problems with high dimensional random inputs, where the traditional polynomial chaos methods, and most of the existing methods, incur prohibitively high simulation cost. Furthermore, the local polynomial chaos method employs a domain decomposition technique to approximate the stochastic solution locally. In each subdomain, a subdomain problem is solved independently and, more importantly, in a much lower dimensional random space. In a postprocesing stage, accurate samples of the original stochastic problems are obtained from the samples of the local solutions by enforcing the correct stochastic structure of the random inputs and the coupling conditions at the interfaces of the subdomains. Overall, the method is able to solve stochastic PDEs in very large dimensions by solving a collection of low dimensional local problems and can be highly efficient. In our paper we present the general mathematical framework of the methodology and use numerical examples to demonstrate the properties of the method.
Numerical Solutions of the Nonlinear Fractional-Order Brusselator System by Bernstein Polynomials
Khan, Rahmat Ali; Tajadodi, Haleh; Johnston, Sarah Jane
2014-01-01
In this paper we propose the Bernstein polynomials to achieve the numerical solutions of nonlinear fractional-order chaotic system known by fractional-order Brusselator system. We use operational matrices of fractional integration and multiplication of Bernstein polynomials, which turns the nonlinear fractional-order Brusselator system to a system of algebraic equations. Two illustrative examples are given in order to demonstrate the accuracy and simplicity of the proposed techniques. PMID:25485293
Polynomial Solutions of Nth Order Non-Homogeneous Differential Equations
ERIC Educational Resources Information Center
Levine, Lawrence E.; Maleh, Ray
2002-01-01
It was shown by Costa and Levine that the homogeneous differential equation (1-x[superscript N])y([superscript N]) + A[subscript N-1]x[superscript N-1)y([superscript N-1]) + A[subscript N-2]x[superscript N-2])y([superscript N-2]) + ... + A[subscript 1]xy[prime] + A[subscript 0]y = 0 has a finite polynomial solution if and only if [for…
Linear precoding based on polynomial expansion: reducing complexity in massive MIMO.
Mueller, Axel; Kammoun, Abla; Björnson, Emil; Debbah, Mérouane
Massive multiple-input multiple-output (MIMO) techniques have the potential to bring tremendous improvements in spectral efficiency to future communication systems. Counterintuitively, the practical issues of having uncertain channel knowledge, high propagation losses, and implementing optimal non-linear precoding are solved more or less automatically by enlarging system dimensions. However, the computational precoding complexity grows with the system dimensions. For example, the close-to-optimal and relatively "antenna-efficient" regularized zero-forcing (RZF) precoding is very complicated to implement in practice, since it requires fast inversions of large matrices in every coherence period. Motivated by the high performance of RZF, we propose to replace the matrix inversion and multiplication by a truncated polynomial expansion (TPE), thereby obtaining the new TPE precoding scheme which is more suitable for real-time hardware implementation and significantly reduces the delay to the first transmitted symbol. The degree of the matrix polynomial can be adapted to the available hardware resources and enables smooth transition between simple maximum ratio transmission and more advanced RZF. By deriving new random matrix results, we obtain a deterministic expression for the asymptotic signal-to-interference-and-noise ratio (SINR) achieved by TPE precoding in massive MIMO systems. Furthermore, we provide a closed-form expression for the polynomial coefficients that maximizes this SINR. To maintain a fixed per-user rate loss as compared to RZF, the polynomial degree does not need to scale with the system, but it should be increased with the quality of the channel knowledge and the signal-to-noise ratio.
Effect upon universal order of Hubble expansion
NASA Astrophysics Data System (ADS)
Frieden, B. R.; Plastino, A.; Plastino, A. R.
2012-01-01
The level of order R in a spherical system of radius r0 with a probability amplitude function ψ(x),x=r,θ,ϕ obeys R=(1/2)r02I, where I=4∫dx| is its Fisher information level. We show that a flat space universe obeying the Robertson-Walker metric has an invariant value of the order as it undergoes either uniform Hubble expansion or contraction. This means that Hubble expansion per se does not cause a loss of universal order as time progresses. Instead, coarse graining processes characterizing decoherence and friction might cause a loss of order. Alternatively, looking backward in time, i.e. under Hubble contraction, as the big bang is approached and the Hubble radius r0 approaches small values, the structure in the amplitude function ψ(x) becomes ever more densely packed, increasing all local slopes ∇ψ and causing the Fisher information I to approach unboundedly large values. As a speculation, this ever-well locates the initial position of the universe in a larger, multiverse. We define a measure of order or complexity proportional to the Fisher information. The measure is applied to our flat-space, dust and gas dominated, universe. Despite the universe’s relentless, ever-accelerating Hubble expansion, its level of order is found to remain constant.
A robust and efficient stepwise regression method for building sparse polynomial chaos expansions
NASA Astrophysics Data System (ADS)
Abraham, Simon; Raisee, Mehrdad; Ghorbaniasl, Ghader; Contino, Francesco; Lacor, Chris
2017-03-01
Polynomial Chaos (PC) expansions are widely used in various engineering fields for quantifying uncertainties arising from uncertain parameters. The computational cost of classical PC solution schemes is unaffordable as the number of deterministic simulations to be calculated grows dramatically with the number of stochastic dimension. This considerably restricts the practical use of PC at the industrial level. A common approach to address such problems is to make use of sparse PC expansions. This paper presents a non-intrusive regression-based method for building sparse PC expansions. The most important PC contributions are detected sequentially through an automatic search procedure. The variable selection criterion is based on efficient tools relevant to probabilistic method. Two benchmark analytical functions are used to validate the proposed algorithm. The computational efficiency of the method is then illustrated by a more realistic CFD application, consisting of the non-deterministic flow around a transonic airfoil subject to geometrical uncertainties. To assess the performance of the developed methodology, a detailed comparison is made with the well established LAR-based selection technique. The results show that the developed sparse regression technique is able to identify the most significant PC contributions describing the problem. Moreover, the most important stochastic features are captured at a reduced computational cost compared to the LAR method. The results also demonstrate the superior robustness of the method by repeating the analyses using random experimental designs.
Numerical solution of DGLAP equations using Laguerre polynomials expansion and Monte Carlo method.
Ghasempour Nesheli, A; Mirjalili, A; Yazdanpanah, M M
2016-01-01
We investigate the numerical solutions of the DGLAP evolution equations at the LO and NLO approximations, using the Laguerre polynomials expansion. The theoretical framework is based on Furmanski et al.'s articles. What makes the content of this paper different from other works, is that all calculations in the whole stages to extract the evolved parton distributions, are done numerically. The employed techniques to do the numerical solutions, based on Monte Carlo method, has this feature that all the results are obtained in a proper wall clock time by computer. The algorithms are implemented in FORTRAN and the employed coding ideas can be used in other numerical computations as well. Our results for the evolved parton densities are in good agreement with some phenomenological models. They also indicate better behavior with respect to the results of similar numerical calculations.
Identification of nonlinear vibrating structures by polynomial expansion in the z-domain
NASA Astrophysics Data System (ADS)
Fasana, Alessandro; Garibaldi, Luigi; Marchesiello, Stefano
2017-02-01
A new method in the frequency domain for the identification of nonlinear vibrating structures is described, by adopting the perspective of nonlinearities as internal feedback forces. The technique is based on a polynomial expansion representation of the frequency response function of the underlying linear system, relying on a z-domain formulation. A least squares approach is adopted to take into account the information of all the frequency response functions but, when large data sets are used, the solution of the resulting system of algebraic linear equations can be a difficult task. A procedure to drastically reduce the matrix dimensions and consequently the computational cost - which largely depends on the number of spectral lines - is adopted, leading to a compact and well conditioned problem. The robustness and numerical performances of the method are demonstrated by its implementation on simulated data from single and two degree of freedom systems with typical nonlinear characteristics.
Special polynomials associated with the fourth order analogue to the Painlevé equations
NASA Astrophysics Data System (ADS)
Kudryashov, Nikolai A.; Demina, Maria V.
2007-04-01
Rational solutions of the fourth order analogue to the Painlevé equations are classified. Special polynomials associated with the rational solutions are introduced. The structure of the polynomials is found. Formulae for their coefficients and degrees are derived. It is shown that special solutions of the Fordy Gibbons, the Caudrey Dodd Gibbon and the Kaup Kupershmidt equations can be expressed through solutions of the equation studied.
Nonlinear first order PDEs reducible to autonomous form polynomially homogeneous in the derivatives
NASA Astrophysics Data System (ADS)
Gorgone, Matteo; Oliveri, Francesco
2017-03-01
It is proved a theorem providing necessary and sufficient conditions enabling one to map a nonlinear system of first order partial differential equations, polynomial in the derivatives, to an equivalent autonomous first order system polynomially homogeneous in the derivatives. The result is intimately related to the symmetry properties of the source system, and the proof, involving the use of the canonical variables associated to the admitted Lie point symmetries, is constructive. First order Monge-Ampère systems, either with constant coefficients or with coefficients depending on the field variables, where the theorem can be successfully applied, are considered.
NASA Astrophysics Data System (ADS)
Wang, Zhengzi
2015-08-01
The influence of ambient temperature is a big challenge to robust infrared face recognition. This paper proposes a new ambient temperature normalization algorithm to improve the performance of infrared face recognition under variable ambient temperatures. Based on statistical regression theory, a second order polynomial model is learned to describe the ambient temperature's impact on infrared face image. Then, infrared image was normalized to reference ambient temperature by the second order polynomial model. Finally, this normalization method is applied to infrared face recognition to verify its efficiency. The experiments demonstrate that the proposed temperature normalization method is feasible and can significantly improve the robustness of infrared face recognition.
High order overlay modeling and APC simulation with Zernike-Legendre polynomials
NASA Astrophysics Data System (ADS)
Ju, JawWuk; Kim, MinGyu; Lee, JuHan; Sherwin, Stuart; Hoo, George; Choi, DongSub; Lee, Dohwa; Jeon, Sanghuck; Lee, Kangsan; Tien, David; Pierson, Bill; Robinson, John C.; Levy, Ady; Smith, Mark D.
2015-03-01
Feedback control of overlay errors to the scanner is a well-established technique in semiconductor manufacturing [1]. Typically, overlay errors are measured, and then modeled by least-squares fitting to an overlay model. Overlay models are typically Cartesian polynomial functions of position within the wafer (Xw, Yw), and of position within the field (Xf, Yf). The coefficients from the data fit can then be fed back to the scanner to reduce overlay errors in future wafer exposures, usually via a historically weighted moving average. In this study, rather than using the standard Cartesian formulation, we examine overlay models using Zernike polynomials to represent the wafer-level terms, and Legendre polynomials to represent the field-level terms. Zernike and Legendre polynomials can be selected to have the same fitting capability as standard polynomials (e.g., second order in X and Y, or third order in X and Y). However, Zernike polynomials have the additional property of being orthogonal over the unit disk, which makes them appropriate for the wafer-level model, and Legendre polynomials are orthogonal over the unit square, which makes them appropriate for the field-level model. We show several benefits of Zernike/Legendre-based models in this investigation in an Advanced Process Control (APC) simulation using highly-sampled fab data. First, the orthogonality property leads to less interaction between the terms, which makes the lot-to-lot variation in the fitted coefficients smaller than when standard polynomials are used. Second, the fitting process itself is less coupled - fitting to a lower-order model, and then fitting the residuals to a higher order model gives very similar results as fitting all of the terms at once. This property makes fitting techniques such as dual pass or cascading [2] unnecessary, and greatly simplifies the options available for the model recipe. The Zernike/Legendre basis gives overlay performance (mean plus 3 sigma of the residuals
Uncertainty propagation of p-boxes using sparse polynomial chaos expansions
NASA Astrophysics Data System (ADS)
Schöbi, Roland; Sudret, Bruno
2017-06-01
In modern engineering, physical processes are modelled and analysed using advanced computer simulations, such as finite element models. Furthermore, concepts of reliability analysis and robust design are becoming popular, hence, making efficient quantification and propagation of uncertainties an important aspect. In this context, a typical workflow includes the characterization of the uncertainty in the input variables. In this paper, input variables are modelled by probability-boxes (p-boxes), accounting for both aleatory and epistemic uncertainty. The propagation of p-boxes leads to p-boxes of the output of the computational model. A two-level meta-modelling approach is proposed using non-intrusive sparse polynomial chaos expansions to surrogate the exact computational model and, hence, to facilitate the uncertainty quantification analysis. The capabilities of the proposed approach are illustrated through applications using a benchmark analytical function and two realistic engineering problem settings. They show that the proposed two-level approach allows for an accurate estimation of the statistics of the response quantity of interest using a small number of evaluations of the exact computational model. This is crucial in cases where the computational costs are dominated by the runs of high-fidelity computational models.
Schmidt-Kalman Filter with Polynomial Chaos Expansion for Orbit Determination of Space Objects
NASA Astrophysics Data System (ADS)
Yang, Y.; Cai, H.; Zhang, K.
2016-09-01
Parameter errors in orbital models can result in poor orbit determination (OD) using a traditional Kalman filter. One approach to account for these errors is to consider them in the so-called Schmidt-Kalman filter (SKF), by augmenting the state covariance matrix (CM) with additional parameter covariance rather than additively estimating these so-called "consider" parameters. This paper introduces a new SKF algorithm with polynomial chaos expansion (PCE-SKF). The PCE approach has been proved to be more efficient than Monte Carlo method for propagating the input uncertainties onto the system response without experiencing any constraints of linear dynamics, or Gaussian distributions of the uncertainty sources. The state and covariance needed in the orbit prediction step are propagated using PCE. An inclined geosynchronous orbit scenario is set up to test the proposed PCE-SKF based OD algorithm. The satellite orbit is propagated based on numerical integration, with the uncertain coefficient of solar radiation pressure considered. The PCE-SKF solutions are compared with extended Kalman filter (EKF), SKF and PCE-EKF (EKF with PCE) solutions. It is implied that the covariance propagation using PCE leads to more precise OD solutions in comparison with those based on linear propagation of covariance.
NASA Astrophysics Data System (ADS)
Kersaudy, Pierric; Sudret, Bruno; Varsier, Nadège; Picon, Odile; Wiart, Joe
2015-04-01
In numerical dosimetry, the recent advances in high performance computing led to a strong reduction of the required computational time to assess the specific absorption rate (SAR) characterizing the human exposure to electromagnetic waves. However, this procedure remains time-consuming and a single simulation can request several hours. As a consequence, the influence of uncertain input parameters on the SAR cannot be analyzed using crude Monte Carlo simulation. The solution presented here to perform such an analysis is surrogate modeling. This paper proposes a novel approach to build such a surrogate model from a design of experiments. Considering a sparse representation of the polynomial chaos expansions using least-angle regression as a selection algorithm to retain the most influential polynomials, this paper proposes to use the selected polynomials as regression functions for the universal Kriging model. The leave-one-out cross validation is used to select the optimal number of polynomials in the deterministic part of the Kriging model. The proposed approach, called LARS-Kriging-PC modeling, is applied to three benchmark examples and then to a full-scale metamodeling problem involving the exposure of a numerical fetus model to a femtocell device. The performances of the LARS-Kriging-PC are compared to an ordinary Kriging model and to a classical sparse polynomial chaos expansion. The LARS-Kriging-PC appears to have better performances than the two other approaches. A significant accuracy improvement is observed compared to the ordinary Kriging or to the sparse polynomial chaos depending on the studied case. This approach seems to be an optimal solution between the two other classical approaches. A global sensitivity analysis is finally performed on the LARS-Kriging-PC model of the fetus exposure problem.
Kersaudy, Pierric; Sudret, Bruno; Varsier, Nadège; Picon, Odile; Wiart, Joe
2015-04-01
In numerical dosimetry, the recent advances in high performance computing led to a strong reduction of the required computational time to assess the specific absorption rate (SAR) characterizing the human exposure to electromagnetic waves. However, this procedure remains time-consuming and a single simulation can request several hours. As a consequence, the influence of uncertain input parameters on the SAR cannot be analyzed using crude Monte Carlo simulation. The solution presented here to perform such an analysis is surrogate modeling. This paper proposes a novel approach to build such a surrogate model from a design of experiments. Considering a sparse representation of the polynomial chaos expansions using least-angle regression as a selection algorithm to retain the most influential polynomials, this paper proposes to use the selected polynomials as regression functions for the universal Kriging model. The leave-one-out cross validation is used to select the optimal number of polynomials in the deterministic part of the Kriging model. The proposed approach, called LARS-Kriging-PC modeling, is applied to three benchmark examples and then to a full-scale metamodeling problem involving the exposure of a numerical fetus model to a femtocell device. The performances of the LARS-Kriging-PC are compared to an ordinary Kriging model and to a classical sparse polynomial chaos expansion. The LARS-Kriging-PC appears to have better performances than the two other approaches. A significant accuracy improvement is observed compared to the ordinary Kriging or to the sparse polynomial chaos depending on the studied case. This approach seems to be an optimal solution between the two other classical approaches. A global sensitivity analysis is finally performed on the LARS-Kriging-PC model of the fetus exposure problem.
NASA Astrophysics Data System (ADS)
Dada, M.; Awojoyogbe, O. B.; Boubaker, K.
2011-04-01
In this study, we have used dimensional analysis to solve the heat equation inside an experimental laser welding setup. The solution for the heat equation is based on the assumption that heat energy diffuses equally on both sides of the laser beam axis and that the temperature along the axis through which the laser beam moves is constant. The amount of heat energy delivered by the laser to the keyhole is analyzed using the Boubaker polynomial expansion scheme BPES.
The accurate solution of Poisson's equation by expansion in Chebyshev polynomials
NASA Technical Reports Server (NTRS)
Haidvogel, D. B.; Zang, T.
1979-01-01
A Chebyshev expansion technique is applied to Poisson's equation on a square with homogeneous Dirichlet boundary conditions. The spectral equations are solved in two ways - by alternating direction and by matrix diagonalization methods. Solutions are sought to both oscillatory and mildly singular problems. The accuracy and efficiency of the Chebyshev approach compare favorably with those of standard second- and fourth-order finite-difference methods.
NASA Astrophysics Data System (ADS)
Qian, Ying-Jing; Yang, Xiao-Dong; Zhai, Guan-Qiao; Zhang, Wei
2017-08-01
Innovated by the nonlinear modes concept in the vibrational dynamics, the vertical periodic orbits around the triangular libration points are revisited for the Circular Restricted Three-body Problem. The ζ -component motion is treated as the dominant motion and the ξ and η -component motions are treated as the slave motions. The slave motions are in nature related to the dominant motion through the approximate nonlinear polynomial expansions with respect to the ζ -position and ζ -velocity during the one of the periodic orbital motions. By employing the relations among the three directions, the three-dimensional system can be transferred into one-dimensional problem. Then the approximate three-dimensional vertical periodic solution can be analytically obtained by solving the dominant motion only on ζ -direction. To demonstrate the effectiveness of the proposed method, an accuracy study was carried out to validate the polynomial expansion (PE) method. As one of the applications, the invariant nonlinear relations in polynomial expansion form are used as constraints to obtain numerical solutions by differential correction. The nonlinear relations among the directions provide an alternative point of view to explore the overall dynamics of periodic orbits around libration points with general rules.
Higher-order numerical methods derived from three-point polynomial interpolation
NASA Technical Reports Server (NTRS)
Rubin, S. G.; Khosla, P. K.
1976-01-01
Higher-order collocation procedures resulting in tridiagonal matrix systems are derived from polynomial spline interpolation and Hermitian finite-difference discretization. The equations generally apply for both uniform and variable meshes. Hybrid schemes resulting from different polynomial approximations for first and second derivatives lead to the nonuniform mesh extension of the so-called compact or Pade difference techniques. A variety of fourth-order methods are described and this concept is extended to sixth-order. Solutions with these procedures are presented for the similar and non-similar boundary layer equations with and without mass transfer, the Burgers equation, and the incompressible viscous flow in a driven cavity. Finally, the interpolation procedure is used to derive higher-order temporal integration schemes and results are shown for the diffusion equation.
A comparison of high-order polynomial and wave-based methods for Helmholtz problems
NASA Astrophysics Data System (ADS)
Lieu, Alice; Gabard, Gwénaël; Bériot, Hadrien
2016-09-01
The application of computational modelling to wave propagation problems is hindered by the dispersion error introduced by the discretisation. Two common strategies to address this issue are to use high-order polynomial shape functions (e.g. hp-FEM), or to use physics-based, or Trefftz, methods where the shape functions are local solutions of the problem (typically plane waves). Both strategies have been actively developed over the past decades and both have demonstrated their benefits compared to conventional finite-element methods, but they have yet to be compared. In this paper a high-order polynomial method (p-FEM with Lobatto polynomials) and the wave-based discontinuous Galerkin method are compared for two-dimensional Helmholtz problems. A number of different benchmark problems are used to perform a detailed and systematic assessment of the relative merits of these two methods in terms of interpolation properties, performance and conditioning. It is generally assumed that a wave-based method naturally provides better accuracy compared to polynomial methods since the plane waves or Bessel functions used in these methods are exact solutions of the Helmholtz equation. Results indicate that this expectation does not necessarily translate into a clear benefit, and that the differences in performance, accuracy and conditioning are more nuanced than generally assumed. The high-order polynomial method can in fact deliver comparable, and in some cases superior, performance compared to the wave-based DGM. In addition to benchmarking the intrinsic computational performance of these methods, a number of practical issues associated with realistic applications are also discussed.
NASA Astrophysics Data System (ADS)
Ben Mahmoud, B. Karem
2009-05-01
In this paper, a temperature dynamical profiling inside vacuum-insulated Hydrogen cryogenic vessels is yielded. The theoretical investigations are based on the similarity between the convective heat equation and the characteristic differential equation of the Boubaker polynomials.
New separated polynomial solutions to the Zernike system on the unit disk and interbasis expansion
NASA Astrophysics Data System (ADS)
Pogosyan, George S.; Wolf, Kurt Bernardo; Yakhno, Alexander
2017-10-01
The differential equation proposed by Frits Zernike to obtain a basis of polynomial orthogonal solutions on the the unit disk to classify wavefront aberrations in circular pupils, is shown to have a set of new orthonormal solution bases, involving Legendre and Gegenbauer polynomials, in non-orthogonal coordinates close to Cartesian ones. We find the overlaps between the original Zernike basis and a representative of the new set, which turn out to be Clebsch-Gordan coefficients.
A survey on orthogonal matrix polynomials satisfying second order differential equations
NASA Astrophysics Data System (ADS)
Duran, Antonio J.; Grunbaum, F. Alberto
2005-06-01
The subject of orthogonal polynomials cuts across a large piece of mathematics and its applications. Two notable examples are mathematical physics in the 19th and 20th centuries, as well as the theory of spherical functions for symmetric spaces. It is also clear that many areas of mathematics grew out of the consideration of problems like the moment problem that are intimately associated to the study of (scalar valued) orthogonal polynomials.Matrix orthogonality on the real line has been sporadically studied during the last half century since Krein devoted some papers to the subject in 1949, see (AMS Translations, Series 2, vol. 97, Providence, Rhode Island, 1971, pp. 75-143, Dokl. Akad. Nauk SSSR 69(2) (1949) 125). In the last decade this study has been made more systematic with the consequence that many basic results of scalar orthogonality have been extended to the matrix case. The most recent of these results is the discovery of important examples of orthogonal matrix polynomials: many families of orthogonal matrix polynomials have been found that (as the classical families of Hermite, Laguerre and Jacobi in the scalar case) satisfy second order differential equations with coefficients independent of n. The aim of this paper is to give an overview of the techniques that have led to these examples, a small sample of the examples themselves and a small step in the challenging direction of finding applications of these new examples.
NASA Astrophysics Data System (ADS)
Martin, Pablo; Zamudio-Cristi, Jorge
1982-12-01
A method is described to obtain fractional approximations for linear first-order differential equations with polynomial coefficients. This approximation can give good accuracy in a large region of the complex variable plane that may include all of the real axis. The parameters of the approximation are solutions of algebraic equations obtained through the coefficients of the higher and lower powers of the variable after the substitution of the fractional approximation in the differential equation. The method is more general than the asymptotical Padé method, and it is not required to determine the power series or asymptotical expansion. A simple approximation for the exponential integral is found, which gives three exact digits for most of the real values of the variable. Approximations of higher accuracy than those of other authors are also obtained.
A Method for Measuring Distortion in Wide-Field Imaging with High Order Polynomials
NASA Astrophysics Data System (ADS)
Yasuda, N.; Okura, Y.; Takata, T.; Furusawa, H.
2011-07-01
In analyzing wide-field images, for example those from the Hyper Suprime-Cam on the Subaru telescope, determining World Coordinate System (WCS) information (for example position and angle) of each CCD is very important. In this paper, we show a method for determining the distortion with high-order polynomials using the TAN-SIP convention, utilizing all of the CCDs in a field of view (FOV) at once.
Two high-order polynomial bendable mirrors in the TLS infrared beamline
NASA Astrophysics Data System (ADS)
Chen, Shean-Jen; Kuan, Chien-Kuang; Perng, Shen-Yaw; Wang, Duan-Jen; Ho, H. C.; Tseng, T. C.; Chen, Chien-Te; Lo, Yi-Chung
1999-10-01
This study presents a Kirkpatrick-Baez mirror system which includes two high-order polynomial bendable mirrors in a Taiwan Light Source (TLS) infrared beamline to create and adjust more accurately collimating images on a focusing point. The source of infrared rays is synchrotron radiation from a TLS bending magnet, so the surface of a vertical focusing mirror (VFM) is designed on an elliptical shape. The Runge-Kutta numerical method is used to compute the optimal high-order polynomial shape of the horizontal focusing mirror (HFM), to focus the horizontal arc source on the point image. The HFM and VFM using 17-4 PH stainless steel substrate without an electroless nickel plate are mechanically bent from planar to the desired fifth-order polynomial shapes with central radii of 3.74 m and 5.43 m by the application of equal couples, respectively. The mirror fabrication process, mechanical design, and the method of adjusting the mirror shape using the Long Trace Profiler measurement system are described. Finally, the roughness of mirrors is 3 angstrom RMS. After the mirrors have been bent, the slope error over 2/3 of clear aperture length (170 mm) was reduced to less than 6.3 (mu) rad RMS.
Zaunders, John; Jing, Junmei; Leipold, Michael; Maecker, Holden; Kelleher, Anthony D; Koch, Inge
2016-01-01
Many methods have been described for automated clustering analysis of complex flow cytometry data, but so far the goal to efficiently estimate multivariate densities and their modes for a moderate number of dimensions and potentially millions of data points has not been attained. We have devised a novel approach to describing modes using second order polynomial histogram estimators (SOPHE). The method divides the data into multivariate bins and determines the shape of the data in each bin based on second order polynomials, which is an efficient computation. These calculations yield local maxima and allow joining of adjacent bins to identify clusters. The use of second order polynomials also optimally uses wide bins, such that in most cases each parameter (dimension) need only be divided into 4-8 bins, again reducing computational load. We have validated this method using defined mixtures of up to 17 fluorescent beads in 16 dimensions, correctly identifying all populations in data files of 100,000 beads in <10 s, on a standard laptop. The method also correctly clustered granulocytes, lymphocytes, including standard T, B, and NK cell subsets, and monocytes in 9-color stained peripheral blood, within seconds. SOPHE successfully clustered up to 36 subsets of memory CD4 T cells using differentiation and trafficking markers, in 14-color flow analysis, and up to 65 subpopulations of PBMC in 33-dimensional CyTOF data, showing its usefulness in discovery research. SOPHE has the potential to greatly increase efficiency of analysing complex mixtures of cells in higher dimensions.
Milgram, A
2011-02-21
This comment addresses critics on the claimed stability of solution to the accelerated-predator-satiety Lotka-Volterra predator-prey problem, proposed by Dubey al. (2010. A solution to the accelerated-predator-satiety Lotka-Volterra predator-prey problem using Boubaker polynomial expansion scheme. Journal of Theoretical Biology 264, 154-160). Critics are based on incompatibilities between the claimed asymptotic behavior and the presumed Malthusian growth of prey population in absence of predator. Copyright Â© 2010 Elsevier Ltd. All rights reserved.
Matrix form of Legendre polynomials for solving linear integro-differential equations of high order
NASA Astrophysics Data System (ADS)
Kammuji, M.; Eshkuvatov, Z. K.; Yunus, Arif A. M.
2017-04-01
This paper presents an effective approximate solution of high order of Fredholm-Volterra integro-differential equations (FVIDEs) with boundary condition. Legendre truncated series is used as a basis functions to estimate the unknown function. Matrix operation of Legendre polynomials is used to transform FVIDEs with boundary conditions into matrix equation of Fredholm-Volterra type. Gauss Legendre quadrature formula and collocation method are applied to transfer the matrix equation into system of linear algebraic equations. The latter equation is solved by Gauss elimination method. The accuracy and validity of this method are discussed by solving two numerical examples and comparisons with wavelet and methods.
Higher-order Multivariable Polynomial Regression to Estimate Human Affective States.
Wei, Jie; Chen, Tong; Liu, Guangyuan; Yang, Jiemin
2016-03-21
From direct observations, facial, vocal, gestural, physiological, and central nervous signals, estimating human affective states through computational models such as multivariate linear-regression analysis, support vector regression, and artificial neural network, have been proposed in the past decade. In these models, linear models are generally lack of precision because of ignoring intrinsic nonlinearities of complex psychophysiological processes; and nonlinear models commonly adopt complicated algorithms. To improve accuracy and simplify model, we introduce a new computational modeling method named as higher-order multivariable polynomial regression to estimate human affective states. The study employs standardized pictures in the International Affective Picture System to induce thirty subjects' affective states, and obtains pure affective patterns of skin conductance as input variables to the higher-order multivariable polynomial model for predicting affective valence and arousal. Experimental results show that our method is able to obtain efficient correlation coefficients of 0.98 and 0.96 for estimation of affective valence and arousal, respectively. Moreover, the method may provide certain indirect evidences that valence and arousal have their brain's motivational circuit origins. Thus, the proposed method can serve as a novel one for efficiently estimating human affective states.
Higher-order Multivariable Polynomial Regression to Estimate Human Affective States
NASA Astrophysics Data System (ADS)
Wei, Jie; Chen, Tong; Liu, Guangyuan; Yang, Jiemin
2016-03-01
From direct observations, facial, vocal, gestural, physiological, and central nervous signals, estimating human affective states through computational models such as multivariate linear-regression analysis, support vector regression, and artificial neural network, have been proposed in the past decade. In these models, linear models are generally lack of precision because of ignoring intrinsic nonlinearities of complex psychophysiological processes; and nonlinear models commonly adopt complicated algorithms. To improve accuracy and simplify model, we introduce a new computational modeling method named as higher-order multivariable polynomial regression to estimate human affective states. The study employs standardized pictures in the International Affective Picture System to induce thirty subjects’ affective states, and obtains pure affective patterns of skin conductance as input variables to the higher-order multivariable polynomial model for predicting affective valence and arousal. Experimental results show that our method is able to obtain efficient correlation coefficients of 0.98 and 0.96 for estimation of affective valence and arousal, respectively. Moreover, the method may provide certain indirect evidences that valence and arousal have their brain’s motivational circuit origins. Thus, the proposed method can serve as a novel one for efficiently estimating human affective states.
Higher-order Multivariable Polynomial Regression to Estimate Human Affective States
Wei, Jie; Chen, Tong; Liu, Guangyuan; Yang, Jiemin
2016-01-01
From direct observations, facial, vocal, gestural, physiological, and central nervous signals, estimating human affective states through computational models such as multivariate linear-regression analysis, support vector regression, and artificial neural network, have been proposed in the past decade. In these models, linear models are generally lack of precision because of ignoring intrinsic nonlinearities of complex psychophysiological processes; and nonlinear models commonly adopt complicated algorithms. To improve accuracy and simplify model, we introduce a new computational modeling method named as higher-order multivariable polynomial regression to estimate human affective states. The study employs standardized pictures in the International Affective Picture System to induce thirty subjects’ affective states, and obtains pure affective patterns of skin conductance as input variables to the higher-order multivariable polynomial model for predicting affective valence and arousal. Experimental results show that our method is able to obtain efficient correlation coefficients of 0.98 and 0.96 for estimation of affective valence and arousal, respectively. Moreover, the method may provide certain indirect evidences that valence and arousal have their brain’s motivational circuit origins. Thus, the proposed method can serve as a novel one for efficiently estimating human affective states. PMID:26996254
Abd-Elhameed, W M
2014-01-01
This paper is concerned with deriving some new formulae expressing explicitly the high-order derivatives of Jacobi polynomials whose parameters difference is one or two of any degree and of any order in terms of their corresponding Jacobi polynomials. The derivatives formulae for Chebyshev polynomials of third and fourth kinds of any degree and of any order in terms of their corresponding Chebyshev polynomials are deduced as special cases. Some new reduction formulae for summing some terminating hypergeometric functions of unit argument are also deduced. As an application, and with the aid of the new introduced derivatives formulae, an algorithm for solving special sixth-order boundary value problems are implemented with the aid of applying Galerkin method. A numerical example is presented hoping to ascertain the validity and the applicability of the proposed algorithms.
Abd-Elhameed, W. M.
2014-01-01
This paper is concerned with deriving some new formulae expressing explicitly the high-order derivatives of Jacobi polynomials whose parameters difference is one or two of any degree and of any order in terms of their corresponding Jacobi polynomials. The derivatives formulae for Chebyshev polynomials of third and fourth kinds of any degree and of any order in terms of their corresponding Chebyshev polynomials are deduced as special cases. Some new reduction formulae for summing some terminating hypergeometric functions of unit argument are also deduced. As an application, and with the aid of the new introduced derivatives formulae, an algorithm for solving special sixth-order boundary value problems are implemented with the aid of applying Galerkin method. A numerical example is presented hoping to ascertain the validity and the applicability of the proposed algorithms. PMID:25386599
Alvarez, Gonzalo; Sen, Cengiz; Furukawa, N.; Motome, Y.; Dagotto, Elbio R
2005-01-01
A software library is presented for the polynomial expansion method (PEM) of the density of states (DOS) introduced in. The library provides all necessary functions for the use of the PEM and its truncated version (TPEM) in a model independent way. The PEM/TPEM replaces the exact diagonalization of the one electron sector in models for fermions coupled to classical fields. The computational cost of the algorithm is O(N)-with N the number of lattice sites-for the TPEM which should be contrasted with the computational cost of the diagonalization technique that scales as O(N4). The method is applied for the first time to a double exchange model with finite Hund coupling and also to diluted spin-fermion models.
Alvarez, Gonzalo; Schulthess, Thomas C
2006-01-01
The calculation of two- and four-particle observables is addressed within the framework of the truncated polynomial expansion method (TPEM). The TPEM replaces the exact diagonalization of the one-electron sector in models for fermions coupled to classical fields such as those used in manganites and diluted magnetic semiconductors. The computational cost of the algorithm is O(N) - with N the number of lattice sites - for the TPEM, which should be contrasted with the computational cost of the diagonalization technique that scales as O(N{sup 4}). By means of the TPEM, the density of states, spectral function, and optical conductivity of a double-exchange model for manganites are calculated on large lattices and compared to previous results and experimental measurements. The ferromagnetic metal becomes an insulator by increasing the direct exchange coupling that competes with the double-exchange mechanism. This metal-insulator transition is investigated in three dimensions.
A weighted ℓ{sub 1}-minimization approach for sparse polynomial chaos expansions
Peng, Ji; Hampton, Jerrad; Doostan, Alireza
2014-06-15
This work proposes a method for sparse polynomial chaos (PC) approximation of high-dimensional stochastic functions based on non-adapted random sampling. We modify the standard ℓ{sub 1}-minimization algorithm, originally proposed in the context of compressive sampling, using a priori information about the decay of the PC coefficients, when available, and refer to the resulting algorithm as weightedℓ{sub 1}-minimization. We provide conditions under which we may guarantee recovery using this weighted scheme. Numerical tests are used to compare the weighted and non-weighted methods for the recovery of solutions to two differential equations with high-dimensional random inputs: a boundary value problem with a random elliptic operator and a 2-D thermally driven cavity flow with random boundary condition.
Third-order polynomial model for analyzing stickup state laminated structure in flexible electronics
NASA Astrophysics Data System (ADS)
Meng, Xianhong; Wang, Zihao; Liu, Boya; Wang, Shuodao
2017-05-01
Laminated hard-soft integrated structures play a significant role in the fabrication and development of flexible electronics devices. Flexible electronics have advantageous characteristics such as soft and light-weight, can be folded, twisted, flipped inside-out, or be pasted onto other surfaces of arbitrary shapes. In this paper, an analytical model is presented to study the mechanics of laminated hard-soft structures in flexible electronics under a stickup state. Third-order polynomials are used to describe the displacement field, and the principle of virtual work is adopted to derive the governing equations and boundary conditions. The normal strain and the shear stress along the thickness direction in the bi-material region are obtained analytically, which agree well with the results from finite element analysis. The analytical model can be used to analyze stickup state laminated structures, and can serve as a valuable reference for the failure prediction and optimal design of flexible electronics in the future.
NASA Astrophysics Data System (ADS)
Zhang, D. H.; Li, F. W.
2011-05-01
An analytic solution for the problem of Williams-Brinkmann axisymmetric steady flow in the vicinity of a stagnation point at a blunt body is proposed. The boundary conditions are embedded in the main system of equations by means of the Boubaker polynomials expansion scheme (BPES). These differential equations are solved analytically and yield continuous and differentiable solutions compared to some published ones.
Fourth-order series expansion of the exchange hole
NASA Astrophysics Data System (ADS)
Wang, Rodrigo; Zhou, Yongxi; Ernzerhof, Matthias
2017-08-01
Approximate functionals for the exchange-correlation energy of electrons often draw on explicit or implicit models for the exchange-correlation hole. Here we focus on the spherically averaged exchange hole ρX(r ,u ) , which depends on the reference point r and on the electron-electron distance u . We extend the well-known [A. D. Becke and M. R. Roussel, Phys. Rev. A 39, 3761 (1989), 10.1103/PhysRevA.39.3761] second-order Taylor-series expansion in u to fourth order and we show that the fourth-order term can add important additional information that is particularly relevant for molecules compared to atoms. Drawing on these findings, we explore exchange functionals that depend on the fourth-order term of the expansion of ρX(r ,u ) . We also find that Gaussian basis set expansions, frequently used in electronic structure codes, result in unsatisfactory representations of the fourth-order term.
NASA Astrophysics Data System (ADS)
Meng, Jin; Li, Heng
2017-07-01
An efficient method for uncertainty quantification for flow in porous media is studied in this paper, where response surface of sparse polynomial chaos expansion (PCE) is constructed with the aid of feature selection method. The number of basis functions in PCE grows exponentially as the random dimensionality increases, which makes the computational cost unaffordable in high-dimensional problems. In this study, a feature selection method is introduced to select major stochastic features for the PCE by running a limited number of simulations, and the resultant PCE is termed as sparse PCE. Specifically, the least absolute shrinkage and selection operator modified least angle regression algorithm (LASSO-LAR) is applied for feature selection and the selected features are assessed by cross-validation (CV). Besides, inherited samples are utilized to make the algorithm self-adaptive. In this study, we test the performance of sparse PCE for uncertainty quantification for flow in heterogeneous media with different spatial variability. The statistical moments and probability density function of the output random field are accurately estimated through the sparse PCE, meanwhile the computational efforts are greatly reduced compared to the Monte Carlo method.
NASA Astrophysics Data System (ADS)
Slim, J.; Rathmann, F.; Nass, A.; Soltner, H.; Gebel, R.; Pretz, J.; Heberling, D.
2017-07-01
For the measurement of the electric dipole moment of protons and deuterons, a novel waveguide RF Wien filter has been designed and will soon be integrated at the COoler SYnchrotron at Jülich. The device operates at the harmonic frequencies of the spin motion. It is based on a waveguide structure that is capable of fulfilling the Wien filter condition (E → ⊥ B →) by design. The full-wave calculations demonstrated that the waveguide RF Wien filter is able to generate high-quality RF electric and magnetic fields. In reality, mechanical tolerances and misalignments decrease the simulated field quality, and it is therefore important to consider them in the simulations. In particular, for the electric dipole moment measurement, it is important to quantify the field errors systematically. Since Monte-Carlo simulations are computationally very expensive, we discuss here an efficient surrogate modeling scheme based on the Polynomial Chaos Expansion method to compute the field quality in the presence of tolerances and misalignments and subsequently to perform the sensitivity analysis at zero additional computational cost.
NASA Astrophysics Data System (ADS)
Mahata, Avik; Mukhopadhyay, Tanmoy; Adhikari, Sondipon
2016-03-01
Nano-twinned structures are mechanically stronger, ductile and stable than its non-twinned form. We have investigated the effect of varying twin spacing and twin boundary width (TBW) on the yield strength of the nano-twinned copper in a probabilistic framework. An efficient surrogate modelling approach based on polynomial chaos expansion has been proposed for the analysis. Effectively utilising 15 sets of expensive molecular dynamics simulations, thousands of outputs have been obtained corresponding to different sets of twin spacing and twin width using virtual experiments based on the surrogates. One of the major outcomes of this work is that there exists an optimal combination of twin boundary spacing and twin width until which the strength can be increased and after that critical point the nanowires weaken. This study also reveals that the yield strength of nano-twinned copper is more sensitive to TBW than twin spacing. Such robust inferences have been possible to be drawn only because of applying the surrogate modelling approach, which makes it feasible to obtain results corresponding to 40 000 combinations of different twin boundary spacing and twin width in a computationally efficient framework.
Polynomial expansion of the planetary secular terms Relativistic and lunar perturbations
NASA Technical Reports Server (NTRS)
Laskar, Jacques
1986-01-01
Relativistic and lunar perturbations must be included in a realistic theory of the secular evolution of planetary elements. The proposed general theory includes the first order of these perturbations. Comparison with more elaborated studies shows that it is sufficient with respect to the accuracy of the present theory.
Second Order Boltzmann-Gibbs Principle for Polynomial Functions and Applications
NASA Astrophysics Data System (ADS)
Gonçalves, Patrícia; Jara, Milton; Simon, Marielle
2017-01-01
In this paper we give a new proof of the second order Boltzmann-Gibbs principle introduced in Gonçalves and Jara (Arch Ration Mech Anal 212(2):597-644, 2014). The proof does not impose the knowledge on the spectral gap inequality for the underlying model and it relies on a proper decomposition of the antisymmetric part of the current of the system in terms of polynomial functions. In addition, we fully derive the convergence of the equilibrium fluctuations towards (1) a trivial process in case of super-diffusive systems, (2) an Ornstein-Uhlenbeck process or the unique energy solution of the stochastic Burgers equation, as defined in Gubinelli and Jara (SPDEs Anal Comput (1):325-350, 2013) and Gubinelli and Perkowski (Arxiv:1508.07764, 2015), in case of weakly asymmetric diffusive systems. Examples and applications are presented for weakly and partial asymmetric exclusion processes, weakly asymmetric speed change exclusion processes and hamiltonian systems with exponential interactions.
NASA Astrophysics Data System (ADS)
Sharma, Dinkar; Singh, Prince; Chauhan, Shubha
2017-06-01
In this paper, a combined form of the Laplace transform method with the homotopy perturbation method is applied to solve nonlinear fifth order Korteweg de Vries (KdV) equations. The method is known as homotopy perturbation transform method (HPTM). The nonlinear terms can be easily handled by the use of He's polynomials. Two test examples are considered to illustrate the present scheme. Further the results are compared with Homotopy perturbation method (HPM).
Time-Ordered Product Expansions for Computational Stochastic Systems Biology
Mjolsness, Eric
2013-01-01
The time-ordered product framework of quantum field theory can also be used to understand salient phenomena in stochastic biochemical networks. It is used here to derive Gillespie’s Stochastic Simulation Algorithm (SSA) for chemical reaction networks; consequently, the SSA can be interpreted in terms of Feynman diagrams. It is also used here to derive other, more general simulation and parameter-learning algorithms including simulation algorithms for networks of stochastic reaction-like processes operating on parameterized objects, and also hybrid stochastic reaction/differential equation models in which systems of ordinary differ-ential equations evolve the parameters of objects that can also undergo stochastic reactions. Thus, the time-ordered product expansion (TOPE) can be used systematically to derive simulation and parameter-fitting algorithms for stochastic systems. PMID:23735739
Time-ordered product expansions for computational stochastic system biology.
Mjolsness, Eric
2013-06-01
The time-ordered product framework of quantum field theory can also be used to understand salient phenomena in stochastic biochemical networks. It is used here to derive Gillespie's stochastic simulation algorithm (SSA) for chemical reaction networks; consequently, the SSA can be interpreted in terms of Feynman diagrams. It is also used here to derive other, more general simulation and parameter-learning algorithms including simulation algorithms for networks of stochastic reaction-like processes operating on parameterized objects, and also hybrid stochastic reaction/differential equation models in which systems of ordinary differential equations evolve the parameters of objects that can also undergo stochastic reactions. Thus, the time-ordered product expansion can be used systematically to derive simulation and parameter-fitting algorithms for stochastic systems.
Time-ordered product expansions for computational stochastic system biology
NASA Astrophysics Data System (ADS)
Mjolsness, Eric
2013-06-01
The time-ordered product framework of quantum field theory can also be used to understand salient phenomena in stochastic biochemical networks. It is used here to derive Gillespie’s stochastic simulation algorithm (SSA) for chemical reaction networks; consequently, the SSA can be interpreted in terms of Feynman diagrams. It is also used here to derive other, more general simulation and parameter-learning algorithms including simulation algorithms for networks of stochastic reaction-like processes operating on parameterized objects, and also hybrid stochastic reaction/differential equation models in which systems of ordinary differential equations evolve the parameters of objects that can also undergo stochastic reactions. Thus, the time-ordered product expansion can be used systematically to derive simulation and parameter-fitting algorithms for stochastic systems.
Quicken, Sjeng; Donders, Wouter P; van Disseldorp, Emiel M J; Gashi, Kujtim; Mees, Barend M E; van de Vosse, Frans N; Lopata, Richard G P; Delhaas, Tammo; Huberts, Wouter
2016-12-01
When applying models to patient-specific situations, the impact of model input uncertainty on the model output uncertainty has to be assessed. Proper uncertainty quantification (UQ) and sensitivity analysis (SA) techniques are indispensable for this purpose. An efficient approach for UQ and SA is the generalized polynomial chaos expansion (gPCE) method, where model response is expanded into a finite series of polynomials that depend on the model input (i.e., a meta-model). However, because of the intrinsic high computational cost of three-dimensional (3D) cardiovascular models, performing the number of model evaluations required for the gPCE is often computationally prohibitively expensive. Recently, Blatman and Sudret (2010, "An Adaptive Algorithm to Build Up Sparse Polynomial Chaos Expansions for Stochastic Finite Element Analysis," Probab. Eng. Mech., 25(2), pp. 183-197) introduced the adaptive sparse gPCE (agPCE) in the field of structural engineering. This approach reduces the computational cost with respect to the gPCE, by only including polynomials that significantly increase the meta-model's quality. In this study, we demonstrate the agPCE by applying it to a 3D abdominal aortic aneurysm (AAA) wall mechanics model and a 3D model of flow through an arteriovenous fistula (AVF). The agPCE method was indeed able to perform UQ and SA at a significantly lower computational cost than the gPCE, while still retaining accurate results. Cost reductions ranged between 70-80% and 50-90% for the AAA and AVF model, respectively.
NASA Astrophysics Data System (ADS)
Zheng, L. C.; Zhang, X. X.; Boubaker, K.; Yücel, U.; Gargouri-Ellouze, E.; Yıldırım, A.
2011-08-01
In this paper, a new model is proposed for the heat transfer characteristics of power law non- Newtonian fluids. The effects of power law viscosity on temperature field were taken into account by assuming that the temperature field is similar to the velocity field with modified Fourier's law of heat conduction for power law fluid media. The solutions obtained by using Boubaker Polynomials Expansion Scheme (BPES) technique are compared with those of the recent related similarity method in the literature with good agreement to verify the protocol exactness.
NASA Astrophysics Data System (ADS)
Novaes, Marcel
2017-02-01
Using the diagrammatic approach to integrals over Gaussian random matrices, we find a representation for polynomial Lie group integrals as infinite sums over certain maps on surfaces. The maps involved satisfy a specific condition: they have some marked vertices, and no closed walks that avoid these vertices. We also formulate our results in terms of permutations, arriving at new kinds of factorization problems.
NASA Astrophysics Data System (ADS)
Isah, Abdulnasir; Chang, Phang
2016-06-01
In this article we propose the wavelet operational method based on shifted Legendre polynomial to obtain the numerical solutions of non-linear systems of fractional order differential equations (NSFDEs). The operational matrix of fractional derivative derived through wavelet-polynomial transformation are used together with the collocation method to turn the NSFDEs to a system of non-linear algebraic equations. Illustrative examples are given in order to demonstrate the accuracy and simplicity of the proposed techniques.
Coherent orthogonal polynomials
Celeghini, E.; Olmo, M.A. del
2013-08-15
We discuss a fundamental characteristic of orthogonal polynomials, like the existence of a Lie algebra behind them, which can be added to their other relevant aspects. At the basis of the complete framework for orthogonal polynomials we include thus–in addition to differential equations, recurrence relations, Hilbert spaces and square integrable functions–Lie algebra theory. We start here from the square integrable functions on the open connected subset of the real line whose bases are related to orthogonal polynomials. All these one-dimensional continuous spaces allow, besides the standard uncountable basis (|x〉), for an alternative countable basis (|n〉). The matrix elements that relate these two bases are essentially the orthogonal polynomials: Hermite polynomials for the line and Laguerre and Legendre polynomials for the half-line and the line interval, respectively. Differential recurrence relations of orthogonal polynomials allow us to realize that they determine an infinite-dimensional irreducible representation of a non-compact Lie algebra, whose second order Casimir C gives rise to the second order differential equation that defines the corresponding family of orthogonal polynomials. Thus, the Weyl–Heisenberg algebra h(1) with C=0 for Hermite polynomials and su(1,1) with C=−1/4 for Laguerre and Legendre polynomials are obtained. Starting from the orthogonal polynomials the Lie algebra is extended both to the whole space of the L{sup 2} functions and to the corresponding Universal Enveloping Algebra and transformation group. Generalized coherent states from each vector in the space L{sup 2} and, in particular, generalized coherent polynomials are thus obtained. -- Highlights: •Fundamental characteristic of orthogonal polynomials (OP): existence of a Lie algebra. •Differential recurrence relations of OP determine a unitary representation of a non-compact Lie group. •2nd order Casimir originates a 2nd order differential equation that defines
A digital-to-analog conversion circuit using third-order polynomial interpolation
NASA Technical Reports Server (NTRS)
Dotson, W. P., Jr.; Wilson, J. H.
1972-01-01
Zero- and third-order digital-to-analog conversion techniques are described, and the theoretical error performances are compared. The design equations and procedures for constructing a third-order digital-to-analog converter by using analog design elements are presented. Both a zero- and a third-order digital-to-analog converter were built, and the performances are compared with various signal inputs.
NASA Astrophysics Data System (ADS)
Othmani, Cherif; Takali, Farid; Njeh, Anouar
2017-06-01
In this paper, the propagation of the Lamb waves in the GaAs-FGPM-AlAs sandwich plate is studied. Based on the orthogonal function, Legendre polynomial series expansion is applied along the thickness direction to obtain the Lamb dispersion curves. The convergence and accuracy of this polynomial method are discussed. In addition, the influences of the volume fraction p and thickness hFGPM of the FGPM middle layer on the Lamb dispersion curves are developed. The numerical results also show differences between the characteristics of Lamb dispersion curves in the sandwich plate for various gradient coefficients of the FGPM middle layer. In fact, if the volume fraction p increases the phase velocity will increases and the number of modes will decreases at a given frequency range. All the developments performed in this paper were implemented in Matlab software. The corresponding results presented in this work may have important applications in several industry areas and developing novel acoustic devices such as sensors, electromechanical transducers, actuators and filters.
Application of a polynomial spline in higher-order accurate viscous-flow computations
NASA Technical Reports Server (NTRS)
Turner, M. G.; Keith, J. S.; Ghia, K. N.; Ghia, U.
1982-01-01
A quartic spline, S(4,2), is proposed which overcomes some of the difficulties associated with the use of splines S(5,3) and S(3,1) and provides fourth-order accurate results with relatively few grid points. The accuracy of spline S(4,2) is comparable to or better than that of the fourth-order box scheme and the compact differencing scheme. The use of spline S(4,2) is suggested as a possible way of obtaining fourth-order accurate solutions to Navier-Stokes equations.
NASA Astrophysics Data System (ADS)
Predescu, Cristian
2004-05-01
In this paper I provide significant mathematical evidence in support of the existence of direct short-time approximations of any polynomial order for the computation of density matrices of physical systems described by arbitrarily smooth and bounded from below potentials. While for Theorem 2, which is “experimental,” I only provide a “physicist’s” proof, I believe the present development is mathematically sound. As a verification, I explicitly construct two short-time approximations to the density matrix having convergence orders 3 and 4, respectively. Furthermore, in Appendix B, I derive the convergence constant for the trapezoidal Trotter path integral technique. The convergence orders and constants are then verified by numerical simulations. While the two short-time approximations constructed are of sure interest to physicists and chemists involved in Monte Carlo path integral simulations, the present paper is also aimed at the mathematical community, who might find the results interesting and worth exploring. I conclude the paper by discussing the implications of the present findings with respect to the solvability of the dynamical sign problem appearing in real-time Feynman path integral simulations.
A higher order non-polynomial spline method for fractional sub-diffusion problems
NASA Astrophysics Data System (ADS)
Li, Xuhao; Wong, Patricia J. Y.
2017-01-01
In this paper we shall develop a numerical scheme for a fractional sub-diffusion problem using parametric quintic spline. The solvability, convergence and stability of the scheme will be established and it is shown that the convergence order is higher than some earlier work done. We also present some numerical examples to illustrate the efficiency of the numerical scheme as well as to compare with other methods.
The exact order of approximation to periodic functions by Bernstein-Stechkin polynomials
Trigub, R M
2013-12-31
The paper concerns the approximation properties of the Bernstein-Stechkin summability method for trigonometric Fourier series. The Jackson-Stechkin theorem is refined. Moreover, for any continuous periodic function not only is the exact upper estimate for approximation found, a lower estimate of the same order is also put forward. To do this special moduli of smoothness and the K-functional are introduced. Bibliography: 16 titles.
Tables of properties of airfoil polynomials
NASA Technical Reports Server (NTRS)
Desmarais, Robert N.; Bland, Samuel R.
1995-01-01
This monograph provides an extensive list of formulas for airfoil polynomials. These polynomials provide convenient expansion functions for the description of the downwash and pressure distributions of linear theory for airfoils in both steady and unsteady subsonic flow.
Synthetic Division, Taylor Polynomials, Partial Fractions.
ERIC Educational Resources Information Center
Lambert, Howard B.
1989-01-01
Reviews the underpinnings of synthetic division. Shows how to quickly obtain the coefficients of the Taylor expansion of a polynomial about a point, and a partial fraction decomposition of a polynomial. (MVL)
High-order Taylor series expansion methods for error propagation in geographic information systems
NASA Astrophysics Data System (ADS)
Xue, Jie; Leung, Yee; Ma, Jiang-Hong
2015-04-01
The quality of modeling results in GIS operations depends on how well we can track error propagating from inputs to outputs. Monte Carlo simulation, moment design and Taylor series expansion have been employed to study error propagation over the years. Among them, first-order Taylor series expansion is popular because error propagation can be analytically studied. Because most operations in GIS are nonlinear, first-order Taylor series expansion generally cannot meet practical needs, and higher-order approximation is thus necessary. In this paper, we employ Taylor series expansion methods of different orders to investigate error propagation when the random error vectors are normally and independently or dependently distributed. We also extend these methods to situations involving multi-dimensional output vectors. We employ these methods to examine length measurement of linear segments, perimeter of polygons and intersections of two line segments basic in GIS operations. Simulation experiments indicate that the fifth-order Taylor series expansion method is most accurate compared with the first-order and third-order method. Compared with the third-order expansion; however, it can only slightly improve the accuracy, but on the expense of substantially increasing the number of partial derivatives that need to be calculated. Striking a balance between accuracy and complexity, the third-order Taylor series expansion method appears to be a more appropriate choice for practical applications.
Normality of different orders for Cantor series expansions
NASA Astrophysics Data System (ADS)
Airey, Dylan; Mance, Bill
2017-10-01
Let S \\subseteq {N} have the property that for each k \\in S the set (S - k) \\cap {N} \\setminus S has asymptotic density 0. We prove that there exists a basic sequence Q where the set of numbers Q-normal of all orders in S but not Q-normal of all orders not in S has full Hausdorff dimension. If the function \
Numerical simulation of stratified shear flow using a higher order Taylor series expansion method
Iwashige, Kengo; Ikeda, Takashi
1995-09-01
A higher order Taylor series expansion method is applied to two-dimensional numerical simulation of stratified shear flow. In the present study, central difference scheme-like method is adopted for an even expansion order, and upwind difference scheme-like method is adopted for an odd order, and the expansion order is variable. To evaluate the effects of expansion order upon the numerical results, a stratified shear flow test in a rectangular channel (Reynolds number = 1.7x10{sup 4}) is carried out, and the numerical velocity and temperature fields are compared with experimental results measured by laser Doppler velocimetry thermocouples. The results confirm that the higher and odd order methods can simulate mean velocity distributions, root-mean-square velocity fluctuations, Reynolds stress, temperature distributions, and root-mean-square temperature fluctuations.
Multipole expansion of acoustical Bessel beams with arbitrary order and location.
Gong, Zhixiong; Marston, Philip L; Li, Wei; Chai, Yingbin
2017-06-01
An exact solution of expansion coefficients for a T-matrix method interacting with acoustic scattering of arbitrary order Bessel beams from an obstacle of arbitrary location is derived analytically. Because of the failure of the addition theorem for spherical harmonics for expansion coefficients of helicoidal Bessel beams, an addition theorem for cylindrical Bessel functions is introduced. Meanwhile, an analytical expression for the integral of products including Bessel and associated Legendre functions is applied to eliminate the integration over the polar angle. Note that this multipole expansion may also benefit other scattering methods and expansions of incident waves, for instance, partial-wave series solutions.
NASA Astrophysics Data System (ADS)
Zhang, Jianzhong; Jiang, Li
2017-08-01
The computation of gravitational vector field of variable-density bodies has been attracting much attention in high precision gravity investigation. In forward and inverse gravity modeling, a 3-D rectangular prism, as a simple building block, is commonly used to approximate 3-D variable-density bodies with irregular shape. In this paper, we first model the density of a rectangular prism using an arbitrary-order polynomial function of depth, and derive the closed-form expressions for computing the vertical and horizontal components of the gravitational vector field of a 3-D rectangular prism with the polynomial density variation. Then, we extend the polynomial density function of depth to a linear combination of three arbitrary-order polynomial functions in x-, y- and z-directions, and derive the analytical expressions of the gravitational vector field of a 3-D rectangular prism with such more general density function. The analytic expressions are numerically validated and compared with other existing methods. It is shown that the new expressions are far superior to the common method based on the stack of a collection of uniform subprisms in computational efficiency. In addition, the numerical error of the expressions is analysed and the results show that the expressions are available for practical applications.
NASA Astrophysics Data System (ADS)
Wang, S.; Huang, G. H.; Baetz, B. W.; Ancell, B. C.
2017-05-01
The particle filtering techniques have been receiving increasing attention from the hydrologic community due to its ability to properly estimate model parameters and states of nonlinear and non-Gaussian systems. To facilitate a robust quantification of uncertainty in hydrologic predictions, it is necessary to explicitly examine the forward propagation and evolution of parameter uncertainties and their interactions that affect the predictive performance. This paper presents a unified probabilistic framework that merges the strengths of particle Markov chain Monte Carlo (PMCMC) and factorial polynomial chaos expansion (FPCE) algorithms to robustly quantify and reduce uncertainties in hydrologic predictions. A Gaussian anamorphosis technique is used to establish a seamless bridge between the data assimilation using the PMCMC and the uncertainty propagation using the FPCE through a straightforward transformation of posterior distributions of model parameters. The unified probabilistic framework is applied to the Xiangxi River watershed of the Three Gorges Reservoir (TGR) region in China to demonstrate its validity and applicability. Results reveal that the degree of spatial variability of soil moisture capacity is the most identifiable model parameter with the fastest convergence through the streamflow assimilation process. The potential interaction between the spatial variability in soil moisture conditions and the maximum soil moisture capacity has the most significant effect on the performance of streamflow predictions. In addition, parameter sensitivities and interactions vary in magnitude and direction over time due to temporal and spatial dynamics of hydrologic processes.
Polynomial interpolation methods for viscous flow calculations
NASA Technical Reports Server (NTRS)
Rubin, S. G.; Khosla, P. K.
1977-01-01
Higher-order collocation procedures which result in block-tridiagonal matrix systems are derived from (1) Taylor series expansions and from (2) polynomial interpolation, and the relationships between the two formulations, called respectively Hermite and spline collocation, are investigated. A Hermite block-tridiagonal system for a nonuniform mesh is derived, and the Hermite approach is extended in order to develop a variable-mesh sixth-order block-tridiagonal procedure. It is shown that all results obtained by Hermite development can be recovered by appropriate spline polynomial interpolation. The additional boundary conditions required for these higher-order procedures are also given. Comparative solutions using second-order accurate finite difference and spline and Hermite formulations are presented for the boundary layer on a flat plate, boundary layers with uniform and variable mass transfer, and the viscous incompressible Navier-Stokes equations describing flow in a driven cavity.
ERIC Educational Resources Information Center
Dobbs, David E.
2010-01-01
This note develops and implements the theory of polynomial asymptotes to (graphs of) rational functions, as a generalization of the classical topics of horizontal asymptotes and oblique/slant asymptotes. Applications are given to hyperbolic asymptotes. Prerequisites include the division algorithm for polynomials with coefficients in the field of…
ERIC Educational Resources Information Center
Dobbs, David E.
2010-01-01
This note develops and implements the theory of polynomial asymptotes to (graphs of) rational functions, as a generalization of the classical topics of horizontal asymptotes and oblique/slant asymptotes. Applications are given to hyperbolic asymptotes. Prerequisites include the division algorithm for polynomials with coefficients in the field of…
Inverse polynomial reconstruction method in DCT domain
NASA Astrophysics Data System (ADS)
Dadkhahi, Hamid; Gotchev, Atanas; Egiazarian, Karen
2012-12-01
The discrete cosine transform (DCT) offers superior energy compaction properties for a large class of functions and has been employed as a standard tool in many signal and image processing applications. However, it suffers from spurious behavior in the vicinity of edge discontinuities in piecewise smooth signals. To leverage the sparse representation provided by the DCT, in this article, we derive a framework for the inverse polynomial reconstruction in the DCT expansion. It yields the expansion of a piecewise smooth signal in terms of polynomial coefficients, obtained from the DCT representation of the same signal. Taking advantage of this framework, we show that it is feasible to recover piecewise smooth signals from a relatively small number of DCT coefficients with high accuracy. Furthermore, automatic methods based on minimum description length principle and cross-validation are devised to select the polynomial orders, as a requirement of the inverse polynomial reconstruction method in practical applications. The developed framework can considerably enhance the performance of the DCT in sparse representation of piecewise smooth signals. Numerical results show that denoising and image approximation algorithms based on the proposed framework indicate significant improvements over wavelet counterparts for this class of signals.
Convergence of high order perturbative expansions in open system quantum dynamics
NASA Astrophysics Data System (ADS)
Xu, Meng; Song, Linze; Song, Kai; Shi, Qiang
2017-02-01
We propose a new method to directly calculate high order perturbative expansion terms in open system quantum dynamics. They are first written explicitly in path integral expressions. A set of differential equations are then derived by extending the hierarchical equation of motion (HEOM) approach. As two typical examples for the bosonic and fermionic baths, specific forms of the extended HEOM are obtained for the spin-boson model and the Anderson impurity model. Numerical results are then presented for these two models. General trends of the high order perturbation terms as well as the necessary orders for the perturbative expansions to converge are analyzed.
Nonperturbative linked-cluster expansions in long-range ordered quantum systems
NASA Astrophysics Data System (ADS)
Ixert, Dominik; Schmidt, Kai Phillip
2016-11-01
We introduce a generic scheme to perform nonperturbative linked cluster expansions in long-range ordered quantum phases. Clusters are considered to be surrounded by an ordered reference state leading to effective edge fields in the exact diagonalization on clusters, which break the associated symmetry of the ordered phase. Two approaches, based either on a self-consistent solution of the order parameter or on minimal sensitivity with respect to the ground-state energy per site, are formulated to find the optimal edge field in each NLCE order. Furthermore, we investigate the scaling behavior of the NLCE data sequences towards the infinite-order limit. We apply our scheme to gapped and gapless ordered phases of XXZ Heisenberg models on various lattices and for spins 1/2 and 1 using several types of cluster expansions ranging from a full-graph decomposition, rectangular clusters, up to more symmetric square clusters. It is found that the inclusion of edge fields allows to regularize nonperturbative linked-cluster expansions in ordered phases yielding convergent data sequences. This includes the long-range spin-ordered ground state of the spin-1/2 and spin-1 Heisenberg model on the square and triangular lattice as well as the trimerized valence bond crystal of the spin-1 Heisenberg model on the kagome lattice.
Role of the U(1) ghost beyond leading order in a large-Nc expansion
Hrayr Matevosyan; Anthony Thomas
2008-09-01
The 1/Nc expansion is one of the very few methods we have for generating a systematic expansion of QCD at the energy scale relevant to hadron structure. The present formulation of this theory relies on 't Hooft's double-line notation for calculating the leading order of a diagram in the 1/Nc expansion, where the local SU(Nc) gauge symmetry is substituted by a U(Nc) symmetry and the associated U(1) ghost field is ignored. In the current work we demonstrate the insufficiency of this formulation for describing certain non-planar diagrams. We derive a more complete set of Feynman rules that include the U(1) ghost field and provide a useful tool for calculating both color factors and 1/Nc orders of given color-singlet diagrams.
An expansion formula with higher-order derivatives for fractional operators of variable order.
Almeida, Ricardo; Torres, Delfim F M
2013-01-01
We obtain approximation formulas for fractional integrals and derivatives of Riemann-Liouville and Marchaud types with a variable fractional order. The approximations involve integer-order derivatives only. An estimation for the error is given. The efficiency of the approximation method is illustrated with examples. As applications, we show how the obtained results are useful to solve differential equations, and problems of the calculus of variations that depend on fractional derivatives of Marchaud type.
An Expansion Formula with Higher-Order Derivatives for Fractional Operators of Variable Order
Almeida, Ricardo
2013-01-01
We obtain approximation formulas for fractional integrals and derivatives of Riemann-Liouville and Marchaud types with a variable fractional order. The approximations involve integer-order derivatives only. An estimation for the error is given. The efficiency of the approximation method is illustrated with examples. As applications, we show how the obtained results are useful to solve differential equations, and problems of the calculus of variations that depend on fractional derivatives of Marchaud type. PMID:24319382
Majewski, Kurt; Heid, Oliver; Kluge, Thomas
2010-06-01
We suggest a polynomial program for the calculation of optimized gradient waveforms for magnetic resonance tomography pulse sequences. Such non-linear mathematical programs can describe gradient system capabilities, meet k-space trajectory specifications, and capture sequence timing conditions. Moreover they allow the incorporation of gradient moment nulling constraints in one or several arbitrary spatial directions, which can reduce flow motion artifacts in the images. We report first experiences in solving such automatic pulse sequence design programs with the interior point solver Ipopt.
Word Order in Spoken German: Syntactic Right-Expansions as an Interactionally Constructed Phenomenon
ERIC Educational Resources Information Center
Schoenfeldt, Juliane
2009-01-01
In real time interaction, the ordering of words is one of the resources participants-to-talk rely on in the negotiation of shared meaning. This dissertation investigates the emergence of syntactic right-expansions in spoken German as a systematic resource in the organization of talk-in-interaction. Employing the methodology of conversation…
The basic function scheme of polynomial type
WU, Wang-yi; Lin, Guang
2009-12-01
A new numerical method---Basic Function Method is proposed. This method can directly discrete differential operator on unstructured grids. By using the expansion of basic function to approach the exact function, the central and upwind schemes of derivative are constructed. By using the second-order polynomial as basic function and applying the technique of flux splitting method and the combination of central and upwind schemes to suppress the non-physical fluctuation near the shock wave, the second-order basic function scheme of polynomial type for solving inviscid compressible flow numerically is constructed in this paper. Several numerical results of many typical examples for two dimensional inviscid compressible transonic and supersonic steady flow illustrate that it is a new scheme with high accuracy and high resolution for shock wave. Especially, combining with the adaptive remeshing technique, the satisfactory results can be obtained by these schemes.
A second-order shock-expansion method applicable to bodies of revolution near zero lift
NASA Technical Reports Server (NTRS)
1957-01-01
A second-order shock-expansion method applicable to bodies of revolution is developed by the use of the predictions of the generalized shock-expansion method in combination with characteristics theory. Equations defining the zero-lift pressure distributions and the normal-force and pitching-moment derivatives are derived. Comparisons with experimental results show that the method is applicable at values of the similarity parameter, the ratio of free-stream Mach number to nose fineness ratio, from about 0.4 to 2.
Superoscillations with arbitrary polynomial shape
NASA Astrophysics Data System (ADS)
Chremmos, Ioannis; Fikioris, George
2015-07-01
We present a method for constructing superoscillatory functions the superoscillatory part of which approximates a given polynomial with arbitrarily small error in a fixed interval. These functions are obtained as the product of the polynomial with a sufficiently flat, bandlimited envelope function whose Fourier transform has at least N-1 continuous derivatives and an Nth derivative of bounded variation, N being the order of the polynomial. Polynomials of arbitrarily high order can be approximated if the Fourier transform of the envelope is smooth, i.e. a bump function.
Petrović, Nikola Z; Belić, Milivoj; Zhong, Wei-Ping
2011-02-01
We obtain exact traveling wave and spatiotemporal soliton solutions to the generalized (3+1)-dimensional nonlinear Schrödinger equation with variable coefficients and polynomial Kerr nonlinearity of an arbitrarily high order. Exact solutions, given in terms of Jacobi elliptic functions, are presented for the special cases of cubic-quintic and septic models. We demonstrate that the widely used method for finding exact solutions in terms of Jacobi elliptic functions is not applicable to the nonlinear Schrödinger equation with saturable nonlinearity.
NASA Astrophysics Data System (ADS)
T'en Taĭ, Nguen
1984-02-01
This paper gives estimates for the orders of zeros of polynomials in a set of analytic functions satisfying a system of linear differential equations with coefficients in \\mathbf{C}(z), in the case when these functions are algebraically dependent over \\mathbf{C}(z). Using the Siegel-Shidlovskiĭ method, these estimates are applied to obtain effective bounds from below for the relative transcendence measure of the values of E-functions in the case when the basic set of E-functions is algebraically dependent over \\mathbf{C}(z).Bibliography: 20 titles.
Numerical constructions involving Chebyshev polynomials
NASA Astrophysics Data System (ADS)
Lyakhovsky, V. D.
2017-02-01
We propose a new algorithm for the character expansion of tensor products of finite-dimensional irreducible representations of simple Lie algebras. The algorithm produces valid results for the algebras B 3, C 3, and D 3. We use the direct correspondence between Weyl anti-invariant functions and multivariate second-kind Chebyshev polynomials. We construct the triangular trigonometric polynomials for the algebra D 3.
Rectangular rotation of spherical harmonic expansion of arbitrary high degree and order
NASA Astrophysics Data System (ADS)
Fukushima, Toshio
2017-08-01
In order to move the polar singularity of arbitrary spherical harmonic expansion to a point on the equator, we rotate the expansion around the y-axis by 90° such that the x-axis becomes a new pole. The expansion coefficients are transformed by multiplying a special value of Wigner D-matrix and a normalization factor. The transformation matrix is unchanged whether the coefficients are 4 π fully normalized or Schmidt quasi-normalized. The matrix is recursively computed by the so-called X-number formulation (Fukushima in J Geodesy 86: 271-285, 2012a). As an example, we obtained 2190× 2190 coefficients of the rectangular rotated spherical harmonic expansion of EGM2008. A proper combination of the original and the rotated expansions will be useful in (i) integrating the polar orbits of artificial satellites precisely and (ii) synthesizing/analyzing the gravitational/geomagnetic potentials and their derivatives accurately in the high latitude regions including the arctic and antarctic area.
Rectangular rotation of spherical harmonic expansion of arbitrary high degree and order
NASA Astrophysics Data System (ADS)
Fukushima, Toshio
2017-02-01
In order to move the polar singularity of arbitrary spherical harmonic expansion to a point on the equator, we rotate the expansion around the y-axis by 90° such that the x-axis becomes a new pole. The expansion coefficients are transformed by multiplying a special value of Wigner D-matrix and a normalization factor. The transformation matrix is unchanged whether the coefficients are 4 π fully normalized or Schmidt quasi-normalized. The matrix is recursively computed by the so-called X-number formulation (Fukushima in J Geodesy 86: 271-285, 2012a). As an example, we obtained 2190× 2190 coefficients of the rectangular rotated spherical harmonic expansion of EGM2008. A proper combination of the original and the rotated expansions will be useful in (i) integrating the polar orbits of artificial satellites precisely and (ii) synthesizing/analyzing the gravitational/geomagnetic potentials and their derivatives accurately in the high latitude regions including the arctic and antarctic area.
Lee, Okkyun; Kappler, Steffen; Polster, Christoph; Taguchi, Katsuyuki
2016-10-26
Photon counting detector (PCD)-based computed tomography exploits spectral information from a transmitted x-ray spectrum to estimate basis line-integrals. The recorded spectrum, however, is distorted and deviates from the transmitted spectrum due to spectral response effect (SRE). Therefore, the SRE needs to be compensated for when estimating basis lineintegrals. One approach is to incorporate the SRE model with an incident spectrum into the PCD measurement model and the other approach is to perform a calibration process that inherently includes both the SRE and the incident spectrum. A maximum likelihood estimator can be used to the former approach, which guarantees asymptotic optimality; however, a heavy computational burden is a concern. Calibration-based estimators are a form of the latter approach. They can be very efficient; however, a heuristic calibration process needs to be addressed. In this paper, we propose a computationally efficient three-step estimator for the former approach using a low-order polynomial approximation of x-ray transmittance. The low-order polynomial approximation can change the original non-linear estimation method to a two-step linearized approach followed by an iterative bias correction step. We show that the calibration process is required only for the bias correction step and prove that it converges to the unbiased solution under practical assumptions. Extensive simulation studies validate the proposed method and show that the estimation results are comparable to those of the ML estimator while the computational time is reduced substantially.
Analytical high-order post-Newtonian expansions for extreme mass ratio binaries
NASA Astrophysics Data System (ADS)
Kavanagh, Chris; Ottewill, Adrian C.; Wardell, Barry
2015-10-01
We present analytic computations of gauge invariant quantities for a point mass in a circular orbit around a Schwarzschild black hole, giving results up to 15.5 post-Newtonian order in this paper and up to 21.5 post-Newtonian order in an online repository. Our calculation is based on the functional series method of Mano, Suzuki, and Takasugi (MST) and a recent series of results by Bini and Damour. We develop an optimized method for generating post-Newtonian expansions of the MST series, enabling significantly faster computations. We also clarify the structure of the expansions for large values of ℓ , and in doing so develop an efficient new method for generating the MST renormalized angular momentum, ν .
Order reduction of z-transfer functions via multipoint Jordan continued-fraction expansion
NASA Technical Reports Server (NTRS)
Lee, Ying-Chin; Hwang, Chyi; Shieh, Leang S.
1992-01-01
The order reduction problem of z-transfer functions is solved by using the multipoint Jordan continued-fraction expansion (MJCFE) technique. An efficient algorithm that does not require the use of complex algebra is presented for obtaining an MJCFE from a stable z-transfer function with expansion points selected from the unit circle and/or the positive real axis of the z-plane. The reduced-order models are exactly the multipoint Pade approximants of the original system and, therefore, they match the (weighted) time-moments of the impulse response and preserve the frequency responses of the system at some characteristic frequencies, such as gain crossover frequency, phase crossover frequency, bandwidth, etc.
Automatic differentiation for Fourier series and the radii polynomial approach
NASA Astrophysics Data System (ADS)
Lessard, Jean-Philippe; Mireles James, J. D.; Ransford, Julian
2016-11-01
In this work we develop a computer-assisted technique for proving existence of periodic solutions of nonlinear differential equations with non-polynomial nonlinearities. We exploit ideas from the theory of automatic differentiation in order to formulate an augmented polynomial system. We compute a numerical Fourier expansion of the periodic orbit for the augmented system, and prove the existence of a true solution nearby using an a-posteriori validation scheme (the radii polynomial approach). The problems considered here are given in terms of locally analytic vector fields (i.e. the field is analytic in a neighborhood of the periodic orbit) hence the computer-assisted proofs are formulated in a Banach space of sequences satisfying a geometric decay condition. In order to illustrate the use and utility of these ideas we implement a number of computer-assisted existence proofs for periodic orbits of the Planar Circular Restricted Three-Body Problem (PCRTBP).
Second-order virial expansion for an atomic gas in a harmonic waveguide
NASA Astrophysics Data System (ADS)
Kristensen, Tom; Leyronas, Xavier; Pricoupenko, Ludovic
2016-06-01
The virial expansion for cold two-component Fermi and Bose atomic gases is considered in the presence of a waveguide and in the vicinity of a Feshbach resonance. The interaction between atoms and the coupling with the Feshbach molecules is modeled using a quantitative separable two-channel model. The scattering phase shift in an atomic waveguide is defined. This permits us to extend the Beth-Uhlenbeck formula for the second-order virial coefficient to this inhomogeneous case.
High order local absorbing boundary conditions for acoustic waves in terms of farfield expansions
NASA Astrophysics Data System (ADS)
Villamizar, Vianey; Acosta, Sebastian; Dastrup, Blake
2017-03-01
We devise a new high order local absorbing boundary condition (ABC) for radiating problems and scattering of time-harmonic acoustic waves from obstacles of arbitrary shape. By introducing an artificial boundary S enclosing the scatterer, the original unbounded domain Ω is decomposed into a bounded computational domain Ω- and an exterior unbounded domain Ω+. Then, we define interface conditions at the artificial boundary S, from truncated versions of the well-known Wilcox and Karp farfield expansion representations of the exact solution in the exterior region Ω+. As a result, we obtain a new local absorbing boundary condition (ABC) for a bounded problem on Ω-, which effectively accounts for the outgoing behavior of the scattered field. Contrary to the low order absorbing conditions previously defined, the error at the artificial boundary induced by this novel ABC can be easily reduced to reach any accuracy within the limits of the computational resources. We accomplish this by simply adding as many terms as needed to the truncated farfield expansions of Wilcox or Karp. The convergence of these expansions guarantees that the order of approximation of the new ABC can be increased arbitrarily without having to enlarge the radius of the artificial boundary. We include numerical results in two and three dimensions which demonstrate the improved accuracy and simplicity of this new formulation when compared to other absorbing boundary conditions.
NASA Astrophysics Data System (ADS)
Canhanga, Betuel; Ni, Ying; Rančić, Milica; Malyarenko, Anatoliy; Silvestrov, Sergei
2017-01-01
After Black-Scholes proposed a model for pricing European Options in 1973, Cox, Ross and Rubinstein in 1979, and Heston in 1993, showed that the constant volatility assumption made by Black-Scholes was one of the main reasons for the model to be unable to capture some market details. Instead of constant volatilities, they introduced stochastic volatilities to the asset dynamic modeling. In 2009, Christoffersen empirically showed "why multifactor stochastic volatility models work so well". Four years later, Chiarella and Ziveyi solved the model proposed by Christoffersen. They considered an underlying asset whose price is governed by two factor stochastic volatilities of mean reversion type. Applying Fourier transforms, Laplace transforms and the method of characteristics they presented a semi-analytical formula to compute an approximate price for American options. The huge calculation involved in the Chiarella and Ziveyi approach motivated the authors of this paper in 2014 to investigate another methodology to compute European Option prices on a Christoffersen type model. Using the first and second order asymptotic expansion method we presented a closed form solution for European option, and provided experimental and numerical studies on investigating the accuracy of the approximation formulae given by the first order asymptotic expansion. In the present paper we will perform experimental and numerical studies for the second order asymptotic expansion and compare the obtained results with results presented by Chiarella and Ziveyi.
Role of the U(1) ghost beyond leading order in a large-Nc expansion
NASA Astrophysics Data System (ADS)
Matevosyan, Hrayr H.; Thomas, Anthony W.
2008-11-01
The 1/Nc expansion is one of the very few methods we have for generating a systematic expansion of QCD at the energy scale relevant to hadron structure. The present formulation of this theory relies on the double-line notation for calculating the leading order of a diagram in the 1/Nc expansion, where the local SU(Nc) gauge symmetry is substituted by a U(Nc) symmetry and the associated U(1) ghost field is ignored. In the current work we demonstrate the insufficiency of this formulation for describing certain non-planar diagrams of interest in current attempts to model QCD. We derive a more complete set of Feynman rules that include the U(1) ghost field and provide a useful tool for calculating both color factors and 1/Nc orders of all color-singlet diagrams. Note: Authored by Jefferson Science Associates, LLC under US DOE contract no DE-AC05-06OR23177. The US Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce this manuscript for US Government purposes.
Analytical high-order post-Newtonian expansions for spinning extreme mass ratio binaries
NASA Astrophysics Data System (ADS)
Kavanagh, Chris; Ottewill, Adrian C.; Wardell, Barry
2016-06-01
We present an analytic computation of Detweiler's redshift invariant for a point mass in a circular orbit around a Kerr black hole, giving results up to 8.5 post-Newtonian order while making no assumptions on the magnitude of the spin of the black hole. Our calculation is based on the functional series method of Mano, Suzuki and Takasugi, and employs a rigorous mode-sum regularization prescription based on the Detweiler-Whiting singular-regular decomposition. The approximations used in our approach are minimal; we use the standard self-force expansion to linear order in the mass ratio, and the standard post-Newtonian expansion in the separation of the binary. A key advantage of this approach is that it produces expressions that include contributions at all orders in the spin of the Kerr black hole. While this work applies the method to the specific case of Detweiler's redshift invariant, it can be readily extended to other gauge-invariant quantities and to higher post-Newtonian orders.
NASA Astrophysics Data System (ADS)
Li, He; Gao, Yi-Tian; Liu, Li-Cai
2015-12-01
The Korteweg-de Vries (KdV)-type equations have been seen in fluid mechanics, plasma physics and lattice dynamics, etc. This paper will address the bilinearization problem for some higher-order KdV equations. Based on the relationship between the bilinear method and Bell-polynomial scheme, with introducing an auxiliary independent variable, we will present the general bilinear forms. By virtue of the symbolic computation, one- and two-soliton solutions are derived. Supported by the National Natural Science Foundation of China under Grant No. 11272023, the Open Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), and by the Fundamental Research Funds for the Central Universities of China under Grant No. 2011BUPTYB02
Generalized quantum kinetic expansion: Higher-order corrections to multichromophoric Förster theory.
Wu, Jianlan; Gong, Zhihao; Tang, Zhoufei
2015-08-21
For a general two-cluster energy transfer network, a new methodology of the generalized quantum kinetic expansion (GQKE) method is developed, which predicts an exact time-convolution equation for the cluster population evolution under the initial condition of the local cluster equilibrium state. The cluster-to-cluster rate kernel is expanded over the inter-cluster couplings. The lowest second-order GQKE rate recovers the multichromophoric Förster theory (MCFT) rate. The higher-order corrections to the MCFT rate are systematically included using the continued fraction resummation form, resulting in the resummed GQKE method. The reliability of the GQKE methodology is verified in two model systems, revealing the relevance of higher-order corrections.
High-order power series expansion of the elastic interaction between conical membrane inclusions.
Fournier, Jean-Baptiste; Galatola, Paolo
2015-08-01
We revisit the problem of the long-range interaction between two conical proteins inserted into a lipid membrane and interacting via the induced deformation of the membrane, first considered by Goulian et al. (M. Goulian, R. Bruinsma, P. Pincus, Europhys. Lett. 22, 145 (1993); Europhys. Lett. 23, 155 (1993)). By means of a complex variables formulation and an iterative solution, we determine analytically an arbitrary high-order expansion of the interaction energy in powers of the inverse distance between two inclusions of different sizes. At leading order and for inclusions of equal sizes, we recover the result obtained by Goulian et al.. We generalize the development to inclusions of different sizes and give explicit formulas that increase the precision by ten orders in the inverse distance.
Nuclear matter properties with nucleon-nucleon forces up to fifth order in the chiral expansion
NASA Astrophysics Data System (ADS)
Hu, Jinniu; Zhang, Ying; Epelbaum, Evgeny; Meißner, Ulf-G.; Meng, Jie
2017-09-01
The properties of nuclear matter are studied using state-of-the-art nucleon-nucleon forces up to fifth order in chiral effective field theory. The equations of state of symmetric nuclear matter and pure neutron matter are calculated in the framework of the Brueckner-Hartree-Fock theory. We discuss in detail the convergence pattern of the chiral expansion and the regulator dependence of the calculated equations of state and provide an estimation of the truncation uncertainty. For all employed values of the regulator, the fifth-order chiral two-nucleon potential is found to generate nuclear saturation properties similar to the available phenomenological high precision potentials. We also extract the symmetry energy of nuclear matter, which is shown to be quite robust with respect to the chiral order and the value of the regulator.
Generalized quantum kinetic expansion: Higher-order corrections to multichromophoric Förster theory
Wu, Jianlan Gong, Zhihao; Tang, Zhoufei
2015-08-21
For a general two-cluster energy transfer network, a new methodology of the generalized quantum kinetic expansion (GQKE) method is developed, which predicts an exact time-convolution equation for the cluster population evolution under the initial condition of the local cluster equilibrium state. The cluster-to-cluster rate kernel is expanded over the inter-cluster couplings. The lowest second-order GQKE rate recovers the multichromophoric Förster theory (MCFT) rate. The higher-order corrections to the MCFT rate are systematically included using the continued fraction resummation form, resulting in the resummed GQKE method. The reliability of the GQKE methodology is verified in two model systems, revealing the relevance of higher-order corrections.
NASA Astrophysics Data System (ADS)
Hsu, Jong-Ping
2014-02-01
A generalization of the usual gauge symmetry leads to fourth-order gauge field equations, which imply a new constant force independent of distances. The force associated with the new U1 gauge symmetry is repulsive among baryons. Such a constant force based on baryon charge conservation gives a field-theoretic understanding of the accelerated cosmic expansion in the observable portion of the universe dominated by baryon galaxies. In consistent with all conservation laws and known forces, a simple rotating "dumbbell model" of the universe is briefly discussed.
Polynomial probability distribution estimation using the method of moments
Mattsson, Lars; Rydén, Jesper
2017-01-01
We suggest a procedure for estimating Nth degree polynomial approximations to unknown (or known) probability density functions (PDFs) based on N statistical moments from each distribution. The procedure is based on the method of moments and is setup algorithmically to aid applicability and to ensure rigor in use. In order to show applicability, polynomial PDF approximations are obtained for the distribution families Normal, Log-Normal, Weibull as well as for a bimodal Weibull distribution and a data set of anonymized household electricity use. The results are compared with results for traditional PDF series expansion methods of Gram–Charlier type. It is concluded that this procedure is a comparatively simple procedure that could be used when traditional distribution families are not applicable or when polynomial expansions of probability distributions might be considered useful approximations. In particular this approach is practical for calculating convolutions of distributions, since such operations become integrals of polynomial expressions. Finally, in order to show an advanced applicability of the method, it is shown to be useful for approximating solutions to the Smoluchowski equation. PMID:28394949
NASA Astrophysics Data System (ADS)
Nikabadze, M. U.
2007-06-01
We consider various forms of equations of motion and heat influx for deformable solids as well as various forms of Hooke's law and Fourier's heat conduction law under the nonclassical parametrization [1-5] of the domain occupied by a thin solid, where the transverse coordinate ranges in the interval [0, 1]. We write out several characteristics inherent in this parametrization. We use the above-mentioned equations and laws to derive the corresponding equations and laws, as well as statements of problems, for thin bodies in moments with respect to Chebyshev polynomials of the second kind. Here the interval [0, 1] is used as the orthogonality interval for the systems of Chebyshev polynomials. For this interval, we write out the basic recursion relations and, in turn, use them to obtain several additional recursion relations, which play an important role in constructing other versions of the theory of thin solids. In particular, we use the recursion relations to obtain the moments of the first and second derivatives of a scalar function, of rank one and two tensors and their components, and of some differential operators of these variables. Moreover, we give the statements of coupled and uncoupled dynamic problems in moments of the ( r, N)th approximation in moment thermomechanics of thin deformable solids. We also state the nonstable temperature problem in moments of the ( r, N)th approximation.
The nucleon-nucleon interaction up to sixth order in the chiral expansion
NASA Astrophysics Data System (ADS)
Machleidt, Ruprecht; Nosyk, Yevgen
2016-09-01
We have calculated the nucleon-nucleon potential up to sixth order (N5LO) of chiral perturbation theory. Previous calculations extended only up to N3LO (fourth order) and typically showed a surplus of attraction, particularly, when the π- N LECs from π- N analysis were applied consistently. Furthermore, the contributions at N2LO and N3LO are both fairly sizeable, thus, raising concerns about the convergence of the chiral expansion. We show that the N4LO contribution is repulsive and, essentially, cancels the excessive attraction of N3LO. The N5LO contribution turns out to be considerably smaller than the N4LO one, hence establishing the desired trend of convergence. The predictions at N5LO are in excellent agreement with the empirical phase shifts of peripheral partial waves. Supported by the US Department of Energy under Grant No. DE-FG02-03ER41270.
A corrected particle method with high-order Taylor expansion for solving the viscoelastic fluid flow
NASA Astrophysics Data System (ADS)
Jiang, T.; Ren, J. L.; Lu, W. G.; Xu, B.
2016-12-01
In this paper, a corrected particle method based on the smoothed particle hydrodynamics (SPH) method with high-order Taylor expansion (CSPH-HT) for solving the viscoelastic flow is proposed and investigated. The validity and merits of the CSPH-HT method are first tested by solving the nonlinear high order Kuramoto-Sivishinsky equation and simulating the drop stretching, respectively. Then the flow behaviors behind two stationary tangential cylinders of polymer melt, which have been received little attention, are investigated by the CSPH-HT method. Finally, the CSPH-HT method is extended to the simulation of the filling process of the viscoelastic fluid. The numerical results show that the CSPH-HT method possesses higher accuracy and stability than other corrected SPH methods and is more reliable than other corrected SPH methods.
Coherent states, 6j symbols and properties of the next to leading order asymptotic expansions
NASA Astrophysics Data System (ADS)
Kamiński, Wojciech; Steinhaus, Sebastian
2013-12-01
We present the first complete derivation of the well-known asymptotic expansion of the SU(2) 6j symbol using a coherent state approach, in particular we succeed in computing the determinant of the Hessian matrix. To do so, we smear the coherent states and perform a partial stationary point analysis with respect to the smearing parameters. This allows us to transform the variables from group elements to dihedral angles of a tetrahedron resulting in an effective action, which coincides with the action of first order Regge calculus associated to a tetrahedron. To perform the remaining stationary point analysis, we compute its Hessian matrix and obtain the correct measure factor. Furthermore, we expand the discussion of the asymptotic formula to next to leading order terms, prove some of their properties and derive a recursion relation for the full 6j symbol.
A corrected particle method with high-order Taylor expansion for solving the viscoelastic fluid flow
NASA Astrophysics Data System (ADS)
Jiang, T.; Ren, J. L.; Lu, W. G.; Xu, B.
2017-02-01
In this paper, a corrected particle method based on the smoothed particle hydrodynamics (SPH) method with high-order Taylor expansion (CSPH-HT) for solving the viscoelastic flow is proposed and investigated. The validity and merits of the CSPH-HT method are first tested by solving the nonlinear high order Kuramoto-Sivishinsky equation and simulating the drop stretching, respectively. Then the flow behaviors behind two stationary tangential cylinders of polymer melt, which have been received little attention, are investigated by the CSPH-HT method. Finally, the CSPH-HT method is extended to the simulation of the filling process of the viscoelastic fluid. The numerical results show that the CSPH-HT method possesses higher accuracy and stability than other corrected SPH methods and is more reliable than other corrected SPH methods.
Coherent states, 6j symbols and properties of the next to leading order asymptotic expansions
Kamiński, Wojciech; Steinhaus, Sebastian
2013-12-15
We present the first complete derivation of the well-known asymptotic expansion of the SU(2) 6j symbol using a coherent state approach, in particular we succeed in computing the determinant of the Hessian matrix. To do so, we smear the coherent states and perform a partial stationary point analysis with respect to the smearing parameters. This allows us to transform the variables from group elements to dihedral angles of a tetrahedron resulting in an effective action, which coincides with the action of first order Regge calculus associated to a tetrahedron. To perform the remaining stationary point analysis, we compute its Hessian matrix and obtain the correct measure factor. Furthermore, we expand the discussion of the asymptotic formula to next to leading order terms, prove some of their properties and derive a recursion relation for the full 6j symbol.
The Gibbs Phenomenon for Series of Orthogonal Polynomials
ERIC Educational Resources Information Center
Fay, T. H.; Kloppers, P. Hendrik
2006-01-01
This note considers the four classes of orthogonal polynomials--Chebyshev, Hermite, Laguerre, Legendre--and investigates the Gibbs phenomenon at a jump discontinuity for the corresponding orthogonal polynomial series expansions. The perhaps unexpected thing is that the Gibbs constant that arises for each class of polynomials appears to be the same…
Constantin, Lucian A; Fabiano, Eduardo; Della Sala, Fabio
2017-09-12
Using the semiclassical neutral atom theory, we developed a modified fourth-order kinetic energy (KE) gradient expansion (GE4m) that keeps unchanged all the linear-response terms of the uniform electron gas and gives a significant improvement with respect to the known semilocal functionals for both large atoms and jellium surfaces. On the other hand, GE4m is not accurate for light atoms; thus, we modified the GE4m coefficients making them dependent on a novel ingredient, the reduced Hartree potential, recently introduced in the Journal of Chemical Physics 2016, 145, 084110, in the context of exchange functionals. The resulting KE gradient expansion functional, named uGE4m, belongs to the novel class of u-meta-generalized-gradient-approximations (uMGGA) whose members depend on the conventional ingredients (i.e., the reduced gradient and Laplacian of the density) as well as on the reduced Hartree potential. To test uGE4m, we defined an appropriate benchmark (including total KE and KE differences for atoms, molecules and jellium clusters) for gradient expansion functionals, that is, including only those systems which are mainly described by a slowly varying density regime. While most of the GGA and meta-GGA KE functionals (we tested 18 of them) are accurate for some properties and inaccurate for others, uGE4m shows a consistently good performance for all the properties considered. This represents a qualitative boost in the KE functional development and highlights the importance of the reduced Hartree potential for the construction of next-generation KE functionals.
NASA Astrophysics Data System (ADS)
Ghosh, Shubhrangshu
2017-09-01
The correlated and coupled dynamics of accretion and outflow around black holes (BHs) are essentially governed by the fundamental laws of conservation as outflow extracts matter, momentum and energy from the accretion region. Here we analyze a robust form of 2.5-dimensional viscous, resistive, advective magnetized accretion-outflow coupling in BH systems. We solve the complete set of coupled MHD conservation equations self-consistently, through invoking a generalized polynomial expansion in two dimensions. We perform a critical analysis of the accretion-outflow region and provide a complete quasi-analytical family of solutions for advective flows. We obtain the physically plausible outflow solutions at high turbulent viscosity parameter α (≳ 0.3), and at a reduced scale-height, as magnetic stresses compress or squeeze the flow region. We found that the value of the large-scale poloidal magnetic field B P is enhanced with the increase of the geometrical thickness of the accretion flow. On the other hand, differential magnetic torque (-{r}2{\\bar{B}}\\varphi {\\bar{B}}z) increases with the increase in \\dot{M}. {\\bar{B}}{{P}}, -{r}2{\\bar{B}}\\varphi {\\bar{B}}z as well as the plasma beta β P get strongly augmented with the increase in the value of α, enhancing the transport of vertical flux outwards. Our solutions indicate that magnetocentrifugal acceleration plausibly plays a dominant role in effusing out plasma from the radial accretion flow in a moderately advective paradigm which is more centrifugally dominated. However in a strongly advective paradigm it is likely that the thermal pressure gradient would play a more contributory role in the vertical transport of plasma.
A multiple-scale Pascal polynomial for 2D Stokes and inverse Cauchy-Stokes problems
NASA Astrophysics Data System (ADS)
Liu, Chein-Shan; Young, D. L.
2016-05-01
The polynomial expansion method is a useful tool for solving both the direct and inverse Stokes problems, which together with the pointwise collocation technique is easy to derive the algebraic equations for satisfying the Stokes differential equations and the specified boundary conditions. In this paper we propose two novel numerical algorithms, based on a third-first order system and a third-third order system, to solve the direct and the inverse Cauchy problems in Stokes flows by developing a multiple-scale Pascal polynomial method, of which the scales are determined a priori by the collocation points. To assess the performance through numerical experiments, we find that the multiple-scale Pascal polynomial expansion method (MSPEM) is accurate and stable against large noise.
On the Beam Functions Spectral Expansions for Fourth-Order Boundary Value Problems
NASA Astrophysics Data System (ADS)
Papanicolaou, N. C.; Christov, C. I.
2007-10-01
In this paper we develop further the Galerkin technique based on the so-called beam functions with application to nonlinear problems. We make use of the formulas expressing a product of two beam functions into a series with respect to the system. First we prove that the overall convergence rate for a fourth-order linear b.v.p is algebraic fifth order, provided that the derivatives of the sought function up to fifth order exist. It is then shown that the inclusion of a quadratic nonlinear term in the equation does not degrade the fifth-order convergence. We validate our findings on a model problem which possesses analytical solution in the linear case. The agreement between the beam-Galerkin solution and the analytical solution for the linear problem is better than 10-12 for 200 terms. We also show that the error introduced by the expansion of the nonlinear term is lesser than 10-9. The beam-Galerkin method outperforms finite differences due to its superior accuracy whilst its advantage over the Chebyshev-tau method is attributed to the smaller condition number of the matrices involved in the former.
SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos
NASA Astrophysics Data System (ADS)
Ahlfeld, R.; Belkouchi, B.; Montomoli, F.
2016-09-01
A new arbitrary Polynomial Chaos (aPC) method is presented for moderately high-dimensional problems characterised by limited input data availability. The proposed methodology improves the algorithm of aPC and extends the method, that was previously only introduced as tensor product expansion, to moderately high-dimensional stochastic problems. The fundamental idea of aPC is to use the statistical moments of the input random variables to develop the polynomial chaos expansion. This approach provides the possibility to propagate continuous or discrete probability density functions and also histograms (data sets) as long as their moments exist, are finite and the determinant of the moment matrix is strictly positive. For cases with limited data availability, this approach avoids bias and fitting errors caused by wrong assumptions. In this work, an alternative way to calculate the aPC is suggested, which provides the optimal polynomials, Gaussian quadrature collocation points and weights from the moments using only a handful of matrix operations on the Hankel matrix of moments. It can therefore be implemented without requiring prior knowledge about statistical data analysis or a detailed understanding of the mathematics of polynomial chaos expansions. The extension to more input variables suggested in this work, is an anisotropic and adaptive version of Smolyak's algorithm that is solely based on the moments of the input probability distributions. It is referred to as SAMBA (PC), which is short for Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos. It is illustrated that for moderately high-dimensional problems (up to 20 different input variables or histograms) SAMBA can significantly simplify the calculation of sparse Gaussian quadrature rules. SAMBA's efficiency for multivariate functions with regard to data availability is further demonstrated by analysing higher order convergence and accuracy for a set of nonlinear test functions with 2, 5 and 10
SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos
Ahlfeld, R. Belkouchi, B.; Montomoli, F.
2016-09-01
A new arbitrary Polynomial Chaos (aPC) method is presented for moderately high-dimensional problems characterised by limited input data availability. The proposed methodology improves the algorithm of aPC and extends the method, that was previously only introduced as tensor product expansion, to moderately high-dimensional stochastic problems. The fundamental idea of aPC is to use the statistical moments of the input random variables to develop the polynomial chaos expansion. This approach provides the possibility to propagate continuous or discrete probability density functions and also histograms (data sets) as long as their moments exist, are finite and the determinant of the moment matrix is strictly positive. For cases with limited data availability, this approach avoids bias and fitting errors caused by wrong assumptions. In this work, an alternative way to calculate the aPC is suggested, which provides the optimal polynomials, Gaussian quadrature collocation points and weights from the moments using only a handful of matrix operations on the Hankel matrix of moments. It can therefore be implemented without requiring prior knowledge about statistical data analysis or a detailed understanding of the mathematics of polynomial chaos expansions. The extension to more input variables suggested in this work, is an anisotropic and adaptive version of Smolyak's algorithm that is solely based on the moments of the input probability distributions. It is referred to as SAMBA (PC), which is short for Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos. It is illustrated that for moderately high-dimensional problems (up to 20 different input variables or histograms) SAMBA can significantly simplify the calculation of sparse Gaussian quadrature rules. SAMBA's efficiency for multivariate functions with regard to data availability is further demonstrated by analysing higher order convergence and accuracy for a set of nonlinear test functions with 2, 5 and 10
High order spatial expansion for the method of characteristics applied to 3-D geometries
Naymeh, L.; Masiello, E.; Sanchez, R.
2013-07-01
The method of characteristics is an efficient and flexible technique to solve the neutron transport equation and has been extensively used in two-dimensional calculations because it permits to deal with complex geometries. However, because of a very fast increase in storage requirements and number of floating operations, its direct application to three-dimensional routine transport calculations it is not still possible. In this work we introduce and analyze several modifications aimed to reduce memory requirements and to diminish the computing burden. We explore high-order spatial approximation, the use of intermediary trajectory-dependent flux expansions and the possibility of dynamic trajectory reconstruction from local tracking for typed subdomains. (authors)
Rotating restricted Schur polynomials
NASA Astrophysics Data System (ADS)
Bornman, Nicholas; de Mello Koch, Robert; Tribelhorn, Laila
2017-09-01
Large N but nonplanar limits of 𝒩 = 4 super-Yang-Mills theory can be described using restricted Schur polynomials. Previous investigations demonstrate that the action of the one loop dilatation operator on restricted Schur operators, with classical dimension of order N and belonging to the su(2) sector, is largely determined by the su(2) ℛ symmetry algebra as well as structural features of perturbative field theory. Studies presented so far have used the form of ℛ symmetry generators when acting on small perturbations of half-BPS operators. In this paper as a first step towards going beyond small perturbations of the half-BPS operators, we explain how the exact action of symmetry generators on restricted Schur polynomials can be determined.
Simplified Phase Diversity algorithm based on a first-order Taylor expansion.
Zhang, Dong; Zhang, Xiaobin; Xu, Shuyan; Liu, Nannan; Zhao, Luoxin
2016-10-01
We present a simplified solution to phase diversity when the observed object is a point source. It utilizes an iterative linearization of the point spread function (PSF) at two or more diverse planes by first-order Taylor expansion to reconstruct the initial wavefront. To enhance the influence of the PSF in the defocal plane which is usually very dim compared to that in the focal plane, we build a new model with the Tikhonov regularization function. The new model cannot only increase the computational speed, but also reduce the influence of the noise. By using the PSFs obtained from Zemax, we reconstruct the wavefront of the Hubble Space Telescope (HST) at the edge of the field of view (FOV) when the telescope is in either the nominal state or the misaligned state. We also set up an experiment, which consists of an imaging system and a deformable mirror, to validate the correctness of the presented model. The result shows that the new model can improve the computational speed with high wavefront detection accuracy.
NASA Astrophysics Data System (ADS)
Guardia, M.; Kaloshin, V.; Zhang, J.
2016-11-01
In this paper we study a so-called separatrix map introduced by Zaslavskii-Filonenko (Sov Phys JETP 27:851-857, 1968) and studied by Treschev (Physica D 116(1-2):21-43, 1998; J Nonlinear Sci 12(1):27-58, 2002), Piftankin (Nonlinearity (19):2617-2644, 2006) Piftankin and Treshchëv (Uspekhi Mat Nauk 62(2(374)):3-108, 2007). We derive a second order expansion of this map for trigonometric perturbations. In Castejon et al. (Random iteration of maps of a cylinder and diffusive behavior. Preprint available at arXiv:1501.03319, 2015), Guardia and Kaloshin (Stochastic diffusive behavior through big gaps in a priori unstable systems (in preparation), 2015), and Kaloshin et al. (Normally Hyperbolic Invariant Laminations and diffusive behavior for the generalized Arnold example away from resonances. Preprint available at http://www.terpconnect.umd.edu/vkaloshi/, 2015), applying the results of the present paper, we describe a class of nearly integrable deterministic systems with stochastic diffusive behavior.
The q-Laguerre matrix polynomials.
Salem, Ahmed
2016-01-01
The Laguerre polynomials have been extended to Laguerre matrix polynomials by means of studying certain second-order matrix differential equation. In this paper, certain second-order matrix q-difference equation is investigated and solved. Its solution gives a generalized of the q-Laguerre polynomials in matrix variable. Four generating functions of this matrix polynomials are investigated. Two slightly different explicit forms are introduced. Three-term recurrence relation, Rodrigues-type formula and the q-orthogonality property are given.
NASA Astrophysics Data System (ADS)
Koçak, H.; Dahong, Z.; Yildirim, A.
2011-05-01
In this study, a range-free method is proposed in order to determine the Antoine constants for a given material (salicylic acid). The advantage of this method is mainly yielding analytical expressions which fit different temperature ranges.
Uniform Asymptotics of Orthogonal Polynomials Arising from Coherent States
NASA Astrophysics Data System (ADS)
Dai, Dan; Hu, Weiying; Wang, Xiang-Sheng
2015-08-01
In this paper, we study a family of orthogonal polynomials {φ_n(z)} arising from nonlinear coherent states in quantum optics. Based on the three-term recurrence relation only, we obtain a uniform asymptotic expansion of φ_n(z) as the polynomial degree n tends to infinity. Our asymptotic results suggest that the weight function associated with the polynomials has an unusual singularity, which has never appeared for orthogonal polynomials in the Askey scheme. Our main technique is the Wang and Wong's difference equation method. In addition, the limiting zero distribution of the polynomials φ_n(z) is provided.
NASA Astrophysics Data System (ADS)
Fukushima, Toshio
2017-07-01
In order to accelerate the spherical harmonic synthesis and/or analysis of arbitrary function on the unit sphere, we developed a pair of procedures to transform between a truncated spherical harmonic expansion and the corresponding two-dimensional Fourier series. First, we obtained an analytic expression of the sine/cosine series coefficient of the 4 π fully normalized associated Legendre function in terms of the rectangle values of the Wigner d function. Then, we elaborated the existing method to transform the coefficients of the surface spherical harmonic expansion to those of the double Fourier series so as to be capable with arbitrary high degree and order. Next, we created a new method to transform inversely a given double Fourier series to the corresponding surface spherical harmonic expansion. The key of the new method is a couple of new recurrence formulas to compute the inverse transformation coefficients: a decreasing-order, fixed-degree, and fixed-wavenumber three-term formula for general terms, and an increasing-degree-and-order and fixed-wavenumber two-term formula for diagonal terms. Meanwhile, the two seed values are analytically prepared. Both of the forward and inverse transformation procedures are confirmed to be sufficiently accurate and applicable to an extremely high degree/order/wavenumber as 2^{30} {≈ } 10^9 . The developed procedures will be useful not only in the synthesis and analysis of the spherical harmonic expansion of arbitrary high degree and order, but also in the evaluation of the derivatives and integrals of the spherical harmonic expansion.
Improved polynomial remainder sequences for Ore polynomials.
Jaroschek, Maximilian
2013-11-01
Polynomial remainder sequences contain the intermediate results of the Euclidean algorithm when applied to (non-)commutative polynomials. The running time of the algorithm is dependent on the size of the coefficients of the remainders. Different ways have been studied to make these as small as possible. The subresultant sequence of two polynomials is a polynomial remainder sequence in which the size of the coefficients is optimal in the generic case, but when taking the input from applications, the coefficients are often larger than necessary. We generalize two improvements of the subresultant sequence to Ore polynomials and derive a new bound for the minimal coefficient size. Our approach also yields a new proof for the results in the commutative case, providing a new point of view on the origin of the extraneous factors of the coefficients.
Chaos, Fractals, and Polynomials.
ERIC Educational Resources Information Center
Tylee, J. Louis; Tylee, Thomas B.
1996-01-01
Discusses chaos theory; linear algebraic equations and the numerical solution of polynomials, including the use of the Newton-Raphson technique to find polynomial roots; fractals; search region and coordinate systems; convergence; and generating color fractals on a computer. (LRW)
Factoring Polynomials and Fibonacci.
ERIC Educational Resources Information Center
Schwartzman, Steven
1986-01-01
Discusses the factoring of polynomials and Fibonacci numbers, offering several challenges teachers can give students. For example, they can give students a polynomial containing large numbers and challenge them to factor it. (JN)
Factoring Polynomials and Fibonacci.
ERIC Educational Resources Information Center
Schwartzman, Steven
1986-01-01
Discusses the factoring of polynomials and Fibonacci numbers, offering several challenges teachers can give students. For example, they can give students a polynomial containing large numbers and challenge them to factor it. (JN)
Chaos, Fractals, and Polynomials.
ERIC Educational Resources Information Center
Tylee, J. Louis; Tylee, Thomas B.
1996-01-01
Discusses chaos theory; linear algebraic equations and the numerical solution of polynomials, including the use of the Newton-Raphson technique to find polynomial roots; fractals; search region and coordinate systems; convergence; and generating color fractals on a computer. (LRW)
NASA Astrophysics Data System (ADS)
McRae, S. M.; Vrscay, E. R.
1992-09-01
The classical hypervirial and Hellmann-Feynman theorems are used to formulate a "perturbation theory without Fourier series" that can be used to generate canonical series expansions for the energies of perturbed periodic orbits for separable classical Hamiltonians. Here, the method is applied to one-dimensional anharmonic oscillators and radial Kepler problems. In all cases, the classical series for energies and expectation values are seen to correspond to the expansions associated with their quantum mechanical counterparts through an appropriate action preserving classical limit. This "action fixing" is inherent in the classical Hellmann-Feynman theorem applied to periodic orbits.
Akbar, M Ali; Hj Mohd Ali, Norhashidah
2014-01-01
The exp(-Ф(η))-expansion method is an ascending method for obtaining exact and solitary wave solutions for nonlinear evolution equations. In this article, we implement the exp(-Ф(η))-expansion method to build solitary wave solutions to the fourth order Boussinesq equation. The procedure is simple, direct and useful with the help of computer algebra. By using this method, we obtain solitary wave solutions in terms of the hyperbolic functions, the trigonometric functions and elementary functions. The results show that the exp(-Ф(η))-expansion method is straightforward and effective mathematical tool for the treatment of nonlinear evolution equations in mathematical physics and engineering. 35C07; 35C08; 35P99.
Expansion of Einstein-Yang-Mills amplitude
NASA Astrophysics Data System (ADS)
Fu, Chih-Hao; Du, Yi-Jian; Huang, Rijun; Feng, Bo
2017-09-01
In this paper, we study from various perspectives the expansion of tree level single trace Einstein-Yang-Mills amplitudes into linear combination of color-ordered Yang-Mills amplitudes. By applying the gauge invariance principle, a programable recursive construction is devised to expand EYM amplitude with arbitrary number of gravitons into EYM amplitudes with fewer gravitons. Based on this recursive technique we write down the complete expansion of any single trace EYM amplitude in the basis of color-order Yang-Mills amplitude. As a byproduct, an algorithm for constructing a polynomial form of the BCJ numerator for Yang-Mills amplitudes is also outlined in this paper. In addition, by applying BCFW recursion relation we show how to arrive at the same EYM amplitude expansion from the on-shell perspective. And we examine the EYM expansion using KLT relations and show how to evaluate the expansion coefficients efficiently.
NASA Astrophysics Data System (ADS)
Wang, Jia Jie; Wriedt, Thomas; Mädler, Lutz; Han, Yi Ping; Hartmann, Peter
2017-03-01
A rigorous, simple and efficient approach is derived in this paper for multipole expansion of a circularly symmetric Bessel beam. Different from the existing rigorous methods which are based on the plane wave spectrum of a Bessel beam, a straight-forward integral procedure is presented in a traditional way to obtain the analytical expressions of the expansion coefficients, also called beam shape coefficients (BSCs). The convergence and correctness of the BSCs are verified numerically in detail for both on-axis and off-axis cases. The results in this paper are useful in various analytical scattering theories, such as the generalized Lorenz-Mie theory and the Null-field method, when a Bessel beam is considered.
NASA Astrophysics Data System (ADS)
Gil, Amparo; Segura, Javier; Temme, Nico M.
2003-04-01
The use of a uniform Airy-type asymptotic expansion for the computation of the modified Bessel functions of the third kind of imaginary orders (Kia(x)) near the transition point x=a, is discussed. In A. Gil et al., Evaluation of the modified Bessel functions of the third kind of imaginary orders, J. Comput. Phys. 17 (2002) 398-411, an algorithm for the evaluation of Kia(x) was presented, which made use of series, a continued fraction method and nonoscillating integral representations. The range of validity of the algorithm was limited by the singularity of the steepest descent paths near the transition point. We show how uniform Airy-type asymptotic expansions fill the gap left by the steepest descent method.
NASA Astrophysics Data System (ADS)
Mohamed, Firdawati binti; Karim, Mohamad Faisal bin Abd
2015-10-01
Modelling physical problems in mathematical form yields the governing equations that may be linear or nonlinear for known and unknown boundaries. The exact solution for those equations may or may not be obtained easily. Hence we seek an analytical approximation solution in terms of asymptotic expansion. In this study, we focus on a singular perturbation in second order ordinary differential equations. Solutions to several perturbed ordinary differential equations are obtained in terms of asymptotic expansion. The aim of this work is to find an approximate analytical solution using the classical method of matched asymptotic expansion (MMAE). The Mathematica computer algebra system is used to perform the algebraic computations. The details procedures will be discussed and the underlying concepts and principles of the MMAE will be clarified. Perturbation problem for linear equation that occurs at one boundary and two boundary layers are discussed. Approximate analytical solution obtained for both cases are illustrated by graph using selected parameter by showing the outer, inner and composite solution separately. Then, the composite solution will be compare to the exact solution to show their accuracy by graph. By comparison, MMAE is found to be one of the best methods to solve singular perturbation problems in second order ordinary differential equation since the results obtained are very close to the exact solution.
NASA Technical Reports Server (NTRS)
Wood, C. A.
1974-01-01
For polynomials of higher degree, iterative numerical methods must be used. Four iterative methods are presented for approximating the zeros of a polynomial using a digital computer. Newton's method and Muller's method are two well known iterative methods which are presented. They extract the zeros of a polynomial by generating a sequence of approximations converging to each zero. However, both of these methods are very unstable when used on a polynomial which has multiple zeros. That is, either they fail to converge to some or all of the zeros, or they converge to very bad approximations of the polynomial's zeros. This material introduces two new methods, the greatest common divisor (G.C.D.) method and the repeated greatest common divisor (repeated G.C.D.) method, which are superior methods for numerically approximating the zeros of a polynomial having multiple zeros. These methods were programmed in FORTRAN 4 and comparisons in time and accuracy are given.
Feenberg, E.; Lee, D.K.
1982-03-01
A study is made of a series-expansion procedure which gives the leading terms of the n-particle distribution function p/sup( n/)(1,2,...,n) as explicit functionals in the radial distribution function g(r). The development of the series is based on the cluster-expansion formalism applied to the Abe form for p/sup( n/) expressed as a product of the generalized Kirkwood superposition approximation P/sup( n/)/sub K/ and a correction factor exp(A/sup( n/)(1,2,...,n)). An ordering parameter ..mu.. is introduced to determine A/sup( n/) and p/sup( n/) in the form of infinite power series in ..mu.., and the postulate of minimal complexity is employed to eliminate an infinite number of possible classes of solutions in a sequential relation connecting A/sup( n/-1) and A/sup( n/). Derivation of the series for p/sup( n/) and many other algebraic manipulations involving a large number of cluster integrals are greatly simplified by the use of a scheme which groups together all cluster terms having, in a certain way, the same source term. In particular, the scheme is useful in demonstrating that the nature of the series structure of p/sup(/sup 3/) is such that its three-point Fourier transform S/sup(/sup 3/)(k/sub 1/,k/sub 2/,k/sub 3/) has as a factor the product of the three liquid-structure functions S(k/sub 1/)S(k/sub 2/)S(k/sub 3/). The results obtained to order ..mu../sup 4/ for A/sup(/sup 3/), p/sup(/sup 3/), and S/sup(/sup 3/) agree with those derived earlier in a more straightforward but tedious approach. The result for p/sup(/sup 4/) shows that the convolution approximation p/sup(/sup 4/)/sub c/, which contains ..mu../sup 3/ terms, must be supplemented by a correction of O(..mu../sup 3/) in order to be accurate through third order. The ..mu..-expansion approach is also examined for the cluster expansion of the correlation function in the Bijl-Dingle-Jastrow description of a many-boson system, and then compared with the number-density expansion formula by using the
NASA Astrophysics Data System (ADS)
Miller, W., Jr.; Li, Q.
2015-04-01
The Wilson and Racah polynomials can be characterized as basis functions for irreducible representations of the quadratic symmetry algebra of the quantum superintegrable system on the 2-sphere, HΨ = EΨ, with generic 3-parameter potential. Clearly, the polynomials are expansion coefficients for one eigenbasis of a symmetry operator L2 of H in terms of an eigenbasis of another symmetry operator L1, but the exact relationship appears not to have been made explicit. We work out the details of the expansion to show, explicitly, how the polynomials arise and how the principal properties of these functions: the measure, 3-term recurrence relation, 2nd order difference equation, duality of these relations, permutation symmetry, intertwining operators and an alternate derivation of Wilson functions - follow from the symmetry of this quantum system. This paper is an exercise to show that quantum mechancal concepts and recurrence relations for Gausian hypergeometrc functions alone suffice to explain these properties; we make no assumptions about the structure of Wilson polynomial/functions, but derive them from quantum principles. There is active interest in the relation between multivariable Wilson polynomials and the quantum superintegrable system on the n-sphere with generic potential, and these results should aid in the generalization. Contracting function space realizations of irreducible representations of this quadratic algebra to the other superintegrable systems one can obtain the full Askey scheme of orthogonal hypergeometric polynomials. All of these contractions of superintegrable systems with potential are uniquely induced by Wigner Lie algebra contractions of so(3, C) and e(2,C). All of the polynomials produced are interpretable as quantum expansion coefficients. It is important to extend this process to higher dimensions.
A nodal expansion method for the neutron diffusion equation in cylindrical geometry
Komlev, O.G.; Suslov, I.R.
1995-12-31
A polynomial nodal expansion method (NEM) is applied to solve multigroup diffusion equations in cylindrical R-Z geometry, Fourth-order polynomials are used to approximate one dimensional (1D) transverse integrated fluxes. The special set of the basis functions is used in R-direction. The transverse integrated leakages are approximated by both constant and quadratic polynomials. Preliminary efficiency evaluation of the NEM is carried out for a fast breeder reactor (FBR) model problem. Results indicate computational efficiency of NEM in comparison with finite-difference method (FDM).
Ubiquity of Kostka Polynomials
NASA Astrophysics Data System (ADS)
Kirillov, Anatol N.
2001-04-01
We report about results revolving around Kostka-Foulkes and parabolic Kostka polynomials and their connections with Representation Theory and Combinatorics. It appears that the set of all parabolic Kostka polynomials forms a semigroup, which we call Liskova semigroup. We show that polynomials frequently appearing in Representation Theory and Combinatorics belong to the Liskova semigroup. Among such polynomials we study rectangular q-Catalan numbers; generalized exponents polynomials; principal specializations of the internal product of Schur functions; generalized q-Gaussian polynomials; parabolic Kostant partition function and its q-analog certain generating functions on the set of transportation matrices. In each case we apply rigged configurations technique to obtain some interesting and new information about Kostka-Foulkes and parabolic Kostka polynomials, Kostant partition function, MacMahon, Gelfand-Tsetlin and Chan-Robbins polytopes. We describe certain connections between generalized saturation and Fulton's conjectures and parabolic Kostka polynomials; domino tableaux and rigged configurations. We study also some properties of l-restricted generalized exponents and the stable behaviour of certain Kostka-Foulkes polynomials.
High-quality two-nucleon potentials up to fifth order of the chiral expansion
NASA Astrophysics Data System (ADS)
Entem, D. R.; Machleidt, R.; Nosyk, Y.
2017-08-01
We present NN potentials through five orders of chiral effective field theory ranging from leading order (LO) to next-to-next-to-next-to-next-to-leading order (N4LO ). The construction may be perceived as consistent in the sense that the same power counting scheme as well as the same cutoff procedures are applied in all orders. Moreover, the long-range parts of these potentials are fixed by the very accurate π N low-energy constants (LECs) as determined in the Roy-Steiner equations analysis by Hoferichter, Ruiz de Elvira, and coworkers. In fact, the uncertainties of these LECs are so small that a variation within the errors leads to effects that are essentially negligible, reducing the error budget of predictions considerably. The NN potentials are fit to the world NN data below the pion-production threshold of the year 2016. The potential of the highest order (N4LO ) reproduces the world NN data with the outstanding χ2/datum of 1.15, which is the highest precision ever accomplished for any chiral NN potential to date. The NN potentials presented may serve as a solid basis for systematic ab initio calculations of nuclear structure and reactions that allow for a comprehensive error analysis. In particular, the consistent order by order development of the potentials will make possible a reliable determination of the truncation error at each order. Our family of potentials is nonlocal and, generally, of soft character. This feature is reflected in the fact that the predictions for the triton binding energy (from two-body forces only) converges to about 8.1 MeV at the highest orders. This leaves room for three-nucleon-force contributions of moderate size.
The Operator Product Expansion Beyond Leading Order for Spin-1/2 Fermions
NASA Astrophysics Data System (ADS)
Emmons, Samuel; Kang, Daekyoung; Platter, Lucas
2016-05-01
Strongly interacting systems of ultracold, two-component fermions have been studied using various techniques for many years. One technique that has been applied is a quantum field theoretical formulation of the zero-range model. In this framework, the Operator Product Expansion was used to derive universal relations for systems with a large scattering length. This corroborated and extended the work of Tan. We calculate finite range corrections to the momentum distribution using the OPE framework and demonstrate the utility of including the 1 /k6 tail from the OPE for the momentum distribution. Corrections to the universal relations for the system are calculated and expressed in terms of the S-wave effective range and an additional quantity D similar to Tan's contact which, in addition to the contact, relates various physical observables. We compare our results with quantum Monte Carlo calculations for the two-component Fermi gas with large scattering length. NSF Grant No. PHY-1516077; U.S. DOE Office of Science, Office of Nuclear Physics Contract Nos. DE-AC52-06NA25396, DE-AC05-00OR22725, an Early Career Research Award; LANL/LDRD Program.
NASA Astrophysics Data System (ADS)
Fuentes, Humberto Peredo
2017-09-01
The application of different mode-shape expansion (MSE) methods to a CFRP based on model order reduction (MOR) and component mode synthesis (CMS) methods is evaluated combining the updated stiffness parameters of the full FE model obtained with a mix-numerical experimental technique (MNET) in a previous work. The eigenvectors and eigenfrequencies of the different MSE methods obtained are compared with respect to the experimental measurements and with a full FE model solutions using the modal assurance criteria (MAC). Furthermore, the stiffness and mass weighted coefficients (K-MAC and M-MAC respectively) are calculated and compared to observe the influence of the different subspace based expansion methods applying the MAC criteria. The K-MAC and M-MAC are basically the MAC coefficients weighted by a partition of the global stiffness and mass matrices respectively. The best K-MAC and M-MAC results per paired mode-sensor are observed in the subspace based expansion MODAL/SEREP and MDRE-WE methods using the updated stiffness parameters. A strong influence of the subspace based on MOR using MSE methods is observed in the K-MAC and M-MAC criteria implemented in SDTools evaluating the stiffness parameters in a contrieved example.
Thermal expansion of the hexagonal (6H) polytype of silicon carbide
NASA Technical Reports Server (NTRS)
Li, Z.; Bradt, R. C.
1986-01-01
X-ray diffraction is presently used to determine the thermal expansion of the hexagonal (6H) polytype of alpha-SiC over the 20-1000 C range. The principal (alpha-11 and alpha-33) axial coefficients of thermal expansion can be expressed by second-order polynomials; the former is noted to be larger than the latter over the entire temperature range, while the thermal expansion anisotropy increases continuously with increasing temperature. The thermal expansion and thermal expansion anisotropy obtained are compared with previously published results for the (6H) polytype, and discussed with respect to the structure.
Magnetic cluster expansion model for random and ordered magnetic face-centered cubic Fe-Ni-Cr alloys
Lavrentiev, M. Yu. Nguyen-Manh, D.; Dudarev, S. L.; Wróbel, J. S.; Ganchenkova, M. G.
2016-07-28
A Magnetic Cluster Expansion model for ternary face-centered cubic Fe-Ni-Cr alloys has been developed, using DFT data spanning binary and ternary alloy configurations. Using this Magnetic Cluster Expansion model Hamiltonian, we perform Monte Carlo simulations and explore magnetic structures of alloys over the entire range of compositions, considering both random and ordered alloy structures. In random alloys, the removal of magnetic collinearity constraint reduces the total magnetic moment but does not affect the predicted range of compositions where the alloys adopt low-temperature ferromagnetic configurations. During alloying of ordered fcc Fe-Ni compounds with Cr, chromium atoms tend to replace nickel rather than iron atoms. Replacement of Ni by Cr in ordered alloys with high iron content increases the Curie temperature of the alloys. This can be explained by strong antiferromagnetic Fe-Cr coupling, similar to that found in bcc Fe-Cr solutions, where the Curie temperature increase, predicted by simulations as a function of Cr concentration, is confirmed by experimental observations. In random alloys, both magnetization and the Curie temperature decrease abruptly with increasing chromium content, in agreement with experiment.
Datta, Nilanjana; Hsieh, Min-Hsiu; Oppenheim, Jonathan
2016-05-15
State redistribution is the protocol in which given an arbitrary tripartite quantum state, with two of the subsystems initially being with Alice and one being with Bob, the goal is for Alice to send one of her subsystems to Bob, possibly with the help of prior shared entanglement. We derive an upper bound on the second order asymptotic expansion for the quantum communication cost of achieving state redistribution with a given finite accuracy. In proving our result, we also obtain an upper bound on the quantum communication cost of this protocol in the one-shot setting, by using the protocol of coherent state merging as a primitive.
Polynomial Graphs and Symmetry
ERIC Educational Resources Information Center
Goehle, Geoff; Kobayashi, Mitsuo
2013-01-01
Most quadratic functions are not even, but every parabola has symmetry with respect to some vertical line. Similarly, every cubic has rotational symmetry with respect to some point, though most cubics are not odd. We show that every polynomial has at most one point of symmetry and give conditions under which the polynomial has rotational or…
Polynomial Graphs and Symmetry
ERIC Educational Resources Information Center
Goehle, Geoff; Kobayashi, Mitsuo
2013-01-01
Most quadratic functions are not even, but every parabola has symmetry with respect to some vertical line. Similarly, every cubic has rotational symmetry with respect to some point, though most cubics are not odd. We show that every polynomial has at most one point of symmetry and give conditions under which the polynomial has rotational or…
NASA Astrophysics Data System (ADS)
Sahoo, S.; Saha Ray, S.
2016-04-01
In the present paper, we construct the analytical exact solutions of a nonlinear evolution equation in mathematical physics; namely time fractional modified KdV equation by using (G‧ / G)-expansion method and improved (G‧ / G)-expansion method. As a result, new types of exact analytical solutions are obtained.
Second-order many-body perturbation expansions of vibrational Dyson self-energies.
Hermes, Matthew R; Hirata, So
2013-07-21
Second-order many-body perturbation theories for anharmonic vibrational frequencies and zero-point energies of molecules are formulated, implemented, and tested. They solve the vibrational Dyson equation self-consistently by taking into account the frequency dependence of the Dyson self-energy in the diagonal approximation, which is expanded in a diagrammatic perturbation series up to second order. Three reference wave functions, all of which are diagrammatically size consistent, are considered: the harmonic approximation and diagrammatic vibrational self-consistent field (XVSCF) methods with and without the first-order Dyson geometry correction, i.e., XVSCF[n] and XVSCF(n), where n refers to the truncation rank of the Taylor-series potential energy surface. The corresponding second-order perturbation theories, XVH2(n), XVMP2[n], and XVMP2(n), are shown to be rigorously diagrammatically size consistent for both total energies and transition frequencies, yield accurate results (typically within a few cm(-1) at n = 4 for water and formaldehyde) for both quantities even in the presence of Fermi resonance, and have access to fundamentals, overtones, and combinations as well as their relative intensities as residues of the vibrational Green's functions. They are implemented into simple algorithms that require only force constants and frequencies of the reference methods (with no basis sets, quadrature, or matrix diagonalization at any stage of the calculation). The rules for enumerating and algebraically interpreting energy and self-energy diagrams are elucidated in detail.
Source emission-pattern polynomial representation
NASA Astrophysics Data System (ADS)
Flores-Hernandez, Ricardo; De Villa, Francisco
1990-12-01
A method to obtain accurate thickness data to characterize the emission patterns of evaporation sources is described. Thickness data is obtained through digital image processing algorithms applied to the monochromatic transmission bands digitized from a set of multilayer Fabry-Perot filters deposited on large flat circular substrates. These computer image-processed taper-thickness patterns are reduced to orthonormal polynomial series expansions in two steps, using Tschebyshev and associated Legendre polynomials. The circular glass substrates employed to characterize each type of evaporation source are kept stationary during the evaporation process of evaporation of each layer to obtain the specific thickness distribution for each type of source.
Parabolic Refined Invariants and Macdonald Polynomials
NASA Astrophysics Data System (ADS)
Chuang, Wu-yen; Diaconescu, Duiliu-Emanuel; Donagi, Ron; Pantev, Tony
2015-05-01
A string theoretic derivation is given for the conjecture of Hausel, Letellier and Rodriguez-Villegas on the cohomology of character varieties with marked points. Their formula is identified with a refined BPS expansion in the stable pair theory of a local root stack, generalizing previous work of the first two authors in collaboration with Pan. Haiman's geometric construction for Macdonald polynomials is shown to emerge naturally in this context via geometric engineering. In particular this yields a new conjectural relation between Macdonald polynomials and refined local orbifold curve counting invariants. The string theoretic approach also leads to a new spectral cover construction for parabolic Higgs bundles in terms of holomorphic symplectic orbifolds.
More on rotations as spin matrix polynomials
Curtright, Thomas L.
2015-09-15
Any nonsingular function of spin j matrices always reduces to a matrix polynomial of order 2j. The challenge is to find a convenient form for the coefficients of the matrix polynomial. The theory of biorthogonal systems is a useful framework to meet this challenge. Central factorial numbers play a key role in the theoretical development. Explicit polynomial coefficients for rotations expressed either as exponentials or as rational Cayley transforms are considered here. Structural features of the results are discussed and compared, and large j limits of the coefficients are examined.
Perturbation expansion and Nth order Fermi golden rule of the nonlinear Schrödinger equations
NASA Astrophysics Data System (ADS)
Zhou, Gang
2007-05-01
In this paper we consider generalized nonlinear Schrödinger equations with external potentials. We find the expressions for the fourth and the sixth order Fermi golden rules (FGRs), conjectured in Gang and Sigal [Rev. Math. Phys. 17, 1143-1207 (2005); Geom. Funct. Anal. 16, No. 7, 1377-1390 (2006)]. The FGR is a key condition in a study of the asymptotic dynamics of trapped solitons.
NASA Technical Reports Server (NTRS)
Gopinath, Ashok
1996-01-01
Analytical and numerical studies are to be carried out to examine time-averaged thermal effects which are induced by the interaction of strong acoustic fields with a rigid boundary (thermoacoustic streaming). Also of interest is the significance of a second-order thermal expansion coefficient that emerges from this analysis. The model problem to be considered is that of a sphere that is acoustically levitated such that it is effectively isolated in a high-intensity standing acoustic field. The solution technique involves matched asymptotic analysis along with numerical solution of the boundary layer equations. The objective of this study is to predict the thermoacoustic streaming behavior and fully understand the role of the associated second-order thermodynamic modulus.
NASA Astrophysics Data System (ADS)
Suzuki, Kimichi; Shiga, Motoyuki; Tachikawa, Masanori
2008-10-01
Path integral molecular dynamics simulation based on the fourth order Trotter expansion has been performed to elucidate the geometrical isotope effect of water dimer anions, H3O2-, D3O2-, and T3O2-, at different temperatures from 50 to 600 K. At low temperatures below 200 K the hydrogen-bonded hydrogen nucleus is near the center of two oxygen atoms with mostly O⋯X⋯O geometry (where X =H, D, or T), while at high temperatures above 400 K, hydrogen becomes more delocalized, showing the coexistence between O⋯X-O and O-X⋯O. The OO distance tends to be shorter as the isotopomer is heavier at low temperatures, while this ordering becomes opposite at high temperatures. It is concluded that the coupling between the OO stretching mode and proton transfer modes is a key to understand such a temperature dependence of a hydrogen-bonded structure.
Some identities of the q-Laguerre polynomials on q-Umbral calculus
NASA Astrophysics Data System (ADS)
Dere, Rahime
2017-07-01
Some interesting identities of Sheffer polynomials was given by Roman [11], [12]. In this paper, we study a q-analogue of Laguerre polynomials which are also q-Sheffer polynomials. Furthermore, we give some new properties and formulas of these q-Laguerre polynomials of higher order by using the theory of the umbral algebra and umbral calculus.
Gauss-Lobatto to Bernstein polynomials transformation
NASA Astrophysics Data System (ADS)
Coluccio, Loredana; Eisinberg, Alfredo; Fedele, Giuseppe
2008-12-01
The aim of this paper is to transform a polynomial expressed as a weighted sum of discrete orthogonal polynomials on Gauss-Lobatto nodes into Bernstein form and vice versa. Explicit formulas and recursion expressions are derived. Moreover, an efficient algorithm for the transformation from Gauss-Lobatto to Bernstein is proposed. Finally, in order to show the robustness of the proposed algorithm, experimental results are reported.
Pakhira, Anindya; Das, Saptarshi; Pan, Indranil; Das, Shantanu
2015-07-01
This paper uses the Continued Fraction Expansion (CFE) method for analog realization of fractional order differ-integrator and few special classes of fractional order (FO) controllers viz. Fractional Order Proportional-Integral-Derivative (FOPID) controller, FO[PD] controller and FO lead-lag compensator. Contemporary researchers have given several formulations for rational approximation of fractional order elements. However, approximation of the controllers studied in this paper, due to having fractional power of a rational transfer function, is not available in analog domain; although its digital realization already exists. This motivates us for applying CFE based analog realization technique for complicated FO controller structures to get equivalent rational transfer functions in terms of the controller tuning parameters. The symbolic expressions for rationalized transfer function in terms of the controller tuning parameters are especially important as ready references, without the need of running CFE algorithm every time and also helps in the synthesis of analog circuits for such FO controllers. Copyright © 2015 ISA. Published by Elsevier Ltd. All rights reserved.
NASA Astrophysics Data System (ADS)
Barabash, Sergey V.; Blum, Volker; Müller, Stefan; Zunger, Alex
2006-07-01
We describe an iterative procedure which yields an accurate cluster expansion for Au-Pd using only a limited number of ab initio formation enthalpies. Our procedure addresses two problems: (a) given the local-density-approximation (LDA) formation energies for a fixed set of structures, it finds the pair and many-body cluster interactions best able to predict the formation energies of new structures, and (b) given such pair and many-body interactions, it augments the LDA set of “input structures” by identifying additional structures that carry most information not yet included in the “input.” Neither step can be done by intuitive selection. Using methods including genetic algorithm and statistical analysis to iteratively solve these problems, we build a cluster expansion able to predict the formation enthalpy of an arbitrary fcc lattice configuration with precision comparable to that of ab initio calculations themselves. We also study possible competing non-fcc structures of Au-Pd, using the results of a “data mining” study. We then address the unresolved problem of bulk ordering in Au-Pd. Experimentally, the phase diagram of Au-Pd shows only a disordered solid solution. Even though the mixing enthalpy is negative, implying ordering, no ordered bulk phases have been detected. Thin film growth shows L12 -ordered structures with composition Au3Pd and AuPd3 and L10 structure with composition AuPd. We find that (i) all the ground states of Au-Pd are fcc structures; (ii) the low- T ordered states of bulk Au-Pd are different from those observed experimentally in thin films; specifically, the ordered bulk Au3Pd is stable in D023 structure and and AuPd in chalcopyritelike Au2Pd2 (201) superlattice structure, whereas thin films are seen in the L12 and L10 structures; (iii) AuPd3 L12 is stable and does not phase separate, contrary to the suggestions of an earlier investigation; (iv) at compositions around Au3Pd , we find several long-period superstructures (LPS
Chen, Weisheng; Li, Wei; Miao, Qiguang
2010-07-01
An adaptive backstepping tracking scheme is developed for a class of strict-feedback systems with unknown periodically time-varying parameters and unknown control gain functions. High-order neural network (HONN) and Fourier series expansion (FSE) are combined into a new function approximator to model each uncertain term in the system. The dynamic surface control (DSC) approach is used to solve the problem of 'explosion of complexity' in the backstepping design procedure. Nussbaum gain function (NGF) is employed to deal with the unknown control gain functions. The uniform boundedness of all closed-loop signals is guaranteed. The tracking error is proved to converge to a small residual set around the origin. Two simulation examples are provided to demonstrate the effectiveness of the control scheme designed in this paper. 2010 ISA. Published by Elsevier Ltd. All rights reserved.
Chemical Reaction Networks for Computing Polynomials.
Salehi, Sayed Ahmad; Parhi, Keshab K; Riedel, Marc D
2017-01-20
Chemical reaction networks (CRNs) provide a fundamental model in the study of molecular systems. Widely used as formalism for the analysis of chemical and biochemical systems, CRNs have received renewed attention as a model for molecular computation. This paper demonstrates that, with a new encoding, CRNs can compute any set of polynomial functions subject only to the limitation that these functions must map the unit interval to itself. These polynomials can be expressed as linear combinations of Bernstein basis polynomials with positive coefficients less than or equal to 1. In the proposed encoding approach, each variable is represented using two molecular types: a type-0 and a type-1. The value is the ratio of the concentration of type-1 molecules to the sum of the concentrations of type-0 and type-1 molecules. The proposed encoding naturally exploits the expansion of a power-form polynomial into a Bernstein polynomial. Molecular encoders for converting any input in a standard representation to the fractional representation as well as decoders for converting the computed output from the fractional to a standard representation are presented. The method is illustrated first for generic CRNs; then chemical reactions designed for an example are mapped to DNA strand-displacement reactions.
Gaussian quadrature for multiple orthogonal polynomials
NASA Astrophysics Data System (ADS)
Coussement, Jonathan; van Assche, Walter
2005-06-01
We study multiple orthogonal polynomials of type I and type II, which have orthogonality conditions with respect to r measures. These polynomials are connected by their recurrence relation of order r+1. First we show a relation with the eigenvalue problem of a banded lower Hessenberg matrix Ln, containing the recurrence coefficients. As a consequence, we easily find that the multiple orthogonal polynomials of type I and type II satisfy a generalized Christoffel-Darboux identity. Furthermore, we explain the notion of multiple Gaussian quadrature (for proper multi-indices), which is an extension of the theory of Gaussian quadrature for orthogonal polynomials and was introduced by Borges. In particular, we show that the quadrature points and quadrature weights can be expressed in terms of the eigenvalue problem of Ln.
Chen, Hongbin; Fitzpatrick, A. Liam; Kaplan, Jared; ...
2017-03-30
Here, one can obtain exact information about Virasoro conformal blocks by analytically continuing the correlators of degenerate operators. We argued in recent work that this technique can be used to explicitly resolve information loss problems in AdS3/CFT2. In this paper we use the technique to perform calculations in the small 1/c ∝ GN expansion: (1) we prove the all-orders resummation of logarithmic factors ∝1/clog z in the Lorentzian regime, demonstrating that 1/c corrections directly shift Lyapunov exponents associated with chaos, as claimed in prior work, (2) we perform another all-orders resummation in the limit of large c with fixed cz,more » interpolating between the early onset of chaos and late time behavior, (3) we explicitly compute the Virasoro vacuum block to order 1/c2 and 1/c3 with external dimensions fixed, corresponding to 2 and 3 loop calculations in AdS3, and (4) we derive the heavy-light vacuum blocks in theories with N = 1, 2 superconformal symmetry.« less
Zinn-Justin, Jean; Jentschura, Ulrich D.
2010-07-15
A long time ago, it has been conjectured that a Hamiltonian with a potential of the form x{sup 2}+ivx{sup 3}, v real, has a real spectrum. This conjecture has been generalized to a class of the so-called PT symmetric Hamiltonians and some proofs have been given. Here, we show by numerical investigation that the divergent perturbation series can be summed efficiently by an order-dependent mapping (ODM) in the whole complex plane of the coupling parameter v{sup 2}, and that some information about the location of level-crossing singularities can be obtained in this way. Furthermore, we discuss to which accuracy the strong-coupling limit can be obtained from the initially weak-coupling perturbative expansion, by the ODM summation method. The basic idea of the ODM summation method is the notion of order-dependent 'local' disk of convergence and analytic continuation by an ODM of the domain of analyticity augmented by the local disk of convergence onto a circle. In the limit of vanishing local radius of convergence, which is the limit of high transformation order, convergence is demonstrated both by numerical evidence as well as by analytic estimates.
NASA Astrophysics Data System (ADS)
Zinn-Justin, Jean; Jentschura, Ulrich D.
2010-07-01
A long time ago, it has been conjectured that a Hamiltonian with a potential of the form x2+ivx3, v real, has a real spectrum. This conjecture has been generalized to a class of the so-called PT symmetric Hamiltonians and some proofs have been given. Here, we show by numerical investigation that the divergent perturbation series can be summed efficiently by an order-dependent mapping (ODM) in the whole complex plane of the coupling parameter v2, and that some information about the location of level-crossing singularities can be obtained in this way. Furthermore, we discuss to which accuracy the strong-coupling limit can be obtained from the initially weak-coupling perturbative expansion, by the ODM summation method. The basic idea of the ODM summation method is the notion of order-dependent "local" disk of convergence and analytic continuation by an ODM of the domain of analyticity augmented by the local disk of convergence onto a circle. In the limit of vanishing local radius of convergence, which is the limit of high transformation order, convergence is demonstrated both by numerical evidence as well as by analytic estimates.
Extending Romanovski polynomials in quantum mechanics
NASA Astrophysics Data System (ADS)
Quesne, C.
2013-12-01
Some extensions of the (third-class) Romanovski polynomials (also called Romanovski/pseudo-Jacobi polynomials), which appear in bound-state wavefunctions of rationally extended Scarf II and Rosen-Morse I potentials, are considered. For the former potentials, the generalized polynomials satisfy a finite orthogonality relation, while for the latter an infinite set of relations among polynomials with degree-dependent parameters is obtained. Both types of relations are counterparts of those known for conventional polynomials. In the absence of any direct information on the zeros of the Romanovski polynomials present in denominators, the regularity of the constructed potentials is checked by taking advantage of the disconjugacy properties of second-order differential equations of Schrödinger type. It is also shown that on going from Scarf I to Scarf II or from Rosen-Morse II to Rosen-Morse I potentials, the variety of rational extensions is narrowed down from types I, II, and III to type III only.
Notes on the Polynomial Identities in Random Overlap Structures
NASA Astrophysics Data System (ADS)
Sollich, Peter; Barra, Adriano
2012-04-01
In these notes we review first in some detail the concept of random overlap structure (ROSt) applied to fully connected and diluted spin glasses. We then sketch how to write down the general term of the expansion of the energy part from the Boltzmann ROSt (for the Sherrington-Kirkpatrick model) and the corresponding term from the RaMOSt, which is the diluted extension suitable for the Viana-Bray model. From the ROSt energy term, a set of polynomial identities (often known as Aizenman-Contucci or AC relations) is shown to hold rigorously at every order because of a recursive structure of these polynomials that we prove. We show also, however, that this set is smaller than the full set of AC identities that is already known. Furthermore, when investigating the RaMOSt energy for the diluted counterpart, at higher orders, combinations of such AC identities appear, ultimately suggesting a crucial role for the entropy in generating these constraints in spin glasses.
Barabash, Sergey V.; Blum, Volker; Zunger, Alex; Mueller, Stefan
2006-07-15
We describe an iterative procedure which yields an accurate cluster expansion for Au-Pd using only a limited number of ab initio formation enthalpies. Our procedure addresses two problems: (a) given the local-density-approximation (LDA) formation energies for a fixed set of structures, it finds the pair and many-body cluster interactions best able to predict the formation energies of new structures, and (b) given such pair and many-body interactions, it augments the LDA set of 'input structures' by identifying additional structures that carry most information not yet included in the 'input'. Neither step can be done by intuitive selection. Using methods including genetic algorithm and statistical analysis to iteratively solve these problems, we build a cluster expansion able to predict the formation enthalpy of an arbitrary fcc lattice configuration with precision comparable to that of ab initio calculations themselves. We also study possible competing non-fcc structures of Au-Pd, using the results of a 'data mining' study. We then address the unresolved problem of bulk ordering in Au-Pd. Experimentally, the phase diagram of Au-Pd shows only a disordered solid solution. Even though the mixing enthalpy is negative, implying ordering, no ordered bulk phases have been detected. Thin film growth shows L1{sub 2}-ordered structures with composition Au{sub 3}Pd and AuPd{sub 3} and L1{sub 0} structure with composition AuPd. We find that (i) all the ground states of Au-Pd are fcc structures; (ii) the low-T ordered states of bulk Au-Pd are different from those observed experimentally in thin films; specifically, the ordered bulk Au{sub 3}Pd is stable in D0{sub 23} structure and and AuPd in chalcopyritelike Au{sub 2}Pd{sub 2} (201) superlattice structure, whereas thin films are seen in the L1{sub 2} and L1{sub 0} structures; (iii) AuPd{sub 3} L1{sub 2} is stable and does not phase separate, contrary to the suggestions of an earlier investigation; (iv) at compositions around
NASA Astrophysics Data System (ADS)
Roquet, F.; Madec, G.; McDougall, Trevor J.; Barker, Paul M.
2015-06-01
A new set of approximations to the standard TEOS-10 equation of state are presented. These follow a polynomial form, making it computationally efficient for use in numerical ocean models. Two versions are provided, the first being a fit of density for Boussinesq ocean models, and the second fitting specific volume which is more suitable for compressible models. Both versions are given as the sum of a vertical reference profile (6th-order polynomial) and an anomaly (52-term polynomial, cubic in pressure), with relative errors of ∼0.1% on the thermal expansion coefficients. A 75-term polynomial expression is also presented for computing specific volume, with a better accuracy than the existing TEOS-10 48-term rational approximation, especially regarding the sound speed, and it is suggested that this expression represents a valuable approximation of the TEOS-10 equation of state for hydrographic data analysis. In the last section, practical aspects about the implementation of TEOS-10 in ocean models are discussed.
Absorption and Fluorescence Lineshape Theory for Polynomial Potentials.
Anda, André; De Vico, Luca; Hansen, Thorsten; Abramavičius, Darius
2016-12-13
The modeling of vibrations in optical spectra relies heavily on the simplifications brought about by using harmonic oscillators. However, realistic molecular systems can deviate substantially from this description. We develop two methods which show that the extension to arbitrarily shaped potential energy surfaces is not only straightforward, but also efficient. These methods are applied to an electronic two-level system with potential energy surfaces of polynomial form and used to study anharmonic features such as the zero-phonon line shape and mirror-symmetry breaking between absorption and fluorescence spectra. The first method, which constructs vibrational wave functions as linear combinations of the harmonic oscillator wave functions, is shown to be extremely robust and can handle large anharmonicities. The second method uses the cumulant expansion, which is readily solved, even at high orders, thanks to an ideally suited matrix theorem.
Interpolation and Polynomial Curve Fitting
ERIC Educational Resources Information Center
Yang, Yajun; Gordon, Sheldon P.
2014-01-01
Two points determine a line. Three noncollinear points determine a quadratic function. Four points that do not lie on a lower-degree polynomial curve determine a cubic function. In general, n + 1 points uniquely determine a polynomial of degree n, presuming that they do not fall onto a polynomial of lower degree. The process of finding such a…
Interpolation and Polynomial Curve Fitting
ERIC Educational Resources Information Center
Yang, Yajun; Gordon, Sheldon P.
2014-01-01
Two points determine a line. Three noncollinear points determine a quadratic function. Four points that do not lie on a lower-degree polynomial curve determine a cubic function. In general, n + 1 points uniquely determine a polynomial of degree n, presuming that they do not fall onto a polynomial of lower degree. The process of finding such a…
High order optimal control of space trajectories with uncertain boundary conditions
NASA Astrophysics Data System (ADS)
Di Lizia, P.; Armellin, R.; Bernelli-Zazzera, F.; Berz, M.
2014-01-01
A high order optimal control strategy is proposed in this work, based on the use of differential algebraic techniques. In the frame of orbital mechanics, differential algebra allows to represent, by high order Taylor polynomials, the dependency of the spacecraft state on initial conditions and environmental parameters. The resulting polynomials can be manipulated to obtain the high order expansion of the solution of two-point boundary value problems. Since the optimal control problem can be reduced to a two-point boundary value problem, differential algebra is used to compute the high order expansion of the solution of the optimal control problem about a reference trajectory. Whenever perturbations in the nominal conditions occur, new optimal control laws for perturbed initial and final states are obtained by the mere evaluation of polynomials. The performances of the method are assessed on lunar landing, rendezvous maneuvers, and a low-thrust Earth-Mars transfer.
Johnson, Matthew G; Malley, Claire; Goffinet, Bernard; Shaw, A Jonathan; Wickett, Norman J
2016-05-01
The pleurocarpous mosses (i.e., Hypnanae) are a species-rich group of land plants comprising about 6,000 species that share the development of female sex organs on short lateral branches, a derived trait within mosses. Many of the families within Hypnales, the largest order of pleurocarpous mosses, trace their origin to a rapid radiation less than 100 million years ago, just after the rise of the angiosperms. As a result, the phylogenetic resolution among families of Hypnales, necessary to test evolutionary hypotheses, has proven difficult using one or few loci. We present the first phylogenetic inference from high-throughput sequence data (transcriptome sequences) for pleurocarpous mosses. To test hypotheses of gene family evolution, we built a species tree of 21 pleurocarpous and six acrocarpous mosses using over one million sites from 659 orthologous genes. We used the species tree to investigate the genomic consequences of the shift to pleurocarpy and to identify whether patterns common to other plant radiations (gene family expansion, whole genome duplication, or changes in the molecular signatures of selection) could be observed. We found that roughly six percent of all gene families have expanded in the pleurocarpous mosses, relative to acrocarpous mosses. These gene families are enriched for several gene ontology (GO) terms, including interaction with other organisms. The increase in copy number coincident with the radiation of Hypnales suggests that a process such as whole genome duplication or a burst of small-scale duplications occurred during the diversification. In over 500 gene families we found evidence of a reduction in purifying selection. These gene families are enriched for several terms in the GO hierarchy related to "tRNA metabolic process." Our results reveal candidate genes and pathways that may be associated with the transition to pleurocarpy, illustrating the utility of phylotranscriptomics for the study of molecular evolution in non
Polynomial chaos representation of databases on manifolds
NASA Astrophysics Data System (ADS)
Soize, C.; Ghanem, R.
2017-04-01
Characterizing the polynomial chaos expansion (PCE) of a vector-valued random variable with probability distribution concentrated on a manifold is a relevant problem in data-driven settings. The probability distribution of such random vectors is multimodal in general, leading to potentially very slow convergence of the PCE. In this paper, we build on a recent development for estimating and sampling from probabilities concentrated on a diffusion manifold. The proposed methodology constructs a PCE of the random vector together with an associated generator that samples from the target probability distribution which is estimated from data concentrated in the neighborhood of the manifold. The method is robust and remains efficient for high dimension and large datasets. The resulting polynomial chaos construction on manifolds permits the adaptation of many uncertainty quantification and statistical tools to emerging questions motivated by data-driven queries.
Quantum Hurwitz numbers and Macdonald polynomials
NASA Astrophysics Data System (ADS)
Harnad, J.
2016-11-01
Parametric families in the center Z(C[Sn]) of the group algebra of the symmetric group are obtained by identifying the indeterminates in the generating function for Macdonald polynomials as commuting Jucys-Murphy elements. Their eigenvalues provide coefficients in the double Schur function expansion of 2D Toda τ-functions of hypergeometric type. Expressing these in the basis of products of power sum symmetric functions, the coefficients may be interpreted geometrically as parametric families of quantum Hurwitz numbers, enumerating weighted branched coverings of the Riemann sphere. Combinatorially, they give quantum weighted sums over paths in the Cayley graph of Sn generated by transpositions. Dual pairs of bases for the algebra of symmetric functions with respect to the scalar product in which the Macdonald polynomials are orthogonal provide both the geometrical and combinatorial significance of these quantum weighted enumerative invariants.
High degree interpolation polynomial in Newton form
NASA Technical Reports Server (NTRS)
Tal-Ezer, Hillel
1988-01-01
Polynomial interpolation is an essential subject in numerical analysis. Dealing with a real interval, it is well known that even if f(x) is an analytic function, interpolating at equally spaced points can diverge. On the other hand, interpolating at the zeroes of the corresponding Chebyshev polynomial will converge. Using the Newton formula, this result of convergence is true only on the theoretical level. It is shown that the algorithm which computes the divided differences is numerically stable only if: (1) the interpolating points are arranged in a different order, and (2) the size of the interval is 4.
Evaluation of multivariate polynomials and their derivatives
NASA Astrophysics Data System (ADS)
Carnicer, J.; Gasca, M.
1990-01-01
An extension of Horner's algorithm to the evaluation of m-variate polynomials and their derivatives is obtained. The schemes of computation are represented by trees because this type of graph describes exactly in which order the computations must be done. Some examples of algorithms for one and two variables are given.
Macdonald Polynomials in Superspace: Conjectural Definition and Positivity Conjectures
NASA Astrophysics Data System (ADS)
Blondeau-Fournier, Olivier; Desrosiers, Patrick; Lapointe, Luc; Mathieu, Pierre
2012-07-01
We introduce a conjectural construction for an extension to superspace of the Macdonald polynomials. The construction, which depends on certain orthogonality and triangularity relations, is tested for high degrees. We conjecture a simple form for the norm of the Macdonald polynomials in superspace and a rather non-trivial expression for their evaluation. We study the limiting cases q = 0 and q = ∞, which lead to two families of Hall-Littlewood polynomials in superspace. We also find that the Macdonald polynomials in superspace evaluated at q = t = 0 or q = t = ∞ seem to generalize naturally the Schur functions. In particular, their expansion coefficients in the corresponding Hall-Littlewood bases appear to be polynomials in t with nonnegative integer coefficients. More strikingly, we formulate a generalization of the Macdonald positivity conjecture to superspace: the expansion coefficients of the Macdonald superpolynomials expanded into a modified version of the Schur superpolynomial basis (the q = t = 0 family) are polynomials in q and t with nonnegative integer coefficients.
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.
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
polynomials which the author has introduced and investigated previously. Bibliography: 13 titles.
Independence polynomial and matching polynomial of the Koch network
NASA Astrophysics Data System (ADS)
Liao, Yunhua; Xie, Xiaoliang
2015-11-01
The lattice gas model and the monomer-dimer model are two classical models in statistical mechanics. It is well known that the partition functions of these two models are associated with the independence polynomial and the matching polynomial in graph theory, respectively. Both polynomials have been shown to belong to the “#P-complete” class, which indicate the problems are computationally “intractable”. We consider these two polynomials of the Koch networks which are scale-free with small-world effects. Explicit recurrences are derived, and explicit formulae are presented for the number of independent sets of a certain type.
A recursive algorithm for Zernike polynomials
NASA Technical Reports Server (NTRS)
Davenport, J. W.
1982-01-01
The analysis of a function defined on a rotationally symmetric system, with either a circular or annular pupil is discussed. In order to numerically analyze such systems it is typical to expand the given function in terms of a class of orthogonal polynomials. Because of their particular properties, the Zernike polynomials are especially suited for numerical calculations. Developed is a recursive algorithm that can be used to generate the Zernike polynomials up to a given order. The algorithm is recursively defined over J where R(J,N) is the Zernike polynomial of degree N obtained by orthogonalizing the sequence R(J), R(J+2), ..., R(J+2N) over (epsilon, 1). The terms in the preceding row - the (J-1) row - up to the N+1 term is needed for generating the (J,N)th term. Thus, the algorith generates an upper left-triangular table. This algorithm was placed in the computer with the necessary support program also included.
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.
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.
NASA Astrophysics Data System (ADS)
Sergeev, A.; Alharbi, F. H.; Jovanovic, R.; Kais, S.
2016-04-01
The gradient expansion of the kinetic energy density functional, when applied to atoms or finite systems, usually grossly overestimates the energy in the fourth order and generally diverges in the sixth order. We avoid the divergence of the integral by replacing the asymptotic series including the sixth order term in the integrand by a rational function. Padé approximants show moderate improvements in accuracy in comparison with partial sums of the series. The results are discussed for atoms and Hooke’s law model for two-electron atoms.
Factoring Polynomials Modulo Composites,
2007-11-02
xA. This is an appropriate time to introduce the Sylvester Matrix . Definition 3.9 Given polynomials f,g as above, the Sylvester matrix off and g is...the coefficient matrix of the above system of equations. We denote this Sylvester matrix as S(f, g) by the following (1 + m) x (1 + m) matrix a, bm a11...bin-i bm S(f,g) = ao at : b- C R(l+m)x(l+m) aq-1 bo ao b0 the empty spaces are filled by zeros. The Sylvester matrix is the coefficient matrix of
Chaves, Rafael
2016-01-08
It is a recent realization that many of the concepts and tools of causal discovery in machine learning are highly relevant to problems in quantum information, in particular quantum nonlocality. The crucial ingredient in the connection between both fields is the mathematical theory of causality, allowing for the representation of arbitrary causal structures and providing a rigorous tool to reason about probabilistic causation. Indeed, Bell's theorem concerns a very particular kind of causal structure and Bell inequalities are a special case of linear constraints following from such models. It is thus natural to look for generalizations involving more complex Bell scenarios. The problem, however, relies on the fact that such generalized scenarios are characterized by polynomial Bell inequalities and no current method is available to derive them beyond very simple cases. In this work, we make a significant step in that direction, providing a new, general, and conceptually clear method for the derivation of polynomial Bell inequalities in a wide class of scenarios. We also show how our construction can be used to allow for relaxations of causal constraints and naturally gives rise to a notion of nonsignaling in generalized Bell networks.
NASA Astrophysics Data System (ADS)
Chaves, Rafael
2016-01-01
It is a recent realization that many of the concepts and tools of causal discovery in machine learning are highly relevant to problems in quantum information, in particular quantum nonlocality. The crucial ingredient in the connection between both fields is the mathematical theory of causality, allowing for the representation of arbitrary causal structures and providing a rigorous tool to reason about probabilistic causation. Indeed, Bell's theorem concerns a very particular kind of causal structure and Bell inequalities are a special case of linear constraints following from such models. It is thus natural to look for generalizations involving more complex Bell scenarios. The problem, however, relies on the fact that such generalized scenarios are characterized by polynomial Bell inequalities and no current method is available to derive them beyond very simple cases. In this work, we make a significant step in that direction, providing a new, general, and conceptually clear method for the derivation of polynomial Bell inequalities in a wide class of scenarios. We also show how our construction can be used to allow for relaxations of causal constraints and naturally gives rise to a notion of nonsignaling in generalized Bell networks.
NASA Astrophysics Data System (ADS)
Butera, P.; Federbush, P.; Pernici, M.
2013-06-01
Recently, an expansion as a power series in 1/d has been presented for the specific entropy of a complete dimer covering of a d-dimensional hypercubic lattice. This paper extends from 3 to 10 the number of terms known in the series. Likewise, an expansion for the entropy, dependent on the dimer density p, of a monomer-dimer system, involving a sum ∑kak(d)pk, has been offered recently. We herein extend the number of known expansion coefficients from 6 to 20 for the hypercubic lattices of general dimensionality d and from 6 to 24 for the hypercubic lattices of dimensionalities d<5. We show that these extensions can lead to accurate numerical estimates of the p-dependent entropy for lattices with dimension d>2. The computations of this paper have led us to make the following marvelous conjecture: In the case of the hypercubic lattices, all the expansion coefficients ak(d) are positive. This paper results from a simple melding of two disparate research programs: one computing to high orders the Mayer series coefficients of a dimer gas and the other studying the development of entropy from these coefficients. An effort is made to make this paper self-contained by including a review of the earlier works.
Quantum chaotic dynamics and random polynomials
Bogomolny, E.; Bohigas, O.; Leboeuf, P.
1996-12-01
We investigate the distribution of roots of polynomials of high degree with random coefficients which, among others, appear naturally in the context of {open_quotes}quantum chaotic dynamics.{close_quotes} It is shown that under quite general conditions their roots tend to concentrate near the unit circle in the complex plane. In order to further increase this tendency, we study in detail the particular case of self-inversive random polynomials and show that for them a finite portion of all roots lies exactly on the unit circle. Correlation functions of these roots are also computed analytically, and compared to the correlations of eigenvalues of random matrices. The problem of ergodicity of chaotic wavefunctions is also considered. For that purpose we introduce a family of random polynomials whose roots spread uniformly over phase space. While these results are consistent with random matrix theory predictions, they provide a new and different insight into the problem of quantum ergodicity Special attention is devoted to the role of symmetries in the distribution of roots of random polynomials.
Polynomials Generated by the Fibonacci Sequence
NASA Astrophysics Data System (ADS)
Garth, David; Mills, Donald; Mitchell, Patrick
2007-06-01
The Fibonacci sequence's initial terms are F_0=0 and F_1=1, with F_n=F_{n-1}+F_{n-2} for n>=2. We define the polynomial sequence p by setting p_0(x)=1 and p_{n}(x)=x*p_{n-1}(x)+F_{n+1} for n>=1, with p_{n}(x)= sum_{k=0}^{n} F_{k+1}x^{n-k}. We call p_n(x) the Fibonacci-coefficient polynomial (FCP) of order n. The FCP sequence is distinct from the well-known Fibonacci polynomial sequence. We answer several questions regarding these polynomials. Specifically, we show that each even-degree FCP has no real zeros, while each odd-degree FCP has a unique, and (for degree at least 3) irrational, real zero. Further, we show that this sequence of unique real zeros converges monotonically to the negative of the golden ratio. Using Rouche's theorem, we prove that the zeros of the FCP's approach the golden ratio in modulus. We also prove a general result that gives the Mahler measures of an infinite subsequence of the FCP sequence whose coefficients are reduced modulo an integer m>=2. We then apply this to the case that m=L_n, the nth Lucas number, showing that the Mahler measure of the subsequence is phi^{n-1}, where phi=(1+sqrt 5)/2.
Thermodynamic characterization of networks using graph polynomials
NASA Astrophysics Data System (ADS)
Ye, Cheng; Comin, César H.; Peron, Thomas K. DM.; Silva, Filipi N.; Rodrigues, Francisco A.; Costa, Luciano da F.; Torsello, Andrea; Hancock, Edwin R.
2015-09-01
In this paper, we present a method for characterizing the evolution of time-varying complex networks by adopting a thermodynamic representation of network structure computed from a polynomial (or algebraic) characterization of graph structure. Commencing from a representation of graph structure based on a characteristic polynomial computed from the normalized Laplacian matrix, we show how the polynomial is linked to the Boltzmann partition function of a network. This allows us to compute a number of thermodynamic quantities for the network, including the average energy and entropy. Assuming that the system does not change volume, we can also compute the temperature, defined as the rate of change of entropy with energy. All three thermodynamic variables can be approximated using low-order Taylor series that can be computed using the traces of powers of the Laplacian matrix, avoiding explicit computation of the normalized Laplacian spectrum. These polynomial approximations allow a smoothed representation of the evolution of networks to be constructed in the thermodynamic space spanned by entropy, energy, and temperature. We show how these thermodynamic variables can be computed in terms of simple network characteristics, e.g., the total number of nodes and node degree statistics for nodes connected by edges. We apply the resulting thermodynamic characterization to real-world time-varying networks representing complex systems in the financial and biological domains. The study demonstrates that the method provides an efficient tool for detecting abrupt changes and characterizing different stages in network evolution.
Factorization of colored knot polynomials at roots of unity
NASA Astrophysics Data System (ADS)
Kononov, Ya.; Morozov, A.
2015-07-01
HOMFLY polynomials are the Wilson-loop averages in Chern-Simons theory and depend on four variables: the closed line (knot) in 3d space-time, representation R of the gauge group SU (N) and exponentiated coupling constant q. From analysis of a big variety of different knots we conclude that at q, which is a 2m-th root of unity, q2m = 1, HOMFLY polynomials in symmetric representations [ r ] satisfy recursion identity: Hr+m =Hr ṡHm for any A =qN, which is a generalization of the property Hr = H1r for special polynomials at m = 1. We conjecture a further generalization to arbitrary representation R, which, however, is checked only for torus knots. Next, Kashaev polynomial, which arises from HR at q2 = e 2 πi / | R |, turns equal to the special polynomial with A substituted by A| R |, provided R is a single-hook representations (including arbitrary symmetric) - what provides a q - A dual to the similar property of Alexander polynomial. All this implies non-trivial relations for the coefficients of the differential expansions, which are believed to provide reasonable coordinates in the space of knots - existence of such universal relations means that these variables are still not unconstrained.
NASA Astrophysics Data System (ADS)
Singh, Mandip
2016-03-01
The series expansion of neutrino evolution matrix “S”, up to first-order in small reactor mixing angle θ13 is very useful formalism to study experiments quantitatively. The formalism has been used especially to investigate CP-violating phase δCP. In order to perform a broad investigation for the possible measurement of δCP phase, we will study small baseline experiments: Chooz (L = 1.03Km), T2K (L = 295Km) and ESS (L = 500Km), medium baseline experiment: NOνA (L = 810Km) and long baseline experiment: LBNE (L = 1300Km).
A general polynomial solution to convection-dispersion equation using boundary layer theory
NASA Astrophysics Data System (ADS)
Wang, Jiao; Shao, Ming'an; Huang, Laiming; Jia, Xiaoxu
2017-04-01
A number of models have been established to simulate the behaviour of solute transport due to chemical pollution, both in croplands and groundwater systems. An approximate polynomial solution to convection-dispersion equation (CDE) based on boundary layer theory has been verified for the use to describe solute transport in semi-infinite systems such as soil column. However, previous studies have only proposed low order polynomial solutions such as parabolic and cubic polynomials. This paper presents a general polynomial boundary layer solution to CDE. Comparison with exact solution suggests the prediction accuracy of the boundary layer solution varies with the order of polynomial expression and soil transport parameters. The results show that prediction accuracy increases with increasing order up to parabolic or cubic polynomial function and with no distinct relationship between accuracy and order for higher order polynomials (n≥slant 3). Comparison of two critical solute transport parameters (i.e., dispersion coefficient and retardation factor), estimated by the boundary layer solution and obtained by CXTFIT curve-fitting, shows a good agreement. The study shows that the general solution can determine the appropriate orders of polynomials for approximate CDE solutions that best describe solute concentration profiles and optimal solute transport parameters. Furthermore, the general polynomial solution to CDE provides a simple approach to solute transport problems, a criterion for choosing the right orders of polynomials for soils with different transport parameters. It is also a potential approach for estimating solute transport parameters of soils in the field.
Graphical Solution of Polynomial Equations
ERIC Educational Resources Information Center
Grishin, Anatole
2009-01-01
Graphing utilities, such as the ubiquitous graphing calculator, are often used in finding the approximate real roots of polynomial equations. In this paper the author offers a simple graphing technique that allows one to find all solutions of a polynomial equation (1) of arbitrary degree; (2) with real or complex coefficients; and (3) possessing…
Estrada index and Chebyshev polynomials
NASA Astrophysics Data System (ADS)
Ginosar, Yuval; Gutman, Ivan; Mansour, Toufik; Schork, Matthias
2008-03-01
Let G be a graph whose eigenvalues are λ1, λ2,…, λn. The Estrada index of G is equal to ∑i=1ne. We point out certain classes of graphs whose characteristic polynomials are closely connected to the Chebyshev polynomials of the second kind. Various relations, in particular approximations, for the Estrada index of these graphs are obtained.
On a Perplexing Polynomial Puzzle
ERIC Educational Resources Information Center
Richmond, Bettina
2010-01-01
It seems rather surprising that any given polynomial p(x) with nonnegative integer coefficients can be determined by just the two values p(1) and p(a), where a is any integer greater than p(1). This result has become known as the "perplexing polynomial puzzle." Here, we address the natural question of what might be required to determine a…
Controlling General Polynomial Networks
NASA Astrophysics Data System (ADS)
Cuneo, N.; Eckmann, J.-P.
2014-06-01
We consider networks of massive particles connected by non-linear springs. Some particles interact with heat baths at different temperatures, which are modeled as stochastic driving forces. The structure of the network is arbitrary, but the motion of each particle is 1D. For polynomial interactions, we give sufficient conditions for Hörmander's "bracket condition" to hold, which implies the uniqueness of the steady state (if it exists), as well as the controllability of the associated system in control theory. These conditions are constructive; they are formulated in terms of inequivalence of the forces (modulo translations) and/or conditions on the topology of the connections. We illustrate our results with examples, including "conducting chains" of variable cross-section. This then extends the results for a simple chain obtained in Eckmann et al. in (Commun Math Phys 201:657-697, 1999).
Polynomial Supertree Methods Revisited
Brinkmeyer, Malte; Griebel, Thasso; Böcker, Sebastian
2011-01-01
Supertree methods allow to reconstruct large phylogenetic trees by combining smaller trees with overlapping leaf sets into one, more comprehensive supertree. The most commonly used supertree method, matrix representation with parsimony (MRP), produces accurate supertrees but is rather slow due to the underlying hard optimization problem. In this paper, we present an extensive simulation study comparing the performance of MRP and the polynomial supertree methods MinCut Supertree, Modified MinCut Supertree, Build-with-distances, PhySIC, PhySIC_IST, and super distance matrix. We consider both quality and resolution of the reconstructed supertrees. Our findings illustrate the tradeoff between accuracy and running time in supertree construction, as well as the pros and cons of voting- and veto-based supertree approaches. Based on our results, we make some general suggestions for supertree methods yet to come. PMID:22229028
Simplified Storm Surge Simulations Using Bernstein Polynomials
NASA Astrophysics Data System (ADS)
Beisiegel, Nicole; Behrens, Jörn
2016-04-01
Storm surge simulations are vital for forecasting, hazard assessment and eventually improving our understanding of Earth system processes. Discontinuous Galerkin (DG) methods have recently been explored in that context, because they are locally mass-conservative and in combination with suitable robust nodal filtering techniques (slope limiters) positivity-preserving and well-balanced for the still water state at rest. These filters manipulate interpolation point values in every time step in order to retain the desirable properties of the scheme. In particular, DG methods are able to represent prognostic variables such as the fluid height at high-order accuracy inside each element (triangle). For simulations that include wetting and drying, however, the high-order accuracy will destabilize the numerical model because point values on quadrature points may become negative during the computation if they do not coincide with interpolation points. This is why the model that we are presenting utilizes Bernstein polynomials as basis functions to model the wetting and drying. This has the advantage that negative pointvalues away from interpolation points are prevented, the model is stabilized and no additional time step restriction is introduced. Numerical tests show that the model is capable of simulating simplified storm surges. Furthermore, a comparison of model results with third-order Bernstein polynomials with results using traditional nodal Lagrange polynomials reveals an improvement in numerical convergence.
NASA Technical Reports Server (NTRS)
Lancaster, J. E.
1973-01-01
Previously published asymptotic solutions for lunar and interplanetery trajectories have been modified and combined to formulate a general analytical solution to the problem of N-bodies. The earlier first-order solutions, derived by the method of matched asymptotic expansions, have been extended to second order for the purpose of obtaining increased accuracy. The complete derivation of the second-order solution, including the application of a regorous matching principle, is given. It is shown that the outer and inner expansions can be matched in a region of order mu to the alpha power, where 2/5 alpha 1/2, and mu (the moon/earth or planet/sun mass ratio) is much less than one. The second-order asymptotic solution has been used as a basis for formulating a number of analytical two-point boundary value solutions. These include earth-to-moon, one- and two-impulse moon-to-Earth, and interplanetary solutions. Each is presented as an explicit analytical solution which does not require iterative steps to satisfy the boundary conditions. The complete derivation of each solution is shown, as well as instructions for numerical evaluation. For Vol. 1, see N73-27738.
Conformal Laplace superintegrable systems in 2D: polynomial invariant subspaces
NASA Astrophysics Data System (ADS)
Escobar-Ruiz, M. A.; Miller, Willard, Jr.
2016-07-01
2nd-order conformal superintegrable systems in n dimensions are Laplace equations on a manifold with an added scalar potential and 2n-1 independent 2nd order conformal symmetry operators. They encode all the information about Helmholtz (eigenvalue) superintegrable systems in an efficient manner: there is a 1-1 correspondence between Laplace superintegrable systems and Stäckel equivalence classes of Helmholtz superintegrable systems. In this paper we focus on superintegrable systems in two-dimensions, n = 2, where there are 44 Helmholtz systems, corresponding to 12 Laplace systems. For each Laplace equation we determine the possible two-variate polynomial subspaces that are invariant under the action of the Laplace operator, thus leading to families of polynomial eigenfunctions. We also study the behavior of the polynomial invariant subspaces under a Stäckel transform. The principal new results are the details of the polynomial variables and the conditions on parameters of the potential corresponding to polynomial solutions. The hidden gl 3-algebraic structure is exhibited for the exact and quasi-exact systems. For physically meaningful solutions, the orthogonality properties and normalizability of the polynomials are presented as well. Finally, for all Helmholtz superintegrable solvable systems we give a unified construction of one-dimensional (1D) and two-dimensional (2D) quasi-exactly solvable potentials possessing polynomial solutions, and a construction of new 2D PT-symmetric potentials is established.
Asymptotic expansions for the reciprocal of the gamma function
NASA Astrophysics Data System (ADS)
Withers, Christopher S.; Nadarajah, Saralees
2014-05-01
Asymptotic expansions are derived for the reciprocal of the gamma function. We show that the coefficients of the expansion are the same, up to a sign change, as the asymptotic expansions for the gamma function obtained by exponentiating the expansions of its logarithm due to Stirling and de Moivre. Expressions for the coefficients are given in terms of Bell polynomials.
NASA Technical Reports Server (NTRS)
Chang, F.-C.; Mott, H.
1974-01-01
This paper presents a technique for the partial-fraction expansion of functions which are ratios of polynomials with real coefficients. The expansion coefficients are determined by writing the polynomials as Taylor's series and obtaining the Laurent series expansion of the function. The general formula for the inverse Laplace transform is also derived.
The number of polynomial solutions of polynomial Riccati equations
NASA Astrophysics Data System (ADS)
Gasull, Armengol; Torregrosa, Joan; Zhang, Xiang
2016-11-01
Consider real or complex polynomial Riccati differential equations a (x) y ˙ =b0 (x) +b1 (x) y +b2 (x)y2 with all the involved functions being polynomials of degree at most η. We prove that the maximum number of polynomial solutions is η + 1 (resp. 2) when η ≥ 1 (resp. η = 0) and that these bounds are sharp. For real trigonometric polynomial Riccati differential equations with all the functions being trigonometric polynomials of degree at most η ≥ 1 we prove a similar result. In this case, the maximum number of trigonometric polynomial solutions is 2η (resp. 3) when η ≥ 2 (resp. η = 1) and, again, these bounds are sharp. Although the proof of both results has the same starting point, the classical result that asserts that the cross ratio of four different solutions of a Riccati differential equation is constant, the trigonometric case is much more involved. The main reason is that the ring of trigonometric polynomials is not a unique factorization domain.
Syromyatnikov, A V
2010-06-02
The spectrum of short-wavelength magnons in a two-dimensional quantum Heisenberg antiferromagnet on a square lattice is calculated to the third order in a 1/S expansion. It is shown that a 1/S series for S = 1/2 converges quickly in the whole Brillouin zone except in the neighborhood of the point k = (π, 0), at which absolute values of the third-and the second-order 1/S-corrections are approximately equal to each other. It is shown that the third-order corrections make deeper the roton-like local minimum at k = (π, 0), improving the agreement with recent experiments and numerical results in the neighborhood of this point. It is suggested that the 1/S series converges slowly near k = (π, 0) also for S = 1 although the spectrum renormalization would be small in this case due to the very small values of high-order 1/S corrections.
Generalized Freud's equation and level densities with polynomial potential
NASA Astrophysics Data System (ADS)
Boobna, Akshat; Ghosh, Saugata
2013-08-01
We study orthogonal polynomials with weight $\\exp[-NV(x)]$, where $V(x)=\\sum_{k=1}^{d}a_{2k}x^{2k}/2k$ is a polynomial of order 2d. We derive the generalised Freud's equations for $d=3$, 4 and 5 and using this obtain $R_{\\mu}=h_{\\mu}/h_{\\mu -1}$, where $h_{\\mu}$ is the normalization constant for the corresponding orthogonal polynomials. Moments of the density functions, expressed in terms of $R_{\\mu}$, are obtained using Freud's equation and using this, explicit results of level densities as $N\\rightarrow\\infty$ are derived.
Higher-order scheme-independent series expansions of γψ ¯ψ ,IR and βIR' in conformal field theories
NASA Astrophysics Data System (ADS)
Ryttov, Thomas A.; Shrock, Robert
2017-05-01
We study a vectorial asymptotically free gauge theory, with gauge group G and Nf massless fermions in a representation R of this group, that exhibits an infrared (IR) zero in its beta function, β , at the coupling α =αIR in the non-Abelian Coulomb phase. For general G and R , we calculate the scheme-independent series expansions of (i) the anomalous dimension of the fermion bilinear, γψ ¯ ψ ,IR , to O (Δf4) and (ii) the derivative β'=d β /d α , to O (Δf5) , both evaluated at αIR, where Δf is an Nf-dependent expansion variable. These are the highest orders to which these expansions have been calculated. We apply these general results to theories with G =SU (Nc) and R equal to the fundamental, adjoint, and symmetric and antisymmetric rank-2 tensor representations. It is shown that for all of these representations, γψ ¯ ψ ,IR , calculated to the order Δfp , with 1 ≤p ≤4 , increases monotonically with decreasing Nf and, for fixed Nf, is a monotonically increasing function of p . Comparisons of our scheme-independent calculations of γψ ¯ ψ ,IR and βIR' are made with our earlier higher n -loop values of these quantities, and with lattice measurements. For R =F , we present results for the limit Nc→∞ and Nf→∞ with Nf/Nc fixed. We also present expansions for αIR calculated to O (Δf4).
Euler’s Theorem for Polynomials
1990-02-09
especially wants to find polynomials over the two element field, GF(2), which are irre- ducible of prime degree p such that L = 2p - 1 is a Mersenne ...relatively prime polynomial, and of the exponent of a polynomial, are investigated. Finally, examples are given which show how to apply these ideas to the...relatively prime polynomial m, and the exponent exp(m) of the polynomial m. We end with some applications of these ideas to the factorization of polynomials
A robust polynomial fitting approach for contact angle measurements.
Atefi, Ehsan; Mann, J Adin; Tavana, Hossein
2013-05-14
Polynomial fitting to drop profile offers an alternative to well-established drop shape techniques for contact angle measurements from sessile drops without a need for liquid physical properties. Here, we evaluate the accuracy of contact angles resulting from fitting polynomials of various orders to drop profiles in a Cartesian coordinate system, over a wide range of contact angles. We develop a differentiator mask to automatically find a range of required number of pixels from a drop profile over which a stable contact angle is obtained. The polynomial order that results in the longest stable regime and returns the lowest standard error and the highest correlation coefficient is selected to determine drop contact angles. We find that, unlike previous reports, a single polynomial order cannot be used to accurately estimate a wide range of contact angles and that a larger order polynomial is needed for drops with larger contact angles. Our method returns contact angles with an accuracy of <0.4° for solid-liquid systems with θ < ~60°. This compares well with the axisymmetric drop shape analysis-profile (ADSA-P) methodology results. Above about 60°, we observe significant deviations from ADSA-P results, most likely because a polynomial cannot trace the profile of drops with close-to-vertical and vertical segments. To overcome this limitation, we implement a new polynomial fitting scheme by transforming drop profiles into polar coordinate system. This eliminates the well-known problem with high curvature drops and enables estimating contact angles in a wide range with a fourth-order polynomial. We show that this approach returns dynamic contact angles with less than 0.7° error as compared to ADSA-P, for the solid-liquid systems tested. This new approach is a powerful alternative to drop shape techniques for estimating contact angles of drops regardless of drop symmetry and without a need for liquid properties.
Gravity Anomaly of Polyhedral Bodies Having a Polynomial Density Contrast
NASA Astrophysics Data System (ADS)
D'Urso, M. G.; Trotta, S.
2017-07-01
We analytically evaluate the gravity anomaly associated with a polyhedral body having an arbitrary geometrical shape and a polynomial density contrast in both the horizontal and vertical directions. The gravity anomaly is evaluated at an arbitrary point that does not necessarily coincide with the origin of the reference frame in which the density function is assigned. Density contrast is assumed to be a third-order polynomial as a maximum but the general approach exploited in the paper can be easily extended to higher-order polynomial functions. Invoking recent results of potential theory, the solution derived in the paper is shown to be singularity-free and is expressed as a sum of algebraic quantities that only depend upon the 3D coordinates of the polyhedron vertices and upon the polynomial density function. The accuracy, robustness and effectiveness of the proposed approach are illustrated by numerical comparisons with examples derived from the existing literature.
Constructing general partial differential equations using polynomial and neural networks.
Zjavka, Ladislav; Pedrycz, Witold
2016-01-01
Sum fraction terms can approximate multi-variable functions on the basis of discrete observations, replacing a partial differential equation definition with polynomial elementary data relation descriptions. Artificial neural networks commonly transform the weighted sum of inputs to describe overall similarity relationships of trained and new testing input patterns. Differential polynomial neural networks form a new class of neural networks, which construct and solve an unknown general partial differential equation of a function of interest with selected substitution relative terms using non-linear multi-variable composite polynomials. The layers of the network generate simple and composite relative substitution terms whose convergent series combinations can describe partial dependent derivative changes of the input variables. This regression is based on trained generalized partial derivative data relations, decomposed into a multi-layer polynomial network structure. The sigmoidal function, commonly used as a nonlinear activation of artificial neurons, may transform some polynomial items together with the parameters with the aim to improve the polynomial derivative term series ability to approximate complicated periodic functions, as simple low order polynomials are not able to fully make up for the complete cycles. The similarity analysis facilitates substitutions for differential equations or can form dimensional units from data samples to describe real-world problems.
A polynomial function of gait performance.
Giaquinto, Salvatore; Galli, Manuela; Nolfe, Giuseppe
2007-01-01
A mathematical data processing method is presented that represents a further step in gait analysis. The proposed polynomial regression analysis is reliable in assessing differences in the same patient and even on the same day. The program also calculates the confidence interval of the whole curve. The procedure was applied to normal subjects in order to collect normative data. When a new subject is tested, the polynomial function obtained is graphically superimposed on control data. Should the new curve fall within the limits described by normative data, it is considered to be equivalent. The procedure can be applied to the same subject, either normal or pathological, for retesting kinematic characteristics. The gait cycle is analyzed as a whole, not point-by-point. Ten normal subjects and two patients, one with recent- and the other with late-onset hemiplegia, were tested. Multiple baseline evaluation is recommended before the start of a rehabilitation program.
Alcocer-Sosa, Mauricio; Gutiérrez, David
2017-04-01
We present a forward modeling solution in the form of an array response kernel for magnetoencephalography. We consider the case when the brain's anatomy is approximated by an ellipsoid and an equivalent current dipole model is used to approximate brain sources. The proposed solution includes the contributions up to the third-order ellipsoidal harmonic terms; hence, we compare this new approximation against the previously available one that only considered up to second-order harmonics. We evaluated the proposed solution when used in the inverse problem of estimating physiologically feasible visual evoked responses from magnetoencephalography data. Our results showed that the contribution of the third-order harmonic terms provides a more realistic representation of the magnetic fields (closer to those generated with a numerical approximation based on the boundary element method) and, subsecuently, the estimated equivalent current dipoles are a better fit to those observed in practice (e.g., in visual evoked potentials). Copyright © 2016 John Wiley & Sons, Ltd.
Nishimoto, Yoshio
2015-09-07
We develop a formalism for the calculation of excitation energies and excited state gradients for the self-consistent-charge density-functional tight-binding method with the third-order contributions of a Taylor series of the density functional theory energy with respect to the fluctuation of electron density (time-dependent density-functional tight-binding (TD-DFTB3)). The formulation of the excitation energy is based on the existing time-dependent density functional theory and the older TD-DFTB2 formulae. The analytical gradient is computed by solving Z-vector equations, and it requires one to calculate the third-order derivative of the total energy with respect to density matrix elements due to the inclusion of the third-order contributions. The comparison of adiabatic excitation energies for selected small and medium-size molecules using the TD-DFTB2 and TD-DFTB3 methods shows that the inclusion of the third-order contributions does not affect excitation energies significantly. A different set of parameters, which are optimized for DFTB3, slightly improves the prediction of adiabatic excitation energies statistically. The application of TD-DFTB for the prediction of absorption and fluorescence energies of cresyl violet demonstrates that TD-DFTB3 reproduced the experimental fluorescence energy quite well.
Orthogonal polynomials and deformed oscillators
NASA Astrophysics Data System (ADS)
Borzov, V. V.; Damaskinsky, E. V.
2015-10-01
In the example of the Fibonacci oscillator, we discuss the construction of oscillator-like systems associated with orthogonal polynomials. We also consider the question of the dimensions of the corresponding Lie algebras.
NASA Astrophysics Data System (ADS)
Cherednichenko, Kirill D.; Smyshlyaev, Valery P.
2004-12-01
We consider a scalar quasilinear equation in the divergence form with periodic rapid oscillations, which may be a model of, e.g., nonlinear conducting, dielectric, or deforming in a restricted way hardening elastic-plastic composites, with “outer” periodicity conditions of a fixed large period. Under some natural growth assumptions on the stored-energy function, we construct for uniformly elliptic problems a full two-scale asymptotic expansion, which has a precise “double-series” structure, separating the slow and the fast variables “in all orders”, so that its “slowly varying” part solves asymptotically an “infinite-order homogenised equation” (cf. Bakhvalov, N.S., Panasenko, G.P.: Homogenisation: Averaging Processes in Periodic Media. Nauka, Moscow, 1984 (in Russian); English translation: Kluwer, 1989), and whose higher-order terms depend on the higher gradients of the slowly varying part. We prove the error bound, i.e., that the truncated asymptotic expansion is “higher-order” close to the actual solution in appropriate norms. The approach is extended to a non-uniformly elliptic case: for two-dimensional power-law potentials we prove the “non-degeneracy” using topological index methods. Examples and explicit formulae for the higher-order terms are given. In particular, we prove that the first term in the higher-order homogenised equations is related to the first-order corrector to the “mean” flux, and has in general the form of a fully nonlinear operator which is quadratic with respect to its highest (second) derivative being a linear combination of the second minors of the Hessian with coefficients depending on the first gradient, and in dimension two is of Monge-Ampère type. We show that this term is present at least for some examples (three-phase power-law laminates).
Tang, Kunkun; Congedo, Pietro M.; Abgrall, Rémi
2016-06-01
The Polynomial Dimensional Decomposition (PDD) is employed in this work for the global sensitivity analysis and uncertainty quantification (UQ) of stochastic systems subject to a moderate to large number of input random variables. Due to the intimate connection between the PDD and the Analysis of Variance (ANOVA) approaches, PDD is able to provide a simpler and more direct evaluation of the Sobol' sensitivity indices, when compared to the Polynomial Chaos expansion (PC). Unfortunately, the number of PDD terms grows exponentially with respect to the size of the input random vector, which makes the computational cost of standard methods unaffordable for real engineering applications. In order to address the problem of the curse of dimensionality, this work proposes essentially variance-based adaptive strategies aiming to build a cheap meta-model (i.e. surrogate model) by employing the sparse PDD approach with its coefficients computed by regression. Three levels of adaptivity are carried out in this paper: 1) the truncated dimensionality for ANOVA component functions, 2) the active dimension technique especially for second- and higher-order parameter interactions, and 3) the stepwise regression approach designed to retain only the most influential polynomials in the PDD expansion. During this adaptive procedure featuring stepwise regressions, the surrogate model representation keeps containing few terms, so that the cost to resolve repeatedly the linear systems of the least-squares regression problem is negligible. The size of the finally obtained sparse PDD representation is much smaller than the one of the full expansion, since only significant terms are eventually retained. Consequently, a much smaller number of calls to the deterministic model is required to compute the final PDD coefficients.
Using Tutte polynomials to analyze the structure of the benzodiazepines
NASA Astrophysics Data System (ADS)
Cadavid Muñoz, Juan José
2014-05-01
Graph theory in general and Tutte polynomials in particular, are implemented for analyzing the chemical structure of the benzodiazepines. Similarity analysis are used with the Tutte polynomials for finding other molecules that are similar to the benzodiazepines and therefore that might show similar psycho-active actions for medical purpose, in order to evade the drawbacks associated to the benzodiazepines based medicine. For each type of benzodiazepines, Tutte polynomials are computed and some numeric characteristics are obtained, such as the number of spanning trees and the number of spanning forests. Computations are done using the computer algebra Maple's GraphTheory package. The obtained analytical results are of great importance in pharmaceutical engineering. As a future research line, the usage of the chemistry computational program named Spartan, will be used to extent and compare it with the obtained results from the Tutte polynomials of benzodiazepines.
Symmetric polynomials in information theory: Entropy and subentropy
Jozsa, Richard; Mitchison, Graeme
2015-06-15
Entropy and other fundamental quantities of information theory are customarily expressed and manipulated as functions of probabilities. Here we study the entropy H and subentropy Q as functions of the elementary symmetric polynomials in the probabilities and reveal a series of remarkable properties. Derivatives of all orders are shown to satisfy a complete monotonicity property. H and Q themselves become multivariate Bernstein functions and we derive the density functions of their Levy-Khintchine representations. We also show that H and Q are Pick functions in each symmetric polynomial variable separately. Furthermore, we see that H and the intrinsically quantum informational quantity Q become surprisingly closely related in functional form, suggesting a special significance for the symmetric polynomials in quantum information theory. Using the symmetric polynomials, we also derive a series of further properties of H and Q.
Design of robust differential microphone arrays with orthogonal polynomials.
Pan, Chao; Benesty, Jacob; Chen, Jingdong
2015-08-01
Differential microphone arrays have the potential to be widely deployed in hands-free communication systems thanks to their frequency-invariant beampatterns, high directivity factors, and small apertures. Traditionally, they are designed and implemented in a multistage way with uniform linear geometries. This paper presents an approach to the design of differential microphone arrays with orthogonal polynomials, more specifically with Jacobi polynomials. It first shows how to express the beampatterns as a function of orthogonal polynomials. Then several differential beamformers are derived and their performance depends on the parameters of the Jacobi polynomials. Simulations show the great flexibility of the proposed method in terms of designing any order differential microphone arrays with different beampatterns and controlling white noise gain.
Herman, Michael F; Wu, Yinghua
2008-03-21
It is shown that a surface hopping expansion of the semiclassical wave function formally satisfies the time independent Schrodinger equation for many-state, multidimensional problems. This wave function includes terms involving hops between different adiabatic quantum states as well as momentum changes without change of state at each point along classical trajectories. The single-state momentum changes correct for the order variant Planck's constant over 2pi(2) errors due to the semiclassical approximation that are present even in single surface problems. A prescription is provided for the direction of this momentum change and the amplitude associated with it. The direction of the momentum change for energy conserving hops between adiabatic states is required to be in the direction of the nonadiabatic coupling vector. The magnitude of the posthop momentum in this direction is determined by the energy, but the sign is not. Hops with both signs of this momentum component are required in order for the wave function to formally satisfy the Schrodinger equation. Numerical results are presented which illustrate how the surface hopping expansion can be implemented and the accuracy that can be obtained.
Properties of Leach-Flessas-Gorringe polynomials
NASA Astrophysics Data System (ADS)
Pursey, D. L.
1990-09-01
A generating function is obtained for the polynomials recently introduced by Leach, Flessas, and Gorringe [J. Math. Phys. 30, 406 (1989)], and is then used to relate the Leach-Flessas-Gorringe (or LFG) polynomials to Hermite polynomials. The generating function is also used to express a number of integrals involving the LFG polynomials as finite sums of parabolic cylinder functions.
Wolfe, Carl E.; Maltman, Kim
2001-01-01
The strong isospin-breaking correction {Omega}{sub st}, which appears in estimates of the standard model value for the direct CP-violating ratio {epsilon}{prime}/{epsilon}, is evaluated to next-to-leading order (NLO) in the chiral expansion using chiral perturbation theory. The relevant linear combinations of the unknown NLO CP-odd weak low-energy constants (LEC's) which, in combination with one-loop and strong LEC contributions, are required for a complete determination at this order, are estimated using two different models. It is found that, to NLO, {Omega}{sub st}=0.08{+-}0.05, significantly reduced from the ''standard'' value, 0.25{+-}0.08, employed in recent analyses. The potentially significant numerical impact of this decrease on standard model predictions for {epsilon}{prime}/{epsilon}, associated with the decreased cancellation between gluonic penguin and electroweak penguin contributions, is also discussed.
2001-11-30
NASA Dryden's new in-house designed Propulsion Flight Test Fixture (PFTF), carried on an F-15B's centerline attachment point, underwent flight envelope expansion in order to verify its design and capabilities.
DIFFERENTIAL CROSS SECTION ANALYSIS IN KAON PHOTOPRODUCTION USING ASSOCIATED LEGENDRE POLYNOMIALS
P. T. P. HUTAURUK, D. G. IRELAND, G. ROSNER
2009-04-01
Angular distributions of differential cross sections from the latest CLAS data sets,6 for the reaction γ + p→K+ + Λ have been analyzed using associated Legendre polynomials. This analysis is based upon theoretical calculations in Ref. 1 where all sixteen observables in kaon photoproduction can be classified into four Legendre classes. Each observable can be described by an expansion of associated Legendre polynomial functions. One of the questions to be addressed is how many associated Legendre polynomials are required to describe the data. In this preliminary analysis, we used data models with different numbers of associated Legendre polynomials. We then compared these models by calculating posterior probabilities of the models. We found that the CLAS data set needs no more than four associated Legendre polynomials to describe the differential cross section data. In addition, we also show the extracted coefficients of the best model.
Chromatic polynomials of random graphs
NASA Astrophysics Data System (ADS)
Van Bussel, Frank; Ehrlich, Christoph; Fliegner, Denny; Stolzenberg, Sebastian; Timme, Marc
2010-04-01
Chromatic polynomials and related graph invariants are central objects in both graph theory and statistical physics. Computational difficulties, however, have so far restricted studies of such polynomials to graphs that were either very small, very sparse or highly structured. Recent algorithmic advances (Timme et al 2009 New J. Phys. 11 023001) now make it possible to compute chromatic polynomials for moderately sized graphs of arbitrary structure and number of edges. Here we present chromatic polynomials of ensembles of random graphs with up to 30 vertices, over the entire range of edge density. We specifically focus on the locations of the zeros of the polynomial in the complex plane. The results indicate that the chromatic zeros of random graphs have a very consistent layout. In particular, the crossing point, the point at which the chromatic zeros with non-zero imaginary part approach the real axis, scales linearly with the average degree over most of the density range. While the scaling laws obtained are purely empirical, if they continue to hold in general there are significant implications: the crossing points of chromatic zeros in the thermodynamic limit separate systems with zero ground state entropy from systems with positive ground state entropy, the latter an exception to the third law of thermodynamics.
Efficient modeling of photonic crystals with local Hermite polynomials
NASA Astrophysics Data System (ADS)
Boucher, C. R.; Li, Zehao; Albrecht, J. D.; Ram-Mohan, L. R.
2014-04-01
Developing compact algorithms for accurate electrodynamic calculations with minimal computational cost is an active area of research given the increasing complexity in the design of electromagnetic composite structures such as photonic crystals, metamaterials, optical interconnects, and on-chip routing. We show that electric and magnetic (EM) fields can be calculated using scalar Hermite interpolation polynomials as the numerical basis functions without having to invoke edge-based vector finite elements to suppress spurious solutions or to satisfy boundary conditions. This approach offers several fundamental advantages as evidenced through band structure solutions for periodic systems and through waveguide analysis. Compared with reciprocal space (plane wave expansion) methods for periodic systems, advantages are shown in computational costs, the ability to capture spatial complexity in the dielectric distributions, the demonstration of numerical convergence with scaling, and variational eigenfunctions free of numerical artifacts that arise from mixed-order real space basis sets or the inherent aberrations from transforming reciprocal space solutions of finite expansions. The photonic band structure of a simple crystal is used as a benchmark comparison and the ability to capture the effects of spatially complex dielectric distributions is treated using a complex pattern with highly irregular features that would stress spatial transform limits. This general method is applicable to a broad class of physical systems, e.g., to semiconducting lasers which require simultaneous modeling of transitions in quantum wells or dots together with EM cavity calculations, to modeling plasmonic structures in the presence of EM field emissions, and to on-chip propagation within monolithic integrated circuits.
Efficient modeling of photonic crystals with local Hermite polynomials
Boucher, C. R.; Li, Zehao; Albrecht, J. D.; Ram-Mohan, L. R.
2014-04-21
Developing compact algorithms for accurate electrodynamic calculations with minimal computational cost is an active area of research given the increasing complexity in the design of electromagnetic composite structures such as photonic crystals, metamaterials, optical interconnects, and on-chip routing. We show that electric and magnetic (EM) fields can be calculated using scalar Hermite interpolation polynomials as the numerical basis functions without having to invoke edge-based vector finite elements to suppress spurious solutions or to satisfy boundary conditions. This approach offers several fundamental advantages as evidenced through band structure solutions for periodic systems and through waveguide analysis. Compared with reciprocal space (plane wave expansion) methods for periodic systems, advantages are shown in computational costs, the ability to capture spatial complexity in the dielectric distributions, the demonstration of numerical convergence with scaling, and variational eigenfunctions free of numerical artifacts that arise from mixed-order real space basis sets or the inherent aberrations from transforming reciprocal space solutions of finite expansions. The photonic band structure of a simple crystal is used as a benchmark comparison and the ability to capture the effects of spatially complex dielectric distributions is treated using a complex pattern with highly irregular features that would stress spatial transform limits. This general method is applicable to a broad class of physical systems, e.g., to semiconducting lasers which require simultaneous modeling of transitions in quantum wells or dots together with EM cavity calculations, to modeling plasmonic structures in the presence of EM field emissions, and to on-chip propagation within monolithic integrated circuits.
NASA Astrophysics Data System (ADS)
Coelho, Rodrigo C. V.; Ilha, Anderson S.; Doria, Mauro M.
2016-10-01
A lattice Boltzmann method is proposed based on the expansion of the equilibrium distribution function in powers of a new set of generalized orthonormal polynomials which are here presented. The new polynomials are orthonormal under the weight defined by the equilibrium distribution function itself. The D-dimensional Hermite polynomials is a sub-case of the present ones, associated to the particular weight of a Gaussian function. The proposed lattice Boltzmann method allows for the treatment of semi-classical fluids, such as electrons in metals under the Drude-Sommerfeld model, which is a particular case that we develop and validate by the Riemann problem.
Realization of couplings in a polynomial type mixed-mode cellular neural network.
Laiho, Mika; Paasio, Ari; Kananen, Asko; Halonen, Kari
2003-12-01
In this paper realization of couplings between cells in a polynomial type mixed-mode cellular neural network (CNN) is analyzed. The choice of the multiplier is discussed and two multiplier types are analyzed. Also, two circuits for generating the second and third order polynomial terms of cell output are described. The accuracy of the multipliers and polynomial circuits at the presence of device mismatch is analyzed.
A nondeterministic shock and vibration application using polynomial chaos expansions
FIELD JR.,RICHARD V.; RED-HORSE,JOHN R.; PAEZ,THOMAS L.
2000-03-28
In the current study, the generality of the key underpinnings of the Stochastic Finite Element (SFEM) method is exploited in a nonlinear shock and vibration application where parametric uncertainty enters through random variables with probabilistic descriptions assumed to be known. The system output is represented as a vector containing Shock Response Spectrum (SRS) data at a predetermined number of frequency points. In contrast to many reliability-based methods, the goal of the current approach is to provide a means to address more general (vector) output entities, to provide this output as a random process, and to assess characteristics of the response which allow one to avoid issues of statistical dependence among its vector components.
Orthonormal aberration polynomials for anamorphic optical imaging systems with circular pupils.
Mahajan, Virendra N
2012-06-20
In a recent paper, we considered the classical aberrations of an anamorphic optical imaging system with a rectangular pupil, representing the terms of a power series expansion of its aberration function. These aberrations are inherently separable in the Cartesian coordinates (x,y) of a point on the pupil. Accordingly, there is x-defocus and x-coma, y-defocus and y-coma, and so on. We showed that the aberration polynomials orthonormal over the pupil and representing balanced aberrations for such a system are represented by the products of two Legendre polynomials, one for each of the two Cartesian coordinates of the pupil point; for example, L(l)(x)L(m)(y), where l and m are positive integers (including zero) and L(l)(x), for example, represents an orthonormal Legendre polynomial of degree l in x. The compound two-dimensional (2D) Legendre polynomials, like the classical aberrations, are thus also inherently separable in the Cartesian coordinates of the pupil point. Moreover, for every orthonormal polynomial L(l)(x)L(m)(y), there is a corresponding orthonormal polynomial L(l)(y)L(m)(x) obtained by interchanging x and y. These polynomials are different from the corresponding orthogonal polynomials for a system with rotational symmetry but a rectangular pupil. In this paper, we show that the orthonormal aberration polynomials for an anamorphic system with a circular pupil, obtained by the Gram-Schmidt orthogonalization of the 2D Legendre polynomials, are not separable in the two coordinates. Moreover, for a given polynomial in x and y, there is no corresponding polynomial obtained by interchanging x and y. For example, there are polynomials representing x-defocus, balanced x-coma, and balanced x-spherical aberration, but no corresponding y-aberration polynomials. The missing y-aberration terms are contained in other polynomials. We emphasize that the Zernike circle polynomials, although orthogonal over a circular pupil, are not suitable for an anamorphic system as
Abelian avalanches and Tutte polynomials
NASA Astrophysics Data System (ADS)
Gabrielov, Andrei
1993-04-01
We introduce a class of deterministic lattice models of failure, Abelian avalanche (AA) models, with continuous phase variables, similar to discrete Abelian sandpile (ASP) models. We investigate analytically the structure of the phase space and statistical properties of avalanches in these models. We show that the distributions of avalanches in AA and ASP models with the same redistribution matrix and loading rate are identical. For an AA model on a graph, statistics of avalanches is linked to Tutte polynomials associated with this graph and its subgraphs. In the general case, statistics of avalanches is linked to an analog of a Tutte polynomial defined for any symmetric matrix.
On Polynomial Solutions of Linear Differential Equations with Polynomial Coefficients
ERIC Educational Resources Information Center
Si, Do Tan
1977-01-01
Demonstrates a method for solving linear differential equations with polynomial coefficients based on the fact that the operators z and D + d/dz are known to be Hermitian conjugates with respect to the Bargman and Louck-Galbraith scalar products. (MLH)
NASA Astrophysics Data System (ADS)
Qi, Tongfei
Ca2RuO4 is a structurally-driven Mott insulator with a metal-insulator (MI) transition at TMI = 357K, followed by a well-separated antiferromagnetic order at T N = 110 K. Slightly substituting Ru with a 3d transition metal ion M effectively shifts TMI and induces exotic magnetic behavior below TN. Moreover, M doping for Ru produces negative thermal expansion in Ca2Ru1-- xMxO4 (M = Cr, Mn, Fe or Cu); the lattice volume expands on cooling with a total volume expansion ratio, DeltaV/V, reaching as high as 1%. The onset of the negative thermal expansion closely tracks TMI and TN, sharply contrasting classic negative thermal expansion that shows no relevance to electronic properties. In addition, the observed negative thermal expansion occurs near room temperature and extends over a wide temperature interval. These findings underscores new physics driven by a complex interplay between orbital, spin and lattice degrees of freedom. These materials constitute a new class of Negative Thermal Expansion (NTE) materials with novel electronic and magnetic functions. KEYWORDS: Transition Metal Oxide, Ruthenate, Negative Thermal Expansion, Single crystal XRD, Invar Effect, Orbital Ordering, Magnetic Ordering, Jahn-Teller Effect.
Entanglement conditions and polynomial identities
Shchukin, E.
2011-11-15
We develop a rather general approach to entanglement characterization based on convexity properties and polynomial identities. This approach is applied to obtain simple and efficient entanglement conditions that work equally well in both discrete as well as continuous-variable environments. Examples of violations of our conditions are presented.
Polynomial Beam Element Analysis Module
Ning, S. Andrew
2013-05-01
pBEAM (Polynomial Beam Element Analysis Module) is a finite element code for beam-like structures. The methodology uses Euler? Bernoulli beam elements with 12 degrees of freedom (3 translation and 3 rotational at each end of the element).
Kropf, Pascal; Shmuel, Amir
2016-07-01
Estimation of current source density (CSD) from the low-frequency part of extracellular electric potential recordings is an unstable linear inverse problem. To make the estimation possible in an experimental setting where recordings are contaminated with noise, it is necessary to stabilize the inversion. Here we present a unified framework for zero- and higher-order singular-value-decomposition (SVD)-based spectral regularization of 1D (linear) CSD estimation from local field potentials. The framework is based on two general approaches commonly employed for solving inverse problems: quadrature and basis function expansion. We first show that both inverse CSD (iCSD) and kernel CSD (kCSD) fall into the category of basis function expansion methods. We then use these general categories to introduce two new estimation methods, quadrature CSD (qCSD), based on discretizing the CSD integral equation with a chosen quadrature rule, and representer CSD (rCSD), an even-determined basis function expansion method that uses the problem's data kernels (representers) as basis functions. To determine the best candidate methods to use in the analysis of experimental data, we compared the different methods on simulations under three regularization schemes (Tikhonov, tSVD, and dSVD), three regularization parameter selection methods (NCP, L-curve, and GCV), and seven different a priori spatial smoothness constraints on the CSD distribution. This resulted in a comparison of 531 estimation schemes. We evaluated the estimation schemes according to their source reconstruction accuracy by testing them using different simulated noise levels, lateral source diameters, and CSD depth profiles. We found that ranking schemes according to the average error over all tested conditions results in a reproducible ranking, where the top schemes are found to perform well in the majority of tested conditions. However, there is no single best estimation scheme that outperforms all others under all tested
NASA Astrophysics Data System (ADS)
Dionissopoulou, S.; Mercouris, Th.; Lyras, A.; Komninos, Y.; Nicolaides, C. A.
1995-04-01
We have computed the above-threshold ionization and the emitted harmonic spectra of H interacting with short laser pulses, with photon energies ranging from 1.16 to 5.44 eV and with peak intensities ranging from 6×1013 to 7×1014 W/cm2, by solving the time-dependent Schrödinger equation (TDSE). The method of solution involves the expansion of the time-dependent wave function Ψ(r-->,t) over the exact wave functions of the discrete and the continuous spectrum, computed numerically, and the subsequent integration of the resulting coupled first-order differential equations by a Taylor series expansion technique. This state-specific approach (SSA) to the solution of the TDSE allows systematic understanding of convergence as a function of the number and type of the field-free states for each value of the laser frequency (ω) and peak intensity (I0). For example, the method allows practical numerical study of the degree of participation of high (n,l) (l=0,1,...,n-1) Rydberg, as well as of high-energy scattering states for each partial wave. For the harmonic spectra, comparisons are made between the results obtained by the SSA and those obtained in recent years by a number of researchers from the application of finite-difference grid methods. As regards economy, a general observation is that in the SSA the necessary number of partial waves is smaller than that required in the grid methods. Predictions are made for the case of ħω=2 eV, I0=2×1014 W/cm2, in the context of a study of the effect of the pulse shape on the harmonic-generation spectrum. It is shown that the number of harmonics and the appearance of the plateau depend on the duration of the peak intensity.
Hydrodynamic-type systems describing 2-dimensional polynomially integrable geodesic flows
NASA Astrophysics Data System (ADS)
Manno, Gianni; Pavlov, Maxim V.
2017-03-01
Starting from a homogeneous polynomial in momenta of arbitrary order we extract multi-component hydrodynamic-type systems which describe 2-dimensional geodesic flows admitting the initial polynomial as integral. All these hydrodynamic-type systems are semi-Hamiltonian, thus implying that they are integrable according to the generalized hodograph method. Moreover, they are integrable in a constructive sense as polynomial first integrals allow to construct generating equations of conservation laws. According to the multiplicity of the roots of the polynomial integral, we separate integrable particular cases.
The Chebyshev Polynomials: Patterns and Derivation
ERIC Educational Resources Information Center
Sinwell, Benjamin
2004-01-01
The Chebyshev polynomials named after a Russian mathematician, Pafnuty Lvovich Chebyshev, have various mathematical applications. A process for obtaining Chebyshev polynomials, and a mathematical inquiry into the patterns they generate, is presented.
Korobov polynomials of the first kind
NASA Astrophysics Data System (ADS)
Dolgy, D. V.; Kim, D. S.; Kim, T.
2017-01-01
In this paper, we study Korobov polynomials of the first kind from the viewpoint of umbral calculus and give new identities for them, associated with special polynomials which are derived from umbral calculus. Bibliography: 12 titles.
NASA Astrophysics Data System (ADS)
Kalagov, G. A.; Kompaniets, M. V.; Nalimov, M. Yu.
2014-11-01
We use quantum-field renormalization group methods to study the phase transition in an equilibrium system of nonrelativistic Fermi particles with the "density-density" interaction in the formalism of temperature Green's functions. We especially attend to the case of particles with spins greater than 1/2 or fermionic fields with additional indices for some reason. In the vicinity of the phase transition point, we reduce this model to a ϕ 4 -type theory with a matrix complex skew-symmetric field. We define a family of instantons of this model and investigate the asymptotic behavior of quantum field expansions in this model. We calculate the β-functions of the renormalization group equation through the third order in the ( 4 ∈)-scheme. In the physical space dimensions D = 2, 3, we resum solutions of the renormalization group equation on trajectories of invariant charges. Our results confirm the previously proposed suggestion that in the system under consideration, there is a first-order phase transition into a superconducting state that occurs at a higher temperature than the classical theory predicts.
Borrel, Guillaume; Gaci, Nadia; Peyret, Pierre; O'Toole, Paul W.; Gribaldo, Simonetta
2014-01-01
Pyrrolysine (Pyl), the 22nd proteogenic amino acid, was restricted until recently to few organisms. Its translational use necessitates the presence of enzymes for synthesizing it from lysine, a dedicated amber stop codon suppressor tRNA, and a specific amino-acyl tRNA synthetase. The three genomes of the recently proposed Thermoplasmata-related 7th order of methanogens contain the complete genetic set for Pyl synthesis and its translational use. Here, we have analyzed the genomic features of the Pyl-coding system in these three genomes with those previously known from Bacteria and Archaea and analyzed the phylogeny of each component. This shows unique peculiarities, notably an amber tRNAPyl with an imperfect anticodon stem and a shortened tRNAPyl synthetase. Phylogenetic analysis indicates that a Pyl-coding system was present in the ancestor of the seventh order of methanogens and appears more closely related to Bacteria than to Methanosarcinaceae, suggesting the involvement of lateral gene transfer in the spreading of pyrrolysine between the two prokaryotic domains. We propose that the Pyl-coding system likely emerged once in Archaea, in a hydrogenotrophic and methanol-H2-dependent methylotrophic methanogen. The close relationship between methanogenesis and the Pyl system provides a possible example of expansion of a still evolving genetic code, shaped by metabolic requirements. PMID:24669202
A New Generalisation of Macdonald Polynomials
NASA Astrophysics Data System (ADS)
Garbali, Alexandr; de Gier, Jan; Wheeler, Michael
2017-01-01
We introduce a new family of symmetric multivariate polynomials, whose coefficients are meromorphic functions of two parameters (q, t) and polynomial in a further two parameters (u, v). We evaluate these polynomials explicitly as a matrix product. At u = v = 0 they reduce to Macdonald polynomials, while at q = 0, u = v = s they recover a family of inhomogeneous symmetric functions originally introduced by Borodin.
A New Generalisation of Macdonald Polynomials
NASA Astrophysics Data System (ADS)
Garbali, Alexandr; de Gier, Jan; Wheeler, Michael
2017-06-01
We introduce a new family of symmetric multivariate polynomials, whose coefficients are meromorphic functions of two parameters ( q, t) and polynomial in a further two parameters ( u, v). We evaluate these polynomials explicitly as a matrix product. At u = v = 0 they reduce to Macdonald polynomials, while at q = 0, u = v = s they recover a family of inhomogeneous symmetric functions originally introduced by Borodin.
Nodal Statistics for the Van Vleck Polynomials
NASA Astrophysics Data System (ADS)
Bourget, Alain
The Van Vleck polynomials naturally arise from the generalized Lamé equation
A Summation Formula for Macdonald Polynomials
NASA Astrophysics Data System (ADS)
de Gier, Jan; Wheeler, Michael
2016-03-01
We derive an explicit sum formula for symmetric Macdonald polynomials. Our expression contains multiple sums over the symmetric group and uses the action of Hecke generators on the ring of polynomials. In the special cases {t = 1} and {q = 0}, we recover known expressions for the monomial symmetric and Hall-Littlewood polynomials, respectively. Other specializations of our formula give new expressions for the Jack and q-Whittaker polynomials.
From Jack to Double Jack Polynomials via the Supersymmetric Bridge
NASA Astrophysics Data System (ADS)
Lapointe, Luc; Mathieu, Pierre
2015-07-01
The Calogero-Sutherland model occurs in a large number of physical contexts, either directly or via its eigenfunctions, the Jack polynomials. The supersymmetric counterpart of this model, although much less ubiquitous, has an equally rich structure. In particular, its eigenfunctions, the Jack superpolynomials, appear to share the very same remarkable combinatorial and structural properties as their non-supersymmetric version. These super-functions are parametrized by superpartitions with fixed bosonic and fermionic degrees. Now, a truly amazing feature pops out when the fermionic degree is sufficiently large: the Jack superpolynomials stabilize and factorize. Their stability is with respect to their expansion in terms of an elementary basis where, in the stable sector, the expansion coefficients become independent of the fermionic degree. Their factorization is seen when the fermionic variables are stripped off in a suitable way which results in a product of two ordinary Jack polynomials (somewhat modified by plethystic transformations), dubbed the double Jack polynomials. Here, in addition to spelling out these results, which were first obtained in the context of Macdonal superpolynomials, we provide a heuristic derivation of the Jack superpolynomial case by performing simple manipulations on the supersymmetric eigen-operators, rendering them independent of the number of particles and of the fermionic degree. In addition, we work out the expression of the Hamiltonian which characterizes the double Jacks. This Hamiltonian, which defines a new integrable system, involves not only the expected Calogero-Sutherland pieces but also combinations of the generators of an underlying affine {widehat{sl}_2} algebra.
Modular polynomial arithmetic in partial fraction decomposition
NASA Technical Reports Server (NTRS)
Abdali, S. K.; Caviness, B. F.; Pridor, A.
1977-01-01
Algorithms for general partial fraction decomposition are obtained by using modular polynomial arithmetic. An algorithm is presented to compute inverses modulo a power of a polynomial in terms of inverses modulo that polynomial. This algorithm is used to make an improvement in the Kung-Tong partial fraction decomposition algorithm.
Restricted Schur polynomials and finite N counting
Collins, Storm
2009-01-15
Restricted Schur polynomials have been posited as orthonormal operators for the change of basis from N=4 SYM to type IIB string theory. In this paper we briefly expound the relationship between the restricted Schur polynomials and the operators forwarded by Brown, Heslop, and Ramgoolam. We then briefly examine the finite N counting of the restricted Schur polynomials.
NASA Astrophysics Data System (ADS)
Espíndola-Heredia, Rodolfo; del Río, Fernando; Malijevsky, Anatol
2009-01-01
The free energy of square-well (SW) systems of hard-core diameter σ with ranges 1≤λ≤3 is expanded in a perturbation series. This interval covers most ranges of interest, from short-ranged SW fluids (λ ≃1.2) used in modeling colloids to long ranges (λ ≃3) where the van der Waals classic approximation holds. The first four terms are evaluated by means of extensive Monte Carlo simulations. The calculations are corrected for the thermodynamic limit and care is taken to evaluate and to control the various sources of error. The results for the first two terms in the series confirm well-known independent results but have an increased estimated accuracy and cover a wider set of well ranges. The results for the third- and fourth-order terms are novel. The free-energy expansion for systems with short and intermediate ranges, 1≤λ≤2, is seen to have properties similar to those of systems with longer ranges, 2≤λ≤3. An equation of state (EOS) is built to represent the free-energy data. The thermodynamics given by this EOS, confronted against independent computer simulations, is shown to predict accurately the internal energy, pressure, specific heat, and chemical potential of the SW fluids considered and for densities 0≤ρσ3≤0.9 including subcritical temperatures. This fourth-order theory is estimated to be accurate except for a small region at high density, ρσ3≈0.9, and low temperature where terms of still higher order might be needed.
HOMFLY polynomials in representation [3, 1] for 3-strand braids
NASA Astrophysics Data System (ADS)
Mironov, A.; Morozov, A.; Morozov, An.; Sleptsov, A.
2016-09-01
This paper is a new step in the project of systematic description of colored knot polynomials started in [1]. In this paper, we managed to explicitly find the inclusive Racah matrix, i.e. the whole set of mixing matrices in channels R ⊗3 -→ Q with all possible Q, for R = [3 , 1]. The calculation is made possible by the use of a newly-developed efficient highest-weight method, still it remains tedious. The result allows one to evaluate and investigate [3 , 1]-colored polynomials for arbitrary 3-strand knots, and this confirms many previous conjectures on various factorizations, universality, and differential expansions. We consider in some detail the next-to-twist-knots three-strand family ( n, -1 | 1 , -1) and deduce its colored HOMFLY. Also confirmed and clarified is the eigenvalue hypothesis for the Racah matrices, which promises to provide a shortcut to generic formulas for arbitrary representations.
Knot polynomials in the first non-symmetric representation
NASA Astrophysics Data System (ADS)
Anokhina, A.; Mironov, A.; Morozov, A.; Morozov, And.
2014-05-01
We describe the explicit form and the hidden structure of the answer for the HOMFLY polynomial for the figure-8 and some other 3-strand knots in representation [21]. This is the first result for non-torus knots beyond (anti)symmetric representations, and its evaluation is far more complicated. We provide a whole variety of different arguments, allowing one to guess the answer for the figure-8 knot, which can be also partly used in more complicated situations. Finally we report the result of exact calculation for figure-8 and some other 3-strand knots based on the previously developed sophisticated technique of multi-strand calculations. We also discuss a formula for the superpolynomial in representation [21] for the figure-8 knot, which heavily relies on the conjectural form of superpolynomial expansion nearby the special polynomial point. Generalizations and details will be presented elsewhere.
Weighted Polynomial Approximation for Automated Detection of Inspiratory Flow Limitation.
Huang, Sheng-Cheng; Jan, Hao-Yu; Fu, Tieh-Cheng; Lin, Wen-Chen; Lin, Geng-Hong; Lin, Wen-Chi; Tsai, Cheng-Lun; Lin, Kang-Ping
2017-01-01
Inspiratory flow limitation (IFL) is a critical symptom of sleep breathing disorders. A characteristic flattened flow-time curve indicates the presence of highest resistance flow limitation. This study involved investigating a real-time algorithm for detecting IFL during sleep. Three categories of inspiratory flow shape were collected from previous studies for use as a development set. Of these, 16 cases were labeled as non-IFL and 78 as IFL which were further categorized into minor level (20 cases) and severe level (58 cases) of obstruction. In this study, algorithms using polynomial functions were proposed for extracting the features of IFL. Methods using first- to third-order polynomial approximations were applied to calculate the fitting curve to obtain the mean absolute error. The proposed algorithm is described by the weighted third-order (w.3rd-order) polynomial function. For validation, a total of 1,093 inspiratory breaths were acquired as a test set. The accuracy levels of the classifications produced by the presented feature detection methods were analyzed, and the performance levels were compared using a misclassification cobweb. According to the results, the algorithm using the w.3rd-order polynomial approximation achieved an accuracy of 94.14% for IFL classification. We concluded that this algorithm achieved effective automatic IFL detection during sleep.
Weighted Polynomial Approximation for Automated Detection of Inspiratory Flow Limitation
Huang, Sheng-Cheng; Jan, Hao-Yu; Fu, Tieh-Cheng; Lin, Geng-Hong; Lin, Wen-Chi; Lin, Kang-Ping
2017-01-01
Inspiratory flow limitation (IFL) is a critical symptom of sleep breathing disorders. A characteristic flattened flow-time curve indicates the presence of highest resistance flow limitation. This study involved investigating a real-time algorithm for detecting IFL during sleep. Three categories of inspiratory flow shape were collected from previous studies for use as a development set. Of these, 16 cases were labeled as non-IFL and 78 as IFL which were further categorized into minor level (20 cases) and severe level (58 cases) of obstruction. In this study, algorithms using polynomial functions were proposed for extracting the features of IFL. Methods using first- to third-order polynomial approximations were applied to calculate the fitting curve to obtain the mean absolute error. The proposed algorithm is described by the weighted third-order (w.3rd-order) polynomial function. For validation, a total of 1,093 inspiratory breaths were acquired as a test set. The accuracy levels of the classifications produced by the presented feature detection methods were analyzed, and the performance levels were compared using a misclassification cobweb. According to the results, the algorithm using the w.3rd-order polynomial approximation achieved an accuracy of 94.14% for IFL classification. We concluded that this algorithm achieved effective automatic IFL detection during sleep. PMID:28634497
Discontinuous Galerkin method based on non-polynomial approximation spaces
Yuan Ling . E-mail: lyuan@dam.brown.edu; Shu Chiwang . E-mail: shu@dam.brown.edu
2006-10-10
In this paper, we develop discontinuous Galerkin (DG) methods based on non-polynomial approximation spaces for numerically solving time dependent hyperbolic and parabolic and steady state hyperbolic and elliptic partial differential equations (PDEs). The algorithm is based on approximation spaces consisting of non-polynomial elementary functions such as exponential functions, trigonometric functions, etc., with the objective of obtaining better approximations for specific types of PDEs and initial and boundary conditions. It is shown that L {sup 2} stability and error estimates can be obtained when the approximation space is suitably selected. It is also shown with numerical examples that a careful selection of the approximation space to fit individual PDE and initial and boundary conditions often provides more accurate results than the DG methods based on the polynomial approximation spaces of the same order of accuracy.
Freeform surface of progressive addition lens represented by Zernike polynomials
NASA Astrophysics Data System (ADS)
Li, Yiyu; Xia, Risheng; Chen, Jiaojie; Feng, Haihua; Yuan, Yimin; Zhu, Dexi; Li, Chaohong
2016-10-01
We used the explicit expression of Zernike polynomials in Cartesian coordinates to fit and describe the freeform surface of progressive addition lens (PAL). The derivatives of Zernike polynomials can easily be calculated from the explicit expression and used to calculate the principal curvatures of freeform surface based on differential geometry. The surface spherical power and surface astigmatism of the freeform surface were successfully derived from the principal curvatures. By comparing with the traditional analytical method, Zernike polynomials with order of 20 is sufficient to represent the freeform surface with nanometer accuracy if dense sampling of the original surface is achieved. Therefore, the data files which contain the massive sampling points of the freeform surface for the generation of the trajectory of diamond tool tip required by diamond machine for PAL manufacture can be simplified by using a few Zernike coefficients.
On adaptive weighted polynomial preconditioning for Hermitian positive definite matrices
NASA Technical Reports Server (NTRS)
Fischer, Bernd; Freund, Roland W.
1992-01-01
The conjugate gradient algorithm for solving Hermitian positive definite linear systems is usually combined with preconditioning in order to speed up convergence. In recent years, there has been a revival of polynomial preconditioning, motivated by the attractive features of the method on modern architectures. Standard techniques for choosing the preconditioning polynomial are based only on bounds for the extreme eigenvalues. Here a different approach is proposed, which aims at adapting the preconditioner to the eigenvalue distribution of the coefficient matrix. The technique is based on the observation that good estimates for the eigenvalue distribution can be derived after only a few steps of the Lanczos process. This information is then used to construct a weight function for a suitable Chebyshev approximation problem. The solution of this problem yields the polynomial preconditioner. In particular, we investigate the use of Bernstein-Szego weights.
NASA Astrophysics Data System (ADS)
Leont'ev, V. K.
2015-11-01
A pseudo-Boolean function is an arbitrary mapping of the set of binary n-tuples to the real line. Such functions are a natural generalization of classical Boolean functions and find numerous applications in various applied studies. Specifically, the Fourier transform of a Boolean function is a pseudo-Boolean function. A number of facts associated with pseudo-Boolean polynomials are presented, and their applications to well-known discrete optimization problems are described.
Weak lensing tomography with orthogonal polynomials
NASA Astrophysics Data System (ADS)
Schäfer, Björn Malte; Heisenberg, Lavinia
2012-07-01
The topic of this paper is weak cosmic shear tomography where the line-of-sight weighting is carried out with a set of specifically constructed orthogonal polynomials, dubbed Tomography with Orthogonal Radial Distance Polynomial Systems (TaRDiS). We investigate the properties of these polynomials and employ weak convergence spectra, which have been obtained by weighting with these polynomials, for the estimation of cosmological parameters. We quantify their power in constraining parameters in a Fisher matrix technique and demonstrate how each polynomial projects out statistically independent information, and how the combination of multiple polynomials lifts degeneracies. The assumption of a reference cosmology is needed for the construction of the polynomials, and as a last point we investigate how errors in the construction with a wrong cosmological model propagate to misestimates in cosmological parameters. TaRDiS performs on a similar level as traditional tomographic methods and some key features of tomography are made easier to understand.
Tutte polynomial of the Apollonian network
NASA Astrophysics Data System (ADS)
Liao, Yunhua; Hou, Yaoping; Shen, Xiaoling
2014-10-01
The Tutte polynomial of a graph, or equivalently the q-state Potts model partition function, is a two-variable polynomial graph invariant of considerable importance in both combinatorics and statistical physics. The computation of this invariant for a graph is, in general, NP-hard. The aim of this paper is to compute the Tutte polynomial of the Apollonian network. Based on the well-known duality property of the Tutte polynomial, we extend the subgraph-decomposition method. In particular, we do not calculate the Tutte polynomial of the Apollonian network directly, instead we calculate the Tutte polynomial of the Apollonian dual graph. By using the close relation between the Apollonian dual graph and the Hanoi graph, we express the Tutte polynomial of the Apollonian dual graph in terms of that of the Hanoi graph. As an application, we also give the number of spanning trees of the Apollonian network.
Grid and basis adaptive polynomial chaos techniques for sensitivity and uncertainty analysis
Perkó, Zoltán Gilli, Luca Lathouwers, Danny Kloosterman, Jan Leen
2014-03-01
The demand for accurate and computationally affordable sensitivity and uncertainty techniques is constantly on the rise and has become especially pressing in the nuclear field with the shift to Best Estimate Plus Uncertainty methodologies in the licensing of nuclear installations. Besides traditional, already well developed methods – such as first order perturbation theory or Monte Carlo sampling – Polynomial Chaos Expansion (PCE) has been given a growing emphasis in recent years due to its simple application and good performance. This paper presents new developments of the research done at TU Delft on such Polynomial Chaos (PC) techniques. Our work is focused on the Non-Intrusive Spectral Projection (NISP) approach and adaptive methods for building the PCE of responses of interest. Recent efforts resulted in a new adaptive sparse grid algorithm designed for estimating the PC coefficients. The algorithm is based on Gerstner's procedure for calculating multi-dimensional integrals but proves to be computationally significantly cheaper, while at the same it retains a similar accuracy as the original method. More importantly the issue of basis adaptivity has been investigated and two techniques have been implemented for constructing the sparse PCE of quantities of interest. Not using the traditional full PC basis set leads to further reduction in computational time since the high order grids necessary for accurately estimating the near zero expansion coefficients of polynomial basis vectors not needed in the PCE can be excluded from the calculation. Moreover the sparse PC representation of the response is easier to handle when used for sensitivity analysis or uncertainty propagation due to the smaller number of basis vectors. The developed grid and basis adaptive methods have been implemented in Matlab as the Fully Adaptive Non-Intrusive Spectral Projection (FANISP) algorithm and were tested on four analytical problems. These show consistent good performance both
Approximating rational triangular Bézier surfaces by polynomial triangular Bézier surfaces
NASA Astrophysics Data System (ADS)
Xu, Hui-Xia; Wang, Guo-Jin
2009-06-01
An attractive method for approximating rational triangular Bézier surfaces by polynomial triangular Bézier surfaces is introduced. The main result is that the arbitrary given order derived vectors of a polynomial triangular surface converge uniformly to those of the approximated rational triangular Bézier surface as the elevated degree tends to infinity. The polynomial triangular surface is constructed as follows. Firstly, we elevate the degree of the approximated rational triangular Bézier surface, then a polynomial triangular Bézier surface is produced, which has the same order and new control points of the degree-elevated rational surface. The approximation method has theoretical significance and application value: it solves two shortcomings-fussy expression and uninsured convergence of the approximation-of Hybrid algorithms for rational polynomial curves and surfaces approximation.
Laricchia, Savio; Constantin, Lucian A; Fabiano, Eduardo; Della Sala, Fabio
2014-01-14
We tested Laplacian-level meta-generalized gradient approximation (meta-GGA) noninteracting kinetic energy functionals based on the fourth-order gradient expansion (GE4). We considered several well-known Laplacian-level meta-GGAs from the literature (bare GE4, modified GE4, and the MGGA functional of Perdew and Constantin (Phys. Rev. B 2007,75, 155109)), as well as two newly designed Laplacian-level kinetic energy functionals (L0.4 and L0.6). First, a general assessment of the different functionals is performed to test them for model systems (one-electron densities, Hooke's atom, and different jellium systems) and atomic and molecular kinetic energies as well as for their behavior with respect to density-scaling transformations. Finally, we assessed, for the first time, the performance of the different functionals for subsystem density functional theory (DFT) calculations on noncovalently interacting systems. We found that the different Laplacian-level meta-GGA kinetic functionals may improve the description of different properties of electronic systems, but no clear overall advantage is found over the best GGA functionals. Concerning the subsystem DFT calculations, the here-proposed L0.4 and L0.6 kinetic energy functionals are competitive with state-of-the-art GGAs, whereas all other Laplacian-level functionals fail badly. The performance of the Laplacian-level functionals is rationalized thanks to a two-dimensional reduced-gradient and reduced-Laplacian decomposition of the nonadditive kinetic energy density.
Thermal expansion of the hexagonal (4H) polytype of SiC
NASA Technical Reports Server (NTRS)
Li, Z.; Bradt, R. C.
1986-01-01
The principal axial coefficients of thermal expansion, alpha(11) and alpha(33), of the (4H) polytype of hexagonal alpha SiC have been determined by X-ray diffraction measurements in the temperature range 20-1000 C. Alpha(11) and alpha(33), derived from the lattice parameter measurements, were expressed as the second-order polynomials in temperature. Alpha(11) was found to be larger than alpha(33) over the entire temperature range, with a thermal expansion anisotropy factor A increasing from 0.04 at room temperature to 0.11 at 1000 C. The thermal expansion results for the (4H) structure were compared with previously published results for the cubic (3C) and the hexagonal (6H) SiC polytypes.
Thermal expansion of the cubic (3C) polytype of SiC
NASA Technical Reports Server (NTRS)
Li, Z.; Bradt, R. C.
1986-01-01
Thermal expansion of the cubic beta or (3C) polytype of SiC was measured from 20 to 1000 C by the X-ray-diffraction technique. Over that temperature range, the coefficient of thermal expansion can be expressed by a second-order polynomial. It increases continuously from about 3.2 x 10 to the -6th/C at room temperature to 5.1 x 10 to the -6th/C at 1000 C, with an average value of 4.45 x 10 to the -6th/C between room temperature and 1000 C. This trend is compared with other published results and is discussed in terms of structural contributions to the thermal expansion.
Thermal expansion of the cubic (3C) polytype of SiC
NASA Technical Reports Server (NTRS)
Li, Z.; Bradt, R. C.
1986-01-01
Thermal expansion of the cubic beta or (3C) polytype of SiC was measured from 20 to 1000 C by the X-ray-diffraction technique. Over that temperature range, the coefficient of thermal expansion can be expressed by a second-order polynomial. It increases continuously from about 3.2 x 10 to the -6th/C at room temperature to 5.1 x 10 to the -6th/C at 1000 C, with an average value of 4.45 x 10 to the -6th/C between room temperature and 1000 C. This trend is compared with other published results and is discussed in terms of structural contributions to the thermal expansion.
Separating and turbulent boundary layer calculations using polynomial interpretation
NASA Technical Reports Server (NTRS)
Rubin, S. G.; Rivera, S.
1977-01-01
Higher order numerical methods derived from polynomial spline interpolation or Hermitian differencing are applied to a separating laminar boundary layer, i.e., the Howarth problem, and the turbulent flat plate boundary layer flow. Preliminary results are presented. It is found that accuracy equal to that of conventional second order accurate finite difference methods is achieved with many fewer mesh points and with reduced computer storage and time requirements.
The stable computation of formal orthogonal polynomials
NASA Astrophysics Data System (ADS)
Beckermann, Bernhard
1996-12-01
For many applications - such as the look-ahead variants of the Lanczos algorithm - a sequence of formal (block-)orthogonal polynomials is required. Usually, one generates such a sequence by taking suitable polynomial combinations of a pair of basis polynomials. These basis polynomials are determined by a look-ahead generalization of the classical three term recurrence, where the polynomial coefficients are obtained by solving a small system of linear equations. In finite precision arithmetic, the numerical orthogonality of the polynomials depends on a good choice of the size of the small systems; this size is usually controlled by a heuristic argument such as the condition number of the small matrix of coefficients. However, quite often it happens that orthogonality gets lost.
NASA Astrophysics Data System (ADS)
Bayad, A.; Kim, T.
2016-04-01
We introduce a special nonlinear differential operator and, using its properties, reduce higher-order Frobenius-Euler Apostol-type polynomials to a finite series of first-order Apostol-type Frobenius-Euler polynomials and Stirling numbers. Interesting identities are established.
A new Arnoldi approach for polynomial eigenproblems
Raeven, F.A.
1996-12-31
In this paper we introduce a new generalization of the method of Arnoldi for matrix polynomials. The new approach is compared with the approach of rewriting the polynomial problem into a linear eigenproblem and applying the standard method of Arnoldi to the linearised problem. The algorithm that can be applied directly to the polynomial eigenproblem turns out to be more efficient, both in storage and in computation.
NASA Astrophysics Data System (ADS)
Baldi, Antonio; Bertolino, Filippo
2013-10-01
It is well known that displacement components estimated using digital image correlation are affected by a systematic error due to the polynomial interpolation required by the numerical algorithm. The magnitude of bias depends on the characteristics of the speckle pattern (i.e., the frequency content of the image), on the fractional part of displacements and on the type of polynomial used for intensity interpolation. In literature, B-Spline polynomials are pointed out as being able to introduce the smaller errors, whereas bilinear and cubic interpolants generally give the worst results. However, the small bias of B-Spline polynomials is partially counterbalanced by a somewhat larger execution time. We will try to improve the accuracy of lower order polynomials by a posteriori correcting their results so as to obtain a faster and more accurate analysis.
Torus HOMFLYPT as the Hall-Littlewood polynomials
NASA Astrophysics Data System (ADS)
Mironov, A.; Morozov, A.; Shakirov, Sh
2012-09-01
We show that the HOMFLYPT polynomials for the torus knots T[m, n] in all fundamental representations are equal to the Hall-Littlewood polynomials in a representation which depends on m, and with a quantum parameter, which depends on n. This makes the long-anticipated interpretation of Wilson averages in 3d Chern-Simons theory as characters precise, at least for the torus knots, and calls for further studies in this direction. This fact is deeply related to the Hall-Littlewood-MacDonald duality of character expansion of superpolynomials found in Mironov et al (2012 J. Geom. Phys. 62 148-55). In fact, the relation continues to hold for extended polynomials, but the symmetry between m and n is broken; then m is the number of strands in the braid. Besides the HOMFLYPT case with q = t, the torus superpolynomials are reduced to the single Hall-Littlewood characters in the two other distinguished cases: q = 0 and t = 0.
Relative risk regression models with inverse polynomials.
Ning, Yang; Woodward, Mark
2013-08-30
The proportional hazards model assumes that the log hazard ratio is a linear function of parameters. In the current paper, we model the log relative risk as an inverse polynomial, which is particularly suitable for modeling bounded and asymmetric functions. The parameters estimated by maximizing the partial likelihood are consistent and asymptotically normal. The advantages of the inverse polynomial model over the ordinary polynomial model and the fractional polynomial model for fitting various asymmetric log relative risk functions are shown by simulation. The utility of the method is further supported by analyzing two real data sets, addressing the specific question of the location of the minimum risk threshold.
Slave mode expansion for obtaining ab initio interatomic potentials
NASA Astrophysics Data System (ADS)
Ai, Xinyuan; Chen, Yue; Marianetti, Chris A.
2014-07-01
Here we propose an approach for performing a Taylor series expansion of the first-principles computed energy of a crystal as a function of the nuclear displacements. We enlarge the dimensionality of the existing displacement space and form new variables (i.e., slave modes) which transform like irreducible representations of the point group and satisfy homogeneity of free space. Standard group theoretical techniques can then be applied to deduce the nonzero expansion coefficients a priori. At a given order, the translation group can be used to contract the products and eliminate terms which are not linearly independent, resulting in a final set of slave mode products. While the expansion coefficients can be computed in a variety of ways, we demonstrate that finite difference is effective up to fourth order. We demonstrate the power of the method in the strongly anharmonic system PbTe. All anharmonic terms within an octahedron are computed up to fourth order. A proper unitary transformation demonstrates that the vast majority of the anharmonicity can be attributed to just two terms, indicating that a minimal model of phonon interactions is achievable. The ability to straightforwardly generate polynomial potentials will allow precise simulations at length and time scales which were previously unrealizable.
The complexity of class polynomial computation via floating point approximations
NASA Astrophysics Data System (ADS)
Enge, Andreas
2009-06-01
We analyse the complexity of computing class polynomials, that are an important ingredient for CM constructions of elliptic curves, via complex floating point approximations of their roots. The heart of the algorithm is the evaluation of modular functions in several arguments. The fastest one of the presented approaches uses a technique devised by Dupont to evaluate modular functions by Newton iterations on an expression involving the arithmetic-geometric mean. Under the heuristic assumption, justified by experiments, that the correctness of the result is not perturbed by rounding errors, the algorithm runs in time O left( sqrt {\\vert D\\vert} log^3 \\vert D\\vert M left( sq... ...arepsilon} \\vert D\\vert right) subseteq O left( h^{2 + \\varepsilon} right) for any \\varepsilon > 0 , where D is the CM discriminant, h is the degree of the class polynomial and M (n) is the time needed to multiply two n -bit numbers. Up to logarithmic factors, this running time matches the size of the constructed polynomials. The estimate also relies on a new result concerning the complexity of enumerating the class group of an imaginary quadratic order and on a rigorously proven upper bound for the height of class polynomials.
New pole placement algorithm - Polynomial matrix approach
NASA Technical Reports Server (NTRS)
Shafai, B.; Keel, L. H.
1990-01-01
A simple and direct pole-placement algorithm is introduced for dynamical systems having a block companion matrix A. The algorithm utilizes well-established properties of matrix polynomials. Pole placement is achieved by appropriately assigning coefficient matrices of the corresponding matrix polynomial. This involves only matrix additions and multiplications without requiring matrix inversion. A numerical example is given for the purpose of illustration.
Notes on Schubert, Grothendieck and Key Polynomials
NASA Astrophysics Data System (ADS)
Kirillov, Anatol N.
2016-03-01
We introduce common generalization of (double) Schubert, Grothendieck, Demazure, dual and stable Grothendieck polynomials, and Di Francesco-Zinn-Justin polynomials. Our approach is based on the study of algebraic and combinatorial properties of the reduced rectangular plactic algebra and associated Cauchy kernels.
Fractal Trigonometric Polynomials for Restricted Range Approximation
NASA Astrophysics Data System (ADS)
Chand, A. K. B.; Navascués, M. A.; Viswanathan, P.; Katiyar, S. K.
2016-05-01
One-sided approximation tackles the problem of approximation of a prescribed function by simple traditional functions such as polynomials or trigonometric functions that lie completely above or below it. In this paper, we use the concept of fractal interpolation function (FIF), precisely of fractal trigonometric polynomials, to construct one-sided uniform approximants for some classes of continuous functions.
On the Waring problem for polynomial rings
Fröberg, Ralf; Ottaviani, Giorgio; Shapiro, Boris
2012-01-01
In this note we discuss an analog of the classical Waring problem for . Namely, we show that a general homogeneous polynomial of degree divisible by k≥2 can be represented as a sum of at most kn k-th powers of homogeneous polynomials in . Noticeably, kn coincides with the number obtained by naive dimension count. PMID:22460787
Percolation critical polynomial as a graph invariant.
Scullard, Christian R
2012-10-01
Every lattice for which the bond percolation critical probability can be found exactly possesses a critical polynomial, with the root in [0,1] providing the threshold. Recent work has demonstrated that this polynomial may be generalized through a definition that can be applied on any periodic lattice. The polynomial depends on the lattice and on its decomposition into identical finite subgraphs, but once these are specified, the polynomial is essentially unique. On lattices for which the exact percolation threshold is unknown, the polynomials provide approximations for the critical probability with the estimates appearing to converge to the exact answer with increasing subgraph size. In this paper, I show how this generalized critical polynomial can be viewed as a graph invariant, similar to the Tutte polynomial. In particular, the critical polynomial is computed on a finite graph and may be found using the recursive deletion-contraction algorithm. This allows calculation on a computer, and I present such results for the kagome lattice using subgraphs of up to 36 bonds. For one of these, I find the prediction p(c)=0.52440572..., which differs from the numerical value, p(c)=0.52440503(5), by only 6.9×10(-7).
Percolation critical polynomial as a graph invariant
NASA Astrophysics Data System (ADS)
Scullard, Christian R.
2012-10-01
Every lattice for which the bond percolation critical probability can be found exactly possesses a critical polynomial, with the root in [0,1] providing the threshold. Recent work has demonstrated that this polynomial may be generalized through a definition that can be applied on any periodic lattice. The polynomial depends on the lattice and on its decomposition into identical finite subgraphs, but once these are specified, the polynomial is essentially unique. On lattices for which the exact percolation threshold is unknown, the polynomials provide approximations for the critical probability with the estimates appearing to converge to the exact answer with increasing subgraph size. In this paper, I show how this generalized critical polynomial can be viewed as a graph invariant, similar to the Tutte polynomial. In particular, the critical polynomial is computed on a finite graph and may be found using the recursive deletion-contraction algorithm. This allows calculation on a computer, and I present such results for the kagome lattice using subgraphs of up to 36 bonds. For one of these, I find the prediction pc=0.52440572⋯, which differs from the numerical value, pc=0.52440503(5), by only 6.9×10-7.
Polynomial solutions of nonlinear integral equations
NASA Astrophysics Data System (ADS)
Dominici, Diego
2009-05-01
We analyze the polynomial solutions of a nonlinear integral equation, generalizing the work of Bender and Ben-Naim (2007 J. Phys. A: Math. Theor. 40 F9, 2008 J. Nonlinear Math. Phys. 15 (Suppl. 3) 73). We show that, in some cases, an orthogonal solution exists and we give its general form in terms of kernel polynomials.
Some Recent Advances in Multivariate Polynomial Interpolation
NASA Astrophysics Data System (ADS)
Carnicer, J. M.; Gasca, M.
2007-09-01
Multivariate polynomial interpolation has received much attention in the last part of the 20th century. In this talk we comment on some recent advances in the last decade, with special emphasis in distributions of points which give rise to unisolvent (or poised) problems in the space of polynomials of a given total degree and simple interpolation formulae.
Polynomial integral for square and inverse-square potential systems
NASA Astrophysics Data System (ADS)
Virdi, Jasvinder Singh; Srivastava, A. K.; Ahmad, Muneer
2017-07-01
Classification and possibility of fourth order dynamical integral for some square and inverse-square potential in two-dimensional dynamical systems is searched out. It is based on the study of the quadratic algebras of the integrals of motion and on the equivalence of different systems under coupling constant metamorphosis. The determining equations for the existence of integrals of motion of arbitrary order are presented and partially solved for the case of fourth-order integrals. A systematic calculation is given of systems in two and higher dimensional pace that allow integrals of fourth order. The algebras of integrals of motions are not necessarily quadratic but close polynomially or rationally.
Tutte polynomial in functional magnetic resonance imaging
NASA Astrophysics Data System (ADS)
García-Castillón, Marlly V.
2015-09-01
Methods of graph theory are applied to the processing of functional magnetic resonance images. Specifically the Tutte polynomial is used to analyze such kind of images. Functional Magnetic Resonance Imaging provide us connectivity networks in the brain which are represented by graphs and the Tutte polynomial will be applied. The problem of computing the Tutte polynomial for a given graph is #P-hard even for planar graphs. For a practical application the maple packages "GraphTheory" and "SpecialGraphs" will be used. We will consider certain diagram which is depicting functional connectivity, specifically between frontal and posterior areas, in autism during an inferential text comprehension task. The Tutte polynomial for the resulting neural networks will be computed and some numerical invariants for such network will be obtained. Our results show that the Tutte polynomial is a powerful tool to analyze and characterize the networks obtained from functional magnetic resonance imaging.
On properties of bi-periodic Fibonacci and Lucas polynomials
NASA Astrophysics Data System (ADS)
Yilmaz, Nazmiye; Coskun, Arzu; Taskara, Necati
2017-07-01
In this paper, we define bi-periodic Fibonacci and Lucas polynomials and investigate properties of these polynomials which generalized of bi-periodic Fibonacci and Lucas numbers. We also obtain some new algebraic properties on these numbers and polynomials.
Generalized Gegenbauer Koornwinder's type polynomials change of bases
NASA Astrophysics Data System (ADS)
AlQudah, Mohammad; AlMheidat, Maalee
2017-07-01
In this paper we characterize the generalized Gegenbauer polynomials using Bernstein basis, and derive the matrix of transformation of the generalized Gegenbauer polynomial basis form into the Bernstein polynomial basis and vice versa.
ERIC Educational Resources Information Center
McArdle, Heather K.
1997-01-01
Describes a week-long activity for general to honors-level students that addresses Hubble's law and the universal expansion theory. Uses a discrepant event-type activity to lead up to the abstract principles of the universal expansion theory. (JRH)
ERIC Educational Resources Information Center
McArdle, Heather K.
1997-01-01
Describes a week-long activity for general to honors-level students that addresses Hubble's law and the universal expansion theory. Uses a discrepant event-type activity to lead up to the abstract principles of the universal expansion theory. (JRH)
NASA Astrophysics Data System (ADS)
Ventura, Guglielmo; Perfetti, Mauro
All solid materials, when cooled to low temperatures experience a change in physical dimensions which called "thermal contraction" and is typically lower than 1 % in volume in the 4-300 K temperature range. Although the effect is small, it can have a heavy impact on the design of cryogenic devices. The thermal contraction of different materials may vary by as much as an order of magnitude: since cryogenic devices are constructed at room temperature with a lot of different materials, one of the major concerns is the effect of the different thermal contraction and the resulting thermal stress that may occur when two dissimilar materials are bonded together. In this chapter, theory of thermal contraction is reported in Sect.
Ordered structure and thermal expansion in tungsten bronze Pb₂K(0.5)Li(0.5)Nb₅O₁₅.
Lin, Kun; Rong, Yangchun; Wu, Hui; Huang, Qingzhen; You, Li; Ren, Yang; Fan, Longlong; Chen, Jun; Xing, Xianran
2014-09-02
The crystal structure and thermal expansion behaviors of a new tetragonal tungsten bronze (TTB) ferroelectric, Pb2K(0.5)Li(0.5)Nb5O15, were systematically investigated by selected-area electron diffraction (SAED), neutron powder diffraction, synchrotron X-ray diffraction (XRD), and high-temperature XRD. SAED and Rietveld refinement reveal that Pb2K(0.5)Li(0.5)Nb5O15 displays a commensurate superstructure of simple orthorhombic TTB structure at room temperature. The structure can be described with space group Bb2₁m. The transition to a paraelectric phase (P4/mbm) occurs at 500 °C. Compared with Pb2KNb5O15 (PKN), the substitution of 0.5K(+) with small 0.5Li(+) into PKN causes the tilting of NbO6 octahedra away from the c axis with Δθ ≈ 10° and raises the Curie temperature by 40 °C, and the negative thermal expansion coefficient along the polar b axis increases more than 50% in the temperature range 25-500 °C. We present that, by introduction of Li(+), the enhanced spontaneous polarization is responsible for the enhanced negative thermal expansion along the b axis, which may be caused by more Pb(2+) in the pentagonal caves.
Lin, J. C.; Tong, P. Lin, S.; Wang, B. S.; Song, W. H.; Tong, W.; Zou, Y. M.; Sun, Y. P.
2015-02-23
The thermal expansion and magnetic properties of antiperovskite manganese nitrides Ag{sub 1−x}NMn{sub 3+x} were reported. The substitution of Mn for Ag effectively broadens the temperature range of negative thermal expansion and drives it to cryogenic temperatures. As x increases, the paramagnetic (PM) to antiferromagnetic (AFM) phase transition temperature decreases. At x ∼ 0.2, the PM-AFM transition overlaps with the AFM to glass-like state transition. Above x = 0.2, two new distinct magnetic transitions were observed: One occurs above room temperature from PM to ferromagnetic (FM), and the other one evolves at a lower temperature (T{sup *}) below which both AFM and FM orderings are involved. Further, electron spin resonance measurement suggests that the broadened volume change near T{sup *} is closely related with the evolution of Γ{sup 5g} AFM ordering.
NASA Astrophysics Data System (ADS)
Lin, J. C.; Tong, P.; Tong, W.; Lin, S.; Wang, B. S.; Song, W. H.; Zou, Y. M.; Sun, Y. P.
2015-02-01
The thermal expansion and magnetic properties of antiperovskite manganese nitrides Ag1-xNMn3+x were reported. The substitution of Mn for Ag effectively broadens the temperature range of negative thermal expansion and drives it to cryogenic temperatures. As x increases, the paramagnetic (PM) to antiferromagnetic (AFM) phase transition temperature decreases. At x ˜ 0.2, the PM-AFM transition overlaps with the AFM to glass-like state transition. Above x = 0.2, two new distinct magnetic transitions were observed: One occurs above room temperature from PM to ferromagnetic (FM), and the other one evolves at a lower temperature (T*) below which both AFM and FM orderings are involved. Further, electron spin resonance measurement suggests that the broadened volume change near T* is closely related with the evolution of Γ5g AFM ordering.
High-order regularization in lattice-Boltzmann equations
NASA Astrophysics Data System (ADS)
Mattila, Keijo K.; Philippi, Paulo C.; Hegele, Luiz A.
2017-04-01
A lattice-Boltzmann equation (LBE) is the discrete counterpart of a continuous kinetic model. It can be derived using a Hermite polynomial expansion for the velocity distribution function. Since LBEs are characterized by discrete, finite representations of the microscopic velocity space, the expansion must be truncated and the appropriate order of truncation depends on the hydrodynamic problem under investigation. Here we consider a particular truncation where the non-equilibrium distribution is expanded on a par with the equilibrium distribution, except that the diffusive parts of high-order non-equilibrium moments are filtered, i.e., only the corresponding advective parts are retained after a given rank. The decomposition of moments into diffusive and advective parts is based directly on analytical relations between Hermite polynomial tensors. The resulting, refined regularization procedure leads to recurrence relations where high-order non-equilibrium moments are expressed in terms of low-order ones. The procedure is appealing in the sense that stability can be enhanced without local variation of transport parameters, like viscosity, or without tuning the simulation parameters based on embedded optimization steps. The improved stability properties are here demonstrated using the perturbed double periodic shear layer flow and the Sod shock tube problem as benchmark cases.
NASA Technical Reports Server (NTRS)
Pototzky, Anthony S.
2008-01-01
A simple matrix polynomial approach is introduced for approximating unsteady aerodynamics in the s-plane and ultimately, after combining matrix polynomial coefficients with matrices defining the structure, a matrix polynomial of the flutter equations of motion (EOM) is formed. A technique of recasting the matrix-polynomial form of the flutter EOM into a first order form is also presented that can be used to determine the eigenvalues near the origin and everywhere on the complex plane. An aeroservoelastic (ASE) EOM have been generalized to include the gust terms on the right-hand side. The reasons for developing the new matrix polynomial approach are also presented, which are the following: first, the "workhorse" methods such as the NASTRAN flutter analysis lack the capability to consistently find roots near the origin, along the real axis or accurately find roots farther away from the imaginary axis of the complex plane; and, second, the existing s-plane methods, such as the Roger s s-plane approximation method as implemented in ISAC, do not always give suitable fits of some tabular data of the unsteady aerodynamics. A method available in MATLAB is introduced that will accurately fit generalized aerodynamic force (GAF) coefficients in a tabular data form into the coefficients of a matrix polynomial form. The root-locus results from the NASTRAN pknl flutter analysis, the ISAC-Roger's s-plane method and the present matrix polynomial method are presented and compared for accuracy and for the number and locations of roots.
The Translated Dowling Polynomials and Numbers
Mangontarum, Mahid M.; Macodi-Ringia, Amila P.; Abdulcarim, Normalah S.
2014-01-01
More properties for the translated Whitney numbers of the second kind such as horizontal generating function, explicit formula, and exponential generating function are proposed. Using the translated Whitney numbers of the second kind, we will define the translated Dowling polynomials and numbers. Basic properties such as exponential generating functions and explicit formula for the translated Dowling polynomials and numbers are obtained. Convexity, integral representation, and other interesting identities are also investigated and presented. We show that the properties obtained are generalizations of some of the known results involving the classical Bell polynomials and numbers. Lastly, we established the Hankel transform of the translated Dowling numbers. PMID:27433494
The Translated Dowling Polynomials and Numbers.
Mangontarum, Mahid M; Macodi-Ringia, Amila P; Abdulcarim, Normalah S
2014-01-01
More properties for the translated Whitney numbers of the second kind such as horizontal generating function, explicit formula, and exponential generating function are proposed. Using the translated Whitney numbers of the second kind, we will define the translated Dowling polynomials and numbers. Basic properties such as exponential generating functions and explicit formula for the translated Dowling polynomials and numbers are obtained. Convexity, integral representation, and other interesting identities are also investigated and presented. We show that the properties obtained are generalizations of some of the known results involving the classical Bell polynomials and numbers. Lastly, we established the Hankel transform of the translated Dowling numbers.
ECG data compression using Jacobi polynomials.
Tchiotsop, Daniel; Wolf, Didier; Louis-Dorr, Valérie; Husson, René
2007-01-01
Data compression is a frequent signal processing operation applied to ECG. We present here a method of ECG data compression utilizing Jacobi polynomials. ECG signals are first divided into blocks that match with cardiac cycles before being decomposed in Jacobi polynomials bases. Gauss quadratures mechanism for numerical integration is used to compute Jacobi transforms coefficients. Coefficients of small values are discarded in the reconstruction stage. For experimental purposes, we chose height families of Jacobi polynomials. Various segmentation approaches were considered. We elaborated an efficient strategy to cancel boundary effects. We obtained interesting results compared with ECG compression by wavelet decomposition methods. Some propositions are suggested to improve the results.
NASA Astrophysics Data System (ADS)
saev, Yu N. I.; Kolchanova, V. A.; Tarasenko, S. S.; Tikhomirova, O. V.
2016-04-01
The paper proposes an original method of calculating the charge distribution on the surface of the conductive plate introduced into the external electrostatic field. The authors managed to obtain the polynomials which allow to solve the integral equation that establishes the relationship between charge distribution of the conductive plate and the potential distribution of the external field and the potential on the surface of the plate. The proposed algorithms solutions are valid in the presence of axial symmetry of the field and the plate. Examples of calculation of conductor charge distribution in the presence of external field by using a polynomial expansion have been presented. The comparisons of results calculated by the polynomial method and by known analytical solutions have been given
A robust polynomial principal component analysis for seismic noise attenuation
NASA Astrophysics Data System (ADS)
Wang, Yuchen; Lu, Wenkai; Wang, Benfeng; Liu, Lei
2016-12-01
Random and coherent noise attenuation is a significant aspect of seismic data processing, especially for pre-stack seismic data flattened by normal moveout correction or migration. Signal extraction is widely used for pre-stack seismic noise attenuation. Principle component analysis (PCA), one of the multi-channel filters, is a common tool to extract seismic signals, which can be realized by singular value decomposition (SVD). However, when applying the traditional PCA filter to seismic signal extraction, the result is unsatisfactory with some artifacts when the seismic data is contaminated by random and coherent noise. In order to directly extract the desired signal and fix those artifacts at the same time, we take into consideration the amplitude variation with offset (AVO) property and thus propose a robust polynomial PCA algorithm. In this algorithm, a polynomial constraint is used to optimize the coefficient matrix. In order to simplify this complicated problem, a series of sub-optimal problems are designed and solved iteratively. After that, the random and coherent noise can be effectively attenuated simultaneously. Applications on synthetic and real data sets note that our proposed algorithm can better suppress random and coherent noise and have a better performance on protecting the desired signals, compared with the local polynomial fitting, conventional PCA and a L1-norm based PCA method.
Evaluation of Piecewise Polynomial Equations for Two Types of Thermocouples
Chen, Andrew; Chen, Chiachung
2013-01-01
Thermocouples are the most frequently used sensors for temperature measurement because of their wide applicability, long-term stability and high reliability. However, one of the major utilization problems is the linearization of the transfer relation between temperature and output voltage of thermocouples. The linear calibration equation and its modules could be improved by using regression analysis to help solve this problem. In this study, two types of thermocouple and five temperature ranges were selected to evaluate the fitting agreement of different-order polynomial equations. Two quantitative criteria, the average of the absolute error values |e|ave and the standard deviation of calibration equation estd, were used to evaluate the accuracy and precision of these calibrations equations. The optimal order of polynomial equations differed with the temperature range. The accuracy and precision of the calibration equation could be improved significantly with an adequate higher degree polynomial equation. The technique could be applied with hardware modules to serve as an intelligent sensor for temperature measurement. PMID:24351627
Uncertainty Quantification in Simulations of Epidemics Using Polynomial Chaos
Santonja, F.; Chen-Charpentier, B.
2012-01-01
Mathematical models based on ordinary differential equations are a useful tool to study the processes involved in epidemiology. Many models consider that the parameters are deterministic variables. But in practice, the transmission parameters present large variability and it is not possible to determine them exactly, and it is necessary to introduce randomness. In this paper, we present an application of the polynomial chaos approach to epidemiological mathematical models based on ordinary differential equations with random coefficients. Taking into account the variability of the transmission parameters of the model, this approach allows us to obtain an auxiliary system of differential equations, which is then integrated numerically to obtain the first-and the second-order moments of the output stochastic processes. A sensitivity analysis based on the polynomial chaos approach is also performed to determine which parameters have the greatest influence on the results. As an example, we will apply the approach to an obesity epidemic model. PMID:22927889
NASA Technical Reports Server (NTRS)
Moore, Craig E.; Cardelino, Beatriz H.; Frazier, Donald O.; Niles, Julian; Wang, Xian-Qiang
1997-01-01
Calculations were performed on the valence contribution to the static molecular third-order polarizabilities (gamma) of thirty carbon-cage fullerenes (C60, C70, five isomers of C78, and twenty-three isomers of C84). The molecular structures were obtained from B3LYP/STO-3G calculations. The values of the tensor elements and an associated numerical uncertainty were obtained using the finite-field approach and polynomial expansions of orders four to eighteen of polarization versus static electric field data. The latter information was obtained from semiempirical calculations using the AM1 hamiltonian.
NASA Technical Reports Server (NTRS)
Moore, Craig E.; Cardelino, Beatriz H.; Frazier, Donald O.; Niles, Julian; Wang, Xian-Qiang
1997-01-01
Calculations were performed on the valence contribution to the static molecular third-order polarizabilities (gamma) of thirty carbon-cage fullerenes (C60, C70, five isomers of C78, and twenty-three isomers of C84). The molecular structures were obtained from B3LYP/STO-3G calculations. The values of the tensor elements and an associated numerical uncertainty were obtained using the finite-field approach and polynomial expansions of orders four to eighteen of polarization versus static electric field data. The latter information was obtained from semiempirical calculations using the AM1 hamiltonian.
Second-order superintegrable quantum systems
Miller, W.; Kalnins, E. G.; Kress, J. M.
2007-03-15
A classical (or quantum) superintegrable system on an n-dimensional Riemannian manifold is an integrable Hamiltonian system with potential that admits 2n - 1 functionally independent constants of the motion that are polynomial in the momenta, the maximum number possible. If these constants of the motion are all quadratic, then the system is second-order superintegrable, the most tractable case and the one we study here. Such systems have remarkable properties: multi-integrability and separability, a quadratic algebra of symmetries whose representation theory yields spectral information about the Schroedinger operator, and deep connections with expansion formulas relating classes of special functions. For n = 2 and for conformally flat spaces when n = 3, we have worked out the structure of the classical systems and shown that the quadratic algebra always closes at order 6. Here, we describe the quantum analogs of these results. We show that, for nondegenerate potentials, each classical system has a unique quantum extension.
Schur Stability Regions for Complex Quadratic Polynomials
ERIC Educational Resources Information Center
Cheng, Sui Sun; Huang, Shao Yuan
2010-01-01
Given a quadratic polynomial with complex coefficients, necessary and sufficient conditions are found in terms of the coefficients such that all its roots have absolute values less than 1. (Contains 3 figures.)
Chromatic polynomials, Potts models and all that
NASA Astrophysics Data System (ADS)
Sokal, Alan D.
2000-04-01
The q-state Potts model can be defined on an arbitrary finite graph, and its partition function encodes much important information about that graph, including its chromatic polynomial, flow polynomial and reliability polynomial. The complex zeros of the Potts partition function are of interest both to statistical mechanicians and to combinatorists. I give a pedagogical introduction to all these problems, and then sketch two recent results: (a) Construction of a countable family of planar graphs whose chromatic zeros are dense in the whole complex q-plane except possibly for the disc | q-1|<1. (b) Proof of a universal upper bound on the q-plane zeros of the chromatic polynomial (or antiferromagnetic Potts-model partition function) in terms of the graph's maximum degree.
Quantum Communication and Quantum Multivariate Polynomial Interpolation
NASA Astrophysics Data System (ADS)
Diep, Do Ngoc; Giang, Do Hoang
2017-09-01
The paper is devoted to the problem of multivariate polynomial interpolation and its application to quantum secret sharing. We show that using quantum Fourier transform one can produce the protocol for quantum secret sharing distribution.
Schur Stability Regions for Complex Quadratic Polynomials
ERIC Educational Resources Information Center
Cheng, Sui Sun; Huang, Shao Yuan
2010-01-01
Given a quadratic polynomial with complex coefficients, necessary and sufficient conditions are found in terms of the coefficients such that all its roots have absolute values less than 1. (Contains 3 figures.)
Efficient isolation of polynomial's real roots
NASA Astrophysics Data System (ADS)
Rouillier, Fabrice; Zimmermann, Paul
2004-01-01
This paper revisits an algorithm isolating the real roots of a univariate polynomial using Descartes' rule of signs. It follows work of Vincent, Uspensky, Collins and Akritas, Johnson, Krandick. Our first contribution is a generic algorithm which enables one to describe all the known algorithms based on Descartes' rule of sign and the bisection strategy in a unified framework. Using that framework, a new algorithm is presented, which is optimal in terms of memory usage and as fast as both Collins and Akritas' algorithm and Krandick's variant, independently of the input polynomial. From this new algorithm, we derive an adaptive semi-numerical version, using multi-precision interval arithmetic. We finally show that these critical optimizations have important consequences since our new algorithm still works with huge polynomials, including orthogonal polynomials of degree 1000 and more, which were out of reach of previous methods.
Tutte Polynomial of Scale-Free Networks
NASA Astrophysics Data System (ADS)
Chen, Hanlin; Deng, Hanyuan
2016-05-01
The Tutte polynomial of a graph, or equivalently the q-state Potts model partition function, is a two-variable polynomial graph invariant of considerable importance in both statistical physics and combinatorics. The computation of this invariant for a graph is NP-hard in general. In this paper, we focus on two iteratively growing scale-free networks, which are ubiquitous in real-life systems. Based on their self-similar structures, we mainly obtain recursive formulas for the Tutte polynomials of two scale-free networks (lattices), one is fractal and "large world", while the other is non-fractal but possess the small-world property. Furthermore, we give some exact analytical expressions of the Tutte polynomial for several special points at ( x, y)-plane, such as, the number of spanning trees, the number of acyclic orientations, etc.
On the linearization problem for ultraspherical polynomials
NASA Astrophysics Data System (ADS)
Bassetti, B.; Montaldi, E.; Raciti, M.
1986-03-01
A direct proof of a formula established by Bressoud in 1981 [D. M. Bressoud, SIAM J. Math. Anal. 12, 161 (1981)], equivalent to the linearization formula for the ultraspherical polynomials, is given. Some related results are briefly discussed.
Distortion theorems for polynomials on a circle
Dubinin, V N
2000-12-31
Inequalities for the derivatives with respect to {phi}=arg z the functions ReP(z), |P(z)|{sup 2} and arg P(z) are established for an algebraic polynomial P(z) at points on the circle |z|=1. These estimates depend, in particular, on the constant term and the leading coefficient of the polynomial P(z) and improve the classical Bernstein and Turan inequalities. The method of proof is based on the techniques of generalized reduced moduli.
The Chinese Remainder Problem and Polynomial Interpolation.
1986-08-01
27709 86a 10 7 16 UNIVERSITY OF WISCONSIN-MADISON MATEMATICS RESEARCH CENTER THE CHINESE REMAINDER PROBLEM AND POLYNOMIAL INTERPOLATION Isaac J...Classifications: lOA10, 41A10 Key Words: Chinese Remainder Theorem, Polynomial Interpolation Work Unit Number 3 (Numerical Analysis and Scientific...Street Wisconsin Numerical Analysis and Madison, Wisconsin 53705 Scientific Computing " 11. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE U. S
Polynomial method for PLL controller optimization.
Wang, Ta-Chung; Lall, Sanjay; Chiou, Tsung-Yu
2011-01-01
The Phase-Locked Loop (PLL) is a key component of modern electronic communication and control systems. PLL is designed to extract signals from transmission channels. It plays an important role in systems where it is required to estimate the phase of a received signal, such as carrier tracking from global positioning system satellites. In order to robustly provide centimeter-level accuracy, it is crucial for the PLL to estimate the instantaneous phase of an incoming signal which is usually buried in random noise or some type of interference. This paper presents an approach that utilizes the recent development in the semi-definite programming and sum-of-squares field. A Lyapunov function will be searched as the certificate of the pull-in range of the PLL system. Moreover, a polynomial design procedure is proposed to further refine the controller parameters for system response away from the equilibrium point. Several simulation results as well as an experiment result are provided to show the effectiveness of this approach.
Polynomial Method for PLL Controller Optimization†
Wang, Ta-Chung; Lall, Sanjay; Chiou, Tsung-Yu
2011-01-01
The Phase-Locked Loop (PLL) is a key component of modern electronic communication and control systems. PLL is designed to extract signals from transmission channels. It plays an important role in systems where it is required to estimate the phase of a received signal, such as carrier tracking from global positioning system satellites. In order to robustly provide centimeter-level accuracy, it is crucial for the PLL to estimate the instantaneous phase of an incoming signal which is usually buried in random noise or some type of interference. This paper presents an approach that utilizes the recent development in the semi-definite programming and sum-of-squares field. A Lyapunov function will be searched as the certificate of the pull-in range of the PLL system. Moreover, a polynomial design procedure is proposed to further refine the controller parameters for system response away from the equilibrium point. Several simulation results as well as an experiment result are provided to show the effectiveness of this approach. PMID:22163973
Possible quantum algorithms for the Bollobas-Riordan-Tutte polynomial of a ribbon graph
NASA Astrophysics Data System (ADS)
Vélez, Mario; Ospina, Juan
2008-04-01
Three possible quantum algorithms, for the computation of the Bollobás-Riordan-Tutte polynomial of a given ribbon graph, are presented and discussed. The first possible algorithm is based on the spanning quasi-trees expansion for generalized Tutte polynomials of generalized graphs and on a quantum version of the Binary Decision Diagram (BDD) for quasi-trees . The second possible algorithm is based on the relation between the Kauffman bracket and the Tutte polynomial; and with an application of the recently introduced Aharonov-Arad-Eban-Landau quantum algorithm. The third possible algorithm is based on the relation between the HOMFLY polynomial and the Tutte polynomial; and with an application of the Wocjan-Yard quantum algorithm. It is claimed that these possible algorithms may be more efficient that the best known classical algorithms. These three algorithms may have interesting applications in computer science at general or in computational biology and bio-informatics in particular. A line for future research based on the categorification project is mentioned.
Online segmentation of time series based on polynomial least-squares approximations.
Fuchs, Erich; Gruber, Thiemo; Nitschke, Jiri; Sick, Bernhard
2010-12-01
The paper presents SwiftSeg, a novel technique for online time series segmentation and piecewise polynomial representation. The segmentation approach is based on a least-squares approximation of time series in sliding and/or growing time windows utilizing a basis of orthogonal polynomials. This allows the definition of fast update steps for the approximating polynomial, where the computational effort depends only on the degree of the approximating polynomial and not on the length of the time window. The coefficients of the orthogonal expansion of the approximating polynomial-obtained by means of the update steps-can be interpreted as optimal (in the least-squares sense) estimators for average, slope, curvature, change of curvature, etc., of the signal in the time window considered. These coefficients, as well as the approximation error, may be used in a very intuitive way to define segmentation criteria. The properties of SwiftSeg are evaluated by means of some artificial and real benchmark time series. It is compared to three different offline and online techniques to assess its accuracy and runtime. It is shown that SwiftSeg-which is suitable for many data streaming applications-offers high accuracy at very low computational costs.
NASA Astrophysics Data System (ADS)
Reuvekamp, Patrick G.; Kremer, Reinhard K.; Köhler, Jürgen; Bussmann-Holder, Annette
2014-09-01
The structural phase transition of EuTiO3 near TS = 282 K is investigated by temperature- and magnetic-field-dependent measurements of the linear thermal expansion (TE) on a polycrystalline sample. At TS a distinct anomaly is seen in the linear coefficient which is similar to the anomaly seen at the cubic-to-tetragonal phase transition in SrTiO3. The anomaly is well described by the thermal displacement correlation function which exhibits an abrupt change at the structural phase transition caused by a double-well potential in the oxygen-oxygen interaction. In accordance with previous data a pronounced magnetic-field effect on the TE is observed which might yield an alternative potential for applications.
Baxter Operator Formalism for Macdonald Polynomials
NASA Astrophysics Data System (ADS)
Gerasimov, Anton; Lebedev, Dimitri; Oblezin, Sergey
2013-11-01
We develop basic constructions of the Baxter operator formalism for the Macdonald polynomials associated with root systems of type A. Precisely, we construct a bispectral pair of mutually commuting Baxter operators such that the Macdonald polynomials are their common eigenfunctions. The bispectral pair of Baxter operators is closely related to the bispectral pair of recursive operators for Macdonald polynomials leading to various families of their integral representations. We also construct the Baxter operator formalism for the q-deformed {{gl}_{ell+1}} -Whittaker functions and the Jack polynomials obtained by degenerations of the Macdonald polynomials associated with the type A ℓ root system. This note provides a generalization of our previous results on the Baxter operator formalism for the Whittaker functions. It was demonstrated previously that Baxter operator formalism for the Whittaker functions has deep connections with representation theory. In particular, the Baxter operators should be considered as elements of appropriate spherical Hecke algebras and their eigenvalues are identified with local Archimedean L-factors associated with admissible representations of reductive groups over {{R}}. We expect that the Baxter operator formalism for the Macdonald polynomials has an interpretation in representation theory over higher-dimensional local/global fields.
On polynomial preconditioning for indefinite Hermitian matrices
NASA Technical Reports Server (NTRS)
Freund, Roland W.
1989-01-01
The minimal residual method is studied combined with polynomial preconditioning for solving large linear systems (Ax = b) with indefinite Hermitian coefficient matrices (A). The standard approach for choosing the polynomial preconditioners leads to preconditioned systems which are positive definite. Here, a different strategy is studied which leaves the preconditioned coefficient matrix indefinite. More precisely, the polynomial preconditioner is designed to cluster the positive, resp. negative eigenvalues of A around 1, resp. around some negative constant. In particular, it is shown that such indefinite polynomial preconditioners can be obtained as the optimal solutions of a certain two parameter family of Chebyshev approximation problems. Some basic results are established for these approximation problems and a Remez type algorithm is sketched for their numerical solution. The problem of selecting the parameters such that the resulting indefinite polynomial preconditioners speeds up the convergence of minimal residual method optimally is also addressed. An approach is proposed based on the concept of asymptotic convergence factors. Finally, some numerical examples of indefinite polynomial preconditioners are given.
Matrix-valued polynomials in Lanczos type methods
Simoncini, V.; Gallopoulos, E.
1994-12-31
It is well known that convergence properties of iterative methods can be derived by studying the behavior of the residual polynomial over a suitable domain of the complex plane. Block Krylov subspace methods for the solution of linear systems A[x{sub 1},{hor_ellipsis}, x{sub s}] = [b{sub 1},{hor_ellipsis}, b{sub s}] lead to the generation of residual polynomials {phi}{sub m} {element_of} {bar P}{sub m,s} where {bar P}{sub m,s} is the subset of matrix-valued polynomials of maximum degree m and size s such that {phi}{sub m}(0) = I{sub s}, R{sub m} := B - AX{sub m} = {phi}{sub m}(A) {circ} R{sub 0}, where {phi}{sub m}(A) {circ} R{sub 0} := R{sub 0} - A{summation}{sub j=0}{sup m-1} A{sup j}R{sub 0}{xi}{sub j}, {xi}{sub j} {element_of} R{sup sxs}. An effective method has to balance adequate approximation with economical computation of iterates defined by the polynomial. Matrix valued polynomials can be used to improve the performance of block methods. Another approach is to solve for a single right-hand side at a time and use the generated information in order to update the approximations of the remaining systems. In light of this, a more general scheme is as follows: A subset of residuals (seeds) is selected and a block short term recurrence method is used to compute approximate solutions for the corresponding systems. At the same time the generated matrix valued polynomial is implicitly applied to the remaining residuals. Subsequently a new set of seeds is selected and the process is continued as above, till convergence of all right-hand sides. The use of a quasi-minimization technique ensures a smooth convergence behavior for all systems. In this talk the authors discuss the implementation of this class of algorithms and formulate strategies for the selection of parameters involved in the computation. Experiments and comparisons with other methods will be presented.
Painlevé V and time-dependent Jacobi polynomials
NASA Astrophysics Data System (ADS)
Basor, Estelle; Chen, Yang; Ehrhardt, Torsten
2010-01-01
In this paper we study the simplest deformation on a sequence of orthogonal polynomials. This in turn induces a deformation on the moment matrix of the polynomials and associated Hankel determinant. We replace the original (or reference) weight w0(x) (supported on \\mathbb {R} or subsets of \\mathbb {R}) by w0(x) e-tx. It is a well-known fact that under such a deformation the recurrence coefficients denoted as αn and βn evolve in t according to the Toda equations, giving rise to the time-dependent orthogonal polynomials and time-dependent determinants, using Sogo's terminology. If w0 is the normal density e^{-x^2},\\;x\\in \\mathbb {R}, or the gamma density xα e-x, x\\in \\mathbb {R}_{+}, α > -1, then the initial value problem of the Toda equations can be trivially solved. This is because under elementary scaling and translation the orthogonality relations reduce to the original ones. However, if w0 is the beta density (1 - x)α(1 + x)β, x in [ - 1, 1], α, β > -1, the resulting 'time-dependent' Jacobi polynomials will again satisfy a linear second-order ode, but no longer in the Sturm-Liouville form, which is to be expected. This deformation induces an irregular singular point at infinity in addition to three regular singular points of the hypergeometric equation satisfied by the Jacobi polynomials. We will show that the coefficients of this ode, as well as the Hankel determinant, are intimately related to a particular Painlevé V. In particular we show that \\\\textsf {p}_1(n,t), where \\\\textsf {p}_1(n,t) is the coefficient of zn-1 of the monic orthogonal polynomials associated with the 'time-dependent' Jacobi weight, satisfies, up to a translation in t, the Jimbo-Miwa σ-form of the same PV; while a recurrence coefficient αn(t) is up to a translation in t and a linear fractional transformation PV(α2/2, - β2/2, 2n + 1 + α + β, - 1/2). These results are found from combining a pair of nonlinear difference equations and a pair of Toda equations. This
Construction of minimum energy high-order Helmholtz bases for structured elements
NASA Astrophysics Data System (ADS)
Rodrigues, Caio F.; Suzuki, Jorge L.; Bittencourt, Marco L.
2016-02-01
We present a construction procedure for high-order expansion bases for structured finite elements specific for the operator under consideration. The procedure aims to obtain bases in such way that the condition numbers for the element matrices are almost constant or have a moderate increase in terms of the polynomial order. The internal modes of the mass and stiffness matrices are made simultaneously diagonal and the minimum energy concept is used to make the boundary modes orthogonal to the internal modes. The performance of the proposed bases is compared to the standard basis using Jacobi polynomials. This is performed through numerical examples for Helmholtz problem and transient linear elasticity employing explicit and implicit time integration algorithms and the conjugate gradient method with diagonal, SSOR and Gauss-Seidel pre-conditioners. The sparsity patterns, conditioning and solution costs are investigated. A significant speedup and reduction in the number of iterations are obtained when compared to the standard basis.
Orthogonality Measure on the Torus for Vector-Valued Jack Polynomials
NASA Astrophysics Data System (ADS)
Dunkl, Charles F.
2016-03-01
For each irreducible module of the symmetric group on N objects there is a set of parametrized nonsymmetric Jack polynomials in N variables taking values in the module. These polynomials are simultaneous eigenfunctions of a commutative set of operators, self-adjoint with respect to certain Hermitian forms. These polynomials were studied by the author and J.-G. Luque using a Yang-Baxter graph technique. This paper constructs a matrix-valued measure on the N-torus for which the polynomials are mutually orthogonal. The construction uses Fourier analysis techniques. Recursion relations for the Fourier-Stieltjes coefficients of the measure are established, and used to identify parameter values for which the construction fails. It is shown that the absolutely continuous part of the measure satisfies a first-order system of differential equations.
A two-dimensional, semi-analytic expansion method for nodal calculations
Palmtag, Scott P.
1995-08-01
Most modern nodal methods used today are based upon the transverse integration procedure in which the multi-dimensional flux shape is integrated over the transverse directions in order to produce a set of coupled one-dimensional flux shapes. The one-dimensional flux shapes are then solved either analytically or by representing the flux shape by a finite polynomial expansion. While these methods have been verified for most light-water reactor applications, they have been found to have difficulty predicting the large thermal flux gradients near the interfaces of highly-enriched MOX fuel assemblies. A new method is presented here in which the neutron flux is represented by a non-seperable, two-dimensional, semi-analytic flux expansion. The main features of this method are (1) the leakage terms from the node are modeled explicitly and therefore, the transverse integration procedure is not used, (2) the corner point flux values for each node are directly edited from the solution method, and a corner-point interpolation is not needed in the flux reconstruction, (3) the thermal flux expansion contains hyperbolic terms representing analytic solutions to the thermal flux diffusion equation, and (4) the thermal flux expansion contains a thermal to fast flux ratio term which reduces the number of polynomial expansion functions needed to represent the thermal flux. This new nodal method has been incorporated into the computer code COLOR2G and has been used to solve a two-dimensional, two-group colorset problem containing uranium and highly-enriched MOX fuel assemblies. The results from this calculation are compared to the results found using a code based on the traditional transverse integration procedure.
On λ-Bell polynomials associated with umbral calculus
NASA Astrophysics Data System (ADS)
Kim, T.; Kim, D. S.
2017-01-01
In this paper, we introduce some new λ-Bell polynomials and Bell polynomials of the second kind and investigate properties of these polynomials. Using our investigation, we derive some new identities for the two kinds of λ-Bell polynomials arising from umbral calculus.
NASA Astrophysics Data System (ADS)
Komatsu, Kazuo; Takata, Hitoshi
2012-11-01
In this paper, we consider an observer design by using a formal linearization based on Fourier expansion for nonlinear dynamic and measurement systems. A non-linear dynamic system is given by a nonlinear ordinary differential equation, and a measurement sysetm is done by a nonlinear equation. Defining a linearization function which consists of the trigonometric functions considered up to the higher-order, a nonlinear dynamic system is transformed into an augmented linear one with respect to this linearization function by using Fourier expansion. Introducing an augmented measurement vector which consists of polynomials of measurement data, a measurement equation is transformed into an augmented linear one with respect to the linearization function in the same way. To these augmented linearized systems, a linear estimation theory is applied to design a new non-linear observer.
Zeros of Jones polynomials for families of knots and links
NASA Astrophysics Data System (ADS)
Chang, S.-C.; Shrock, R.
2001-12-01
We calculate Jones polynomials VL( t) for several families of alternating knots and links by computing the Tutte polynomials T( G, x, y) for the associated graphs G and then obtaining VL( t) as a special case of the Tutte polynomial. For each of these families we determine the zeros of the Jones polynomial, including the accumulation set in the limit of infinitely many crossings. A discussion is also given of the calculation of Jones polynomials for non-alternating links.
Uncertainty Analysis via Failure Domain Characterization: Polynomial Requirement Functions
NASA Technical Reports Server (NTRS)
Crespo, Luis G.; Munoz, Cesar A.; Narkawicz, Anthony J.; Kenny, Sean P.; Giesy, Daniel P.
2011-01-01
This paper proposes an uncertainty analysis framework based on the characterization of the uncertain parameter space. This characterization enables the identification of worst-case uncertainty combinations and the approximation of the failure and safe domains with a high level of accuracy. Because these approximations are comprised of subsets of readily computable probability, they enable the calculation of arbitrarily tight upper and lower bounds to the failure probability. A Bernstein expansion approach is used to size hyper-rectangular subsets while a sum of squares programming approach is used to size quasi-ellipsoidal subsets. These methods are applicable to requirement functions whose functional dependency on the uncertainty is a known polynomial. Some of the most prominent features of the methodology are the substantial desensitization of the calculations from the uncertainty model assumed (i.e., the probability distribution describing the uncertainty) as well as the accommodation for changes in such a model with a practically insignificant amount of computational effort.
Estimation of the entropy based on its polynomial representation.
Vinck, Martin; Battaglia, Francesco P; Balakirsky, Vladimir B; Vinck, A J Han; Pennartz, Cyriel M A
2012-05-01
Estimating entropy from empirical samples of finite size is of central importance for information theory as well as the analysis of complex statistical systems. Yet, this delicate task is marred by intrinsic statistical bias. Here we decompose the entropy function into a polynomial approximation function and a remainder function. The approximation function is based on a Taylor expansion of the logarithm. Given n observations, we give an unbiased, linear estimate of the first n power series terms based on counting sets of k coincidences. For the remainder function we use nonlinear Bayesian estimation with a nearly flat prior distribution on the entropy that was developed by Nemenman, Shafee, and Bialek. Our simulations show that the combined entropy estimator has reduced bias in comparison to other available estimators.
Estimation of the entropy based on its polynomial representation
NASA Astrophysics Data System (ADS)
Vinck, Martin; Battaglia, Francesco P.; Balakirsky, Vladimir B.; Vinck, A. J. Han; Pennartz, Cyriel M. A.
2012-05-01
Estimating entropy from empirical samples of finite size is of central importance for information theory as well as the analysis of complex statistical systems. Yet, this delicate task is marred by intrinsic statistical bias. Here we decompose the entropy function into a polynomial approximation function and a remainder function. The approximation function is based on a Taylor expansion of the logarithm. Given n observations, we give an unbiased, linear estimate of the first n power series terms based on counting sets of k coincidences. For the remainder function we use nonlinear Bayesian estimation with a nearly flat prior distribution on the entropy that was developed by Nemenman, Shafee, and Bialek. Our simulations show that the combined entropy estimator has reduced bias in comparison to other available estimators.
Construction and properties of non-polynomial spline-curves
NASA Astrophysics Data System (ADS)
Laksâ, Arne
2014-12-01
Bézier curves can be expressed by using De Casteljau's corner cutting algorithm. This can also be formulated using factorization matrices. Each matrix reduce the coefficient vector with 1, and each line in each matrix represent a linear interpolation describing the corner cutting. This matrix formulation may also be used for B-splines. One just have to introduce a linear transformation from the local domains of the basis function, to the interval [0;1]. This leads to a further expansion where the linear transformation is "deformed" by a perturbation function. The result is a non-polynomial spline. We will se that the typical properties of B-splines are preserved or even improved in the nonpolynomial case. Further, we describes the construction of the B-spline, and provide some practical examples.
Extending a Property of Cubic Polynomials to Higher-Degree Polynomials
ERIC Educational Resources Information Center
Miller, David A.; Moseley, James
2012-01-01
In this paper, the authors examine a property that holds for all cubic polynomials given two zeros. This property is discovered after reviewing a variety of ways to determine the equation of a cubic polynomial given specific conditions through algebra and calculus. At the end of the article, they will connect the property to a very famous method…
Extending a Property of Cubic Polynomials to Higher-Degree Polynomials
ERIC Educational Resources Information Center
Miller, David A.; Moseley, James
2012-01-01
In this paper, the authors examine a property that holds for all cubic polynomials given two zeros. This property is discovered after reviewing a variety of ways to determine the equation of a cubic polynomial given specific conditions through algebra and calculus. At the end of the article, they will connect the property to a very famous method…
Nielsen, Jon F; Ghugre, Nilesh R; Panigrahy, Ashok
2004-11-01
We have investigated the use of two different image coregistration algorithms for identifying local regions of erroneously high fractional anisotropy (FA) as derived from diffusion tensor imaging (DTI) data sets in newborns. The first algorithm uses conventional affine registration of each of the diffusion-weighted images to the unweighted (b = 0) image for each slice, while the second algorithm uses second-order polynomial warping. Similarity between images was determined using the mutual information (MI) criterion, which is the preferred 'cost' criterion for coregistration of images with significantly different image intensity distributions. We have found that subtle differences exist in the FA values resulting from affine and second-order polynomial coregistration and demonstrate that nonlinear distortions introduce artifacts of spatial extent similar to real white matter structures in the newborn subcortex. We show that polynomial coregistration systematically reduces the presence of erroneous regions of high FA and that such artifacts can be identified by visual inspection of FA maps resulting from affine and polynomial coregistrations. Furthermore, we show that nonlinear distortions may be particularly pronounced when acquiring image slices of axial orientation at the height of the nasal cavity. Finally, we show that third-order polynomial MI coregistration (using the images resulting from second-order coregistration as input) has no observable effect on the resulting FA maps.
A wavelet-optimized, very high order adaptive grid and order numerical method
NASA Technical Reports Server (NTRS)
Jameson, Leland
1996-01-01
Differencing operators of arbitrarily high order can be constructed by interpolating a polynomial through a set of data followed by differentiation of this polynomial and finally evaluation of the polynomial at the point where a derivative approximation is desired. Furthermore, the interpolating polynomial can be constructed from algebraic, trigonometric, or, perhaps exponential polynomials. This paper begins with a comparison of such differencing operator construction. Next, the issue of proper grids for high order polynomials is addressed. Finally, an adaptive numerical method is introduced which adapts the numerical grid and the order of the differencing operator depending on the data. The numerical grid adaptation is performed on a Chebyshev grid. That is, at each level of refinement the grid is a Chebvshev grid and this grid is refined locally based on wavelet analysis.
Nishimura, S.; Sugama, H.; Maassberg, H.; Beidler, C. D.; Murakami, S.; Nakamura, Y.; Hirooka, S.
2010-08-15
The dependence of neoclassical parallel flow calculations on the maximum order of Laguerre polynomial expansions is investigated in a magnetic configuration of the Large Helical Device [S. Murakami, A. Wakasa, H. Maassberg, et al., Nucl. Fusion 42, L19 (2002)] using the monoenergetic coefficient database obtained by an international collaboration. On the basis of a previous generalization (the so-called Sugama-Nishimura method [H. Sugama and S. Nishimura, Phys. Plasmas 15, 042502 (2008)]) to an arbitrary order of the expansion, the 13 M, 21 M, and 29 M approximations are compared. In a previous comparison, only the ion distribution function in the banana collisionality regime of single-ion-species plasmas in tokamak configurations was investigated. In this paper, the dependence of the problems including electrons and impurities in the general collisionality regime in an actual nonsymmetric toroidal configuration is reported. In particular, qualities of approximations for the electron distribution function are investigated in detail.
Self-similarity of higher-order moving averages
NASA Astrophysics Data System (ADS)
Arianos, Sergio; Carbone, Anna; Türk, Christian
2011-10-01
In this work, higher-order moving average polynomials are defined by straightforward generalization of the standard moving average. The self-similarity of the polynomials is analyzed for fractional Brownian series and quantified in terms of the Hurst exponent H by using the detrending moving average method. We prove that the exponent H of the fractional Brownian series and of the detrending moving average variance asymptotically agree for the first-order polynomial. Such asymptotic values are compared with the results obtained by the simulations. The higher-order polynomials correspond to trend estimates at shorter time scales as the degree of the polynomial increases. Importantly, the increase of polynomial degree does not require to change the moving average window. Thus trends at different time scales can be obtained on data sets with the same size. These polynomials could be interesting for those applications relying on trend estimates over different time horizons (financial markets) or on filtering at different frequencies (image analysis).
Gabor-based kernel PCA with fractional power polynomial models for face recognition.
Liu, Chengjun
2004-05-01
This paper presents a novel Gabor-based kernel Principal Component Analysis (PCA) method by integrating the Gabor wavelet representation of face images and the kernel PCA method for face recognition. Gabor wavelets first derive desirable facial features characterized by spatial frequency, spatial locality, and orientation selectivity to cope with the variations due to illumination and facial expression changes. The kernel PCA method is then extended to include fractional power polynomial models for enhanced face recognition performance. A fractional power polynomial, however, does not necessarily define a kernel function, as it might not define a positive semidefinite Gram matrix. Note that the sigmoid kernels, one of the three classes of widely used kernel functions (polynomial kernels, Gaussian kernels, and sigmoid kernels), do not actually define a positive semidefinite Gram matrix either. Nevertheless, the sigmoid kernels have been successfully used in practice, such as in building support vector machines. In order to derive real kernel PCA features, we apply only those kernel PCA eigenvectors that are associated with positive eigenvalues. The feasibility of the Gabor-based kernel PCA method with fractional power polynomial models has been successfully tested on both frontal and pose-angled face recognition, using two data sets from the FERET database and the CMU PIE database, respectively. The FERET data set contains 600 frontal face images of 200 subjects, while the PIE data set consists of 680 images across five poses (left and right profiles, left and right half profiles, and frontal view) with two different facial expressions (neutral and smiling) of 68 subjects. The effectiveness of the Gabor-based kernel PCA method with fractional power polynomial models is shown in terms of both absolute performance indices and comparative performance against the PCA method, the kernel PCA method with polynomial kernels, the kernel PCA method with fractional power
Accelerating the loop expansion
Ingermanson, R.
1986-07-29
This thesis introduces a new non-perturbative technique into quantum field theory. To illustrate the method, I analyze the much-studied phi/sup 4/ theory in two dimensions. As a prelude, I first show that the Hartree approximation is easy to obtain from the calculation of the one-loop effective potential by a simple modification of the propagator that does not affect the perturbative renormalization procedure. A further modification then susggests itself, which has the same nice property, and which automatically yields a convex effective potential. I then show that both of these modifications extend naturally to higher orders in the derivative expansion of the effective action and to higher orders in the loop-expansion. The net effect is to re-sum the perturbation series for the effective action as a systematic ''accelerated'' non-perturbative expansion. Each term in the accelerated expansion corresponds to an infinite number of terms in the original series. Each term can be computed explicitly, albeit numerically. Many numerical graphs of the various approximations to the first two terms in the derivative expansion are given. I discuss the reliability of the results and the problem of spontaneous symmetry-breaking, as well as some potential applications to more interesting field theories. 40 refs.
NASA Astrophysics Data System (ADS)
Recchioni, Maria Cristina
2001-12-01
This paper investigates the application of the method introduced by L. Pasquini (1989) for simultaneously approaching the zeros of polynomial solutions to a class of second-order linear homogeneous ordinary differential equations with polynomial coefficients to a particular case in which these polynomial solutions have zeros symmetrically arranged with respect to the origin. The method is based on a family of nonlinear equations which is associated with a given class of differential equations. The roots of the nonlinear equations are related to the roots of the polynomial solutions of differential equations considered. Newton's method is applied to find the roots of these nonlinear equations. In (Pasquini, 1994) the nonsingularity of the roots of these nonlinear equations is studied. In this paper, following the lines in (Pasquini, 1994), the nonsingularity of the roots of these nonlinear equations is studied. More favourable results than the ones in (Pasquini, 1994) are proven in the particular case of polynomial solutions with symmetrical zeros. The method is applied to approximate the roots of Hermite-Sobolev type polynomials and Freud polynomials. A lower bound for the smallest positive root of Hermite-Sobolev type polynomials is given via the nonlinear equation. The quadratic convergence of the method is proven. A comparison with a classical method that uses the Jacobi matrices is carried out. We show that the algorithm derived by the proposed method is sometimes preferable to the classical QR type algorithms for computing the eigenvalues of the Jacobi matrices even if these matrices are real and symmetric.
Minimal residual method stronger than polynomial preconditioning
Faber, V.; Joubert, W.; Knill, E.
1994-12-31
Two popular methods for solving symmetric and nonsymmetric systems of equations are the minimal residual method, implemented by algorithms such as GMRES, and polynomial preconditioning methods. In this study results are given on the convergence rates of these methods for various classes of matrices. It is shown that for some matrices, such as normal matrices, the convergence rates for GMRES and for the optimal polynomial preconditioning are the same, and for other matrices such as the upper triangular Toeplitz matrices, it is at least assured that if one method converges then the other must converge. On the other hand, it is shown that matrices exist for which restarted GMRES always converges but any polynomial preconditioning of corresponding degree makes no progress toward the solution for some initial error. The implications of these results for these and other iterative methods are discussed.
Fitting parametrized polynomials with scattered surface data.
van Ruijven, L J; Beek, M; van Eijden, T M
1999-07-01
Currently used joint-surface models require the measurements to be structured according to a grid. With the currently available tracking devices a large quantity of unstructured surface points can be measured in a relatively short time. In this paper a method is presented to fit polynomial functions to three-dimensional unstructured data points. To test the method spherical, cylindrical, parabolic, hyperbolic, exponential, logarithmic, and sellar surfaces with different undulations were used. The resulting polynomials were compared with the original shapes. The results show that even complex joint surfaces can be modelled with polynomial functions. In addition, the influence of noise and the number of data points was also analyzed. From a surface (diam: 20 mm) which is measured with a precision of 0.2 mm a model can be constructed with a precision of 0.02 mm.
Supersymmetric pairing of kinks for polynomial nonlinearities
Rosu, H.C.; Cornejo-Perez, O.
2005-04-01
We show how one can obtain kink solutions of ordinary differential equations with polynomial nonlinearities by an efficient factorization procedure directly related to the factorization of their nonlinear polynomial part. We focus on reaction-diffusion equations in the traveling frame and damped-anharmonic-oscillator equations. We also report an interesting pairing of the kink solutions, a result obtained by reversing the factorization brackets in the supersymmetric quantum-mechanical style. In this way, one gets ordinary differential equations with a different polynomial nonlinearity possessing kink solutions of different width but propagating at the same velocity as the kinks of the original equation. This pairing of kinks could have many applications. We illustrate the mathematical procedure with several important cases, among which are the generalized Fisher equation, the FitzHugh-Nagumo equation, and the polymerization fronts of microtubules.
Polynomial solution of quantum Grassmann matrices
NASA Astrophysics Data System (ADS)
Tierz, Miguel
2017-05-01
We study a model of quantum mechanical fermions with matrix-like index structure (with indices N and L) and quartic interactions, recently introduced by Anninos and Silva. We compute the partition function exactly with q-deformed orthogonal polynomials (Stieltjes-Wigert polynomials), for different values of L and arbitrary N. From the explicit evaluation of the thermal partition function, the energy levels and degeneracies are determined. For a given L, the number of states of different energy is quadratic in N, which implies an exponential degeneracy of the energy levels. We also show that at high-temperature we have a Gaussian matrix model, which implies a symmetry that swaps N and L, together with a Wick rotation of the spectral parameter. In this limit, we also write the partition function, for generic L and N, in terms of a single generalized Hermite polynomial.
Smooth polynomial approximation of spiral arcs
NASA Astrophysics Data System (ADS)
Cripps, R. J.; Hussain, M. Z.; Zhu, S.
2010-03-01
Constructing fair curve segments using parametric polynomials is difficult due to the oscillatory nature of polynomials. Even NURBS curves can exhibit unsatisfactory curvature profiles. Curve segments with monotonic curvature profiles, for example spiral arcs, exist but are intrinsically non-polynomial in nature and thus difficult to integrate into existing CAD systems. A method of constructing an approximation to a generalised Cornu spiral (GCS) arc using non-rational quintic Bézier curves matching end points, end slopes and end curvatures is presented. By defining an objective function based on the relative error between the curvature profiles of the GCS and its Bézier approximation, a curve segment is constructed that has a monotonic curvature profile within a specified tolerance.
Constructing Polynomial Spectral Models for Stars
NASA Astrophysics Data System (ADS)
Rix, Hans-Walter; Ting, Yuan-Sen; Conroy, Charlie; Hogg, David W.
2016-08-01
Stellar spectra depend on the stellar parameters and on dozens of photospheric elemental abundances. Simultaneous fitting of these { N } ˜ 10-40 model labels to observed spectra has been deemed unfeasible because the number of ab initio spectral model grid calculations scales exponentially with { N }. We suggest instead the construction of a polynomial spectral model (PSM) of order { O } for the model flux at each wavelength. Building this approximation requires a minimum of only ≤ft(≥nfrac{}{}{0em}{}{{ N }+{ O }}{{ O }}\\right) calculations: e.g., a quadratic spectral model ({ O }=2) to fit { N }=20 labels simultaneously can be constructed from as few as 231 ab initio spectral model calculations; in practice, a somewhat larger number (˜300-1000) of randomly chosen models lead to a better performing PSM. Such a PSM can be a good approximation only over a portion of label space, which will vary case-by-case. Yet, taking the APOGEE survey as an example, a single quadratic PSM provides a remarkably good approximation to the exact ab initio spectral models across much of this survey: for random labels within that survey the PSM approximates the flux to within 10-3 and recovers the abundances to within ˜0.02 dex rms of the exact models. This enormous speed-up enables the simultaneous many-label fitting of spectra with computationally expensive ab initio models for stellar spectra, such as non-LTE models. A PSM also enables the simultaneous fitting of observational parameters, such as the spectrum’s continuum or line-spread function.
Lattice harmonics expansion revisited
NASA Astrophysics Data System (ADS)
Kontrym-Sznajd, G.; Holas, A.
2017-04-01
The main subject of the work is to provide the most effective way of determining the expansion of some quantities into orthogonal polynomials, when these quantities are known only along some limited number of sampling directions. By comparing the commonly used Houston method with the method based on the orthogonality relation, some relationships, which define the applicability and correctness of these methods, are demonstrated. They are verified for various sets of sampling directions applicable for expanding quantities having the full symmetry of the Brillouin zone of cubic and non-cubic lattices. All results clearly show that the Houston method is always better than the orthogonality-relation one. For the cubic symmetry we present a few sets of special directions (SDs) showing how their construction and, next, a proper application depend on the choice of various sets of lattice harmonics. SDs are important mainly for experimentalists who want to reconstruct anisotropic quantities from their measurements, performed at a limited number of sampling directions.
NASA Astrophysics Data System (ADS)
Zhdanov, K. R.; Kameneva, M. Yu.; Kozeeva, L. P.; Lavrov, A. N.
2016-08-01
Layered cobaltates YBaCo2O5 + x have been investigated in the oxygen concentration range 0.23 ≤ x ≤ 0.52. It has been revealed that the oxygen ordering plays the key role in the appearance of anomalies in temperature dependences of structural parameters and electron transport. It has been shown that the orthorhombic lattice distortion caused by oxygen chain ordering is a necessary "trigger" for the phase transition from the insulating state to the metallic state at T ≈ 290-295 K, after which the orthorhombic distortion is significantly more pronounced. In the boundary region of the cobaltate compositions, where the oxygen ordering has a partial or local character, there are additional low-temperature (100-240 K) structural and resistive features with a large hysteresis. The observed anomalies can be explained by a change in the spin state of the cobalt ions, which is extremely sensitive to parameters of the crystal field acting on the ions, as well as by the spin-transition-induced delocalization of electrons.
On the modular structure of the genus-one Type II superstring low energy expansion
NASA Astrophysics Data System (ADS)
D'Hoker, Eric; Green, Michael B.; Vanhove, Pierre
2015-08-01
The analytic contribution to the low energy expansion of Type II string amplitudes at genus-one is a power series in space-time derivatives with coefficients that are determined by integrals of modular functions over the complex structure modulus of the world-sheet torus. These modular functions are associated with world-sheet vacuum Feynman diagrams and given by multiple sums over the discrete momenta on the torus. In this paper we exhibit exact differential and algebraic relations for a certain infinite class of such modular functions by showing that they satisfy Laplace eigenvalue equations with inhomogeneous terms that are polynomial in non-holomorphic Eisenstein series. Furthermore, we argue that the set of modular functions that contribute to the coefficients of interactions up to order are linear sums of functions in this class and quadratic polynomials in Eisenstein series and odd Riemann zeta values. Integration over the complex structure results in coefficients of the low energy expansion that are rational numbers multiplying monomials in odd Riemann zeta values.
On convergence rates of mixtures of polynomial experts.
Mendes, Eduardo F; Jiang, Wenxin
2012-11-01
In this letter, we consider a mixture-of-experts structure where m experts are mixed, with each expert being related to a polynomial regression model of order k. We study the convergence rate of the maximum likelihood estimator in terms of how fast the Hellinger distance of the estimated density converges to the true density, when the sample size n increases. The convergence rate is found to be dependent on both m and k, while certain choices of m and k are found to produce near-optimal convergence rates.
The Potts model and the Tutte polynomial
NASA Astrophysics Data System (ADS)
Welsh, D. J. A.; Merino, C.
2000-03-01
This is an invited survey on the relation between the partition function of the Potts model and the Tutte polynomial. On the assumption that the Potts model is more familiar we have concentrated on the latter and its interpretations. In particular we highlight the connections with Abelian sandpiles, counting problems on random graphs, error correcting codes, and the Ehrhart polynomial of a zonotope. Where possible we use the mean field and square lattice as illustrations. We also discuss in some detail the complexity issues involved.
Using Taylor Expansions to Prepare Students for Calculus
ERIC Educational Resources Information Center
Lutzer, Carl V.
2011-01-01
We propose an alternative to the standard introduction to the derivative. Instead of using limits of difference quotients, students develop Taylor expansions of polynomials. This alternative allows students to develop many of the central ideas about the derivative at an intuitive level, using only skills and concepts from precalculus, and…
Using Taylor Expansions to Prepare Students for Calculus
ERIC Educational Resources Information Center
Lutzer, Carl V.
2011-01-01
We propose an alternative to the standard introduction to the derivative. Instead of using limits of difference quotients, students develop Taylor expansions of polynomials. This alternative allows students to develop many of the central ideas about the derivative at an intuitive level, using only skills and concepts from precalculus, and…
Location of Zeros of Wiener and Distance Polynomials
Dehmer, Matthias; Ilić, Aleksandar
2012-01-01
The geometry of polynomials explores geometrical relationships between the zeros and the coefficients of a polynomial. A classical problem in this theory is to locate the zeros of a given polynomial by determining disks in the complex plane in which all its zeros are situated. In this paper, we infer bounds for general polynomials and apply classical and new results to graph polynomials namely Wiener and distance polynomials whose zeros have not been yet investigated. Also, we examine the quality of such bounds by considering four graph classes and interpret the results. PMID:22438861
Slave Mode Expansion for Obtaining Ab Initio Interatomic Potentials and its Applications
NASA Astrophysics Data System (ADS)
Ai, Xinyuan
Having an interatomic potential overcomes limitations within DFT since it has a negligible cost in computing material properties while DFT is severely restricted by its computational cost when carrying out such tasks. In this thesis, we propose a new approach for creating an interatomic potential based on the Taylor series expansion of the crystal energy as a function of its nuclear displacements. We enlarge the dimensionality of the existing displacement space and form new variables (ie. slave modes) which transform like irreducible representations of the point group and satisfy homogeneity of free space. Standard group theoretical techniques can then be applied to deduce the non-zero expansion coefficients a priori. At a given order, the translation group can be used to contract the products and eliminate terms which are not linearly independent, resulting in a final set of slave mode products. By the end of the day, one ends up with an expansion that satisfies lattice symmetry and its number of coefficients is much smaller than that of a common Taylor series expansion. While the expansion coefficients can be computed in a variety of ways, we demonstrate that finite difference is effective up to fifth order. On the other hand, we demonstrate the power of the method in the strongly anharmonic systems PbTe and graphene. All anharmonic terms within an octahedron are computed up to fourth order for PbTe, while those within a hexagon are computed up to fourth order and dimer terms are computed at fifth order for graphene. In addition, for PbTe, a proper unitary transformation of its potential demonstrates that the vast majority of the anharmonicity can be attributed to just two terms, indicating that a minimal model of phonon interactions is achievable. The ability to straightforwardly generate polynomial potentials will allow precise simulations at length and time scales which were previously unrealizable.
Improved Convergence for Two-Component Activity Expansions
DeWitt, H E; Rogers, F J; Sonnad, V
2007-03-06
It is well known that an activity expansion of the grand canonical partition function works well for attractive interactions, but works poorly for repulsive interactions, such as occur between atoms and molecules. The virial expansion of the canonical partition function shows just the opposite behavior. This poses a problem for applications that involve both types of interactions, such as occur in the outer layers of low-mass stars. We show that it is possible to obtain expansions for repulsive systems that convert the poorly performing Mayer activity expansion into a series of rational polynomials that converge uniformly to the virial expansion. In the current work we limit our discussion to the second virial approximation. In contrast to the Mayer activity expansion the activity expansion presented herein converges for both attractive and repulsive systems.
Predicting physical time series using dynamic ridge polynomial neural networks.
Al-Jumeily, Dhiya; Ghazali, Rozaida; Hussain, Abir
2014-01-01
Forecasting naturally occurring phenomena is a common problem in many domains of science, and this has been addressed and investigated by many scientists. The importance of time series prediction stems from the fact that it has wide range of applications, including control systems, engineering processes, environmental systems and economics. From the knowledge of some aspects of the previous behaviour of the system, the aim of the prediction process is to determine or predict its future behaviour. In this paper, we consider a novel application of a higher order polynomial neural network architecture called Dynamic Ridge Polynomial Neural Network that combines the properties of higher order and recurrent neural networks for the prediction of physical time series. In this study, four types of signals have been used, which are; The Lorenz attractor, mean value of the AE index, sunspot number, and heat wave temperature. The simulation results showed good improvements in terms of the signal to noise ratio in comparison to a number of higher order and feedforward neural networks in comparison to the benchmarked techniques.
Predicting Physical Time Series Using Dynamic Ridge Polynomial Neural Networks
Al-Jumeily, Dhiya; Ghazali, Rozaida; Hussain, Abir
2014-01-01
Forecasting naturally occurring phenomena is a common problem in many domains of science, and this has been addressed and investigated by many scientists. The importance of time series prediction stems from the fact that it has wide range of applications, including control systems, engineering processes, environmental systems and economics. From the knowledge of some aspects of the previous behaviour of the system, the aim of the prediction process is to determine or predict its future behaviour. In this paper, we consider a novel application of a higher order polynomial neural network architecture called Dynamic Ridge Polynomial Neural Network that combines the properties of higher order and recurrent neural networks for the prediction of physical time series. In this study, four types of signals have been used, which are; The Lorenz attractor, mean value of the AE index, sunspot number, and heat wave temperature. The simulation results showed good improvements in terms of the signal to noise ratio in comparison to a number of higher order and feedforward neural networks in comparison to the benchmarked techniques. PMID:25157950
Relativistic Sommerfeld Low Temperature Expansion
NASA Astrophysics Data System (ADS)
Lourenço, O.; Dutra, M.; Delfino, A.; Sá Martins, J. S.
We derive a relativistic Sommerfeld expansion for thermodynamic quantities in many-body fermionic systems. The expansion is used to generate the equation of state of the Walecka model and its isotherms. We find that these results are in good agreement with numerical calculations, even when the expansion is truncated at its lowest order, in the low temperature regime, defined by T/xf ≪ 1. Although the interesting region near the liquid-gas phase transition is excluded by this criterion, the expansion may still find usefulness in the study of very cold nuclear matter systems, such as neutron stars.
Convergent series for lattice models with polynomial interactions
NASA Astrophysics Data System (ADS)
Ivanov, Aleksandr S.; Sazonov, Vasily K.
2017-01-01
The standard perturbative weak-coupling expansions in lattice models are asymptotic. The reason for this is hidden in the incorrect interchange of the summation and integration. However, substituting the Gaussian initial approximation of the perturbative expansions by a certain interacting model or regularizing original lattice integrals, one can construct desired convergent series. In this paper we develop methods, which are based on the joint and separate utilization of the regularization and new initial approximation. We prove, that the convergent series exist and can be expressed as re-summed standard perturbation theory for any model on the finite lattice with the polynomial interaction of even degree. We discuss properties of such series and study their applicability to practical computations on the example of the lattice ϕ4-model. We calculate <ϕn2 > expectation value using the convergent series, the comparison of the results with the Borel re-summation and Monte Carlo simulations shows a good agreement between all these methods.
NASA Astrophysics Data System (ADS)
Tate, Stephen James
2013-10-01
In the 1960s, the technique of using cluster expansion bounds in order to achieve bounds on the virial expansion was developed by Lebowitz and Penrose (J. Math. Phys. 5:841, 1964) and Ruelle (Statistical Mechanics: Rigorous Results. Benjamin, Elmsford, 1969). This technique is generalised to more recent cluster expansion bounds by Poghosyan and Ueltschi (J. Math. Phys. 50:053509, 2009), which are related to the work of Procacci (J. Stat. Phys. 129:171, 2007) and the tree-graph identity, detailed by Brydges (Phénomènes Critiques, Systèmes Aléatoires, Théories de Jauge. Les Houches 1984, pp. 129-183, 1986). The bounds achieved by Lebowitz and Penrose can also be sharpened by doing the actual optimisation and achieving expressions in terms of the Lambert W-function. The different bound from the cluster expansion shows some improvements for bounds on the convergence of the virial expansion in the case of positive potentials, which are allowed to have a hard core.
Classroom Aids for Mathematics, Volume 1: Polynomials.
ERIC Educational Resources Information Center
Holden, Herbert L.
The goal of this pamphlet is to provide instructors of various scientific disciplines with mathematically accurate graphs of elementary polynomial functions. The figures in this pamphlet are intended to provide suitable material for the preparation of classroom handouts and overhead transparencies. In addition, sample sets of exercises are…
On solvable Dirac equation with polynomial potentials
Stachowiak, Tomasz
2011-01-15
One-dimensional Dirac equation is analyzed with regard to the existence of exact (or closed-form) solutions for polynomial potentials. The notion of Liouvillian functions is used to define solvability, and it is shown that except for the linear potentials the equation in question is not solvable.
Simulating Nonequilibrium Radiation via Orthogonal Polynomial Refinement
2015-01-07
resolution orthogonal polynomial refinement technique for this multi-disciplinary science. Through the computational mathematics basic research, a...thus the phenomenon must be modeled [1-4]. In addition, the chemical species concentrations and its associated thermodynamic states of an inhomogeneous... thermodynamic state and compositions of the flow medium. The required optical parameters for the nonequilibrium phenomena simulation need to be determined
Optimization of Cubic Polynomial Functions without Calculus
ERIC Educational Resources Information Center
Taylor, Ronald D., Jr.; Hansen, Ryan
2008-01-01
In algebra and precalculus courses, students are often asked to find extreme values of polynomial functions in the context of solving an applied problem; but without the notion of derivative, something is lost. Either the functions are reduced to quadratics, since students know the formula for the vertex of a parabola, or solutions are…
Polynomial Asymptotes of the Second Kind
ERIC Educational Resources Information Center
Dobbs, David E.
2011-01-01
This note uses the analytic notion of asymptotic functions to study when a function is asymptotic to a polynomial function. Along with associated existence and uniqueness results, this kind of asymptotic behaviour is related to the type of asymptote that was recently defined in a more geometric way. Applications are given to rational functions and…
Polynomial Asymptotes of the Second Kind
ERIC Educational Resources Information Center
Dobbs, David E.
2011-01-01
This note uses the analytic notion of asymptotic functions to study when a function is asymptotic to a polynomial function. Along with associated existence and uniqueness results, this kind of asymptotic behaviour is related to the type of asymptote that was recently defined in a more geometric way. Applications are given to rational functions and…
Least squares polynomial fits and their accuracy
NASA Technical Reports Server (NTRS)
Lear, W. M.
1977-01-01
Equations are presented which attempt to fit least squares polynomials to tables of date. It is concluded that much data are needed to reduce the measurement error standard deviation by a significant amount, however at certain points great accuracy is attained.
On Arithmetic-Geometric-Mean Polynomials
ERIC Educational Resources Information Center
Griffiths, Martin; MacHale, Des
2017-01-01
We study here an aspect of an infinite set "P" of multivariate polynomials, the elements of which are associated with the arithmetic-geometric-mean inequality. In particular, we show in this article that there exist infinite subsets of probability "P" for which every element may be expressed as a finite sum of squares of real…
On Arithmetic-Geometric-Mean Polynomials
ERIC Educational Resources Information Center
Griffiths, Martin; MacHale, Des
2017-01-01
We study here an aspect of an infinite set "P" of multivariate polynomials, the elements of which are associated with the arithmetic-geometric-mean inequality. In particular, we show in this article that there exist infinite subsets of probability "P" for which every element may be expressed as a finite sum of squares of real…
Multivariate polynomial interpolation under projectivities part I
NASA Astrophysics Data System (ADS)
Mühlbach, G.; Gasca, M.
1991-10-01
In this note interpolation by real polynomials of several real variables is treated. Existence and unicity of the interpolant for knot systems being the perspective images of certain regular knot systems is discussed. Moreover, for such systems a Newton interpolation formula is derived allowing a recursive computation of the interpolant via multivariate divided differences. A numerical example is given.
On the history of multivariate polynomial interpolation
NASA Astrophysics Data System (ADS)
Gasca, Mariano; Sauer, Thomas
2000-10-01
Multivariate polynomial interpolation is a basic and fundamental subject in Approximation Theory and Numerical Analysis, which has received and continues receiving not deep but constant attention. In this short survey, we review its development in the first 75 years of this century, including a pioneering paper by Kronecker in the 19th century.
A dynamical polynomial chaos approach for long-time evolution of SPDEs
NASA Astrophysics Data System (ADS)
Ozen, H. Cagan; Bal, Guillaume
2017-08-01
We propose a Dynamical generalized Polynomial Chaos (DgPC) method to solve time-dependent stochastic partial differential equations (SPDEs) with white noise forcing. The long-time simulation of SPDE solutions by Polynomial Chaos (PC) methods is notoriously difficult as the dimension of the stochastic variables increases linearly with time. Exploiting the Markovian property of white noise, DgPC [1] implements a restart procedure that allows us to expand solutions at future times in terms of orthogonal polynomials of the measure describing the solution at a given time and the future white noise. The dimension of the representation is kept minimal by application of a Karhunen-Loeve (KL) expansion. Using frequent restarts and low degree polynomials on sparse multi-index sets, the method allows us to perform long time simulations, including the calculation of invariant measures for systems which possess one. We apply the method to the numerical simulation of stochastic Burgers and Navier-Stokes equations with white noise forcing. Our method also allows us to incorporate time-independent random coefficients such as a random viscosity. We propose several numerical simulations and show that the algorithm compares favorably with standard Monte Carlo methods.
NASA Astrophysics Data System (ADS)
Tao, Jianmin; Li, Jianmin
1996-11-01
Lower bounds for the first-order gradient corrections to the kinetic- and exchange-energy functionals in atoms are derived using Benson's inequality. They are formulated in terms of the expectation values and the momentum expectation value in a simple manner, namely, T2[ρ]=1/72 ∫ \\|∇ρ(r)\\|2/ρ(r) dr>=n2/72
Polynomial Eigenvalue Solutions to Minimal Problems in Computer Vision.
Kukelova, Zuzana; Bujnak, Martin; Pajdla, Tomas
2012-07-01
We present a method for solving systems of polynomial equations appearing in computer vision. This method is based on polynomial eigenvalue solvers and is more straightforward and easier to implement than the state-of-the-art Gröbner basis method since eigenvalue problems are well studied, easy to understand, and efficient and robust algorithms for solving these problems are available. We provide a characterization of problems that can be efficiently solved as polynomial eigenvalue problems (PEPs) and present a resultant-based method for transforming a system of polynomial equations to a polynomial eigenvalue problem. We propose techniques that can be used to reduce the size of the computed polynomial eigenvalue problems. To show the applicability of the proposed polynomial eigenvalue method, we present the polynomial eigenvalue solutions to several important minimal relative pose problems.
NASA Astrophysics Data System (ADS)
Qiu, Yiheng; Henderson, Thomas M.; Scuseria, Gustavo E.
2016-09-01
Spin-projected Hartree-Fock is written as a particle-hole excitation ansatz over a symmetry-adapted reference determinant. Remarkably, this expansion has an analytic expression that we were able to decipher. While the form of the polynomial expansion is universal, the excitation amplitudes need to be optimized. This is equivalent to the optimization of orbitals in the conventional projected Hartree-Fock framework of non-orthogonal determinants. Using the inverse of the particle-hole expansion, we similarity transform the Hamiltonian in a coupled-cluster style theory. The left eigenvector of the non-Hermitian Hamiltonian is constructed in a similar particle-hole expansion fashion, and we show that to numerically reproduce variational projected Hartree-Fock results, one needs as many pair excitations in the bra as the number of strongly correlated entangled pairs in the system. This single-excitation polynomial similarity transformation theory is an alternative to our recently presented double excitation theory, but supports projected Hartree-Fock and coupled cluster simultaneously rather than interpolating between them.
Khan, Waseem A; Haroon, Hiba
2016-01-01
In 2008, Liu and Wang established various symmetric identities for Bernoulli, Euler and Genocchi polynomials. In this paper, we extend these identities in a unified and generalized form to families of Hermite-Bernoulli, Euler and Genocchi polynomials. The procedure followed is that of generating functions. Some relevant connections of the general theory developed here with the results obtained earlier by Pathan and Khan are also pointed out.
A slave cluster expansion for obtaining ab-initio interatomic potentials
NASA Astrophysics Data System (ADS)
Ai, Xinyuan; Marianetti, Chris
2014-03-01
Here we propose a new approach for performing a Taylor series expansion of the first-principles computed energy of a crystal as a function of the nuclear displacements. We enlarge the dimensionality of the existing displacement space and form new variables (i.e. slave clusters) which transform like irreducible representations of the point group and satisfy homogeneity of free space. Standard group theoretical techniques can then be applied to deduce the non-zero expansion coefficients apriori at a given order, and the translation group can be used to contract the products and eliminate terms which are not linearly independent. While the expansion coefficients could likely be computed in a variety of ways, we demonstrate that finite difference is effective up to fourth order. We demonstrate the power of the method in the strongly anharmonic system PbTe. All anharmonic terms within an octahedron are computed up to fourth order. A proper linear transform demonstrates that the vast majority of the anharmonicity can be attributed to just two terms, indicating that a minimal model of phonon interactions is achievable. The ability to straightforwardly generate polynomial potentials will allow precise simulations at length and time scales which were previously unrealizable.
An Analytic Formula for the A_2 Jack Polynomials
NASA Astrophysics Data System (ADS)
Mangazeev, Vladimir V.
2007-01-01
In this letter I shall review my joint results with Vadim Kuznetsov and Evgeny Sklyanin [Indag. Math. 14 (2003), 451-482] on separation of variables (SoV) for the An Jack polynomials. This approach originated from the work [RIMS Kokyuroku 919 (1995), 27-34] where the integral representations for the A2 Jack polynomials was derived. Using special polynomial bases I shall obtain a more explicit expression for the A2 Jack polynomials in terms of generalised hypergeometric functions.
Cubic spline functions and polynomials for calculation of absorption rate.
Popović, J
1998-01-01
A model-independent method for calculation of the absorption rate based on an exact mathematical solution to the deconvolution problem of systems with linear pharmacokinetics and a polyexponential impulse responses has been examined. Theoretical analysis shows how a noninteracting primary input can be precisely evaluated when data on blood levels from a known source such as an i.v. bolus or zero-order infusion are available. This work compares the use of a Lagrange 3rd degree polynomial with that of a cubic spline function (special 3rd degree polynomial) for calculation of the absorption rate. The method is compared to another using simulated data (12 data points) containing various degrees of random noise.The accuracy of the methods is determined by how well the estimates represent the true values. It was found that the accuracy of the two methods was not significantly different, and that it was of the same order of magnitude as the noise level of the data.
Approximating Exponential and Logarithmic Functions Using Polynomial Interpolation
ERIC Educational Resources Information Center
Gordon, Sheldon P.; Yang, Yajun
2017-01-01
This article takes a closer look at the problem of approximating the exponential and logarithmic functions using polynomials. Either as an alternative to or a precursor to Taylor polynomial approximations at the precalculus level, interpolating polynomials are considered. A measure of error is given and the behaviour of the error function is…
Approximating Exponential and Logarithmic Functions Using Polynomial Interpolation
ERIC Educational Resources Information Center
Gordon, Sheldon P.; Yang, Yajun
2017-01-01
This article takes a closer look at the problem of approximating the exponential and logarithmic functions using polynomials. Either as an alternative to or a precursor to Taylor polynomial approximations at the precalculus level, interpolating polynomials are considered. A measure of error is given and the behaviour of the error function is…
Inverse of polynomial matrices in the irreducible form
NASA Technical Reports Server (NTRS)
Chang, Fan R.; Shieh, Leang S.; Mcinnis, Bayliss C.
1987-01-01
An algorithm is developed for finding the inverse of polynomial matrices in the irreducible form. The computational method involves the use of the left (right) matrix division method and the determination of linearly dependent vectors of the remainders. The obtained transfer function matrix has no nontrivial common factor between the elements of the numerator polynomial matrix and the denominator polynomial.
Tutte polynomial of a small-world Farey graph
NASA Astrophysics Data System (ADS)
Liao, Yunhua; Hou, Yaoping; Shen, Xiaoling
2013-11-01
In this paper, we find recursive formulas for the Tutte polynomials of a family of small-world Farey graphs, which are modular, and has an exponential degree hierarchy. As applications of the recursive formula, the exact expressions for the chromatic polynomial and the reliability polynomial of Fare graphs are derived and the number of connected spanning subgraphs is also obtained.
Exact Potts/Tutte polynomials for polygon chain graphs
NASA Astrophysics Data System (ADS)
Shrock, Robert
2011-04-01
We present exact calculations of Potts model partition functions and the equivalent Tutte polynomials for polygon chain graphs with open and cyclic boundary conditions. Special cases of the results that yield flow and reliability polynomials are discussed. We also analyze special cases of the Tutte polynomials that determine various quantities of graph-theoretic interest.
Multiple Representations and the Understanding of Taylor Polynomials
ERIC Educational Resources Information Center
Habre, Samer
2009-01-01
The study of Maclaurin and Taylor polynomials entails the comprehension of various new mathematical ideas. Those polynomials are initially discussed at the college level in a calculus class and then again in a course on numerical methods. This article investigates the understanding of these polynomials by students taking a numerical methods class…
Herman's Condition and Siegel Disks of Bi-Critical Polynomials
NASA Astrophysics Data System (ADS)
Chéritat, Arnaud; Roesch, Pascale
2016-06-01
We extend a theorem of Herman from the case of unicritical polynomials to the case of polynomials with two finite critical values. This theorem states that Siegel disks of such polynomials, under a diophantine condition (called Herman's condition) on the rotation number, must have a critical point on their boundaries.
Thermal expansion coefficient of single-crystal silicon from 7 K to 293 K
NASA Astrophysics Data System (ADS)
Middelmann, Thomas; Walkov, Alexander; Bartl, Guido; Schödel, René
2015-11-01
We measured the absolute lengths of three single-crystal silicon samples by means of an imaging Twyman-Green interferometer in the temperature range from 7 K to 293 K with uncertainties of about 1 nm. From these measurements we extracted the coefficient of thermal expansion with uncertainties on the order of 1 ×10-9/K . To access the functional dependence of the length on the temperature, usually polynomials are fitted to the data. Instead we used a physically motivated model equation with seven fit parameters for the whole temperature range. The coefficient of thermal expansion is obtained from the derivative of the best fit. The measurements conducted in 2012 and 2014 demonstrate a high reproducibility, and the agreement of two independently produced samples supports single-crystal silicon as a reference material for thermal expansion. Although the results for all three samples agree with each other and with measurements performed at other institutes, they significantly differ from the currently recommended values for the thermal expansion of crystalline silicon.
Chen, Fei; Tillberg, Paul W.; Boyden, Edward S.
2014-01-01
In optical microscopy, fine structural details are resolved by using refraction to magnify images of a specimen. Here we report the discovery that, by synthesizing a swellable polymer network within a specimen, it can be physically expanded, resulting in physical magnification. By covalently anchoring specific labels located within the specimen directly to the polymer network, labels spaced closer than the optical diffraction limit can be isotropically separated and optically resolved, a process we call expansion microscopy (ExM). Thus, this process can be used to perform scalable super-resolution microscopy with diffraction-limited microscopes. We demonstrate ExM with effective ~70 nm lateral resolution in both cultured cells and brain tissue, performing three-color super-resolution imaging of ~107 μm3 of the mouse hippocampus with a conventional confocal microscope. PMID:25592419
Bigravity from gradient expansion
Yamashita, Yasuho; Tanaka, Takahiro
2016-05-04
We discuss how the ghost-free bigravity coupled with a single scalar field can be derived from a braneworld setup. We consider DGP two-brane model without radion stabilization. The bulk configuration is solved for given boundary metrics, and it is substituted back into the action to obtain the effective four-dimensional action. In order to obtain the ghost-free bigravity, we consider the gradient expansion in which the brane separation is supposed to be sufficiently small so that two boundary metrics are almost identical. The obtained effective theory is shown to be ghost free as expected, however, the interaction between two gravitons takes the Fierz-Pauli form at the leading order of the gradient expansion, even though we do not use the approximation of linear perturbation. We also find that the radion remains as a scalar field in the four-dimensional effective theory, but its coupling to the metrics is non-trivial.
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
Bounding the Failure Probability Range of Polynomial Systems Subject to P-box Uncertainties
NASA Technical Reports Server (NTRS)
Crespo, Luis G.; Kenny, Sean P.; Giesy, Daniel P.
2012-01-01
This paper proposes a reliability analysis framework for systems subject to multiple design requirements that depend polynomially on the uncertainty. Uncertainty is prescribed by probability boxes, also known as p-boxes, whose distribution functions have free or fixed functional forms. An approach based on the Bernstein expansion of polynomials and optimization is proposed. In particular, we search for the elements of a multi-dimensional p-box that minimize (i.e., the best-case) and maximize (i.e., the worst-case) the probability of inner and outer bounding sets of the failure domain. This technique yields intervals that bound the range of failure probabilities. The offset between this bounding interval and the actual failure probability range can be made arbitrarily tight with additional computational effort.
Multivariate Polynomials Estimation Based on GradientBoost in Multimodal Biometrics
NASA Astrophysics Data System (ADS)
Parviz, Mehdi; Moin, M. Shahram
One of the traditional criteria to estimate the value of coefficients of multivariate polynomials in regression applications is MSE, which is known as OWM in classifier combination literature. In this paper, we address the use of GradientBoost algorithm to estimate coefficients of multivariate polynomials for score fusion level in multimodal biometric systems. Our experiments on NIST-bssr1 score database showed an improvement in verification accuracy and also reduction of number of coefficients, which increased the memory efficiency. In addition, we examined combination of OWM, and GradientBoost which showed better ROC performance and lower model order compared to OWM alone.
Soare, S.; Cazacu, O.; Yoon, J. W.
2007-05-17
With few exceptions, non-quadratic homogeneous polynomials have received little attention as possible candidates for yield functions. One reason might be that not every such polynomial is a convex function. In this paper we show that homogeneous polynomials can be used to develop powerful anisotropic yield criteria, and that imposing simple constraints on the identification process leads, aposteriori, to the desired convexity property. It is shown that combinations of such polynomials allow for modeling yielding properties of metallic materials with any crystal structure, i.e. both cubic and hexagonal which display strength differential effects. Extensions of the proposed criteria to 3D stress states are also presented. We apply these criteria to the description of the aluminum alloy AA2090T3. We prove that a sixth order orthotropic homogeneous polynomial is capable of a satisfactory description of this alloy. Next, applications to the deep drawing of a cylindrical cup are presented. The newly proposed criteria were implemented as UMAT subroutines into the commercial FE code ABAQUS. We were able to predict six ears on the AA2090T3 cup's profile. Finally, we show that a tension/compression asymmetry in yielding can have an important effect on the earing profile.
Polynomial Modeling of Child and Adult Intonation in German Spontaneous Speech
ERIC Educational Resources Information Center
de Ruiter, Laura E.
2011-01-01
In a data set of 291 spontaneous utterances from German 5-year-olds, 7-year-olds and adults, nuclear pitch contours were labeled manually using the GToBI annotation system. Ten different contour types were identified.The fundamental frequency (F0) of these contours was modeled using third-order orthogonal polynomials, following an approach similar…
Concentration of the L{sub 1}-norm of trigonometric polynomials and entire functions
Malykhin, Yu V; Ryutin, K S
2014-11-30
For any sufficiently large n, the minimal measure of a subset of [−π,π] on which some nonzero trigonometric polynomial of order ≤n gains half of the L{sub 1}-norm is shown to be π/(n+1). A similar result for entire functions of exponential type is established. Bibliography: 13 titles.
Jack polynomial fractional quantum Hall states and their generalizations
NASA Astrophysics Data System (ADS)
Baratta, Wendy; Forrester, Peter J.
2011-02-01
In the study of fractional quantum Hall states, a certain clustering condition involving up to four integers has been identified. We give a simple proof that particular Jack polynomials with α=-(r-1)/(k+1), (r-1) and (k+1) relatively prime, and with partition given in terms of its frequencies by [n0k0k0k⋯0m] satisfy this clustering condition. Our proof makes essential use of the fact that these Jack polynomials are translationally invariant. We also consider nonsymmetric Jack polynomials, symmetric and nonsymmetric generalized Hermite and Laguerre polynomials, and Macdonald polynomials from the viewpoint of the clustering.
On computing factors of cyclotomic polynomials
NASA Astrophysics Data System (ADS)
Brent, Richard P.
1993-07-01
For odd square-free n > 1 the cyclotomic polynomial {Φ_n}(x) satisfies the identity of Gauss, 4{Φ_n}(x) = A_n^2 - {( - 1)^{(n - 1)/2}}nB_n^2. A similar identity of Aurifeuille, Le Lasseur, and Lucas is {Φ_n}({( - 1)^{(n - 1)/2}}x) = C_n^2 - nxD_n^2 or, in the case that n is even and square-free, ± {Φ_{n/2}}( - {x^2}) = C_n^2 - nxD_n^2. Here, {A_n}(x), ldots ,{D_n}(x) are polynomials with integer coefficients. We show how these coefficients can be computed by simple algorithms which require O({n^2}) arithmetic operations and work over the integers. We also give explicit formulae and generating functions for {A_n}(x), ldots ,{D_n}(x) , and illustrate the application to integer factorization with some numerical examples.
A Non-Polynomial Gravity Formulation for Loop Quantum Cosmology Bounce
NASA Astrophysics Data System (ADS)
Chinaglia, Stefano; Colléaux, Aimeric; Zerbini, Sergio
2017-09-01
Recently the so-called mimetic gravity approach has been used to obtain corrections to Friedmann equation of General Relativity similar to the ones present in loop quantum cosmology. In this paper, we propose an alternative way to derive this modified Friedmann equation via the so-called non-polynomial gravity approach, which consists in adding geometric non-polynomial higher derivative terms to Hilbert-Einstein action, which are nonetheless polynomials and lead to second order differential equation in Friedmann-Lema\\^itre-Robertson-Walker spacetimes. Our explicit action turns out to be a realization of the Helling proposal of effective action with infinite number of terms. The model is investigated also in presence of non vanishing cosmological constant and a new exact bounce solution is found and studied.
Transfer matrix computation of critical polynomials for two-dimensional Potts models
Jacobsen, Jesper Lykke; Scullard, Christian R.
2013-02-04
We showed, In our previous work, that critical manifolds of the q-state Potts model can be studied by means of a graph polynomial PB(q, v), henceforth referred to as the critical polynomial. This polynomial may be defined on any periodic two-dimensional lattice. It depends on a finite subgraph B, called the basis, and the manner in which B is tiled to construct the lattice. The real roots v = eK — 1 of PB(q, v) either give the exact critical points for the lattice, or provide approximations that, in principle, can be made arbitrarily accurate by increasing the size ofmore » B in an appropriate way. In earlier work, PB(q, v) was defined by a contraction-deletion identity, similar to that satisfied by the Tutte polynomial. Here, we give a probabilistic definition of PB(q, v), which facilitates its computation, using the transfer matrix, on much larger B than was previously possible.We present results for the critical polynomial on the (4, 82), kagome, and (3, 122) lattices for bases of up to respectively 96, 162, and 243 edges, compared to the limit of 36 edges with contraction-deletion. We discuss in detail the role of the symmetries and the embedding of B. The critical temperatures vc obtained for ferromagnetic (v > 0) Potts models are at least as precise as the best available results from Monte Carlo simulations or series expansions. For instance, with q = 3 we obtain vc(4, 82) = 3.742 489 (4), vc(kagome) = 1.876 459 7 (2), and vc(3, 122) = 5.033 078 49 (4), the precision being comparable or superior to the best simulation results. More generally, we trace the critical manifolds in the real (q, v) plane and discuss the intricate structure of the phase diagram in the antiferromagnetic (v < 0) region.« less
Transfer matrix computation of critical polynomials for two-dimensional Potts models
NASA Astrophysics Data System (ADS)
Lykke Jacobsen, Jesper; Scullard, Christian R.
2013-02-01
In our previous work [1] we have shown that critical manifolds of the q-state Potts model can be studied by means of a graph polynomial PB(q, v), henceforth referred to as the critical polynomial. This polynomial may be defined on any periodic two-dimensional lattice. It depends on a finite subgraph B, called the basis, and the manner in which B is tiled to construct the lattice. The real roots v = eK - 1 of PB(q, v) either give the exact critical points for the lattice, or provide approximations that, in principle, can be made arbitrarily accurate by increasing the size of B in an appropriate way. In earlier work, PB(q, v) was defined by a contraction-deletion identity, similar to that satisfied by the Tutte polynomial. Here, we give a probabilistic definition of PB(q, v), which facilitates its computation, using the transfer matrix, on much larger B than was previously possible. We present results for the critical polynomial on the (4, 82), kagome, and (3, 122) lattices for bases of up to respectively 96, 162, and 243 edges, compared to the limit of 36 edges with contraction-deletion. We discuss in detail the role of the symmetries and the embedding of B. The critical temperatures vc obtained for ferromagnetic (v > 0) Potts models are at least as precise as the best available results from Monte Carlo simulations or series expansions. For instance, with q = 3 we obtain vc(4, 82) = 3.742 489 (4), vc(kagome) = 1.876 459 7 (2), and vc(3, 122) = 5.033 078 49 (4), the precision being comparable or superior to the best simulation results. More generally, we trace the critical manifolds in the real (q, v) plane and discuss the intricate structure of the phase diagram in the antiferromagnetic (v < 0) region.
Trigonometric Polynomials For Estimation Of Spectra
NASA Technical Reports Server (NTRS)
Greenhall, Charles A.
1990-01-01
Orthogonal sets of trigonometric polynomials used as suboptimal substitutes for discrete prolate-spheroidal "windows" of Thomson method of estimation of spectra. As used here, "windows" denotes weighting functions used in sampling time series to obtain their power spectra within specified frequency bands. Simplified windows designed to require less computation than do discrete prolate-spheroidal windows, albeit at price of some loss of accuracy.
Detecting Prime Numbers via Roots of Polynomials
ERIC Educational Resources Information Center
Dobbs, David E.
2012-01-01
It is proved that an integer n [greater than or equal] 2 is a prime (resp., composite) number if and only if there exists exactly one (resp., more than one) nth-degree monic polynomial f with coefficients in Z[subscript n], the ring of integers modulo n, such that each element of Z[subscript n] is a root of f. This classroom note could find use in…
Detecting prime numbers via roots of polynomials
NASA Astrophysics Data System (ADS)
Dobbs, David E.
2012-04-01
It is proved that an integer n ≥ 2 is a prime (resp., composite) number if and only if there exists exactly one (resp., more than one) nth-degree monic polynomial f with coefficients in Z n , the ring of integers modulo n, such that each element of Z n is a root of f. This classroom note could find use in any introductory course on abstract algebra or elementary number theory.
Polynomials with Restricted Coefficients and Their Applications
1987-01-01
JAMES S . BYRNES, PRINCIPAL INVESTIGATOR DONALD. J. NEWMAN, PRINCIPAL SCIENTIST MARTIN GOLDSTEIN, SENIOR SCIENTIST AIR FORCE OFFICE OF SCIENTIFIC...lacludd SecuntrY ClaaIflcationJ Polynomials with Restricted Coefficienks a@,d Thei• Applic tions 12. PERSONAL AUTHOR( S ) JAI.MES S . Byrnes 13a, TYPE OF...COEFFICIENTS AND THEIR APPLICATIONS PREPARED BY: JAMES S . BYRNES, PRINCIPAL INVESTIGATOR DONALD J. NEWMAN, PRINCIPAL SCIENTIST S " MARTIN GOLDSTEIN
Polynomial complexity despite the fermionic sign
NASA Astrophysics Data System (ADS)
Rossi, R.; Prokof'ev, N.; Svistunov, B.; Van Houcke, K.; Werner, F.
2017-04-01
It is commonly believed that in unbiased quantum Monte Carlo approaches to fermionic many-body problems, the infamous sign problem generically implies prohibitively large computational times for obtaining thermodynamic-limit quantities. We point out that for convergent Feynman diagrammatic series evaluated with a recently introduced Monte Carlo algorithm (see Rossi R., arXiv:1612.05184), the computational time increases only polynomially with the inverse error on thermodynamic-limit quantities.
Multivariate polynomial interpolation under projectivities II
NASA Astrophysics Data System (ADS)
Gasca, M.; Mühlbach, G.
1992-10-01
This is the second part of a note on interpolation by real polynomials of several real variables. For certain regular knot systems (geometric or regular meshes, tensor product grids), Neville-Aitken algorithms are derived explicitly. By application of a projectivity they can be extended in a simple way to arbitrary (k+1)-pencil lattices as recently introduced by Lee and Phillips. A numerical example is given.
Detecting Prime Numbers via Roots of Polynomials
ERIC Educational Resources Information Center
Dobbs, David E.
2012-01-01
It is proved that an integer n [greater than or equal] 2 is a prime (resp., composite) number if and only if there exists exactly one (resp., more than one) nth-degree monic polynomial f with coefficients in Z[subscript n], the ring of integers modulo n, such that each element of Z[subscript n] is a root of f. This classroom note could find use in…
Nested Canalyzing, Unate Cascade, and Polynomial Functions.
Jarrah, Abdul Salam; Raposa, Blessilda; Laubenbacher, Reinhard
2007-09-15
This paper focuses on the study of certain classes of Boolean functions that have appeared in several different contexts. Nested canalyzing functions have been studied recently in the context of Boolean network models of gene regulatory networks. In the same context, polynomial functions over finite fields have been used to develop network inference methods for gene regulatory networks. Finally, unate cascade functions have been studied in the design of logic circuits and binary decision diagrams. This paper shows that the class of nested canalyzing functions is equal to that of unate cascade functions. Furthermore, it provides a description of nested canalyzing functions as a certain type of Boolean polynomial function. Using the polynomial framework one can show that the class of nested canalyzing functions, or, equivalently, the class of unate cascade functions, forms an algebraic variety which makes their analysis amenable to the use of techniques from algebraic geometry and computational algebra. As a corollary of the functional equivalence derived here, a formula in the literature for the number of unate cascade functions provides such a formula for the number of nested canalyzing functions.
The bivariate Rogers Szegö polynomials
NASA Astrophysics Data System (ADS)
Chen, William Y. C.; Saad, Husam L.; Sun, Lisa H.
2007-06-01
We present an operator approach to deriving Mehler's formula and the Rogers formula for the bivariate Rogers-Szegö polynomials hn(x, y|q). The proof of Mehler's formula can be considered as a new approach to the nonsymmetric Poisson kernel formula for the continuous big q-Hermite polynomials Hn(x; a|q) due to Askey, Rahman and Suslov. Mehler's formula for hn(x, y|q) involves a 3phi2 sum and the Rogers formula involves a 2phi1 sum. The proofs of these results are based on parameter augmentation with respect to the q-exponential operator and the homogeneous q-shift operator in two variables. By extending recent results on the Rogers-Szegö polynomials hn(x|q) due to Hou, Lascoux and Mu, we obtain another Rogers-type formula for hn(x, y|q). Finally, we give a change of base formula for Hn(x; a|q) which can be used to evaluate some integrals by using the Askey-Wilson integral.
Maximal aggregation of polynomial dynamical systems.
Cardelli, Luca; Tribastone, Mirco; Tschaikowski, Max; Vandin, Andrea
2017-09-19
Ordinary differential equations (ODEs) with polynomial derivatives are a fundamental tool for understanding the dynamics of systems across many branches of science, but our ability to gain mechanistic insight and effectively conduct numerical evaluations is critically hindered when dealing with large models. Here we propose an aggregation technique that rests on two notions of equivalence relating ODE variables whenever they have the same solution (backward criterion) or if a self-consistent system can be written for describing the evolution of sums of variables in the same equivalence class (forward criterion). A key feature of our proposal is to encode a polynomial ODE system into a finitary structure akin to a formal chemical reaction network. This enables the development of a discrete algorithm to efficiently compute the largest equivalence, building on approaches rooted in computer science to minimize basic models of computation through iterative partition refinements. The physical interpretability of the aggregation is shown on polynomial ODE systems for biochemical reaction networks, gene regulatory networks, and evolutionary game theory.
Classification by using Prony's method with a polynomial model
NASA Astrophysics Data System (ADS)
Mueller, R.; Lee, W.; Okamitsu, J.
2012-06-01
Prony's Method with a Polynomial Model (PMPM) is a novel way of doing classification. Given a number of training samples with features and labels, it assumes a Gaussian mixture model for each feature, and uses Prony's method to determine a method of moments solution for the means and priors of the Gaussian distributions in the Gaussian mixture model. The features are then sorted in descending order by their relative performance. Based on the Gaussian mixture model of the first feature, training samples are partitioned into clusters by determining which Gaussian distribution each training sample is most likely from. Then with the training samples in each cluster, a new Gaussian mixture model is built for the next most powerful feature. This process repeats until a Gaussian mixture model is built for each feature, and a tree is thus grown with the training data partitioned into several final clusters. A "leaf" model for each final cluster is the weighted least squares solution (regression) for approximating a polynomial function of the features to the truth labels. Testing consists of determining for each testing sample a likelihood that the testing sample belongs to each cluster, and then regressions are weighted by their likelihoods and averaged to produce the test confidence. Evaluation of PMPM is done by extracting features from data collected by both Ground Penetrating Radar and Metal Detector of a robot-mounted land-mine detection system, training PMPM models, and testing in a cross-validation fashion.
Ladder operators for solvable potentials connected with exceptional orthogonal polynomials
NASA Astrophysics Data System (ADS)
Quesne, C.
2015-04-01
Exceptional orthogonal polynomials constitute the main part of the bound-state wavefunctions of some solvable quantum potentials, which are rational extensions of well-known shape-invariant ones. The former potentials are most easily built from the latter by using higher-order supersymmetric quantum mechanics (SUSYQM) or Darboux method. They may in general belong to three different types (or a mixture of them): types I and II, which are strictly isospectral, and type III, for which k extra bound states are created below the starting potential spectrum. A well-known SUSYQM method enables one to construct ladder operators for the extended potentials by combining the supercharges with the ladder operators of the starting potential. The resulting ladder operators close a polynomial Heisenberg algebra (PHA) with the corresponding Hamiltonian. In the special case of type III extended potentials, for this PHA the k extra bound states form k singlets isolated from the higher excited states. Some alternative constructions of ladder operators are reviewed. Among them, there is one that combines the state-adding and state-deleting approaches to type III extended potentials (or so-called Darboux-Crum and Krein-Adler transformations) and mixes the k extra bound states with the higher excited states. This novel approach can be used for building integrals of motion for two-dimensional superintegrable systems constructed from rationally-extended potentials.
Maximum of the Characteristic Polynomial of Random Unitary Matrices
NASA Astrophysics Data System (ADS)
Arguin, Louis-Pierre; Belius, David; Bourgade, Paul
2017-01-01
It was recently conjectured by Fyodorov, Hiary and Keating that the maximum of the characteristic polynomial on the unit circle of a {N× N} random unitary matrix sampled from the Haar measure grows like {CN/(log N)^{3/4}} for some random variable C. In this paper, we verify the leading order of this conjecture, that is, we prove that with high probability the maximum lies in the range {[N^{1 - ɛ},N^{1 + ɛ}]}, for arbitrarily small ɛ. The method is based on identifying an approximate branching random walk in the Fourier decomposition of the characteristic polynomial, and uses techniques developed to describe the extremes of branching random walks and of other log-correlated random fields. A key technical input is the asymptotic analysis of Toeplitz determinants with dimension-dependent symbols. The original argument for these asymptotics followed the general idea that the statistical mechanics of 1/ f-noise random energy models is governed by a freezing transition. We also prove the conjectured freezing of the free energy for random unitary matrices.
NASA Astrophysics Data System (ADS)
Mirzaee, Farshid; Bimesl, Saeed
This article presents a new reliable solver based on polynomial approximation, using the Euler polynomials to construct the approximate solutions of the second-order linear hyperbolic partial differential equations with two variables and constant coefficients. Also, a formula expressing explicitly the Euler expansion coefficients of a function with one or two variables is proved. Another explicit formula, which expresses the two dimensional Euler operational matrix of differentiation is also given. Application of these formulae for reducing the problem to a system of linear algebraic equations with the unknown Euler coefficients, is explained. Hence, the result system can be solved and the unknown Euler coefficients can be found approximately. Illustrative examples with comparisons are given to confirm the reliability of the proposed method. The results show the efficiency and accuracy of the present work.
NASA Astrophysics Data System (ADS)
Huang, Li
2016-11-01
Inspired by the recently proposed Legendre orthogonal polynomial representation for imaginary-time Green’s functions G(τ), we develop an alternate and superior representation for G(τ) and implement it in the hybridization expansion continuous-time quantum Monte Carlo impurity solver. This representation is based on the kernel polynomial method, which introduces some integral kernel functions to filter the numerical fluctuations caused by the explicit truncations of polynomial expansion series and can improve the computational precision significantly. As an illustration of the new representation, we re-examine the imaginary-time Green’s functions of the single-band Hubbard model in the framework of dynamical mean-field theory. The calculated results suggest that with carefully chosen integral kernel functions, whether the system is metallic or insulating, the Gibbs oscillations found in the previous Legendre orthogonal polynomial representation have been vastly suppressed and remarkable corrections to the measured Green’s functions have been obtained. Project supported by the National Natural Science Foundation of China (Grant No. 11504340).
Statistics of Data Fitting: Flaws and Fixes of Polynomial Analysis of Channeled Spectra
NASA Astrophysics Data System (ADS)
Karstens, William; Smith, David
2013-03-01
Starting from general statistical principles, we have critically examined Baumeister's procedure* for determining the refractive index of thin films from channeled spectra. Briefly, the method assumes that the index and interference fringe order may be approximated by polynomials quadratic and cubic in photon energy, respectively. The coefficients of the polynomials are related by differentiation, which is equivalent to comparing energy differences between fringes. However, we find that when the fringe order is calculated from the published IR index for silicon* and then analyzed with Baumeister's procedure, the results do not reproduce the original index. This problem has been traced to 1. Use of unphysical powers in the polynomials (e.g., time-reversal invariance requires that the index is an even function of photon energy), and 2. Use of insufficient terms of the correct parity. Exclusion of unphysical terms and addition of quartic and quintic terms to the index and order polynomials yields significantly better fits with fewer parameters. This represents a specific example of using statistics to determine if the assumed fitting model adequately captures the physics contained in experimental data. The use of analysis of variance (ANOVA) and the Durbin-Watson statistic to test criteria for the validity of least-squares fitting will be discussed. *D.F. Edwards and E. Ochoa, Appl. Opt. 19, 4130 (1980). Supported in part by the US Department of Energy, Office of Nuclear Physics under contract DE-AC02-06CH11357.
NASA Astrophysics Data System (ADS)
Cantero, M. J.; Ferrer, M. P.; Moral, L.; Velázquez, L.
2003-05-01
Szego's procedure to connect orthogonal polynomials on the unit circle and orthogonal polynomials on [-1,1] is generalized to nonsymmetric measures. It generates the so-called semi-orthogonal functions on the linear space of Laurent polynomials [Lambda], and leads to a new orthogonality structure in the module [Lambda]×[Lambda]. This structure can be interpreted in terms of a 2×2 matrix measure on [-1,1], and semi-orthogonal functions provide the corresponding sequence of orthogonal matrix polynomials. This gives a connection between orthogonal polynomials on the unit circle and certain classes of matrix orthogonal polynomials on [-1,1]. As an application, the strong asymptotics of these matrix orthogonal polynomials is derived, obtaining an explicit expression for the corresponding Szego's matrix function.
Orthogonal polynomial approximation in higher dimensions: Applications in astrodynamics
NASA Astrophysics Data System (ADS)
Bani Younes, Ahmad Hani Abd Alqader
We propose novel methods to utilize orthogonal polynomial approximation in higher dimension spaces, which enable us to modify classical differential equation solvers to perform high precision, long-term orbit propagation. These methods have immediate application to efficient propagation of catalogs of Resident Space Objects (RSOs) and improved accounting for the uncertainty in the ephemeris of these objects. More fundamentally, the methodology promises to be of broad utility in solving initial and two point boundary value problems from a wide class of mathematical representations of problems arising in engineering, optimal control, physical sciences and applied mathematics. We unify and extend classical results from function approximation theory and consider their utility in astrodynamics. Least square approximation, using the classical Chebyshev polynomials as basis functions, is reviewed for discrete samples of the to-be-approximated function. We extend the orthogonal approximation ideas to n-dimensions in a novel way, through the use of array algebra and Kronecker operations. Approximation of test functions illustrates the resulting algorithms and provides insight into the errors of approximation, as well as the associated errors arising when the approximations are differentiated or integrated. Two sets of applications are considered that are challenges in astrodynamics. The first application addresses local approximation of high degree and order geopotential models, replacing the global spherical harmonic series by a family of locally precise orthogonal polynomial approximations for efficient computation. A method is introduced which adapts the approximation degree radially, compatible with the truth that the highest degree approximations (to ensure maximum acceleration error < 10-9 ms-2, globally) are required near the Earths surface, whereas lower degree approximations are required as radius increases. We show that a four order of magnitude speedup is
Computing the roots of complex orthogonal and kernel polynomials
Saylor, P.E.; Smolarski, D.C.
1988-01-01
A method is presented to compute the roots of complex orthogonal and kernel polynomials. An important application of complex kernel polynomials is the acceleration of iterative methods for the solution of nonsymmetric linear equations. In the real case, the roots of orthogonal polynomials coincide with the eigenvalues of the Jacobi matrix, a symmetric tridiagonal matrix obtained from the defining three-term recurrence relationship for the orthogonal polynomials. In the real case kernel polynomials are orthogonal. The Stieltjes procedure is an algorithm to compute the roots of orthogonal and kernel polynomials bases on these facts. In the complex case, the Jacobi matrix generalizes to a Hessenberg matrix, the eigenvalues of which are roots of either orthogonal or kernel polynomials. The resulting algorithm generalizes the Stieljes procedure. It may not be defined in the case of kernel polynomials, a consequence of the fact that they are orthogonal with respect to a nonpositive bilinear form. (Another consequence is that kernel polynomials need not be of exact degree.) A second algorithm that is always defined is presented for kernel polynomials. Numerical examples are described.
FEDOROVA,A.; ZEITLIN,M.; PARSA,Z.
2000-03-31
In this paper the authors present applications of methods from wavelet analysis to polynomial approximations for a number of accelerator physics problems. According to a variational approach in the general case they have the solution as a multiresolution (multiscales) expansion on the base of compactly supported wavelet basis. They give an extension of their results to the cases of periodic orbital particle motion and arbitrary variable coefficients. Then they consider more flexible variational method which is based on a biorthogonal wavelet approach. Also they consider a different variational approach, which is applied to each scale.
The Gravity Anomaly of a 2D Polygonal Body Having Density Contrast Given by Polynomial Functions
NASA Astrophysics Data System (ADS)
D'Urso, M. G.
2015-05-01
An analytical solution is presented for the gravity anomaly produced by a 2D body whose geometrical shape is arbitrary and where the density contrast is a polynomial function in both the horizontal and vertical directions. Approximating the real shape of the body by a polygon, the solution is expressed as sum of algebraic quantities that depend only upon the coordinates of the vertices of the polygon and upon the polynomial density function. The solution presented in the paper, which refers to a third-order polynomial function as a maximum, exhibits an intrinsic symmetry that naturally suggests its extension to the case of higher-order polynomials describing the density contrast. Furthermore, the gravity anomaly is evaluated at an arbitrary point that does not necessarily coincide with the origin of the reference frame in which the density function is assigned. Invoking recent results of potential theory, the solution derived in the paper is shown to be singularity-free and numerically robust. The accuracy and effectiveness of the proposed approach is witnessed by the numerical comparisons with examples derived from the existing literature.
Representation of videokeratoscopic height data with Zernike polynomials
NASA Astrophysics Data System (ADS)
Schwiegerling, Jim; Greivenkamp, John E.; Miller, Joseph M.
1995-10-01
Videokeratoscopic data are generally displayed as a color-coded map of corneal refractive power, corneal curvature, or surface height. Although the merits of the refractive power and curvature methods have been extensively debated, the display of corneal surface height demands further investigation. A significant drawback to viewing corneal surface height is that the spherical and cylindrical components of the cornea obscure small variations in the surface. To overcome this drawback, a methodology for decomposing corneal height data into a unique set of Zernike polynomials is presented. Repeatedly removing the low-order Zernike terms reveals the hidden height variations. Examples of the decomposition-and-display technique are shown for cases of astigmatism, keratoconus, and radial keratotomy. Copyright (c) 1995 Optical Society of America
Polynomial Local Improvement Algorithms in Combinatorial Optimization.
1981-11-01
NUMBER SOL 81- 21 IIS -J O 15 14. TITLE (am#Su&Utl & YEO RPR ERO OEE Polynomial Local Improvement Algorithms in TcnclRpr Combinatorial Optimization 6...Stanford, CA 94305 II . CONTROLLING OFFICE NAME AND ADDRESS It. REPORT DATE Office of Naval Research - Dept. of the Navy November 1981 800 N. Qu~incy Street...corresponds to a node of the tree. ii ) The father of a vertex is its optimal adjacent vertex; if a vertex is a local optimum, it has no father. The tree is
A polynomial f(R) inflation model
Huang, Qing-Guo
2014-02-01
Motivated by the ultraviolet complete theory of quantum gravity, for example the string theory, we investigate a polynomial f(R) inflation model in detail. We calculate the spectral index and tensor-to-scalar ratio in the f(R) inflation model with the form of f(R) = R + (R{sup 2})/6M{sup 2} + (λn)/2n (R{sup n})/(3M{sup 2}){sup n-1}. Compared to Planck 2013, we find that R{sup n} term should be exponentially suppressed, i.e. |λ{sub n}|∼<10{sup −2n+2.6}.
Tetromino tilings and the Tutte polynomial
NASA Astrophysics Data System (ADS)
Lykke Jacobsen, Jesper
2007-02-01
We consider tiling rectangles of size 4m × 4n by T-shaped tetrominoes. Each tile is assigned a weight that depends on its orientation and position on the lattice. For a particular choice of the weights, the generating function of tilings is shown to be the evaluation of the multivariate Tutte polynomial ZG(Q, v) (known also to physicists as the partition function of the Q-state Potts model) on an (m - 1) × (n - 1) rectangle G, where the parameter Q and the edge weights v can take arbitrary values depending on the tile weights.
A polynomial f(R) inflation model
Huang, Qing-Guo
2014-02-19
Motivated by the ultraviolet complete theory of quantum gravity, for example the string theory, we investigate a polynomial f(R) inflation model in detail. We calculate the spectral index and tensor-to-scalar ratio in the f(R) inflation model with the form of f(R)=R+((R{sup 2})/(6M{sup 2}))+((λ{sub n})/(2n))((R{sup n})/((3M{sup 2}){sup n−1})). Compared to Planck 2013, we find that R{sup n} term should be exponentially suppressed, i.e. |λ{sub n}|≲10{sup −2n+2.6}.
Piecewise polynomial control in mechanical systems
NASA Astrophysics Data System (ADS)
Alesova, Irina M.; Babadzanjanz, Levon K.; Pototskaya, Irina Yu.; Pupysheva, Yulia Yu.; Saakyan, Artur T.
2017-07-01
The controlled motion of a mechanical system is represented by the linear ODE system with constant coefficients. The admissible control is a piecewise polynomial function that blanks selected frequency components of the solution of linear equations at the moment T. As "the expenditure" functional we use the integral of the sum of the modules of coordinates of the control along the interval [0, T]. The problem under consideration is to construct an admissible control which minimizes the Expenditure. To solve this problem the method that consists of analytical and numerical parts is proposed. All results of research are formulated as the theorem.
Polynomial solutions of the Monge-Ampère equation
Aminov, Yu A
2014-11-30
The question of the existence of polynomial solutions to the Monge-Ampère equation z{sub xx}z{sub yy}−z{sub xy}{sup 2}=f(x,y) is considered in the case when f(x,y) is a polynomial. It is proved that if f is a polynomial of the second degree, which is positive for all values of its arguments and has a positive squared part, then no polynomial solution exists. On the other hand, a solution which is not polynomial but is analytic in the whole of the x, y-plane is produced. Necessary and sufficient conditions for the existence of polynomial solutions of degree up to 4 are found and methods for the construction of such solutions are indicated. An approximation theorem is proved. Bibliography: 10 titles.
Parallel multigrid smoothing: polynomial versus Gauss-Seidel
NASA Astrophysics Data System (ADS)
Adams, Mark; Brezina, Marian; Hu, Jonathan; Tuminaro, Ray
2003-07-01
Gauss-Seidel is often the smoother of choice within multigrid applications. In the context of unstructured meshes, however, maintaining good parallel efficiency is difficult with multiplicative iterative methods such as Gauss-Seidel. This leads us to consider alternative smoothers. We discuss the computational advantages of polynomial smoothers within parallel multigrid algorithms for positive definite symmetric systems. Two particular polynomials are considered: Chebyshev and a multilevel specific polynomial. The advantages of polynomial smoothing over traditional smoothers such as Gauss-Seidel are illustrated on several applications: Poisson's equation, thin-body elasticity, and eddy current approximations to Maxwell's equations. While parallelizing the Gauss-Seidel method typically involves a compromise between a scalable convergence rate and maintaining high flop rates, polynomial smoothers achieve parallel scalable multigrid convergence rates without sacrificing flop rates. We show that, although parallel computers are the main motivation, polynomial smoothers are often surprisingly competitive with Gauss-Seidel smoothers on serial machines.
Polynomial compensation, inversion, and approximation of discrete time linear systems
NASA Technical Reports Server (NTRS)
Baram, Yoram
1987-01-01
The least-squares transformation of a discrete-time multivariable linear system into a desired one by convolving the first with a polynomial system yields optimal polynomial solutions to the problems of system compensation, inversion, and approximation. The polynomial coefficients are obtained from the solution to a so-called normal linear matrix equation, whose coefficients are shown to be the weighting patterns of certain linear systems. These, in turn, can be used in the recursive solution of the normal equation.
Conventional modeling of the multilayer perceptron using polynomial basis functions
NASA Technical Reports Server (NTRS)
Chen, Mu-Song; Manry, Michael T.
1993-01-01
A technique for modeling the multilayer perceptron (MLP) neural network, in which input and hidden units are represented by polynomial basis functions (PBFs), is presented. The MLP output is expressed as a linear combination of the PBFs and can therefore be expressed as a polynomial function of its inputs. Thus, the MLP is isomorphic to conventional polynomial discriminant classifiers or Volterra filters. The modeling technique was successfully applied to several trained MLP networks.
Nishimura, Shin
2015-12-15
The spherical coordinates expressions of the Rosenbluth potentials are applied to the field particle portion in the linearized Coulomb collision operator. The Sonine (generalized Laguerre) polynomial expansion formulas for this operator allowing general field particles' velocity distributions are derived. An important application area of these formulas is the study of flows of thermalized particles in NBI-heated or burning plasmas since the energy space structure of the fast ions' slowing down velocity distribution cannot be expressed by usual orthogonal polynomial expansions, and since the Galilean invariant property and the momentum conservation of the collision must be distinguished there.
Tutte Polynomial of Pseudofractal Scale-Free Web
NASA Astrophysics Data System (ADS)
Peng, Junhao; Xiong, Jian; Xu, Guoai
2015-06-01
The Tutte polynomial of a graph is a 2-variable polynomial which is quite important in both Combinatorics and Statistical physics. It contains various numerical invariants and polynomial invariants, such as the number of spanning trees, the number of spanning forests, the number of acyclic orientations, the reliability polynomial, chromatic polynomial and flow polynomial. In this paper, we study and obtain a recursive formula for the Tutte polynomial of pseudofractal scale-free web (PSFW), and thus logarithmic complexity algorithm to calculate the Tutte polynomial of the PSFW is obtained, although it is NP-hard for general graph. By solving the recurrence relations derived from the Tutte polynomial, the rigorous solution for the number of spanning trees of the PSFW is obtained. Therefore, an alternative approach to determine explicitly the number of spanning trees of the PSFW is given. Furthermore, we analyze the all-terminal reliability of the PSFW and compare the results with those of the Sierpinski gasket which has the same number of nodes and edges as the PSFW. In contrast with the well-known conclusion that inhomogeneous networks (e.g., scale-free networks) are more robust than homogeneous networks (i.e., networks in which each node has approximately the same number of links) with respect to random deletion of nodes, the Sierpinski gasket (which is a homogeneous network), as our results show, is more robust than the PSFW (which is an inhomogeneous network) with respect to random edge failures.
New Polynomial-Based Molecular Descriptors with Low Degeneracy
Dehmer, Matthias; Mueller, Laurin A. J.; Graber, Armin
2010-01-01
In this paper, we introduce a novel graph polynomial called the ‘information polynomial’ of a graph. This graph polynomial can be derived by using a probability distribution of the vertex set. By using the zeros of the obtained polynomial, we additionally define some novel spectral descriptors. Compared with those based on computing the ordinary characteristic polynomial of a graph, we perform a numerical study using real chemical databases. We obtain that the novel descriptors do have a high discrimination power. PMID:20689599
Some properties of multiple orthogonal polynomials associated with Macdonald functions
NASA Astrophysics Data System (ADS)
Coussement, Els; van Assche, Walter
2001-08-01
Multiple orthogonal polynomials corresponding to two weights on [0,[infinity]) associated with modified Bessel functions (Macdonald functions) K[nu] and K[nu]+1 were introduced in Van Assche, Yakubovich (Integral Transforms Special Funct. 9 (2000) 229-244) and recently also studied by Ben Cheikh, Douak (Meth. Appl. Anal., to appear). We obtain explicit formulas for type I vector polynomials (An,n,Bn,n) and (An+1,n,Bn+1,n) and for type II polynomials Pn,n and Pn+1,n. We also obtain generating functions for types I and II polynomials.
Quantum algorithms for virtual Jones polynomials via Thistlethwaite theorems
NASA Astrophysics Data System (ADS)
Vélez, Mario; Ospina, Juan
2010-04-01
Recently a quantum algorithm for the Jones polynomial of virtual links was proposed by Kauffman and Dye via the implementation of the virtual braid group in anyonic topological quantum computation when the virtual crossings are considered as generalized swap gates. Also recently, a mathematical method for the computation of the Jones polynomial of a given virtual link in terms of the relative Tuttle polynomial of its face (Tait) graph with some suitable variable substitutions was proposed by Diao and Hetyei. The method of Diao and Hetyei is offered as an alternative to the ribbon graph approach according to which the Tutte polynomial of a given virtual link is computed in terms of the Bollobás- Riordan polynomial of the corresponding ribbon graph. The method of Diao and Hetyei can be considered as an extension of the celebrated Thistlethwaite theorem according to which invariant polynomials for knots and links are derived from invariant polynomials for graphs. Starting from these ideas we propose a quantum algorithm for the Jones polynomial of a given virtual link in terms of the generalized Tutte polynomials by exploiting the Thistlethwaite theorem and the Kauffman algorithm . Our method is claimed as the quantum version of the Diao-Hetyei method. Possible supersymmetric implementations of our algortihm are discussed jointly with its formulations using topological quantum lambda calculus.
Using Tutte polynomials to characterize sexual contact networks
NASA Astrophysics Data System (ADS)
Cadavid Muñoz, Juan José
2014-06-01
Tutte polynomials are used to characterize the dynamic and topology of the sexual contact networks, in which pathogens are transmitted as an epidemic. Tutte polynomials provide an algebraic characterization of the sexual contact networks and allow the projection of spread control strategies for sexual transmission diseases. With the usage of Tutte polynomials, it allows obtaining algebraic expressions for the basic reproductive number of different pathogenic agents. Computations are done using the computer algebra software Maple, and it's GraphTheory Package. The topological complexity of a contact network is represented by the algebraic complexity of the correspondent polynomial. The change in the topology of the contact network is represented as a change in the algebraic form of the associated polynomial. With the usage of the Tutte polynomials, the number of spanning trees for each contact network can be obtained. From the obtained results in the polynomial form, it can be said that Tutte polynomials are of great importance for designing and implementing control measures for slowing down the propagation of sexual transmitted pathologies. As a future research line, the analysis of weighted sexual contact networks using weighted Tutte polynomials is considered.
Return times of polynomials as meta-Fibonacci numbers
NASA Astrophysics Data System (ADS)
Emerson, Nathaniel D.
We consider generalized closest return times of a complex polynomial of degree at least two. Most previous studies on this subject have focused on the properties of polynomials with particular return times, especially the Fibonacci numbers. We study the general form of these closest return times. The main result of this paper is that these closest return times are meta-Fibonacci numbers. In particular, this result applies to the return times of a principal nest of a polynomial. Furthermore, we show that an analogous result holds in a tree with dynamics that is associated with a polynomial.
Chern-Simons matrix models and Stieltjes-Wigert polynomials
Dolivet, Yacine; Tierz, Miguel
2007-02-15
Employing the random matrix formulation of Chern-Simons theory on Seifert manifolds, we show how the Stieltjes-Wigert orthogonal polynomials are useful in exact computations in Chern-Simons matrix models. We construct a biorthogonal extension of the Stieltjes-Wigert polynomials, not available in the literature, necessary to study Chern-Simons matrix models when the geometry is a lens space. We also study the relationship between Stieltjes-Wigert and Rogers-Szegoe polynomials and the corresponding equivalence with a unitary matrix model. Finally, we give a detailed proof of a result that relates quantum dimensions with averages of Schur polynomials in the Stieltjes-Wigert ensemble.
Multi-indexed Jacobi polynomials and Maya diagrams
NASA Astrophysics Data System (ADS)
Takemura, Kouichi
2014-11-01
Multi-indexed Jacobi polynomials are defined by the Wronskian of four types of eigenfunctions of the Pöschl-Teller Hamiltonian. We give a correspondence between multi-indexed Jacobi polynomials and pairs of Maya diagrams, and we show that any multi-indexed Jacobi polynomial is essentially equal to some multi-indexed Jacobi polynomial of two types of eigenfunction. As an application, we show a Wronskian-type formula of some special eigenstates of the deformed Pöschl-Teller Hamiltonian.
Application of field dependent polynomial model
NASA Astrophysics Data System (ADS)
Janout, Petr; Páta, Petr; Skala, Petr; Fliegel, Karel; Vítek, Stanislav; Bednář, Jan
2016-09-01
Extremely wide-field imaging systems have many advantages regarding large display scenes whether for use in microscopy, all sky cameras, or in security technologies. The Large viewing angle is paid by the amount of aberrations, which are included with these imaging systems. Modeling wavefront aberrations using the Zernike polynomials is known a longer time and is widely used. Our method does not model system aberrations in a way of modeling wavefront, but directly modeling of aberration Point Spread Function of used imaging system. This is a very complicated task, and with conventional methods, it was difficult to achieve the desired accuracy. Our optimization techniques of searching coefficients space-variant Zernike polynomials can be described as a comprehensive model for ultra-wide-field imaging systems. The advantage of this model is that the model describes the whole space-variant system, unlike the majority models which are partly invariant systems. The issue that this model is the attempt to equalize the size of the modeled Point Spread Function, which is comparable to the pixel size. Issues associated with sampling, pixel size, pixel sensitivity profile must be taken into account in the design. The model was verified in a series of laboratory test patterns, test images of laboratory light sources and consequently on real images obtained by an extremely wide-field imaging system WILLIAM. Results of modeling of this system are listed in this article.
Tabulating knot polynomials for arborescent knots
NASA Astrophysics Data System (ADS)
Mironov, A.; Morozov, A.; Morozov, A.; Ramadevi, P.; Singh, Vivek Kumar; Sleptsov, A.
2017-02-01
Arborescent knots are those which can be represented in terms of double fat graphs or equivalently as tree Feynman diagrams. This is the class of knots for which the present knowledge is sufficient for lifting topological description to the level of effective analytical formulas. The paper describes the origin and structure of the new tables of colored knot polynomials, which will be posted at the dedicated site (http://knotebook.org). Even if formal expressions are known in terms of modular transformation matrices, the computation in finite time requires additional ideas. We use the ‘family’ approach, suggested in Mironov and Morozov (2015 Nucl. Phys. B 899 395–413), and apply it to arborescent knots in the Rolfsen table by developing a Feynman diagram technique, associated with an auxiliary matrix model field theory. Gauge invariance in this theory helps to provide meaning to Racah matrices in the case of non-trivial multiplicities and explains the need for peculiar sign prescriptions in the calculation of [21]-colored HOMFLY-PT polynomials.
On factorization of generalized Macdonald polynomials
NASA Astrophysics Data System (ADS)
Kononov, Ya.; Morozov, A.
2016-08-01
A remarkable feature of Schur functions—the common eigenfunctions of cut-and-join operators from W_∞ —is that they factorize at the peculiar two-parametric topological locus in the space of time variables, which is known as the hook formula for quantum dimensions of representations of U_q(SL_N) and which plays a big role in various applications. This factorization survives at the level of Macdonald polynomials. We look for its further generalization to generalized Macdonald polynomials (GMPs), associated in the same way with the toroidal Ding-Iohara-Miki algebras, which play the central role in modern studies in Seiberg-Witten-Nekrasov theory. In the simplest case of the first-coproduct eigenfunctions, where GMP depend on just two sets of time variables, we discover a weak factorization—on a one- (rather than four-) parametric slice of the topological locus, which is already a very non-trivial property, calling for proof and better understanding.
Seizure prediction using polynomial SVM classification.
Zisheng Zhang; Parhi, Keshab K
2015-08-01
This paper presents a novel patient-specific algorithm for prediction of seizures in epileptic patients with low hardware complexity and low power consumption. In the proposed approach, we first compute the spectrogram of the input fragmented EEG signals from a few electrodes. Each fragmented data clip is ten minutes in duration. Band powers, relative spectral powers and ratios of spectral powers are extracted as features. The features are then subjected to electrode selection and feature selection using classification and regression tree. The baseline experiment uses all features from selected electrodes and these features are then subjected to a radial basis function kernel support vector machine (RBF-SVM) classifier. The proposed method further selects a small number features from the selected electrodes and train a polynomial support vector machine (SVM) classifier with degree of 2 on these features. Prediction performances are compared between the baseline experiment and the proposed method. The algorithm is tested using intra-cranial EEG (iEEG) from the American Epilepsy Society Seizure Prediction Challenge database. The baseline experiment using a large number of features and RBF-SVM achieves a 100% sensitivity and an average AUC of 0.9985, while the proposed algorithm using only a small number of features and polynomial SVM with degree of 2 can achieve a sensitivity of 100.0%, an average area under curve (AUC) of 0.9795. For both experiments, only 10% of the available training data are used for training.
Three-dimensional reconstruction of microscopic images using different order intensity derivatives
NASA Astrophysics Data System (ADS)
Wang, Yu; Jiang, Huan
2015-02-01
Fluorescence microscopic image three-dimensional (3-D) reconstruction is a challenging topic in image processing and computer vision, and can be widely applied to life science, biology, and medicine. A microscopic images 3-D reconstruction method is proposed for transparent or partially transparent microscopic samples, which is based on the Taylor expansion theorem and polynomial fitting. First, the image stack of the specimen is divided into several groups in an overlapping or nonoverlapping way along the optical axis, and the first image of every group is regarded as the reference image. Then, different order intensity derivatives are calculated using all the images of every group and a polynomial fitting method. Subsequently, a new image can be generated by means of Taylor expansion theorem and the calculated different order intensity derivatives and for which the distance to the reference image is Δz along the optical axis. Finally, the microscopic specimen can be reconstructed in 3-D form using deconvolution technology and all the images including both the observed and the generated images. The experimental results show the superior performance via processing simulated and real fluorescence microscopic degraded images.
Fock expansion of multimode pure Gaussian states
Cariolaro, Gianfranco; Pierobon, Gianfranco
2015-12-15
The Fock expansion of multimode pure Gaussian states is derived starting from their representation as displaced and squeezed multimode vacuum states. The approach is new and appears to be simpler and more general than previous ones starting from the phase-space representation given by the characteristic or Wigner function. Fock expansion is performed in terms of easily evaluable two-variable Hermite–Kampé de Fériet polynomials. A relatively simple and compact expression for the joint statistical distribution of the photon numbers in the different modes is obtained. In particular, this result enables one to give a simple characterization of separable and entangled states, as shown for two-mode and three-mode Gaussian states.
On simulation and verification of the atmospheric turbulent phase screen with Zernike polynomials
NASA Astrophysics Data System (ADS)
Dai, Liming; Tong, Shoufeng; Zhang, Lei; Wang, Yinhuan
2015-04-01
Atmospheric turbulence is one of the main factors that influence the spread of laser communication in the atmosphere affect, which will change the random distribution of the refractive index of air, and affect the image quality of the beam through the atmosphere seriously. To study atmospheric turbulence in order to grasp changes in atmospheric turbulence, by taking the appropriate methods to control and reduce the effects of atmospheric turbulence on the beam quality. In addition to studying atmospheric turbulence using experimental methods and theoretical analysis. Numerical simulation is an effective means to study the problem of turbulence. Zernike polynomials were used to produce atmospheric turbulence phase screen in this article. The phase structure function and the atmospheric coherence length were used to check whether the atmospheric turbulence phase screen is right or not. Simulation results were studied show that, the atmospheric turbulence phase screen generated with Zernike polynomial method was consistent with the theoretical values in the low spatial frequency components, but, the simulation results had big difference with the theoretical values in the high spatial frequency components. The reason is that Zernike polynomials method has some limitations. In addition, the distribution of turbulence in the atmospheric turbulence phase screen can be changed by increasing the Zernike polynomials of orders or changing the receiving apertures, but which involves great and complex calculation. Therefore, in the specific application of the laser communication system, the best experimental program should be considered. Statistical properties of atmospheric turbulence phase can be described by the phase structure function. Therefore, the structure of the function will be used to determine the phase screen simulation phase screen is accurate. To give a better understanding of both methods the difference between simulation results, the simulation results of
High order weighted essentially nonoscillatory WENO-η schemes for hyperbolic conservation laws
NASA Astrophysics Data System (ADS)
Fan, Ping
2014-07-01
In [8], the authors have designed a new fifth-order WENO finite-difference scheme (named WENO-η) by introducing a new local smoothness indicator which is defined based on the Lagrangian interpolation polynomials and has a more succinct form compared with the classical one proposed by Jiang and Shu [12]. With this new local smoothness indicator, higher order global smoothness indicators were able to be devised and the corresponding scheme (named WENO-Zη) displayed less numerical dissipations than the classic fifth-order WENO schemes, including WENO-JS [12] and WENO-Z [5,6]. In this paper, a close look is taken at Taylor expansions of the Lagrangian interpolation polynomials of the WENO sub-stencils and the related inherited symmetries of the local smoothness indicators, which allows the extension of the WENO-η scheme to higher orders of accuracy. Furthermore, general formulae for the global smoothness indicators are derived with which the WENO-Zη schemes can be extended to all odd-orders of accuracy. Numerical experiments are conducted to demonstrate the performance of the proposed schemes.
NASA Astrophysics Data System (ADS)
Liu, Jie; Sun, Xingsheng; Han, Xu; Jiang, Chao; Yu, Dejie
2015-05-01
Based on the Gegenbauer polynomial expansion theory and regularization method, an analytical method is proposed to identify dynamic loads acting on stochastic structures. Dynamic loads are expressed as functions of time and random parameters in time domain and the forward model of dynamic load identification is established through the discretized convolution integral of loads and the corresponding unit-pulse response functions of system. Random parameters are approximated through the random variables with λ-probability density function (PDFs) or their derivative PDFs. For this kind of random variables, Gegenbauer polynomial expansion is the unique correct choice to transform the problem of load identification for a stochastic structure into its equivalent deterministic system. Just via its equivalent deterministic system, the load identification problem of a stochastic structure can be solved by any available deterministic methods. With measured responses containing noise, the improved regularization operator is adopted to overcome the ill-posedness of load reconstruction and to obtain the stable and approximate solutions of certain inverse problems and the valid assessments of the statistics of identified loads. Numerical simulations demonstrate that with regard to stochastic structures, the identification and assessment of dynamic loads are achieved steadily and effectively by the presented method.
Decidability of classes of algebraic systems in polynomial time
Anokhin, M I
2002-02-28
For some classes of algebraic systems several kinds of polynomial-time decidability are considered, which use an oracle performing signature operations and computing predicates. Relationships between various kinds of decidability are studied. Several results on decidability and undecidability in polynomial time are proved for some finitely based varieties of universal algebras.
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…
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…