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
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.
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.
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.
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.
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.
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.
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.
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
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.
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.
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.
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.
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
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.
Karajan, N; Otto, D; Oladyshkin, S; Ehlers, W
2014-10-01
A possibility to simulate the mechanical behaviour of the human spine is given by modelling the stiffer structures, i.e. the vertebrae, as a discrete multi-body system (MBS), whereas the softer connecting tissue, i.e. the softer intervertebral discs (IVD), is represented in a continuum-mechanical sense using the finite-element method (FEM). From a modelling point of view, the mechanical behaviour of the IVD can be included into the MBS in two different ways. They can either be computed online in a so-called co-simulation of a MBS and a FEM or offline in a pre-computation step, where a representation of the discrete mechanical response of the IVD needs to be defined in terms of the applied degrees of freedom (DOF) of the MBS. For both methods, an appropriate homogenisation step needs to be applied to obtain the discrete mechanical response of the IVD, i.e. the resulting forces and moments. The goal of this paper was to present an efficient method to approximate the mechanical response of an IVD in an offline computation. In a previous paper (Karajan et al. in Biomech Model Mechanobiol 12(3):453-466, 2012), it was proven that a cubic polynomial for the homogenised forces and moments of the FE model is a suitable choice to approximate the purely elastic response as a coupled function of the DOF of the MBS. In this contribution, the polynomial chaos expansion (PCE) is applied to generate these high-dimensional polynomials. Following this, the main challenge is to determine suitable deformation states of the IVD for pre-computation, such that the polynomials can be constructed with high accuracy and low numerical cost. For the sake of a simple verification, the coupling method and the PCE are applied to the same simplified motion segment of the spine as was used in the previous paper, i.e. two cylindrical vertebrae and a cylindrical IVD in between. In a next step, the loading rates are included as variables in the polynomial response functions to account for a more
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.
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.
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
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)
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.
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.
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.
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.
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.
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.
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.
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.
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.
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. 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.
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.
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.
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.
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)
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.
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.
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.
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.
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.
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.
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.
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
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.
Multi-particle dynamical systems and polynomials
NASA Astrophysics Data System (ADS)
Demina, Maria V.; Kudryashov, Nikolai A.
2016-05-01
Polynomial dynamical systems describing interacting particles in the plane are studied. A method replacing integration of a polynomial multi-particle dynamical system by finding polynomial solutions of partial differential equations is introduced. The method enables one to integrate a wide class of polynomial multi-particle dynamical systems. The general solutions of certain dynamical systems related to linear second-order partial differential equations are found. As a by-product of our results, new families of orthogonal polynomials are derived.
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.
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.
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.
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.
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 .
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.
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.
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).
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.
Momentum space orthogonal polynomial projection quantization
NASA Astrophysics Data System (ADS)
Handy, C. R.; Vrinceanu, D.; Marth, C. B.; Gupta, R.
2016-04-01
The orthogonal polynomial projection quantization (OPPQ) is an algebraic method for solving Schrödinger’s equation by representing the wave function as an expansion {{\\Psi }}(x)={\\displaystyle \\sum }n{{{Ω }}}n{P}n(x)R(x) in terms of polynomials {P}n(x) orthogonal with respect to a suitable reference function R(x), which decays asymptotically not faster than the bound state wave function. The expansion coefficients {{{Ω }}}n are obtained as linear combinations of power moments {μ }{{p}}=\\int {x}p{{\\Psi }}(x) {{d}}x. In turn, the {μ }{{p}}'s are generated by a linear recursion relation derived from Schrödinger’s equation from an initial set of low order moments. It can be readily argued that for square integrable wave functions representing physical states {{lim}}n\\to ∞ {{{Ω }}}n=0. Rapidly converging discrete energies are obtained by setting Ω coefficients to zero at arbitrarily high order. This paper introduces an extention of OPPQ in momentum space by using the representation {{Φ }}(k)={\\displaystyle \\sum }n{{{\\Xi }}}n{Q}n(k)T(k), where Q n (k) are polynomials orthogonal with respect to a suitable reference function T(k). The advantage of this new representation is that it can help solving problems for which there is no coordinate space moment equation. This is because the power moments in momentum space are the Taylor expansion coefficients, which are recursively calculated via Schrödinger’s equation. We show the convergence of this new method for the sextic anharmonic oscillator and an algebraic treatment of Gross-Pitaevskii nonlinear equation.
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.
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.
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.
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.
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
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.
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.
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.
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…
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.
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.
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)
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
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.
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.
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.
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2012-11-26
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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)
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.
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)
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.
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 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.
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.
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.
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.
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.
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…
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.
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.
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.
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.
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.
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.
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.
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
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.
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.
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
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.
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.
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
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.
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.
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.
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
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.
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)
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).
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.
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)
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.
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.
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…
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…
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.
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).
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.
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.
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.
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.
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.
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.
Alcocer-Sosa, Mauricio; Gutiérrez, David
2016-06-25
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.
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).
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.
Federal Register 2010, 2011, 2012, 2013, 2014
2012-09-24
... Land Policy and Management Act of 1976; 43 U.S.C. 1714, it is ordered as follows: 1. Subject to valid existing rights, the following described lands are hereby withdrawn from all forms of appropriation under...\\, lot 2 of NW\\1/4\\, and S\\1/2\\S\\1/2\\; Sec. 4, lots 1 and 2 of NE\\1/4\\, lots 1 and 2 of NW\\1/4\\, SW\\1/...
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.
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.
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.
Stochastic Estimation via Polynomial Chaos
2015-10-01
Program Manager Lethality, Vulnerability and Survivability Branch This report is published in the interest of scientific and technical...initial conditions for partial differential equations. Here, the elementary theory of the polynomial chaos is presented followed by the details of a...the elementary theory of the polynomial chaos is presented followed by the details of a number of example calculations where the statistical mean and
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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
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)
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.
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.
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).
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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
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 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.
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 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.
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.
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.
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.
Spectral likelihood expansions for Bayesian inference
NASA Astrophysics Data System (ADS)
Nagel, Joseph B.; Sudret, Bruno
2016-03-01
A spectral approach to Bayesian inference is presented. It pursues the emulation of the posterior probability density. The starting point is a series expansion of the likelihood function in terms of orthogonal polynomials. From this spectral likelihood expansion all statistical quantities of interest can be calculated semi-analytically. The posterior is formally represented as the product of a reference density and a linear combination of polynomial basis functions. Both the model evidence and the posterior moments are related to the expansion coefficients. This formulation avoids Markov chain Monte Carlo simulation and allows one to make use of linear least squares instead. The pros and cons of spectral Bayesian inference are discussed and demonstrated on the basis of simple applications from classical statistics and inverse modeling.
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.
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.
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.
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.
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.
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.
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.
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.
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
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.
Laguerre-Polynomial-Weighted Two-Mode Squeezed State
NASA Astrophysics Data System (ADS)
He, Rui; Fan, Hong-Yi; Song, Jun; Zhou, Jun
2016-07-01
We propose a new optical field named Laguerre-polynomial-weighted two-mode squeezed state. We find that such a state can be generated by passing the l-photon excited two-mode squeezed vacuum state C l a † l S 2|00> through an single-mode amplitude damping channel. Physically, this paper actually is concerned what happens when both excitation and damping of photons co-exist for a two-mode squeezed state, e.g., dessipation of photon-added two-mode squeezed vacuum state. We employ the summation method within ordered product of operators and a new generating function formula about two-variable Hermite polynomials to proceed our discussion.
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.
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.
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.
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.
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
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.
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
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.
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.
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
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.
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.
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.
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.
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.
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…
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.
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.
1970-10-01
either burned simultaneously with a portland ce4nt or !r;terground with portland cement clinker ; Type M - a mixture of portland cement, calcium-aluminate... clinker that is interground with portland clinker or blended with portland cement or, alternately, it may be formed simul- taneously vrith the portland ... clinker compounds during the burning process. 3. Expansive cement, Type M is either a mixture of portland cement, calcium aluminate cement, and calcium
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.
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.
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.
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…
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.
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.
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.
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.
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.
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.
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.
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.
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
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.
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…
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.
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.
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
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.
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.
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.
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.
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…
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.
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…
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.
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.
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.
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…
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
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…
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.
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 polynomial form of the scattering equations
NASA Astrophysics Data System (ADS)
Dolan, Louise; Goddard, Peter
2014-07-01
The scattering equations, recently proposed by Cachazo, He and Yuan as providing a kinematic basis for describing tree amplitudes for massless particles in arbitrary space-time dimension (including scalars, gauge bosons and gravitons), are reformulated in polynomial form. The scattering equations for N particles are shown to be equivalent to a Möbius invariant system of N - 3 equations, m = 0, 2 ≤ m ≤ N - 2, in N variables, where m is a homogeneous polynomial of degree m, with the exceptional property of being linear in each variable taken separately. Fixing the Möbius invariance appropriately, yields polynomial equations h m = 0, 1 ≤ m ≤ N - 3, in N - 3 variables, where h m has degree m. The linearity of the equations in the individual variables facilitates computation, e.g. the elimination of variables to obtain single variable equations determining the solutions. Expressions are given for the tree amplitudes in terms of the m and h m . The extension to the massive case for scalar particles is described and the special case of four dimensional space-time is discussed.
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.
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.
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.
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.
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.
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
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}.
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}.
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
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Lectures on the asymptotic expansion of a Hermitian matrix integral
NASA Astrophysics Data System (ADS)
Mulase, Motohico
In these lectures three different methods of computing the asymptotic expansion of a Hermitian matrix integral is presented. The first one is a combinatorial method using Feynman diagrams. This leads us to the generating function of the reciprocal of the order of the automorphism group of a tiling of a Riemann surface. The second method is based on the classical analysis of orthogonal polynomials. A rigorous asymptotic method is established, and a special case of the matrix integral is computed in terms of the Riemann ζ-function. The third method is derived from a formula for the τ-function solution to the KP equations. This method leads us to a new class of solutions of the KP equations that are transcendental, in the sense that they cannot be obtained by the celebrated Krichever construction and its generalizations based on algebraic geometry of vector bundles on Riemann surfaces. In each case a mathematically rigorous way of dealing with asymptotic series in an infinite number of variables is established.
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…
The evolution of piecewise polynomial wave functions
NASA Astrophysics Data System (ADS)
Andrews, Mark
2017-01-01
For a non-relativistic particle, we consider the evolution of wave functions that consist of polynomial segments, usually joined smoothly together. These spline wave functions are compact (that is, they are initially zero outside a finite region), but they immediately extend over all available space as they evolve. The simplest splines are the square and triangular wave functions in one dimension, but very complicated splines have been used in physics. In general the evolution of such spline wave functions can be expressed in terms of antiderivatives of the propagator; in the case of a free particle or an oscillator, all the evolutions are expressed exactly in terms of Fresnel integrals. Some extensions of these methods to two and three dimensions are discussed.
Polynomial Monogamy Relations for Entanglement Negativity
NASA Astrophysics Data System (ADS)
Allen, Grant W.; Meyer, David A.
2017-02-01
The notion of nonclassical correlations is a powerful contrivance for explaining phenomena exhibited in quantum systems. It is well known, however, that quantum systems are not free to explore arbitrary correlations—the church of the smaller Hilbert space only accepts monogamous congregants. We demonstrate how to characterize the limits of what is quantum mechanically possible with a computable measure, entanglement negativity. We show that negativity only saturates the standard linear monogamy inequality in trivial cases implied by its monotonicity under local operations and classical communication, and derive a necessary and sufficient inequality which, for the first time, is a nonlinear higher degree polynomial. For very large quantum systems, we prove that the negativity can be distributed at least linearly for the tightest constraint and conjecture that it is at most linear.
Schur polynomials and biorthogonal random matrix ensembles
NASA Astrophysics Data System (ADS)
Tierz, Miguel
2010-06-01
The study of the average of Schur polynomials over a Stieltjes-Wigert ensemble has been carried out by Dolivet and Tierz [J. Math. Phys. 48, 023507 (2007); e-print arXiv:hep-th/0609167], where it was shown that it is equal to quantum dimensions. Using the same approach, we extend the result to the biorthogonal case. We also study, using the Littlewood-Richardson rule, some particular cases of the quantum dimension result. Finally, we show that the notion of Giambelli compatibility of Schur averages, introduced by Borodin et al. [Adv. Appl. Math. 37, 209 (2006); e-print arXiv:math-ph/0505021], also holds in the biorthogonal setting.
A Dialogue of Multipoles: Matched Asymptotic Expansion for Caged Black Holes
NASA Astrophysics Data System (ADS)
Gorbonos, Dan; Kol, Barak
2004-06-01
No analytic solution is known to date for a black hole in a compact dimension. We develop an analytic perturbation theory where the small parameter is the size of the black hole relative to the size of the compact dimension. We set up a general procedure for an arbitrary order in the perturbation series based on an asymptotic matched expansion between two coordinate patches: the near horizon zone and the asymptotic zone. The procedure is ordinary perturbation expansion in each zone, where additionally some boundary data comes from the other zone, and so the procedure alternates between the zones. It can be viewed as a dialogue of multipoles where the black hole changes its shape (mass multipoles) in response to the field (multipoles) created by its periodic ``mirrors'', and that in turn changes its field and so on. We present the leading correction to the full metric including the first correction to the area-temperature relation, the leading term for black hole eccentricity and the ``Archimedes effect''. The next order corrections will appear in a sequel. On the way we determine independently the static perturbations of the Schwarzschild black hole in dimension d geq 5, where the system of equations can be reduced to ``a master equation'' — a single ordinary differential equation. The solutions are hypergeometric functions which in some cases reduce to polynomials.
Properties of the zeros of generalized basic hypergeometric polynomials
NASA Astrophysics Data System (ADS)
Bihun, Oksana; Calogero, Francesco
2015-11-01
We define the generalized basic hypergeometric polynomial of degree N in terms of the generalized basic hypergeometric function, by choosing one of its parameters to allow the termination of the series after a finite number of summands. In this paper, we obtain a set of nonlinear algebraic equations satisfied by the N zeros of the polynomial. Moreover, we obtain an N × N matrix M defined in terms of the zeros of the polynomial, which, in turn, depend on the parameters of the polynomial. The eigenvalues of this remarkable matrix M are given by neat expressions that depend only on some of the parameters of the polynomial; that is, the matrix M is isospectral. Moreover, in case the parameters that appear in the expressions for the eigenvalues of M are rational, the matrix M has rational eigenvalues, a Diophantine property.
Multiple Meixner polynomials and non-Hermitian oscillator Hamiltonians
NASA Astrophysics Data System (ADS)
Ndayiragije, F.; Van Assche, W.
2013-12-01
Multiple Meixner polynomials are polynomials in one variable which satisfy orthogonality relations with respect to r > 1 different negative binomial distributions (Pascal distributions). There are two kinds of multiple Meixner polynomials, depending on the selection of the parameters in the negative binomial distribution. We recall their definition and some formulas and give generating functions and explicit expressions for the coefficients in the nearest neighbor recurrence relation. Following a recent construction of Miki, Tsujimoto, Vinet and Zhedanov (for multiple Meixner polynomials of the first kind), we construct r > 1 non-Hermitian oscillator Hamiltonians in r dimensions which are simultaneously diagonalizable and for which the common eigenstates are expressed in terms of multiple Meixner polynomials of the second kind.
Han, Mangui
2004-01-01
Thermal expansion (TE) and magnetostriction (MS) measurements have been conducted for Gd_{5}(Si_{x}Ge_{1-x})_{4} with a series of x values to study its critical behavior in the vicinity of transition temperatures. It was found that the Curie temperature of Gd_{5}(Si_{x}Ge_{1-x})_{4} for x 0 ~ 0.5 is dependent on magnetic field, direction of change of temperature (Tc on cooling was lower than Tc on heating), purity of Gd starting material, compositions, material preparation methods, and also can be triggered by the external magnetic field with a different dT/dB rate for different x values. For Gd_{5}(Si_{1.95}Ge_{2.05}), Gd_{5}(Si_{2}Ge_{2}), Gd_{5}(Si_{2.09}Ge_{1.91}), it was also found that the transition is a first order magneto-structural transition, which means the magnetic transition and crystalline structure transition occur simultaneously, and completely reversible. Temperature hysteresis and phase coexistence have been found to confirm that it is a first order transformation. While for Gd_{5}(Si_{0.15}Ge_{3.85}), it is partially reversible at some temperature range between the antiferromagnetic and the ferromagnetic state. For Gd_{5}(Si_{2.3}Ge_{1.7}) and Gd_{5}(Si_{3}Ge_{1}), it was a second order transformation between the paramagnetic and ferromagnetic state, because no ΔT have been found. Giant magnetostriction was only found on Gd_{5}(Si_{1.95}Ge_{2.05}), Gd_{5}(Si_{2}Ge_{2}), Gd_{5}(Si_{2.09}Ge_{1.91}) in their vicinity of first order transformation. MFM images have also been taken on polycrystal sample Gd_{5}(Si_{2.09}Ge_{1.91}) to investigate the transformation process. The results also indicates that the Curie temperature was lower and the thermally
NASA Technical Reports Server (NTRS)
Belcastro, Christine M.
1998-01-01
Robust control system analysis and design is based on an uncertainty description, called a linear fractional transformation (LFT), which separates the uncertain (or varying) part of the system from the nominal system. These models are also useful in the design of gain-scheduled control systems based on Linear Parameter Varying (LPV) methods. Low-order LFT models are difficult to form for problems involving nonlinear parameter variations. This paper presents a numerical computational method for constructing and LFT model for a given LPV model. The method is developed for multivariate polynomial problems, and uses simple matrix computations to obtain an exact low-order LFT representation of the given LPV system without the use of model reduction. Although the method is developed for multivariate polynomial problems, multivariate rational problems can also be solved using this method by reformulating the rational problem into a polynomial form.
Intricacies of cosmological bounce in polynomial metric f(R) gravity for flat FLRW spacetime
Bhattacharya, Kaushik; Chakrabarty, Saikat E-mail: snilch@iitk.ac.in
2016-02-01
In this paper we present the techniques for computing cosmological bounces in polynomial f(R) theories, whose order is more than two, for spatially flat FLRW spacetime. In these cases the conformally connected Einstein frame shows up multiple scalar potentials predicting various possibilities of cosmological evolution in the Jordan frame where the f(R) theory lives. We present a reasonable way in which one can associate the various possible potentials in the Einstein frame, for cubic f(R) gravity, to the cosmological development in the Jordan frame. The issue concerning the energy conditions in f(R) theories is presented. We also point out the very important relationships between the conformal transformations connecting the Jordan frame and the Einstein frame and the various instabilities of f(R) theory. All the calculations are done for cubic f(R) gravity but we hope the results are sufficiently general for higher order polynomial gravity.
NASA Astrophysics Data System (ADS)
Notaris, Sotirios
1995-03-01
Given a fixed n≥1, and a (monic) orthogonal polynomial πn(·)Dπn(·;dσ) relative to a positive measuredσ on the interval [a, b], one can define the nonnegative measure , to which correspond the (monic) orthogonal polynomials . The coefficients in the three-term recurrence relation for , whendσ is a Chebyshev measure of any of the four kinds, were obtained analytically in closed form by Gautschi and Li. Here, we give explicit formulae for the Stieltjes polynomials whendσ is any of the four Chebyshev measures. In addition, we show that the corresponding Gauss-Kronrod quadrature formulae for each of these , based on the zeros of and , have all the desirable properties of the interlacing of nodes, their inclusion in [-1, 1], and the positivity of all quadrature weights. Exceptions occur only for the Chebyshev measuredσ of the third or fourth kind andn even, in which case the inclusion property fails. The precise degree of exactness for each of these formulae is also determined.
Dobrev, Veselin A.; Kolev, Tzanio V.; Rieben, Robert N.
2012-09-20
The numerical approximation of the Euler equations of gas dynamics in a movingLagrangian frame is at the heart of many multiphysics simulation algorithms. Here, we present a general framework for high-order Lagrangian discretization of these compressible shock hydrodynamics equations using curvilinear finite elements. This method is an extension of the approach outlined in [Dobrev et al., Internat. J. Numer. Methods Fluids, 65 (2010), pp. 1295--1310] and can be formulated for any finite dimensional approximation of the kinematic and thermodynamic fields, including generic finite elements on two- and three-dimensional meshes with triangular, quadrilateral, tetrahedral, or hexahedral zones. We discretize the kinematic variables of position and velocity using a continuous high-order basis function expansion of arbitrary polynomial degree which is obtained via a corresponding high-order parametric mapping from a standard reference element. This enables the use of curvilinear zone geometry, higher-order approximations for fields within a zone, and a pointwise definition of mass conservation which we refer to as strong mass conservation. Moreover, we discretize the internal energy using a piecewise discontinuous high-order basis function expansion which is also of arbitrary polynomial degree. This facilitates multimaterial hydrodynamics by treating material properties, such as equations of state and constitutive models, as piecewise discontinuous functions which vary within a zone. To satisfy the Rankine--Hugoniot jump conditions at a shock boundary and generate the appropriate entropy, we introduce a general tensor artificial viscosity which takes advantage of the high-order kinematic and thermodynamic information available in each zone. Finally, we apply a generic high-order time discretization process to the semidiscrete equations to develop the fully discrete numerical algorithm. Our method can be viewed as the high-order generalization of the so-called staggered
ElaStic: A tool for calculating second-order elastic constants from first principles
NASA Astrophysics Data System (ADS)
Golesorkhtabar, Rostam; Pavone, Pasquale; Spitaler, Jürgen; Puschnig, Peter; Draxl, Claudia
2013-08-01
Elastic properties play a key role in materials science and technology. The elastic tensors at any order are defined by the Taylor expansion of the elastic energy or stress in terms of the applied strain. In this paper, we present ElaStic, a tool that is able to calculate the full second-order elastic stiffness tensor for any crystal structure from ab initio total-energy and/or stress calculations. This tool also provides the elastic compliances tensor and applies the Voigt and Reuss averaging procedure in order to obtain an evaluation of the bulk, shear, and Young moduli as well as the Poisson ratio of poly-crystalline samples. In a first step, the space-group is determined. Then, a set of deformation matrices is selected, and the corresponding structure files are produced. In a next step, total-energy or stress calculations for each deformed structure are performed by a chosen density-functional theory code. The computed energies/stresses are fitted as polynomial functions of the applied strain in order to get derivatives at zero strain. The knowledge of these derivatives allows for the determination of all independent components of the elastic tensor. In this context, the accuracy of the elastic constants critically depends on the polynomial fit. Therefore, we carefully study how the order of the polynomial fit and the deformation range influence the numerical derivatives, and we propose a new approach to obtain the most reliable results. We have applied ElaStic to representative materials for each crystal system, using total energies and stresses calculated with the full-potential all-electron codes exciting and WIEN2k as well as the pseudo-potential code Quantum ESPRESSO.
NASA Astrophysics Data System (ADS)
Monnin, P.; Bosmans, H.; Verdun, F. R.; Marshall, N. W.
2014-10-01
Given the adverse impact of image noise on the perception of important clinical details in digital mammography, routine quality control measurements should include an evaluation of noise. The European Guidelines, for example, employ a second-order polynomial fit of pixel variance as a function of detector air kerma (DAK) to decompose noise into quantum, electronic and fixed pattern (FP) components and assess the DAK range where quantum noise dominates. This work examines the robustness of the polynomial method against an explicit noise decomposition method. The two methods were applied to variance and noise power spectrum (NPS) data from six digital mammography units. Twenty homogeneously exposed images were acquired with PMMA blocks for target DAKs ranging from 6.25 to 1600 µGy. Both methods were explored for the effects of data weighting and squared fit coefficients during the curve fitting, the influence of the additional filter material (2 mm Al versus 40 mm PMMA) and noise de-trending. Finally, spatial stationarity of noise was assessed. Data weighting improved noise model fitting over large DAK ranges, especially at low detector exposures. The polynomial and explicit decompositions generally agreed for quantum and electronic noise but FP noise fraction was consistently underestimated by the polynomial method. Noise decomposition as a function of position in the image showed limited noise stationarity, especially for FP noise; thus the position of the region of interest (ROI) used for noise decomposition may influence fractional noise composition. The ROI area and position used in the Guidelines offer an acceptable estimation of noise components. While there are limitations to the polynomial model, when used with care and with appropriate data weighting, the method offers a simple and robust means of examining the detector noise components as a function of detector exposure.
NASA Astrophysics Data System (ADS)
Ellis, Richard S.; Machta, Jonathan; Otto, Peter Tak-Hun
2008-10-01
The purpose of this paper is to prove connections among the asymptotic behavior of the magnetization, the structure of the phase transitions, and a class of polynomials that we call the Ginzburg-Landau polynomials. The model under study is a mean-field version of a lattice spin model due to Blume and Capel. It is defined by a probability distribution that depends on the parameters β and K, which represent, respectively, the inverse temperature and the interaction strength. Our main focus is on the asymptotic behavior of the magnetization m( β n , K n ) for appropriate sequences ( β n , K n ) that converge to a second-order point or to the tricritical point of the model and that lie inside various subsets of the phase-coexistence region. The main result states that as ( β n , K n ) converges to one of these points ( β, K), m(βn,Kn)˜ bar{x}|β -βn|^{γ}→ 0 . In this formula γ is a positive constant, and bar{x} is the unique positive, global minimum point of a certain polynomial g. We call g the Ginzburg-Landau polynomial because of its close connection with the Ginzburg-Landau phenomenology of critical phenomena. For each sequence the structure of the set of global minimum points of the associated Ginzburg-Landau polynomial mirrors the structure of the set of global minimum points of the free-energy functional in the region through which ( β n , K n ) passes and thus reflects the phase-transition structure of the model in that region. This paper makes rigorous the predictions of the Ginzburg-Landau phenomenology of critical phenomena and the tricritical scaling theory for the mean-field Blume-Capel model.
A Simple Algorithm for Computing Partial Fraction Expansions with Multiple Poles
ERIC Educational Resources Information Center
Man, Yiu-Kwong
2007-01-01
A simple algorithm for computing the partial fraction expansions of proper rational functions with multiple poles is presented. The main idea is to use the Heaviside's cover-up technique to determine the numerators of the partial fractions and polynomial divisions to reduce the multiplicities of the poles involved successively, without the use of…
A Closed Formula for the Asymptotic Expansion of the Bergman Kernel
NASA Astrophysics Data System (ADS)
Xu, Hao
2012-09-01
We prove a graph theoretic closed formula for coefficients in the Tian-Yau-Zelditch asymptotic expansion of the Bergman kernel. The formula is expressed in terms of the characteristic polynomial of the directed graphs representing Weyl invariants. The proof relies on a combinatorial interpretation of a recursive formula due to M. Engliš and A. Loi.
On the Characteristic Polynomial of a Random Unitary Matrix
NASA Astrophysics Data System (ADS)
Hughes, C. P.; Keating, J. P.; O'Connell, Neil
We present a range of fluctuation and large deviations results for the logarithm of the characteristic polynomial Z of a random N×N unitary matrix, as N-->∞. First we show that , evaluated at a finite set of distinct points, is asymptotically a collection of i.i.d. complex normal random variables. This leads to a refinement of a recent central limit theorem due to Keating and Snaith, and also explains the covariance structure of the eigenvalue counting function. Next we obtain a central limit theorem for ln Z in a Sobolev space of generalised functions on the unit circle. In this limiting regime, lower-order terms which reflect the global covariance structure are no longer negligible and feature in the covariance structure of the limiting Gaussian measure. Large deviations results for ln Z/A, evaluated at a finite set of distinct points, can be obtained for . For higher-order scalings we obtain large deviations results for ln Z/A evaluated at a single point. There is a phase transition at A= ln N (which only applies to negative deviations of the real part) reflecting a switch from global to local conspiracy.
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 P_{B}(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 = e^{K} — 1 of P_{B}(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, P_{B}(q, v) was defined by a contraction-deletion identity, similar to that satisfied by the Tutte polynomial. Here, we give a probabilistic definition of P_{B}(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, 8^{2}), kagome, and (3, 12^{2}) 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 v_{c }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 v_{c}(4, 8^{2}) = 3.742 489 (4), v_{c}(kagome) = 1.876 459 7 (2), and v_{c}(3, 12^{2}) = 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.
Polynomial force approximations and multifrequency atomic force microscopy.
Platz, Daniel; Forchheimer, Daniel; Tholén, Erik A; Haviland, David B
2013-01-01
We present polynomial force reconstruction from experimental intermodulation atomic force microscopy (ImAFM) data. We study the tip-surface force during a slow surface approach and compare the results with amplitude-dependence force spectroscopy (ADFS). Based on polynomial force reconstruction we generate high-resolution surface-property maps of polymer blend samples. The polynomial method is described as a special example of a more general approximative force reconstruction, where the aim is to determine model parameters that best approximate the measured force spectrum. This approximative approach is not limited to spectral data, and we demonstrate how it can be adapted to a force quadrature picture.
Rupert, C.P.; Miller, C.T.
2008-01-01
We examine a variety of polynomial-chaos-motivated approximations to a stochastic form of a steady state groundwater flow model. We consider approaches for truncating the infinite dimensional problem and producing decoupled systems. We discuss conditions under which such decoupling is possible and show that to generalize the known decoupling by numerical cubature, it would be necessary to find new multivariate cubature rules. Finally, we use the acceleration of Monte Carlo to compare the quality of polynomial models obtained for all approaches and find that in general the methods considered are more efficient than Monte Carlo for the relatively small domains considered in this work. A curse of dimensionality in the series expansion of the log-normal stochastic random field used to represent hydraulic conductivity provides a significant impediment to efficient approximations for large domains for all methods considered in this work, other than the Monte Carlo method. PMID:18836519
NASA Astrophysics Data System (ADS)
Le Roy, Robert J.
2017-01-01
This paper describes computer program RKR1, which implements the first-order semiclassical Rydberg-Klein-Rees procedure for determining the potential energy function for a diatomic molecule from a knowledge of the dependence of the molecular vibrational energies Gv and inertial rotation constants Bv on the vibrational quantum number v. RKR1 allows the vibrational energies and rotational constants to be defined in terms of: (i) conventional Dunham polynomial expansions, (ii) near-dissociation expansions (NDE's), or (iii) the mixed Dunham/NDE "MXR" functions introduced by Tellinghuisen [J Chem Phys 2003; 118: 3532]. Internal convergence tests ascertain and report on the precision of the resulting turning points. For cases in which only vibrational data are available, RKR1 also allows an overall potential to be constructed by combining directly-calculated well widths with inner turning points generated from a Morse function. It can also automatically smooth over irregular or unphysical behavior of the steep inner wall of the potential.
SO(N) restricted Schur polynomials
Kemp, Garreth
2015-02-15
We focus on the 1/4-BPS sector of free super Yang-Mills theory with an SO(N) gauge group. This theory has an AdS/CFT (an equivalence between a conformal field theory in d-1 dimensions and type II string theory defined on an AdS space in d-dimensions) dual in the form of type IIB string theory with AdS{sub 5}×RP{sup 5} geometry. With the aim of studying excited giant graviton dynamics, we construct an orthogonal basis for this sector of the gauge theory in this work. First, we demonstrate that the counting of states, as given by the partition function, and the counting of restricted Schur polynomials match by restricting to a particular class of Young diagram labels. We then give an explicit construction of these gauge invariant operators and evaluate their two-point function exactly. This paves the way to studying the spectral problem of these operators and their D-brane duals.
Prediction of zeolite-cement-sand unconfined compressive strength using polynomial neural network
NASA Astrophysics Data System (ADS)
MolaAbasi, H.; Shooshpasha, I.
2016-04-01
The improvement of local soils with cement and zeolite can provide great benefits, including strengthening slopes in slope stability problems, stabilizing problematic soils and preventing soil liquefaction. Recently, dosage methodologies are being developed for improved soils based on a rational criterion as it exists in concrete technology. There are numerous earlier studies showing the possibility of relating Unconfined Compressive Strength (UCS) and Cemented sand (CS) parameters (voids/cement ratio) as a power function fits. Taking into account the fact that the existing equations are incapable of estimating UCS for zeolite cemented sand mixture (ZCS) well, artificial intelligence methods are used for forecasting them. Polynomial-type neural network is applied to estimate the UCS from more simply determined index properties such as zeolite and cement content, porosity as well as curing time. In order to assess the merits of the proposed approach, a total number of 216 unconfined compressive tests have been done. A comparison is carried out between the experimentally measured UCS with the predictions in order to evaluate the performance of the current method. The results demonstrate that generalized polynomial-type neural network has a great ability for prediction of the UCS. At the end sensitivity analysis of the polynomial model is applied to study the influence of input parameters on model output. The sensitivity analysis reveals that cement and zeolite content have significant influence on predicting UCS.
Ridge Polynomial Neural Network with Error Feedback for Time Series Forecasting.
Waheeb, Waddah; Ghazali, Rozaida; Herawan, Tutut
2016-01-01
Time series forecasting has gained much attention due to its many practical applications. Higher-order neural network with recurrent feedback is a powerful technique that has been used successfully for time series forecasting. It maintains fast learning and the ability to learn the dynamics of the time series over time. Network output feedback is the most common recurrent feedback for many recurrent neural network models. However, not much attention has been paid to the use of network error feedback instead of network output feedback. In this study, we propose a novel model, called Ridge Polynomial Neural Network with Error Feedback (RPNN-EF) that incorporates higher order terms, recurrence and error feedback. To evaluate the performance of RPNN-EF, we used four univariate time series with different forecasting horizons, namely star brightness, monthly smoothed sunspot numbers, daily Euro/Dollar exchange rate, and Mackey-Glass time-delay differential equation. We compared the forecasting performance of RPNN-EF with the ordinary Ridge Polynomial Neural Network (RPNN) and the Dynamic Ridge Polynomial Neural Network (DRPNN). Simulation results showed an average 23.34% improvement in Root Mean Square Error (RMSE) with respect to RPNN and an average 10.74% improvement with respect to DRPNN. That means that using network errors during training helps enhance the overall forecasting performance for the network.
Ridge Polynomial Neural Network with Error Feedback for Time Series Forecasting
Ghazali, Rozaida; Herawan, Tutut
2016-01-01
Time series forecasting has gained much attention due to its many practical applications. Higher-order neural network with recurrent feedback is a powerful technique that has been used successfully for time series forecasting. It maintains fast learning and the ability to learn the dynamics of the time series over time. Network output feedback is the most common recurrent feedback for many recurrent neural network models. However, not much attention has been paid to the use of network error feedback instead of network output feedback. In this study, we propose a novel model, called Ridge Polynomial Neural Network with Error Feedback (RPNN-EF) that incorporates higher order terms, recurrence and error feedback. To evaluate the performance of RPNN-EF, we used four univariate time series with different forecasting horizons, namely star brightness, monthly smoothed sunspot numbers, daily Euro/Dollar exchange rate, and Mackey-Glass time-delay differential equation. We compared the forecasting performance of RPNN-EF with the ordinary Ridge Polynomial Neural Network (RPNN) and the Dynamic Ridge Polynomial Neural Network (DRPNN). Simulation results showed an average 23.34% improvement in Root Mean Square Error (RMSE) with respect to RPNN and an average 10.74% improvement with respect to DRPNN. That means that using network errors during training helps enhance the overall forecasting performance for the network. PMID:27959927
ISAR Imaging of Maneuvering Targets Based on the Modified Discrete Polynomial-Phase Transform
Wang, Yong; Abdelkader, Ali Cherif; Zhao, Bin; Wang, Jinxiang
2015-01-01
Inverse synthetic aperture radar (ISAR) imaging of a maneuvering target is a challenging task in the field of radar signal processing. The azimuth echo can be characterized as a multi-component polynomial phase signal (PPS) after the translational compensation, and the high quality ISAR images can be obtained by the parameters estimation of it combined with the Range-Instantaneous-Doppler (RID) technique. In this paper, a novel parameters estimation algorithm of the multi-component PPS with order three (cubic phase signal-CPS) based on the modified discrete polynomial-phase transform (MDPT) is proposed, and the corresponding new ISAR imaging algorithm is presented consequently. This algorithm is efficient and accurate to generate a focused ISAR image, and the results of real data demonstrate the effectiveness of it. PMID:26404299
NASA Astrophysics Data System (ADS)
Prastyaningrum, I.; Cari, C.; Suparmi, A.
2016-11-01
The approximation analytical solution of Dirac equation for Modified Poschl Teller plus Trigonometric Scarf Potential are investigated numerically in terms of finite Romanovsky Polynomial. The combination of two potentials are substituted into Dirac Equation then the variables are separated into radial and angular parts. The Dirac equation is solved by using Romanovsky Polynomial Method. The equation that can reduce from the second order of differential equation into the differential equation of hypergeometry type by substituted variable method. The energy spectrum is numerically solved using Matlab 2011. Where the increase in the radial quantum number nr and variable of modified Poschl Teller Potential causes the energy to decrease. The radial and the angular part of the wave function also visualized with Matlab 2011. The results show, by the disturbance of a combination between this potential can change the wave function of the radial and angular part.
Ding, A Adam; Wu, Hulin
2014-10-01
We propose a new method to use a constrained local polynomial regression to estimate the unknown parameters in ordinary differential equation models with a goal of improving the smoothing-based two-stage pseudo-least squares estimate. The equation constraints are derived from the differential equation model and are incorporated into the local polynomial regression in order to estimate the unknown parameters in the differential equation model. We also derive the asymptotic bias and variance of the proposed estimator. Our simulation studies show that our new estimator is clearly better than the pseudo-least squares estimator in estimation accuracy with a small price of computational cost. An application example on immune cell kinetics and trafficking for influenza infection further illustrates the benefits of the proposed new method.
NASA Astrophysics Data System (ADS)
Boreskov, K. G.; Turbiner, A. V.; López Vieyra, J. C.; García, M. A. G.
It is shown that the E8 trigonometric Olshanetsky-Perelomov Hamiltonian, when written in terms of the fundamental trigonometric invariants, is in algebraic form, i.e. it has polynomial coefficients, and preserves two infinite flags of polynomial spaces marked by the Weyl (co)-vector and E8 highest root (both in the basis of simple roots) as characteristic vectors. The explicit form of the Hamiltonian in new variables has been obtained both by direct calculation and by means of the orbit function technique. It is shown the triangularity of the Hamiltonian in the bases of orbit functions and of algebraic monomials ordered through Weyl heights. Examples of first eigenfunctions are presented.
ISAR Imaging of Maneuvering Targets Based on the Modified Discrete Polynomial-Phase Transform.
Wang, Yong; Abdelkader, Ali Cherif; Zhao, Bin; Wang, Jinxiang
2015-09-03
Inverse synthetic aperture radar (ISAR) imaging of a maneuvering target is a challenging task in the field of radar signal processing. The azimuth echo can be characterized as a multi-component polynomial phase signal (PPS) after the translational compensation, and the high quality ISAR images can be obtained by the parameters estimation of it combined with the Range-Instantaneous-Doppler (RID) technique. In this paper, a novel parameters estimation algorithm of the multi-component PPS with order three (cubic phase signal-CPS) based on the modified discrete polynomial-phase transform (MDPT) is proposed, and the corresponding new ISAR imaging algorithm is presented consequently. This algorithm is efficient and accurate to generate a focused ISAR image, and the results of real data demonstrate the effectiveness of it.
A divide-and-inner product parallel algorithm for polynomial evaluation
Hu, Jie; Li, Lei; Nakamura, Tadao
1994-12-31
In this paper, a divide-and-inner product parallel algorithm for evaluating a polynomial of degree N (N+1=KL) on a MIMD computer is presented. It needs 2K + log{sub 2}L steps to evaluate a polynomial of degree N in parallel on L+1 processors (L{<=}2K-2log{sub 2}K) which is a decrease of log{sub 2}L steps as compared with the L-order Homer`s method, and which is a decrease of (2log{sub 2}L){sup 1/2} steps as compared with the some MIMD algorithms. The new algorithm is simple in structure and easy to be realized.
Mathematical analysis of lead field expansions.
Taylor, J G; Ioannides, A A; Müller-Gärtner, H W
1999-02-01
The solution to the bioelectromagnetic inverse problem is discussed in terms of a generalized lead field expansion, extended to weights depending polynomially on the current strength. The expansion coefficients are obtained from the resulting system of equations which relate the lead field expansion to the data. The framework supports a family of algorithms which include the class of minimum norm solutions and those of weighted minimum norm, including FOCUSS (suitably modified to conform to requirements of rotational invariance). The weighted-minimum-norm family is discussed in some detail, making explicit the dependence (or independence) of the weighting scheme on the modulus of the unknown current density vector. For all but the linear case, and with a single power in the weight, a highly nonlinear system of equations results. These are analyzed and their solution reduced to tractable problems for a finite number of degrees of freedom. In the simplest magnetic field tomography (MFT) case, this is shown to possess expected properties for localized distributed sources. A sensitivity analysis supports this conclusion.
NASA Astrophysics Data System (ADS)
Shamasundar, R.; Mulder, W. A.
2016-10-01
Finite-element discretizations of the acoustic wave equation in the time domain often employ mass lumping to avoid the cost of inverting a large sparse mass matrix. For the second-order formulation of the wave equation, mass lumping on Legendre-Gauss-Lobatto points does not harm the accuracy. Here, we consider a first-order formulation of the wave equation. In that case, the numerical dispersion for odd-degree polynomials exhibits super-convergence with a consistent mass matrix but mass lumping destroys that property. We consider defect correction as a means to restore the accuracy, in which the consistent mass matrix is approximately inverted using the lumped one as preconditioner. For the lowest-degree element on a uniform mesh, fourth-order accuracy in 1D can be obtained with just a single iteration of defect correction. The numerical dispersion curve describes the error in the eigenvalues of the discrete set of equations. However, the error in the eigenvectors also play a role, in two ways. For polynomial degrees above one and when considering a 1-D mesh with constant element size and constant material properties, a number of modes, equal to the maximum polynomial degree, are coupled. One of these is the correct physical mode that should approximate the true eigenfunction of the operator, the other are spurious and should have a small amplitude when the true eigenfunction is projected onto them. We analyze the behaviour of this error as a function of the normalized wavenumber in the form of the leading terms in its series expansion and find that this error exceeds the dispersion error, except for the lowest degree where the eigenvector error is zero. Numerical 1-D tests confirm this behaviour. We briefly analyze the 2-D case, where the lowest-degree polynomial also appears to provide fourth-order accuracy with defect correction, if the grid of squares or triangles is highly regular and material properties are constant.
Generalized Rayleigh and Jacobi Processes and Exceptional Orthogonal Polynomials
NASA Astrophysics Data System (ADS)
Chou, C.-I.; Ho, C.-L.
2013-09-01
We present four types of infinitely many exactly solvable Fokker-Planck equations, which are related to the newly discovered exceptional orthogonal polynomials. They represent the deformed versions of the Rayleigh process and the Jacobi process.
On polynomial integrability of the Euler equations on so(4)
NASA Astrophysics Data System (ADS)
Llibre, Jaume; Yu, Jiang; Zhang, Xiang
2015-10-01
In this paper we prove that the Euler equations on the Lie algebra so(4) with a diagonal quadratic Hamiltonian either satisfy the Manakov condition, or have at most four functionally independent polynomial first integrals.
Fitting discrete aspherical surface sag data using orthonormal polynomials.
Hilbig, David; Ceyhan, Ufuk; Henning, Thomas; Fleischmann, Friedrich; Knipp, Dietmar
2015-08-24
Characterizing real-life optical surfaces usually involves finding the best-fit of an appropriate surface model to a set of discrete measurement data. This process can be greatly simplified by choosing orthonormal polynomials for the surface description. In case of rotationally symmetric aspherical surfaces, new sets of orthogonal polynomials were introduced by Forbes to replace the numerical unstable standard description. From these, for the application of surface retrieval using experimental ray tracing, the sag orthogonal Q(con)-polynomials are of particular interest. However, these are by definition orthogonal over continuous data and may not be orthogonal for discrete data. In this case, the simplified solution is not valid. Hence, a Gram-Schmidt orthonormalization of these polynomials over the discrete data set is proposed to solve this problem. The resulting difference will be presented by a performance analysis and comparison to the direct matrix inversion method.
Quantization of gauge fields, graph polynomials and graph homology
Kreimer, Dirk; Sars, Matthias; Suijlekom, Walter D. van
2013-09-15
We review quantization of gauge fields using algebraic properties of 3-regular graphs. We derive the Feynman integrand at n loops for a non-abelian gauge theory quantized in a covariant gauge from scalar integrands for connected 3-regular graphs, obtained from the two Symanzik polynomials. The transition to the full gauge theory amplitude is obtained by the use of a third, new, graph polynomial, the corolla polynomial. This implies effectively a covariant quantization without ghosts, where all the relevant signs of the ghost sector are incorporated in a double complex furnished by the corolla polynomial–we call it cycle homology–and by graph homology. -- Highlights: •We derive gauge theory Feynman from scalar field theory with 3-valent vertices. •We clarify the role of graph homology and cycle homology. •We use parametric renormalization and the new corolla polynomial.
Complex Chebyshev-polynomial-based unified model (CCPBUM) neural networks
NASA Astrophysics Data System (ADS)
Jeng, Jin-Tsong; Lee, Tsu-Tian
1998-03-01
In this paper, we propose complex Chebyshev Polynomial Based unified model neural network for the approximation of complex- valued function. Based on this approximate transformable technique, we have derived the relationship between the single-layered neural network and multi-layered perceptron neural network. It is shown that the complex Chebyshev Polynomial Based unified model neural network can be represented as a functional link network that are based on Chebyshev polynomial. We also derived a new learning algorithm for the proposed network. It turns out that the complex Chebyshev Polynomial Based unified model neural network not only has the same capability of universal approximator, but also has faster learning speed than conventional complex feedforward/recurrent neural network.
Orthogonal sets of data windows constructed from trigonometric polynomials
NASA Technical Reports Server (NTRS)
Greenhall, C. A.
1989-01-01
Suboptimal, easily computable substitutes for the discrete prolate-spheroidal windows used by Thomson for spectral estimation are given. Trigonometric coefficients and energy leakages of the window polynomials are tabulated.
Cubic Polynomials with Rational Roots and Critical Points
ERIC Educational Resources Information Center
Gupta, Shiv K.; Szymanski, Waclaw
2010-01-01
If you want your students to graph a cubic polynomial, it is best to give them one with rational roots and critical points. In this paper, we describe completely all such cubics and explain how to generate them.
Damon, Bruce M; Heemskerk, Anneriet M; Ding, Zhaohua
2012-06-01
Fiber curvature is a functionally significant muscle structural property, but its estimation from diffusion-tensor magnetic resonance imaging fiber tracking data may be confounded by noise. The purpose of this study was to investigate the use of polynomial fitting of fiber tracts for improving the accuracy and precision of fiber curvature (κ) measurements. Simulated image data sets were created in order to provide data with known values for κ and pennation angle (θ). Simulations were designed to test the effects of increasing inherent fiber curvature (3.8, 7.9, 11.8 and 15.3 m(-1)), signal-to-noise ratio (50, 75, 100 and 150) and voxel geometry (13.8- and 27.0-mm(3) voxel volume with isotropic resolution; 13.5-mm(3) volume with an aspect ratio of 4.0) on κ and θ measurements. In the originally reconstructed tracts, θ was estimated accurately under most curvature and all imaging conditions studied; however, the estimates of κ were imprecise and inaccurate. Fitting the tracts to second-order polynomial functions provided accurate and precise estimates of κ for all conditions except very high curvature (κ=15.3 m(-1)), while preserving the accuracy of the θ estimates. Similarly, polynomial fitting of in vivo fiber tracking data reduced the κ values of fitted tracts from those of unfitted tracts and did not change the θ values. Polynomial fitting of fiber tracts allows accurate estimation of physiologically reasonable values of κ, while preserving the accuracy of θ estimation.
Damon, Bruce M.; Heemskerk, Anneriet M.; Ding, Zhaohua
2012-01-01
Fiber curvature is a functionally significant muscle structural property, but its estimation from diffusion-tensor MRI fiber tracking data may be confounded by noise. The purpose of this study was to investigate the use of polynomial fitting of fiber tracts for improving the accuracy and precision of fiber curvature (κ) measurements. Simulated image datasets were created in order to provide data with known values for κ and pennation angle (θ). Simulations were designed to test the effects of increasing inherent fiber curvature (3.8, 7.9, 11.8, and 15.3 m−1), signal-to-noise ratio (50, 75, 100, and 150), and voxel geometry (13.8 and 27.0 mm3 voxel volume with isotropic resolution; 13.5 mm3 volume with an aspect ratio of 4.0) on κ and θ measurements. In the originally reconstructed tracts, θ was estimated accurately under most curvature and all imaging conditions studied; however, the estimates of κ were imprecise and inaccurate. Fitting the tracts to 2nd order polynomial functions provided accurate and precise estimates of κ for all conditions except very high curvature (κ=15.3 m−1), while preserving the accuracy of the θ estimates. Similarly, polynomial fitting of in vivo fiber tracking data reduced the κ values of fitted tracts from those of unfitted tracts and did not change the θ values. Polynomial fitting of fiber tracts allows accurate estimation of physiologically reasonable values of κ, while preserving the accuracy of θ estimation. PMID:22503094
Performance comparison of polynomial representations for optimizing optical freeform systems
NASA Astrophysics Data System (ADS)
Brömel, A.; Gross, H.; Ochse, D.; Lippmann, U.; Ma, C.; Zhong, Y.; Oleszko, M.
2015-09-01
Optical systems can benefit strongly from freeform surfaces, however the choice of the right representation isn`t an easy one. Classical representations like X-Y-polynomials, as well as Zernike-polynomials are often used for such systems, but should have some disadvantage regarding their orthogonality, resulting in worse convergence and reduced quality in final results compared to newer representations like the Q-polynomials by Forbes. Additionally the supported aperture is a circle, which can be a huge drawback in case of optical systems with rectangular aperture. In this case other representations like Chebyshev-or Legendre-polynomials come into focus. There are a larger number of possibilities; however the experience with these newer representations is rather limited. Therefore in this work the focus is on investigating the performance of four widely used representations in optimizing two ambitious systems with very different properties: Three-Mirror-Anastigmat and an anamorphic System. The chosen surface descriptions offer support for circular or rectangular aperture, as well as different grades of departure from rotational symmetry. The basic shapes are for example a conic or best-fit-sphere and the polynomial set is non-, spatial or slope-orthogonal. These surface representations were chosen to evaluate the impact of these aspects on the performance optimization of the two example systems. Freeform descriptions investigated here were XY-polynomials, Zernike in Fringe representation, Q-polynomials by Forbes, as well as 2-dimensional Chebyshev-polynomials. As a result recommendations for the right choice of freeform surface representations for practical issues in the optimization of optical systems can be given.
Polynomial Interpolation and Sums of Powers of Integers
ERIC Educational Resources Information Center
Cereceda, José Luis
2017-01-01
In this note, we revisit the problem of polynomial interpolation and explicitly construct two polynomials in n of degree k + 1, P[subscript k](n) and Q[subscript k](n), such that P[subscript k](n) = Q[subscript k](n) = f[subscript k](n) for n = 1, 2,… , k, where f[subscript k](1), f[subscript k](2),… , f[subscript k](k) are k arbitrarily chosen…
Polynomials for crystal frameworks and the rigid unit mode spectrum
Power, S. C.
2014-01-01
To each discrete translationally periodic bar-joint framework in , we associate a matrix-valued function defined on the d-torus. The rigid unit mode (RUM) spectrum of is defined in terms of the multi-phases of phase-periodic infinitesimal flexes and is shown to correspond to the singular points of the function and also to the set of wavevectors of harmonic excitations which have vanishing energy in the long wavelength limit. To a crystal framework in Maxwell counting equilibrium, which corresponds to being square, the determinant of gives rise to a unique multi-variable polynomial . For ideal zeolites, the algebraic variety of zeros of on the d-torus coincides with the RUM spectrum. The matrix function is related to other aspects of idealized framework rigidity and flexibility, and in particular leads to an explicit formula for the number of supercell-periodic floppy modes. In the case of certain zeolite frameworks in dimensions two and three, direct proofs are given to show the maximal floppy mode property (order N). In particular, this is the case for the cubic symmetry sodalite framework and some other idealized zeolites. PMID:24379422
On spline and polynomial interpolation of low earth orbiter data: GRACE example
NASA Astrophysics Data System (ADS)
Uz, Metehan; Ustun, Aydin
2016-04-01
GRACE satellites, which are equipped with specific science instruments such as K/Ka band ranging system, have still orbited around the earth since 17 March 2002. In this study the kinematic and reduced-dynamic orbits of GRACE-A/B were determined to 10 seconds interval by using Bernese 5.2 GNSS software during May, 2010 and also daily orbit solutions were validated with GRACE science orbit, GNV1B. The RMS values of kinematic and reduced-dynamic orbit validations were about 2.5 and 1.5 cm, respectively. Throughout the time period of interest, more or less data gaps were encountered in the kinematic orbits due to lack of GPS measurements and satellite manoeuvres. Thus, the least square polynomial and the cubic spline approaches (natural, not-a-knot and clamped) were tested to interpolate both small data gaps and 5 second interval on precise orbits. The latter is necessary for example in case of data densification in order to use the K / Ka band observations. The interpolated coordinates to 5 second intervals were also validated with GNV1B orbits. The validation results show that spline approaches have delivered approximately 1 cm RMS values and are better than those of least square polynomial interpolation. When data gaps occur on daily orbit, the spline validation results became worse depending on the size of the data gaps. Hence, the daily orbits were fragmented into small arcs including 30, 40 or 50 knots to evaluate effect of the least square polynomial interpolation on data gaps. From randomly selected daily arc sets, which are belonging to different times, 5, 10, 15 and 20 knots were removed, independently. While 30-knot arcs were evaluated with fifth-degree polynomial, sixth-degree polynomial was employed to interpolate artificial gaps over 40- and 50-knot arcs. The differences of interpolated and removed coordinates were tested with each other by considering GNV1B validation RMS result, 2.5 cm. With 95% confidence level, data gaps up to 5 and 10 knots can
One-parameter extension of the Doi-Peliti formalism and its relation with orthogonal polynomials.
Ohkubo, Jun
2012-10-01
An extension of the Doi-Peliti formalism for stochastic chemical kinetics is proposed. Using the extension, path-integral expressions consistent with previous studies are obtained. In addition, the extended formalism is naturally connected to orthogonal polynomials. We show that two different orthogonal polynomials, i.e., Charlier polynomials and Hermite polynomials, can be used to express the Doi-Peliti formalism explicitly.
X-ray spectrum estimation from transmission measurements by an exponential of a polynomial model
NASA Astrophysics Data System (ADS)
Perkhounkov, Boris; Stec, Jessika; Sidky, Emil Y.; Pan, Xiaochuan
2016-04-01
There has been much recent research effort directed toward spectral computed tomography (CT). An important step in realizing spectral CT is determining the spectral response of the scanning system so that the relation between material thicknesses and X-ray transmission intensity is known. We propose a few parameter spectrum model that can accurately model the X-ray transmission curves and has a form which is amenable to simultaneous spectral CT image reconstruction and CT system spectrum calibration. While the goal is to eventually realize the simultaneous image reconstruction/spectrum estimation algorithm, in this work we investigate the effectiveness of the model on spectrum estimation from simulated transmission measurements through known thicknesses of known materials. The simulated transmission measurements employ a typical X-ray spectrum used for CT and contain noise due to the randomness in detecting finite numbers of photons. The proposed model writes the X-ray spectrum as the exponential of a polynomial (EP) expansion. The model parameters are obtained by use of a standard software implementation of the Nelder-Mead simplex algorithm. The performance of the model is measured by the relative error between the predicted and simulated transmission curves. The estimated spectrum is also compared with the model X-ray spectrum. For reference, we also employ a polynomial (P) spectrum model and show performance relative to the proposed EP model.
Shao, Yan-Lin Faltinsen, Odd M.
2014-10-01
We propose a new efficient and accurate numerical method based on harmonic polynomials to solve boundary value problems governed by 3D Laplace equation. The computational domain is discretized by overlapping cells. Within each cell, the velocity potential is represented by the linear superposition of a complete set of harmonic polynomials, which are the elementary solutions of Laplace equation. By its definition, the method is named as Harmonic Polynomial Cell (HPC) method. The characteristics of the accuracy and efficiency of the HPC method are demonstrated by studying analytical cases. Comparisons will be made with some other existing boundary element based methods, e.g. Quadratic Boundary Element Method (QBEM) and the Fast Multipole Accelerated QBEM (FMA-QBEM) and a fourth order Finite Difference Method (FDM). To demonstrate the applications of the method, it is applied to some studies relevant for marine hydrodynamics. Sloshing in 3D rectangular tanks, a fully-nonlinear numerical wave tank, fully-nonlinear wave focusing on a semi-circular shoal, and the nonlinear wave diffraction of a bottom-mounted cylinder in regular waves are studied. The comparisons with the experimental results and other numerical results are all in satisfactory agreement, indicating that the present HPC method is a promising method in solving potential-flow problems. The underlying procedure of the HPC method could also be useful in other fields than marine hydrodynamics involved with solving Laplace equation.
Cluster expansions for correlated finite nuclei: The normalization integral
NASA Astrophysics Data System (ADS)
Guardiola, R.; Polls, A.
1980-06-01
Cluster expansions of various kinds are studied with regard to the calculation of the normalization integral in 16O. The factor Aviles-Hartog-Tolhoek cluster expansion is shown to be very satisfactory at third order. A diagrammatic analysis of this cluster expansion is also included and numerically tested.
Application of overlay modeling and control with Zernike polynomials in an HVM environment
NASA Astrophysics Data System (ADS)
Ju, JaeWuk; Kim, MinGyu; Lee, JuHan; Nabeth, Jeremy; Jeon, Sanghuck; Heo, Hoyoung; Robinson, John C.; Pierson, Bill
2016-03-01
Shrinking technology nodes and smaller process margins require improved photolithography overlay control. Generally, overlay measurement results are modeled with Cartesian polynomial functions for both intra-field and inter-field models and the model coefficients are sent to an advanced process control (APC) system operating in an XY Cartesian basis. Dampened overlay corrections, typically via exponentially or linearly weighted moving average in time, are then retrieved from the APC system to apply on the scanner in XY Cartesian form for subsequent lot exposure. The goal of the above method is to process lots with corrections that target the least possible overlay misregistration in steady state as well as in change point situations. In this study, we model overlay errors on product using Zernike polynomials with same fitting capability as the process of reference (POR) to represent the wafer-level terms, and use the standard Cartesian polynomials to represent the field-level terms. APC calculations for wafer-level correction are performed in Zernike basis while field-level calculations use standard XY Cartesian basis. Finally, weighted wafer-level correction terms are converted to XY Cartesian space in order to be applied on the scanner, along with field-level corrections, for future wafer exposures. Since Zernike polynomials have the property of being orthogonal in the unit disk we are able to reduce the amount of collinearity between terms and improve overlay stability. Our real time Zernike modeling and feedback evaluation was performed on a 20-lot dataset in a high volume manufacturing (HVM) environment. The measured on-product results were compared to POR and showed a 7% reduction in overlay variation including a 22% terms variation. This led to an on-product raw overlay Mean + 3Sigma X&Y improvement of 5% and resulted in 0.1% yield improvement.
Quasi-Fibonacci Numbers of the Seventh Order
NASA Astrophysics Data System (ADS)
Witula, Roman; Slota, Damian; Warzynski, Adam
2006-09-01
In this paper we introduce and investigate the so-called quasi-Fibonacci numbers of the seventh order. We discover many surprising relations and identities, and study some applications to polynomials.
Lin, Chih-Hong
2016-09-01
Because the V-belt continuously variable transmission system spurred by permanent magnet (PM) synchronous motor has much unknown nonlinear and time-varying characteristics, the better control performance design for the linear control design is a time consuming procedure. In order to overcome difficulties for design of the linear controllers, the composite recurrent Laguerre orthogonal polynomials modified particle swarm optimization (PSO) neural network (NN) control system which has online learning capability to come back to the nonlinear and time-varying of system, is developed for controlling PM synchronous motor servo-driven V-belt continuously variable transmission system with the lumped nonlinear load disturbances. The composite recurrent Laguerre orthogonal polynomials NN control system consists of an inspector control, a recurrent Laguerre orthogonal polynomials NN control with adaptation law and a recouped control with estimation law. Moreover, the adaptation law of online parameters in the recurrent Laguerre orthogonal polynomials NN is originated from Lyapunov stability theorem. Additionally, two optimal learning rates of the parameters by means of modified PSO are posed in order to achieve better convergence. At last, comparative studies shown by experimental results are illustrated to demonstrate the control performance of the proposed control scheme.
NASA Astrophysics Data System (ADS)
Gelman, L.; Gould, J. D.
2007-11-01
The new technique, the time-frequency chirp-Wigner transform has been proposed recently. This technique is further investigated for the general case of higher order chirps, i.e. non-stationary signals with any nonlinear polynomial variation of the instantaneous frequency in time. Analytical and numerical comparison of the chirp-Wigner transform and the classical Wigner distribution was performed for processing of single-component and multi-component higher order chirps. It is shown for the general case of single component higher order chirps that the chirp-Wigner transform has an essential advantage in comparison with the traditional Wigner distribution: the chirp-Wigner transform ideally follows the nonlinear polynomial frequency variation without amplitude errors. It is shown for multi-component signal where each component is a higher order chirp, that the chirp-Wigner transform adjusted to a single component will follow the instantaneous frequency of the component without amplitude errors. It is also shown that the classical Wigner distribution is unable to estimate component amplitudes of single component and multi-component higher order chirps.
Magnetic Clouds: Global and local expansion
NASA Astrophysics Data System (ADS)
Gulisano, Adriana; Demoulin, Pascal; Soledad Nakwacki, Ms Maria; Dasso, Sergio; Emilia Ruiz, Maria
Magnetic clouds (MCs) are magnetized objects forming flux ropes, which are expelled from the Sun and travel through the heliosphere, transporting important amounts of energy, mass, magnetic flux, and magnetic helicity from the Sun to the interplanetary medium. To know the detailed dynamical evolution of MCs is very useful to improve the knowledge of solar processes, for instance from linking a transient solar source with its interplanetary manifestation. During its travel, and mainly due to the decrease of the total (magnetic plus thermal) pressure in the surrounding solar wind, MCs are objects in expansion. However, the detailed magnetic structure and the dynamical evolution of MCs is still not fully known. Even the identification of their boundaries is an open question in some cases. In a previous work we have shown that from onepoint observations of the bulk velocity profile, it is possible to infer the 'local' expansion rate for a given MC, i.e., the expansion rate while the MC is observed by the spacecraft. By the another hand, and from the comparison of sizes for different MCs observed at different heliodistances, it is possible to quantify an 'average' expansion law (i.e., a global expansion). In this work, in order to study the variability of the 'local' expansion with respect to the 'average' expansion of MCs during their travel, we present results and a comparison between both approaches. We make a detailed study of one-point observations (magnetic and bulk velocity) using a set of MCs and we get the 'local' expansion rate for each studied event. We compare the obtained 'local' expansion rates with the 'average' expansion law, and also with the expansion rates for the stationary solar wind.
Giant negative thermal expansion in magnetic nanocrystals.
Zheng, X G; Kubozono, H; Yamada, H; Kato, K; Ishiwata, Y; Xu, C N
2008-12-01
Most solids expand when they are heated, but a property known as negative thermal expansion has been observed in a number of materials, including the oxide ZrW2O8 (ref. 1) and the framework material ZnxCd1-x(CN)2 (refs 2,3). This unusual behaviour can be understood in terms of low-energy phonons, while the colossal values of both positive and negative thermal expansion recently observed in another framework material, Ag3[Co(CN)6], have been explained in terms of the geometric flexibility of its metal-cyanide-metal linkages. Thermal expansion can also be stopped in some magnetic transition metal alloys below their magnetic ordering temperature, a phenomenon known as the Invar effect, and the possibility of exploiting materials with tuneable positive or negative thermal expansion in industrial applications has led to intense interest in both the Invar effect and negative thermal expansion. Here we report the results of thermal expansion experiments on three magnetic nanocrystals-CuO, MnF2 and NiO-and find evidence for negative thermal expansion in both CuO and MnF2 below their magnetic ordering temperatures, but not in NiO. Larger particles of CuO and MnF2 also show prominent magnetostriction (that is, they change shape in response to an applied magnetic field), which results in significantly reduced thermal expansion below their magnetic ordering temperatures; this behaviour is not observed in NiO. We propose that the negative thermal expansion effect in CuO (which is four times larger than that observed in ZrW2O8) and MnF2 is a general property of nanoparticles in which there is strong coupling between magnetism and the crystal lattice.
Efficient computer algebra algorithms for polynomial matrices in control design
NASA Technical Reports Server (NTRS)
Baras, J. S.; Macenany, D. C.; Munach, R.
1989-01-01
The theory of polynomial matrices plays a key role in the design and analysis of multi-input multi-output control and communications systems using frequency domain methods. Examples include coprime factorizations of transfer functions, cannonical realizations from matrix fraction descriptions, and the transfer function design of feedback compensators. Typically, such problems abstract in a natural way to the need to solve systems of Diophantine equations or systems of linear equations over polynomials. These and other problems involving polynomial matrices can in turn be reduced to polynomial matrix triangularization procedures, a result which is not surprising given the importance of matrix triangularization techniques in numerical linear algebra. Matrices with entries from a field and Gaussian elimination play a fundamental role in understanding the triangularization process. In the case of polynomial matrices, matrices with entries from a ring for which Gaussian elimination is not defined and triangularization is accomplished by what is quite properly called Euclidean elimination. Unfortunately, the numerical stability and sensitivity issues which accompany floating point approaches to Euclidean elimination are not very well understood. New algorithms are presented which circumvent entirely such numerical issues through the use of exact, symbolic methods in computer algebra. The use of such error-free algorithms guarantees that the results are accurate to within the precision of the model data--the best that can be hoped for. Care must be taken in the design of such algorithms due to the phenomenon of intermediate expressions swell.
Isothermal titration calorimetry: general formalism using binding polynomials.
Freire, Ernesto; Schön, Arne; Velazquez-Campoy, Adrian
2009-01-01
The theory of the binding polynomial constitutes a very powerful formalism by which many experimental biological systems involving ligand binding can be analyzed under a unified framework. The analysis of isothermal titration calorimetry (ITC) data for systems possessing more than one binding site has been cumbersome because it required the user to develop a binding model to fit the data. Furthermore, in many instances, different binding models give rise to identical binding isotherms, making it impossible to discriminate binding mechanisms using binding data alone. One of the main advantages of the binding polynomials is that experimental data can be analyzed by employing a general model-free methodology that provides essential information about the system behavior (e.g., whether there exists binding cooperativity, whether the cooperativity is positive or negative, and the magnitude of the cooperative energy). Data analysis utilizing binding polynomials yields a set of binding association constants and enthalpy values that conserve their validity after the correct model has been determined. In fact, once the correct model is validated, the binding polynomial parameters can be immediately translated into the model specific constants. In this chapter, we describe the general binding polynomial formalism and provide specific theoretical and experimental examples of its application to isothermal titration calorimetry.
Multipole expansions and intense fields
NASA Astrophysics Data System (ADS)
Reiss, Howard R.
1984-02-01
In the context of two-body bound-state systems subjected to a plane-wave electromagnetic field, it is shown that high field intensity introduces a distinction between long-wavelength approximation and electric dipole approximation. This distinction is gauge dependent, since it is absent in Coulomb gauge, whereas in "completed" gauges of Göppert-Mayer type the presence of high field intensity makes electric quadrupole and magnetic dipole terms of importance equal to electric dipole at long wavelengths. Another consequence of high field intensity is that multipole expansions lose their utility in view of the equivalent importance of a number of low-order multipole terms and the appearance of large-magnitude terms which defy multipole categorization. This loss of the multipole expansion is gauge independent. Also gauge independent is another related consequence of high field intensity, which is the intimate coupling of center-of-mass and relative coordinate motions in a two-body system.
Euler polynomials and identities for non-commutative operators
NASA Astrophysics Data System (ADS)
De Angelis, Valerio; Vignat, Christophe
2015-12-01
Three kinds of identities involving non-commutating operators and Euler and Bernoulli polynomials are studied. The first identity, as given by Bender and Bettencourt [Phys. Rev. D 54(12), 7710-7723 (1996)], expresses the nested commutator of the Hamiltonian and momentum operators as the commutator of the momentum and the shifted Euler polynomial of the Hamiltonian. The second one, by Pain [J. Phys. A: Math. Theor. 46, 035304 (2013)], links the commutators and anti-commutators of the monomials of the position and momentum operators. The third appears in a work by Figuieira de Morisson and Fring [J. Phys. A: Math. Gen. 39, 9269 (2006)] in the context of non-Hermitian Hamiltonian systems. In each case, we provide several proofs and extensions of these identities that highlight the role of Euler and Bernoulli polynomials.
A Formally Verified Conflict Detection Algorithm for Polynomial Trajectories
NASA Technical Reports Server (NTRS)
Narkawicz, Anthony; Munoz, Cesar
2015-01-01
In air traffic management, conflict detection algorithms are used to determine whether or not aircraft are predicted to lose horizontal and vertical separation minima within a time interval assuming a trajectory model. In the case of linear trajectories, conflict detection algorithms have been proposed that are both sound, i.e., they detect all conflicts, and complete, i.e., they do not present false alarms. In general, for arbitrary nonlinear trajectory models, it is possible to define detection algorithms that are either sound or complete, but not both. This paper considers the case of nonlinear aircraft trajectory models based on polynomial functions. In particular, it proposes a conflict detection algorithm that precisely determines whether, given a lookahead time, two aircraft flying polynomial trajectories are in conflict. That is, it has been formally verified that, assuming that the aircraft trajectories are modeled as polynomial functions, the proposed algorithm is both sound and complete.
Asymptotic formulae for the zeros of orthogonal polynomials
NASA Astrophysics Data System (ADS)
Badkov, V. M.
2012-09-01
Let p_n(t) be an algebraic polynomial that is orthonormal with weight p(t) on the interval \\lbrack -1, 1 \\rbrack . When p(t) is a perturbation (in certain limits) of the Chebyshev weight of the first kind, the zeros of the polynomial p_n(\\cos\\tau) and the differences between pairs of (not necessarily consecutive) zeros are shown to satisfy asymptotic formulae as n\\to\\infty, which hold uniformly with respect to the indices of the zeros. Similar results are also obtained for perturbations of the Chebyshev weight of the second kind. First, some preliminary results on the asymptotic behaviour of the difference between two zeros of an orthogonal trigonometric polynomial, which are needed, are established. Bibliography: 15 titles.
Model Fractional Quantum Hall States and Jack Polynomials
Bernevig, B. Andrei; Haldane, F. D. M.
2008-06-20
We describe an occupation-number-like picture of fractional quantum Hall states in terms of polynomial wave functions characterized by a dominant occupation-number configuration. The bosonic variants of single-component Abelian and non-Abelian fractional quantum Hall states are modeled by Jack symmetric polynomials (Jacks), characterized by dominant occupation-number configurations satisfying a generalized Pauli principle. In a series of well-known quantum Hall states, including the Laughlin, Read-Moore, and Read-Rezayi, the Jack polynomials naturally implement a ''squeezing rule'' that constrains allowed configurations to be restricted to those obtained by squeezing the dominant configuration. The Jacks presented in this Letter describe new trial uniform states, but it is yet to be determined to which actual experimental fractional quantum Hall effect states they apply.
Error estimates of Lagrange interpolation and orthonormal expansions for Freud weights
NASA Astrophysics Data System (ADS)
Kwon, K. H.; Lee, D. W.
2001-08-01
Let Sn[f] be the nth partial sum of the orthonormal polynomials expansion with respect to a Freud weight. Then we obtain sufficient conditions for the boundedness of Sn[f] and discuss the speed of the convergence of Sn[f] in weighted Lp space. We also find sufficient conditions for the boundedness of the Lagrange interpolation polynomial Ln[f], whose nodal points are the zeros of orthonormal polynomials with respect to a Freud weight. In particular, if W(x)=e-(1/2)x2 is the Hermite weight function, then we obtain sufficient conditions for the inequalities to hold:andwhere and k=0,1,2...,r.
Polynomial approximation of functions in Sobolev spaces
Dupont, T.; Scott, R.
1980-04-01
Constructive proofs and several generalizations of approximation results of J. H. Bramble and S. R. Hilbert are presented. Using an averaged Taylor series, we represent a function as a polynomical plus a remainder. The remainder can be manipulated in many ways to give different types of bounds. Approximation of functions in fractional order Sobolev spaces is treated as well as the usual integer order spaces and several nonstandard Sobolev-like spaces.
A comparison of companion matrix methods to find roots of a trigonometric polynomial
NASA Astrophysics Data System (ADS)
Boyd, John P.
2013-08-01
A trigonometric polynomial is a truncated Fourier series of the form fN(t)≡∑j=0Naj cos(jt)+∑j=1N bj sin(jt). It has been previously shown by the author that zeros of such a polynomial can be computed as the eigenvalues of a companion matrix with elements which are complex valued combinations of the Fourier coefficients, the "CCM" method. However, previous work provided no examples, so one goal of this new work is to experimentally test the CCM method. A second goal is introduce a new alternative, the elimination/Chebyshev algorithm, and experimentally compare it with the CCM scheme. The elimination/Chebyshev matrix (ECM) algorithm yields a companion matrix with real-valued elements, albeit at the price of usefulness only for real roots. The new elimination scheme first converts the trigonometric rootfinding problem to a pair of polynomial equations in the variables (c,s) where c≡cos(t) and s≡sin(t). The elimination method next reduces the system to a single univariate polynomial P(c). We show that this same polynomial is the resultant of the system and is also a generator of the Groebner basis with lexicographic ordering for the system. Both methods give very high numerical accuracy for real-valued roots, typically at least 11 decimal places in Matlab/IEEE 754 16 digit floating point arithmetic. The CCM algorithm is typically one or two decimal places more accurate, though these differences disappear if the roots are "Newton-polished" by a single Newton's iteration. The complex-valued matrix is accurate for complex-valued roots, too, though accuracy decreases with the magnitude of the imaginary part of the root. The cost of both methods scales as O(N3) floating point operations. In spite of intimate connections of the elimination/Chebyshev scheme to two well-established technologies for solving systems of equations, resultants and Groebner bases, and the advantages of using only real-valued arithmetic to obtain a companion matrix with real-valued elements
Orthogonal polynomials, Laguerre Fock space, and quasi-classical asymptotics
NASA Astrophysics Data System (ADS)
Engliš, Miroslav; Ali, S. Twareque
2015-07-01
Continuing our earlier investigation of the Hermite case [S. T. Ali and M. Engliš, J. Math. Phys. 55, 042102 (2014)], we study an unorthodox variant of the Berezin-Toeplitz quantization scheme associated with Laguerre polynomials. In particular, we describe a "Laguerre analogue" of the classical Fock (Segal-Bargmann) space and the relevant semi-classical asymptotics of its Toeplitz operators; the former actually turns out to coincide with the Hilbert space appearing in the construction of the well-known Barut-Girardello coherent states. Further extension to the case of Legendre polynomials is likewise discussed.
A novel computational approach to approximate fuzzy interpolation polynomials.
Jafarian, Ahmad; Jafari, Raheleh; Mohamed Al Qurashi, Maysaa; Baleanu, Dumitru
2016-01-01
This paper build a structure of fuzzy neural network, which is well sufficient to gain a fuzzy interpolation polynomial of the form [Formula: see text] where [Formula: see text] is crisp number (for [Formula: see text], which interpolates the fuzzy data [Formula: see text]. Thus, a gradient descent algorithm is constructed to train the neural network in such a way that the unknown coefficients of fuzzy polynomial are estimated by the neural network. The numeral experimentations portray that the present interpolation methodology is reliable and efficient.
On solutions of polynomial growth of ordinary differential equations
NASA Astrophysics Data System (ADS)
van den Berg, I. P.
We present a theorem on the existence of solutions of polynomial growth of ordinary differential equations of type E: {dY}/{dX} = F(X, Y) , where F is of class C1. We show that the asymptotic behaviour of these solutions and the variation of neighbouring solutions are obtained by solving an asymptotic functional equation related to E, and that this method has practical value. The theorem is standard; its nonstandard proof uses macroscope and microscope techniques. The result is an extension of results by F. and M. Diener and G. Reeb on solutions of polynomial growth of rational differential equations.
Polynomial approximation of Poincare maps for Hamiltonian system
NASA Technical Reports Server (NTRS)
Froeschle, Claude; Petit, Jean-Marc
1992-01-01
Different methods are proposed and tested for transforming a non-linear differential system, and more particularly a Hamiltonian one, into a map without integrating the whole orbit as in the well-known Poincare return map technique. We construct piecewise polynomial maps by coarse-graining the phase-space surface of section into parallelograms and using either only values of the Poincare maps at the vertices or also the gradient information at the nearest neighbors to define a polynomial approximation within each cell. The numerical experiments are in good agreement with both the real symplectic and Poincare maps.
Riemann hypothesis for period polynomials of modular forms
Jin, Seokho; Ma, Wenjun; Ono, Ken; Soundararajan, Kannan
2016-01-01
The period polynomial rf(z) for an even weight k≥4 newform f∈Sk(Γ0(N)) is the generating function for the critical values of L(f,s). It has a functional equation relating rf(z) to rf(−1Nz). We prove the Riemann hypothesis for these polynomials: that the zeros of rf(z) lie on the circle |z|=1/N. We prove that these zeros are equidistributed when either k or N is large. PMID:26903628
Optimal Electric Utility Expansion
1989-10-10
SAGE-WASP is designed to find the optimal generation expansion policy for an electrical utility system. New units can be automatically selected from a user-supplied list of expansion candidates which can include hydroelectric and pumped storage projects. The existing system is modeled. The calculational procedure takes into account user restrictions to limit generation configurations to an area of economic interest. The optimization program reports whether the restrictions acted as a constraint on the solution. All expansion configurations considered are required to pass a user supplied reliability criterion. The discount rate and escalation rate are treated separately for each expansion candidate and for each fuel type. All expenditures are separated into local and foreign accounts, and a weighting factor can be applied to foreign expenditures.
Weakly relativistic plasma expansion
Fermous, Rachid Djebli, Mourad
2015-04-15
Plasma expansion is an important physical process that takes place in laser interactions with solid targets. Within a self-similar model for the hydrodynamical multi-fluid equations, we investigated the expansion of both dense and under-dense plasmas. The weakly relativistic electrons are produced by ultra-intense laser pulses, while ions are supposed to be in a non-relativistic regime. Numerical investigations have shown that relativistic effects are important for under-dense plasma and are characterized by a finite ion front velocity. Dense plasma expansion is found to be governed mainly by quantum contributions in the fluid equations that originate from the degenerate pressure in addition to the nonlinear contributions from exchange and correlation potentials. The quantum degeneracy parameter profile provides clues to set the limit between under-dense and dense relativistic plasma expansions at a given density and temperature.
Nelson, E.A.; Christensen, E.J.; Mackey, H.E.; Sharitz, R.R.; Jensen, J.R.; Hodgson, M.E.
1984-02-01
Since 1954, cooling water discharges from K Reactor ({anti X} = 370 cfs {at} 59 C) to Pen Branch have altered vegetation and deposited sediment in the Savannah River Swamp forming the Pen Branch delta. Currently, the delta covers over 300 acres and continues to expand at a rate of about 16 acres/yr. Examination of delta expansion can provide important information on environmental impacts to wetlands exposed to elevated temperature and flow conditions. To assess the current status and predict future expansion of the Pen Branch delta, historic aerial photographs were analyzed using both basic photo interpretation and computer techniques to provide the following information: (1) past and current expansion rates; (2) location and changes of impacted areas; (3) total acreage presently affected. Delta acreage changes were then compared to historic reactor discharge temperature and flow data to see if expansion rate variations could be related to reactor operations.
ERIC Educational Resources Information Center
Fakhruddin, Hasan
1993-01-01
Describes a paradox in the equation for thermal expansion. If the calculations for heating a rod and subsequently cooling a rod are determined, the new length of the cool rod is shorter than expected. (PR)
Karagiannis, Georgios; Lin, Guang
2014-02-15
Generalized polynomial chaos (gPC) expansions allow the representation of the solution of a stochastic system as a series of polynomial terms. The number of gPC terms increases dramatically with the dimension of the random input variables. When the number of the gPC terms is larger than that of the available samples, a scenario that often occurs if the evaluations of the system are expensive, the evaluation of the gPC expansion can be inaccurate due to over-fitting. We propose a fully Bayesian approach that allows for global recovery of the stochastic solution, both in spacial and random domains, by coupling Bayesian model uncertainty and regularization regression methods. It allows the evaluation of the PC coefficients on a grid of spacial points via (1) Bayesian model average or (2) medial probability model, and their construction as functions on the spacial domain via spline interpolation. The former accounts the model uncertainty and provides Bayes-optimal predictions; while the latter, additionally, provides a sparse representation of the solution by evaluating the expansion on a subset of dominating gPC bases when represented as a gPC expansion. Moreover, the method quantifies the importance of the gPC bases through inclusion probabilities. We design an MCMC sampler that evaluates all the unknown quantities without the need of ad-hoc techniques. The proposed method is suitable for, but not restricted to, problems whose stochastic solution is sparse at the stochastic level with respect to the gPC bases while the deterministic solver involved is expensive. We demonstrate the good performance of the proposed method and make comparisons with others on 1D, 14D and 40D in random space elliptic stochastic partial differential equations.
Rybkin, Vladimir V; Ekström, Ulf; Helgaker, Trygve
2013-08-05
In geometry optimizations and molecular dynamics calculations, it is often necessary to transform a geometry step that has been determined in internal coordinates to Cartesian coordinates. A new method for performing such transformations, the high-order path-expansion (HOPE) method, is here presented. The new method treats the nonlinear relation between internal and Cartesian coordinates by means of automatic differentiation. The method is reliable, applicable to any system of internal coordinates, and computationally more efficient than the traditional method of iterative back transformations. As a bonus, the HOPE method determines not just the Cartesian step vector but also a continuous step path expressed in the form of a polynomial, which is useful for determining reaction coordinates, for integrating trajectories, and for visualization.
Celeste, Ricardo; Maringolo, Milena P; Comar, Moacyr; Viana, Rommel B; Guimarães, Amanda R; Haiduke, Roberto L A; da Silva, Albérico B F
2015-10-01
Accurate Gaussian basis sets for atoms from H to Ba were obtained by means of the generator coordinate Hartree-Fock (GCHF) method based on a polynomial expansion to discretize the Griffin-Wheeler-Hartree-Fock equations (GWHF). The discretization of the GWHF equations in this procedure is based on a mesh of points not equally distributed in contrast with the original GCHF method. The results of atomic Hartree-Fock energies demonstrate the capability of these polynomial expansions in designing compact and accurate basis sets to be used in molecular calculations and the maximum error found when compared to numerical values is only 0.788 mHartree for indium. Some test calculations with the B3LYP exchange-correlation functional for N2, F2, CO, NO, HF, and HCN show that total energies within 1.0 to 2.4 mHartree compared to the cc-pV5Z basis sets are attained with our contracted bases with a much smaller number of polarization functions (2p1d and 2d1f for hydrogen and heavier atoms, respectively). Other molecular calculations performed here are also in very good accordance with experimental and cc-pV5Z results. The most important point to be mentioned here is that our generator coordinate basis sets required only a tiny fraction of the computational time when compared to B3LYP/cc-pV5Z calculations.
An extended UTD analysis for the scattering and diffraction from cubic polynomial strips
NASA Technical Reports Server (NTRS)
Constantinides, E. D.; Marhefka, R. J.
1993-01-01
Spline and polynomial type surfaces are commonly used in high frequency modeling of complex structures such as aircraft, ships, reflectors, etc. It is therefore of interest to develop an efficient and accurate solution to describe the scattered fields from such surfaces. An extended Uniform Geometrical Theory of Diffraction (UTD) solution for the scattering and diffraction from perfectly conducting cubic polynomial strips is derived and involves the incomplete Airy integrals as canonical functions. This new solution is universal in nature and can be used to effectively describe the scattered fields from flat, strictly concave or convex, and concave convex boundaries containing edges. The classic UTD solution fails to describe the more complicated field behavior associated with higher order phase catastrophes and therefore a new set of uniform reflection and first-order edge diffraction coefficients is derived. Also, an additional diffraction coefficient associated with a zero-curvature (inflection) point is presented. Higher order effects such as double edge diffraction, creeping waves, and whispering gallery modes are not examined. The extended UTD solution is independent of the scatterer size and also provides useful physical insight into the various scattering and diffraction processes. Its accuracy is confirmed via comparison with some reference moment method results.
Visualizing higher order finite elements. Final report
Thompson, David C; Pebay, Philippe Pierre
2005-11-01
This report contains an algorithm for decomposing higher-order finite elements into regions appropriate for isosurfacing and proves the conditions under which the algorithm will terminate. Finite elements are used to create piecewise polynomial approximants to the solution of partial differential equations for which no analytical solution exists. These polynomials represent fields such as pressure, stress, and momentum. In the past, these polynomials have been linear in each parametric coordinate. Each polynomial coefficient must be uniquely determined by a simulation, and these coefficients are called degrees of freedom. When there are not enough degrees of freedom, simulations will typically fail to produce a valid approximation to the solution. Recent work has shown that increasing the number of degrees of freedom by increasing the order of the polynomial approximation (instead of increasing the number of finite elements, each of which has its own set of coefficients) can allow some types of simulations to produce a valid approximation with many fewer degrees of freedom than increasing the number of finite elements alone. However, once the simulation has determined the values of all the coefficients in a higher-order approximant, tools do not exist for visual inspection of the solution. This report focuses on a technique for the visual inspection of higher-order finite element simulation results based on decomposing each finite element into simplicial regions where existing visualization algorithms such as isosurfacing will work. The requirements of the isosurfacing algorithm are enumerated and related to the places where the partial derivatives of the polynomial become zero. The original isosurfacing algorithm is then applied to each of these regions in turn.
Tsallis p, q-deformed Touchard polynomials and Stirling numbers
NASA Astrophysics Data System (ADS)
Herscovici, O.; Mansour, T.
2017-01-01
In this paper, we develop and investigate a new two-parametrized deformation of the Touchard polynomials, based on the definition of the NEXT q-exponential function of Tsallis. We obtain new generalizations of the Stirling numbers of the second kind and of the binomial coefficients and represent two new statistics for the set partitions.
Two-variable Hermite Polynomial State and Its Wigner Function
NASA Astrophysics Data System (ADS)
Meng, Xiang-Guo; Wang, Ji-Suo; Liang, Bao-Long
2009-08-01
In this paper we obtain the Wigner functions of two-variable Hermite polynomial states (THPS) and their marginal distribution using the entangled state |ξ> representation. Also we obtain tomogram of THPS by virtue of the Radon transformation between the Wigner operator and the projection operator of another entangled state |η,τ 1,τ 2>.
Cubic Polynomials with Real or Complex Coefficients: The Full Picture
ERIC Educational Resources Information Center
Bardell, Nicholas S.
2016-01-01
The cubic polynomial with real coefficients has a rich and interesting history primarily associated with the endeavours of great mathematicians like del Ferro, Tartaglia, Cardano or Vieta who sought a solution for the roots (Katz, 1998; see Chapter 12.3: The Solution of the Cubic Equation). Suffice it to say that since the times of renaissance…
Least-Squares Adaptive Control Using Chebyshev Orthogonal Polynomials
NASA Technical Reports Server (NTRS)
Nguyen, Nhan T.; Burken, John; Ishihara, Abraham
2011-01-01
This paper presents a new adaptive control approach using Chebyshev orthogonal polynomials as basis functions in a least-squares functional approximation. The use of orthogonal basis functions improves the function approximation significantly and enables better convergence of parameter estimates. Flight control simulations demonstrate the effectiveness of the proposed adaptive control approach.
Segmented Polynomial Models in Quasi-Experimental Research.
ERIC Educational Resources Information Center
Wasik, John L.
1981-01-01
The use of segmented polynomial models is explained. Examples of design matrices of dummy variables are given for the least squares analyses of time series and discontinuity quasi-experimental research designs. Linear combinations of dummy variable vectors appear to provide tests of effects in the two quasi-experimental designs. (Author/BW)
Central suboptimal H ∞ control design for nonlinear polynomial systems
NASA Astrophysics Data System (ADS)
Basin, Michael V.; Shi, Peng; Calderon-Alvarez, Dario
2011-05-01
This article presents the central finite-dimensional H ∞ regulator for nonlinear polynomial systems, which is suboptimal for a given threshold γ with respect to a modified Bolza-Meyer quadratic criterion including the attenuation control term with the opposite sign. In contrast to the previously obtained results, the article reduces the original H ∞ control problem to the corresponding optimal H 2 control problem, using this technique proposed in Doyle et al. [Doyle, J.C., Glover, K., Khargonekar, P.P., and Francis, B.A. (1989), 'State-space Solutions to Standard H 2 and H ∞ Control Problems', IEEE Transactions on Automatic Control, 34, 831-847]. This article yields the central suboptimal H ∞ regulator for nonlinear polynomial systems in a closed finite-dimensional form, based on the optimal H 2 regulator obtained in Basin and Calderon-Alvarez [Basin, M.V., and Calderon-Alvarez, D. (2008b), 'Optimal Controller for Uncertain Stochastic Polynomial Systems', Journal of the Franklin Institute, 345, 293-302]. Numerical simulations are conducted to verify performance of the designed central suboptimal regulator for nonlinear polynomial systems against the central suboptimal H ∞ regulator available for the corresponding linearised system.
New Bernstein type inequalities for polynomials on ellipses
NASA Technical Reports Server (NTRS)
Freund, Roland; Fischer, Bernd
1990-01-01
New and sharp estimates are derived for the growth in the complex plane of polynomials known to have a curved majorant on a given ellipse. These so-called Bernstein type inequalities are closely connected with certain constrained Chebyshev approximation problems on ellipses. Also presented are some new results for approximation problems of this type.
Computational Technique for Teaching Mathematics (CTTM): Visualizing the Polynomial's Resultant
ERIC Educational Resources Information Center
Alves, Francisco Regis Vieira
2015-01-01
We find several applications of the Dynamic System Geogebra--DSG related predominantly to the basic mathematical concepts at the context of the learning and teaching in Brasil. However, all these works were developed in the basic level of Mathematics. On the other hand, we discuss and explore, with DSG's help, some applications of the polynomial's…
Chemical Equilibrium and Polynomial Equations: Beware of Roots.
ERIC Educational Resources Information Center
Smith, William R.; Missen, Ronald W.
1989-01-01
Describes two easily applied mathematical theorems, Budan's rule and Rolle's theorem, that in addition to Descartes's rule of signs and intermediate-value theorem, are useful in chemical equilibrium. Provides examples that illustrate the use of all four theorems. Discusses limitations of the polynomial equation representation of chemical…
XXZ-type Bethe ansatz equations and quasi-polynomials
NASA Astrophysics Data System (ADS)
Li, Jian Rong; Tarasov, Vitaly
2013-01-01
We study solutions of the Bethe ansatz equation for the XXZ-type integrable model associated with the Lie algebra fraktur sfraktur lN. We give a correspondence between solutions of the Bethe ansatz equations and collections of quasi-polynomials. This extends the results of E. Mukhin and A. Varchenko for the XXX-type model and the trigonometric Gaudin model.
Effects of Polynomial Trends on Detrending Moving Average Analysis
NASA Astrophysics Data System (ADS)
Shao, Ying-Hui; Gu, Gao-Feng; Jiang, Zhi-Qiang; Zhou, Wei-Xing
2015-07-01
The detrending moving average (DMA) algorithm is one of the best performing methods to quantify the long-term correlations in nonstationary time series. As many long-term correlated time series in real systems contain various trends, we investigate the effects of polynomial trends on the scaling behaviors and the performances of three widely used DMA methods including backward algorithm (BDMA), centered algorithm (CDMA) and forward algorithm (FDMA). We derive a general framework for polynomial trends and obtain analytical results for constant shifts and linear trends. We find that the behavior of the CDMA method is not influenced by constant shifts. In contrast, linear trends cause a crossover in the CDMA fluctuation functions. We also find that constant shifts and linear trends cause crossovers in the fluctuation functions obtained from the BDMA and FDMA methods. When a crossover exists, the scaling behavior at small scales comes from the intrinsic time series while that at large scales is dominated by the constant shifts or linear trends. We also derive analytically the expressions of crossover scales and show that the crossover scale depends on the strength of the polynomial trends, the Hurst index, and in some cases (linear trends for BDMA and FDMA) the length of the time series. In all cases, the BDMA and the FDMA behave almost the same under the influence of constant shifts or linear trends. Extensive numerical experiments confirm excellently the analytical derivations. We conclude that the CDMA method outperforms the BDMA and FDMA methods in the presence of polynomial trends.
Computing Tutte polynomials of contact networks in classrooms
NASA Astrophysics Data System (ADS)
Hincapié, Doracelly; Ospina, Juan
2013-05-01
Objective: The topological complexity of contact networks in classrooms and the potential transmission of an infectious disease were analyzed by sex and age. Methods: The Tutte polynomials, some topological properties and the number of spanning trees were used to algebraically compute the topological complexity. Computations were made with the Maple package GraphTheory. Published data of mutually reported social contacts within a classroom taken from primary school, consisting of children in the age ranges of 4-5, 7-8 and 10-11, were used. Results: The algebraic complexity of the Tutte polynomial and the probability of disease transmission increases with age. The contact networks are not bipartite graphs, gender segregation was observed especially in younger children. Conclusion: Tutte polynomials are tools to understand the topology of the contact networks and to derive numerical indexes of such topologies. It is possible to establish relationships between the Tutte polynomial of a given contact network and the potential transmission of an infectious disease within such network
Low temperature thermal expansion of soda-borate glasses
NASA Astrophysics Data System (ADS)
Piñango, Ester S.; Vieira, S.; Villar, R.
1983-10-01
The thermal expansion of glassy (B 2O 3) 1- x(Na 2O) x, for x = 0.06, 0.16 and 0.25 has been measured in the temperature range 4 K < T < 20 K. The results are analysed in terms of a polynomial α = aT + bT3 + cT5 + dT7 and the values of the coefficients are discussed. The linear term a is small and positive in the three glasses. This yields a small and positive Grüneisen parameter for the two level systems. The cubic term is negative and is not affected by change in coordination, phonon dispersion being responsible for the fast increase in thermal expansion on increasing the temperature.
M-Interval Orthogonal Polynomial Estimators with Applications
NASA Astrophysics Data System (ADS)
Jaroszewicz, Boguslaw Emanuel
In this dissertation, adaptive estimators of various statistical nonlinearities are constructed and evaluated. The estimators are based on classical orthogonal polynomials which allows an exact computation of convergence rates. The first part of the dissertation is devoted to the estimation of one- and multi-dimensional probability density functions. The most attractive computationally is the Legendre estimator, which corresponds to the mean square segmented polynomial approximation of a pdf. Exact bounds for two components of the estimation error--deterministic bias and random error--are derived for all the polynomial estimators. The bounds on the bias are functions of the "smoothness" of the estimated pdf as measured by the number of continuous derivatives the pdf possesses. Adaptively estimated the optimum number of orthonormal polynomials minimizes the total error. In the second part, the theory of polynomial estimators is applied to the estimation of derivatives of pdf and regression functions. The optimum detectors for small signals in nongaussian noise, as well as any kind of statistical filtering involving likelihood function, are based on the nonlinearity which is a ratio of the derivative of the pdf and the pdf itself. Several different polynomial estimators of this nonlinearity are developed and compared. The theory of estimation is then extended to the multivariable case. The partial derivative nonlinearity is used for detection of signals in dependent noise. When the dimensionality of the nonlinearity is very large, the transformed Hermite estimators are particularly useful. The estimators can be viewed as two-stage filters: the first stage is a pre -whitening filter optimum in gaussian noise and the second stage is a nonlinear filter, which improves performance in nongaussian noise. Filtering of this type can be applied to predictive coding, nonlinear identification and other estimation problems involving a conditional expected value. In the third
Henn, Brenna M; Cavalli-Sforza, L L; Feldman, Marcus W
2012-10-30
Genetic and paleoanthropological evidence is in accord that today's human population is the result of a great demic (demographic and geographic) expansion that began approximately 45,000 to 60,000 y ago in Africa and rapidly resulted in human occupation of almost all of the Earth's habitable regions. Genomic data from contemporary humans suggest that this expansion was accompanied by a continuous loss of genetic diversity, a result of what is called the "serial founder effect." In addition to genomic data, the serial founder effect model is now supported by the genetics of human parasites, morphology, and linguistics. This particular population history gave rise to the two defining features of genetic variation in humans: genomes from the substructured populations of Africa retain an exceptional number of unique variants, and there is a dramatic reduction in genetic diversity within populations living outside of Africa. These two patterns are relevant for medical genetic studies mapping genotypes to phenotypes and for inferring the power of natural selection in human history. It should be appreciated that the initial expansion and subsequent serial founder effect were determined by demographic and sociocultural factors associated with hunter-gatherer populations. How do we reconcile this major demic expansion with the population stability that followed for thousands years until the inventions of agriculture? We review advances in understanding the genetic diversity within Africa and the great human expansion out of Africa and offer hypotheses that can help to establish a more synthetic view of modern human evolution.
Factorization of differential expansion for antiparallel double-braid knots
NASA Astrophysics Data System (ADS)
Morozov, A.
2016-09-01
Continuing the quest for exclusive Racah matrices, which are needed for evaluation of colored arborescent-knot polynomials in Chern-Simons theory, we suggest to extract them from a new kind of a double-evolution — that of the antiparallel double-braids, which is a simple two-parametric family of two-bridge knots, generalizing the one-parametric family of twist knots. In the case of rectangular representations R = [ r s ] we found an evidence that the corresponding differential expansion miraculously factorizes and can be obtained from that for the twist knots. This reduces the problem of rectangular exclusive Racah to constructing the answers for just a few twist knots. We develop a recent conjecture on the structure of differential expansion for the simplest members of this family (the trefoil and the figure-eight knot) and provide the exhaustive answer for the first unknown case of R = [33]. The answer includes HOMFLY of arbitrary twist and double-braid knots and Racah matrices overline{S} and S — what allows to calculate [33]-colored polynomials for arbitrary arborescent (double-fat) knots. For generic rectangular representations fully described are only the contributions of the single-floor pyramids. One step still remains to be done.
Design and Use of a Learning Object for Finding Complex Polynomial Roots
ERIC Educational Resources Information Center
Benitez, Julio; Gimenez, Marcos H.; Hueso, Jose L.; Martinez, Eulalia; Riera, Jaime
2013-01-01
Complex numbers are essential in many fields of engineering, but students often fail to have a natural insight of them. We present a learning object for the study of complex polynomials that graphically shows that any complex polynomials has a root and, furthermore, is useful to find the approximate roots of a complex polynomial. Moreover, we…
A note on the zeros of Freud-Sobolev orthogonal polynomials
NASA Astrophysics Data System (ADS)
Moreno-Balcazar, Juan J.
2007-10-01
We prove that the zeros of a certain family of Sobolev orthogonal polynomials involving the Freud weight function e-x4 on are real, simple, and interlace with the zeros of the Freud polynomials, i.e., those polynomials orthogonal with respect to the weight function e-x4. Some numerical examples are shown.
From Chebyshev to Bernstein: A Tour of Polynomials Small and Large
ERIC Educational Resources Information Center
Boelkins, Matthew; Miller, Jennifer; Vugteveen, Benjamin
2006-01-01
Consider the family of monic polynomials of degree n having zeros at -1 and +1 and all their other real zeros in between these two values. This article explores the size of these polynomials using the supremum of the absolute value on [-1, 1], showing that scaled Chebyshev and Bernstein polynomials give the extremes.
Calabi-Yau three-folds:. Poincaré polynomials and fractals
NASA Astrophysics Data System (ADS)
Ashmore, Anthony; He, Yang-Hui
2013-10-01
We study the Poincaré polynomials of all known Calabi-Yau three-folds as constrained polynomials of Littlewood type, thus generalising the wellknown investigation into the distribution of the Euler characteristic and Hodge numbers. We find interesting fractal behaviour in the roots of these polynomials, in relation to the existence of isometries, distribution versus typicality, and mirror symmetry.
Quasi-Fibonacci Numbers of Order 11
NASA Astrophysics Data System (ADS)
Wituła, Roman; Słota, Damian
2007-08-01
In this paper we introduce and investigate the so-called quasi-Fibonacci numbers of order 11 . These numbers are defined by five conjugate recurrence equations of order five. We study some relations and identities concerning these numbers. We present some applications to the decomposition of some polynomials. Many of the identities presented here are the generalizations of the identities characteristic for general recurrence sequences of order three given by Rabinowitz.
Karagiannis, Georgios Lin, Guang
2014-02-15
Generalized polynomial chaos (gPC) expansions allow us to represent the solution of a stochastic system using a series of polynomial chaos basis functions. The number of gPC terms increases dramatically as the dimension of the random input variables increases. When the number of the gPC terms is larger than that of the available samples, a scenario that often occurs when the corresponding deterministic solver is computationally expensive, evaluation of the gPC expansion can be inaccurate due to over-fitting. We propose a fully Bayesian approach that allows for global recovery of the stochastic solutions, in both spatial and random domains, by coupling Bayesian model uncertainty and regularization regression methods. It allows the evaluation of the PC coefficients on a grid of spatial points, via (1) the Bayesian model average (BMA) or (2) the median probability model, and their construction as spatial functions on the spatial domain via spline interpolation. The former accounts for the model uncertainty and provides Bayes-optimal predictions; while the latter provides a sparse representation of the stochastic solutions by evaluating the expansion on a subset of dominating gPC bases. Moreover, the proposed methods quantify the importance of the gPC bases in the probabilistic sense through inclusion probabilities. We design a Markov chain Monte Carlo (MCMC) sampler that evaluates all the unknown quantities without the need of ad-hoc techniques. The proposed methods are suitable for, but not restricted to, problems whose stochastic solutions are sparse in the stochastic space with respect to the gPC bases while the deterministic solver involved is expensive. We demonstrate the accuracy and performance of the proposed methods and make comparisons with other approaches on solving elliptic SPDEs with 1-, 14- and 40-random dimensions.
Physics suggests that the interplay of momentum, continuity, and geometry in outward radial flow must produce density and concomitant pressure reductions. In other words, this flow is intrinsically auto-expansive. It has been proposed that this process is the key to understanding...
Guzek, J.C.; Lujan, R.A.
1984-01-01
Disclosed is a cooler for television cameras and other temperature sensitive equipment. The cooler uses compressed gas ehich is accelerated to a high velocity by passing it through flow passageways having nozzle portions which expand the gas. This acceleration and expansion causes the gas to undergo a decrease in temperature thereby cooling the cooler body and adjacent temperature sensitive equipment.
For the Long Island, New Jersey, and southern New England region, one facet of marsh drowning as a result of accelerated sea level rise is the expansion of salt marsh ponds and pannes. Over the past century, marsh ponds and pannes have formed and expanded in areas of poor drainag...
ERIC Educational Resources Information Center
Ayoub, Ayoub B.
2006-01-01
In this article, the author takes up the special trinomial (1 + x + x[squared])[superscript n] and shows that the coefficients of its expansion are entries of a Pascal-like triangle. He also shows how to calculate these entries recursively and explicitly. This article could be used in the classroom for enrichment. (Contains 1 table.)
NASA Technical Reports Server (NTRS)
1985-01-01
Under an Egyptian government contract, PADCO studies urban growth in the Nile Area. They were assisted by LANDSAT survey maps and measurements provided by TAC. TAC had classified the raw LANDSAT data and processed it into various categories to detail urban expansion. PADCO crews spot checked the results, and correlations were established.
Fuchs, Erich; Gruber, Christian; Reitmaier, Tobias; Sick, Bernhard
2009-09-01
Neural networks are often used to process temporal information, i.e., any kind of information related to time series. In many cases, time series contain short-term and long-term trends or behavior. This paper presents a new approach to capture temporal information with various reference periods simultaneously. A least squares approximation of the time series with orthogonal polynomials will be used to describe short-term trends contained in a signal (average, increase, curvature, etc.). Long-term behavior will be modeled with the tapped delay lines of a time-delay neural network (TDNN). This network takes the coefficients of the orthogonal expansion of the approximating polynomial as inputs such considering short-term and long-term information efficiently. The advantages of the method will be demonstrated by means of artificial data and two real-world application examples, the prediction of the user number in a computer network and online tool wear classification in turning.
Henn, Brenna M.; Cavalli-Sforza, L. L.; Feldman, Marcus W.
2012-01-01
Genetic and paleoanthropological evidence is in accord that today’s human population is the result of a great demic (demographic and geographic) expansion that began approximately 45,000 to 60,000 y ago in Africa and rapidly resulted in human occupation of almost all of the Earth’s habitable regions. Genomic data from contemporary humans suggest that this expansion was accompanied by a continuous loss of genetic diversity, a result of what is called the “serial founder effect.” In addition to genomic data, the serial founder effect model is now supported by the genetics of human parasites, morphology, and linguistics. This particular population history gave rise to the two defining features of genetic variation in humans: genomes from the substructured populations of Africa retain an exceptional number of unique variants, and there is a dramatic reduction in genetic diversity within populations living outside of Africa. These two patterns are relevant for medical genetic studies mapping genotypes to phenotypes and for inferring the power of natural selection in human history. It should be appreciated that the initial expansion and subsequent serial founder effect were determined by demographic and sociocultural factors associated with hunter-gatherer populations. How do we reconcile this major demic expansion with the population stability that followed for thousands years until the inventions of agriculture? We review advances in understanding the genetic diversity within Africa and the great human expansion out of Africa and offer hypotheses that can help to establish a more synthetic view of modern human evolution. PMID:23077256
Estimation of asymptotic stability regions via composite homogeneous polynomial Lyapunov functions
NASA Astrophysics Data System (ADS)
Pang, Guochen; Zhang, Kanjian
2015-03-01
In this article, we present a new method to estimate the asymptotic stability regions for a class of nonlinear systems via composite homogeneous polynomial Lyapunov functions, where these nonlinear systems are approximated as a convex hull of some linear systems. Since level set of the composite homogeneous polynomial Lyapunov functions is a union set of several homogeneous polynomial functions, the composite homogeneous polynomial Lyapunov functions are nonconservative compared with quadratic or homogeneous polynomial Lyapunov functions. Numerical examples are used to illustrate the effectiveness of our method.
NASA Astrophysics Data System (ADS)
Shen, Xiang; Liu, Bin; Li, Qing-Quan
2017-03-01
The Rational Function Model (RFM) has proven to be a viable alternative to the rigorous sensor models used for geo-processing of high-resolution satellite imagery. Because of various errors in the satellite ephemeris and instrument calibration, the Rational Polynomial Coefficients (RPCs) supplied by image vendors are often not sufficiently accurate, and there is therefore a clear need to correct the systematic biases in order to meet the requirements of high-precision topographic mapping. In this paper, we propose a new RPC bias-correction method using the thin-plate spline modeling technique. Benefiting from its excellent performance and high flexibility in data fitting, the thin-plate spline model has the potential to remove complex distortions in vendor-provided RPCs, such as the errors caused by short-period orbital perturbations. The performance of the new method was evaluated by using Ziyuan-3 satellite images and was compared against the recently developed least-squares collocation approach, as well as the classical affine-transformation and quadratic-polynomial based methods. The results show that the accuracies of the thin-plate spline and the least-squares collocation approaches were better than the other two methods, which indicates that strong non-rigid deformations exist in the test data because they cannot be adequately modeled by simple polynomial-based methods. The performance of the thin-plate spline method was close to that of the least-squares collocation approach when only a few Ground Control Points (GCPs) were used, and it improved more rapidly with an increase in the number of redundant observations. In the test scenario using 21 GCPs (some of them located at the four corners of the scene), the correction residuals of the thin-plate spline method were about 36%, 37%, and 19% smaller than those of the affine transformation method, the quadratic polynomial method, and the least-squares collocation algorithm, respectively, which demonstrates
Using the network reliability polynomial to characterize and design networks
EUBANK, STEPHEN; YOUSSEF, MINA; KHORRAMZADEH, YASAMIN
2015-01-01
We consider methods for solving certain network characterization and design problems that arise in network epidemiology. We argue that the network reliability polynomial introduced by Moore and Shannon is a useful framework in which to reason about these problems. Specifically, we show how efficient estimation of the polynomial permits characterizing and distinguishing very large networks in ways that are are dynamically relevant. Furthermore, a generalization of flows and cuts to structures that determine the reliability suggests a new measure of edge or vertex centrality that we call criticality. We describe how criticality is related to the more common notion of betweenness and illustrate its application to targeting interventions to control outbreaks of infectious disease. Although our applications are to infectious disease outbreaks, the methods we develop are applicable to many other diffusive dynamical systems over complex networks. PMID:26085930
Multivariable Hermite polynomials and phase-space dynamics
NASA Technical Reports Server (NTRS)
Dattoli, G.; Torre, Amalia; Lorenzutta, S.; Maino, G.; Chiccoli, C.
1994-01-01
The phase-space approach to classical and quantum systems demands for advanced analytical tools. Such an approach characterizes the evolution of a physical system through a set of variables, reducing to the canonically conjugate variables in the classical limit. It often happens that phase-space distributions can be written in terms of quadratic forms involving the above quoted variables. A significant analytical tool to treat these problems may come from the generalized many-variables Hermite polynomials, defined on quadratic forms in R(exp n). They form an orthonormal system in many dimensions and seem the natural tool to treat the harmonic oscillator dynamics in phase-space. In this contribution we discuss the properties of these polynomials and present some applications to physical problems.
Fast complex memory polynomial-based adaptive digital predistorter
NASA Astrophysics Data System (ADS)
Singh Sappal, Amandeep; Singh Patterh, Manjeet; Sharma, Sanjay
2011-07-01
Today's 3G wireless systems require both high linearity and high power amplifier (PA) efficiency. The high peak-to-average ratios of the digital modulation schemes used in 3G wireless systems require that the RF PA maintain high linearity over a large range while maintaining this high efficiency; these two requirements are often at odds with each other with many of the traditional amplifier architectures. In this article, a fast and easy-to-implement adaptive digital predistorter has been presented for Wideband Code Division Multiplexed signals using complex memory polynomial work function. The proposed algorithm has been implemented to test a Motorola LDMOSFET PA. The proposed technique also takes care of the memory effects of the PA, which have been ignored in many proposed techniques in the literature. The results show that the new complex memory polynomial-based adaptive digital predistorter has better linearisation performance than conventional predistortion techniques.
Piecewise quartic polynomial curves with a local shape parameter
NASA Astrophysics Data System (ADS)
Han, Xuli
2006-10-01
Piecewise quartic polynomial curves with a local shape parameter are presented in this paper. The given blending function is an extension of the cubic uniform B-splines. The changes of a local shape parameter will only change two curve segments. With the increase of the value of a shape parameter, the curves approach a corresponding control point. The given curves possess satisfying shape-preserving properties. The given curve can also be used to interpolate locally the control points with GC2 continuity. Thus, the given curves unify the representation of the curves for interpolating and approximating the control polygon. As an application, the piecewise polynomial curves can intersect an ellipse at different knot values by choosing the value of the shape parameter. The given curve can approximate an ellipse from the both sides and can then yield a tight envelope for an ellipse. Some computing examples for curve design are given.
Teichert, Gregory H.; Gunda, N. S. Harsha; Rudraraju, Shiva; ...
2016-12-18
Free energies play a central role in many descriptions of equilibrium and non-equilibrium properties of solids. Continuum partial differential equations (PDEs) of atomic transport, phase transformations and mechanics often rely on first and second derivatives of a free energy function. The stability, accuracy and robustness of numerical methods to solve these PDEs are sensitive to the particular functional representations of the free energy. In this communication we investigate the influence of different representations of thermodynamic data on phase field computations of diffusion and two-phase reactions in the solid state. First-principles statistical mechanics methods were used to generate realistic free energymore » data for HCP titanium with interstitially dissolved oxygen. While Redlich-Kister polynomials have formed the mainstay of thermodynamic descriptions of multi-component solids, they require high order terms to fit oscillations in chemical potentials around phase transitions. Here, we demonstrate that high fidelity fits to rapidly fluctuating free energy functions are obtained with spline functions. As a result, spline functions that are many degrees lower than Redlich-Kister polynomials provide equal or superior fits to chemical potential data and, when used in phase field computations, result in solution times approaching an order of magnitude speed up relative to the use of Redlich-Kister polynomials.« less
Teichert, Gregory H.; Gunda, N. S. Harsha; Rudraraju, Shiva; Natarajan, Anirudh Raju; Puchala, Brian; Van der Ven, Anton; Garikipati, Krishna
2016-12-18
Free energies play a central role in many descriptions of equilibrium and non-equilibrium properties of solids. Continuum partial differential equations (PDEs) of atomic transport, phase transformations and mechanics often rely on first and second derivatives of a free energy function. The stability, accuracy and robustness of numerical methods to solve these PDEs are sensitive to the particular functional representations of the free energy. In this communication we investigate the influence of different representations of thermodynamic data on phase field computations of diffusion and two-phase reactions in the solid state. First-principles statistical mechanics methods were used to generate realistic free energy data for HCP titanium with interstitially dissolved oxygen. While Redlich-Kister polynomials have formed the mainstay of thermodynamic descriptions of multi-component solids, they require high order terms to fit oscillations in chemical potentials around phase transitions. Here, we demonstrate that high fidelity fits to rapidly fluctuating free energy functions are obtained with spline functions. As a result, spline functions that are many degrees lower than Redlich-Kister polynomials provide equal or superior fits to chemical potential data and, when used in phase field computations, result in solution times approaching an order of magnitude speed up relative to the use of Redlich-Kister polynomials.
Lee, Y.-G.; Zou, W.-N.; Pan, E.
2015-01-01
This paper presents a closed-form solution for the arbitrary polygonal inclusion problem with polynomial eigenstrains of arbitrary order in an anisotropic magneto-electro-elastic full plane. The additional displacements or eigendisplacements, instead of the eigenstrains, are assumed to be a polynomial with general terms of order M+N. By virtue of the extended Stroh formulism, the induced fields are expressed in terms of a group of basic functions which involve boundary integrals of the inclusion domain. For the special case of polygonal inclusions, the boundary integrals are carried out explicitly, and their averages over the inclusion are also obtained. The induced fields under quadratic eigenstrains are mostly analysed in terms of figures and tables, as well as those under the linear and cubic eigenstrains. The connection between the present solution and the solution via the Green's function method is established and numerically verified. The singularity at the vertices of the arbitrary polygon is further analysed via the basic functions. The general solution and the numerical results for the constant, linear, quadratic and cubic eigenstrains presented in this paper enable us to investigate the features of the inclusion and inhomogeneity problem concerning polynomial eigenstrains in semiconductors and advanced composites, while the results can further serve as benchmarks for future analyses of Eshelby's inclusion problem. PMID:26345141
Falk, Carl F; Cai, Li
2016-06-01
We present a semi-parametric approach to estimating item response functions (IRF) useful when the true IRF does not strictly follow commonly used functions. Our approach replaces the linear predictor of the generalized partial credit model with a monotonic polynomial. The model includes the regular generalized partial credit model at the lowest order polynomial. Our approach extends Liang's (A semi-parametric approach to estimate IRFs, Unpublished doctoral dissertation, 2007) method for dichotomous item responses to the case of polytomous data. Furthermore, item parameter estimation is implemented with maximum marginal likelihood using the Bock-Aitkin EM algorithm, thereby facilitating multiple group analyses useful in operational settings. Our approach is demonstrated on both educational and psychological data. We present simulation results comparing our approach to more standard IRF estimation approaches and other non-parametric and semi-parametric alternatives.
Quaternionic polynomials with multiple zeros: A numerical point of view
NASA Astrophysics Data System (ADS)
Falcão, M. I.; Miranda, F.; Severino, R.; Soares, M. J.
2017-01-01
In the ring of quaternionic polynomials there is no easy solution to the problem of finding a suitable definition of multiplicity of a zero. In this paper we discuss different notions of multiple zeros available in the literature and add a computational point of view to this problem, by taking into account the behavior of the well known Newton's method in the presence of such roots.
Evaluation of the tensor polynomial failure criterion for composite materials
NASA Technical Reports Server (NTRS)
Tennyson, R. C.; Macdonald, D.; Nanyaro, A. P.
1978-01-01
A comprehensive experimental and analytical evaluation of the tensor polynomial failure criterion was undertaken to determine its capability for predicting the ultimate strength of laminated composite structures subject to a plane stress state. Results are presented demonstrating that a quadratic formulation is too conservative and a cubic representation is required. Strength comparisons with test data derived from glass/epoxy and graphite/epoxy tubular specimens are also provided to validate the cubic strength criterion.
Invariance of Conjunctions of Polynomial Equalities for Algebraic Differential Equations
2014-07-01
non- linear hybrid systems by linear algebraic methods. In Radhia Cousot and Matthieu Martel, editors, SAS, volume 6337 of LNCS, pages 373–389. Springer...Tarski. A decision method for elementary algebra and geometry. Bulletin of the American Mathematical Society, 59, 1951. [36] Wolfgang Walter. Ordinary...Invariance of Conjunctions of Polynomial Equalities for Algebraic Differential Equations Khalil Ghorbal1 Andrew Sogokon2 André Platzer1 July 2014
The partially closed Griffith crack. [under polynomial loads
NASA Technical Reports Server (NTRS)
Thresher, R. W.; Smith, F. W.
1973-01-01
A solution is presented for a Griffith crack subjected to an arbitrary polynomial loading function which causes one end of the crack to remain closed. Closed form expressions are presented for the crack opening length and for the stress and displacements in the plane of the crack. The special case of pure bending is presented as an example and for this case the stress intensity factor is computed.
Fibonacci chain polynomials: Identities from self-similarity
NASA Technical Reports Server (NTRS)
Lang, Wolfdieter
1995-01-01
Fibonacci chains are special diatomic, harmonic chains with uniform nearest neighbor interaction and two kinds of atoms (mass-ratio r) arranged according to the self-similar binary Fibonacci sequence ABAABABA..., which is obtained by repeated substitution of A yields AB and B yields A. The implications of the self-similarity of this sequence for the associated orthogonal polynomial systems which govern these Fibonacci chains with fixed mass-ratio r are studied.
An exponential polynomial observer for synchronization of chaotic systems
NASA Astrophysics Data System (ADS)
Mata-Machuca, J. L.; Martínez-Guerra, R.; Aguilar-López, R.
2010-12-01
In this paper, we consider the synchronization problem via nonlinear observer design. A new exponential polynomial observer for a class of nonlinear oscillators is proposed, which is robust against output noises. A sufficient condition for synchronization is derived analytically with the help of Lyapunov stability theory. The proposed technique has been applied to synchronize chaotic systems (Rikitake and Rössler systems) by means of numerical simulation.
Modeling a Temporally Evolving Atmosphere with Zernike Polynomials
2012-09-01
Systems, SPIE Press, 2010 5. J.W. Goodman , Statistical Optics , John Wiley & Sons, Inc., New York, NY, 1985 6. R. J. Noll, "Zernike Polynomials and...temporal model of phase screen generation. The long standing Fourier transform (FT) based method assumes the frozen flow hypothesis holds, where... optical tilt. 1. INTRODUCTION For conventional imaging systems, Geosynchronous Earth Orbit (GEO) space objects cannot be resolved due to
The $\\hbar$ Expansion in Quantum Field Theory
Brodsky, Stanley J.; Hoyer, Paul; /Southern Denmark U., CP3-Origins /Helsinki U. /Helsinki Inst. of Phys.
2010-10-27
We show how expansions in powers of Planck's constant {h_bar} = h = 2{pi} can give new insights into perturbative and nonperturbative properties of quantum field theories. Since {h_bar} is a fundamental parameter, exact Lorentz invariance and gauge invariance are maintained at each order of the expansion. The physics of the {h_bar} expansion depends on the scheme; i.e., different expansions are obtained depending on which quantities (momenta, couplings and masses) are assumed to be independent of {h_bar}. We show that if the coupling and mass parameters appearing in the Lagrangian density are taken to be independent of {h_bar}, then each loop in perturbation theory brings a factor of {h_bar}. In the case of quantum electrodynamics, this scheme implies that the classical charge e, as well as the fine structure constant are linear in {h_bar}. The connection between the number of loops and factors of {h_bar} is more subtle for bound states since the binding energies and bound-state momenta themselves scale with {h_bar}. The {h_bar} expansion allows one to identify equal-time relativistic bound states in QED and QCD which are of lowest order in {h_bar} and transform dynamically under Lorentz boosts. The possibility to use retarded propagators at the Born level gives valence-like wave-functions which implicitly describe the sea constituents of the bound states normally present in its Fock state representation.
NASA Astrophysics Data System (ADS)
Dragomirescu, Florica Ioana
2012-11-01
The main motivation for a temporal stability investigation of initially localized perturbations in a swirling flow stability problem consists in pointing out the critical frequencies at which instability can sets in, an important key in predicting and understanding the flow particularities. The linearized disturbance equations define a second order ordinary differential equation with non-constant coefficients which we solve in order to determine the critical frequency in different physical parameters spaces. A non-classical polynomials based spectral method is proposed for the numerical treatment of the resulting generalized eigenvalue problem governing the stability of the flow. Numerical investigation are performed in the inviscid case for a moderate level of swirl and dominant temporal instability modes are retrieved for each Fourier component pair. The obtained values of the growth rate associated with the most amplified wavenumber are compared with existing inviscid temporal instability evaluations and good agreements are found.
Krishnamoorthi, R; Anna Poorani, G
2016-01-01
Iris normalization is an important stage in any iris biometric, as it has a propensity to trim down the consequences of iris distortion. To indemnify the variation in size of the iris owing to the action of stretching or enlarging the pupil in iris acquisition process and camera to eyeball distance, two normalization schemes has been proposed in this work. In the first method, the iris region of interest is normalized by converting the iris into the variable size rectangular model in order to avoid the under samples near the limbus border. In the second method, the iris region of interest is normalized by converting the iris region into a fixed size rectangular model in order to avoid the dimensional discrepancies between the eye images. The performance of the proposed normalization methods is evaluated with orthogonal polynomials based iris recognition in terms of FAR, FRR, GAR, CRR and EER.
Pulse transmission transmitter including a higher order time derivate filter
Dress, Jr., William B.; Smith, Stephen F.
2003-09-23
Systems and methods for pulse-transmission low-power communication modes are disclosed. A pulse transmission transmitter includes: a clock; a pseudorandom polynomial generator coupled to the clock, the pseudorandom polynomial generator having a polynomial load input; an exclusive-OR gate coupled to the pseudorandom polynomial generator, the exclusive-OR gate having a serial data input; a programmable delay circuit coupled to both the clock and the exclusive-OR gate; a pulse generator coupled to the programmable delay circuit; and a higher order time derivative filter coupled to the pulse generator. The systems and methods significantly reduce lower-frequency emissions from pulse transmission spread-spectrum communication modes, which reduces potentially harmful interference to existing radio frequency services and users and also simultaneously permit transmission of multiple data bits by utilizing specific pulse shapes.
Equations on knot polynomials and 3d/5d duality
Mironov, A.; Morozov, A.
2012-09-24
We briefly review the current situation with various relations between knot/braid polynomials (Chern-Simons correlation functions), ordinary and extended, considered as functions of the representation and of the knot topology. These include linear skein relations, quadratic Plucker relations, as well as 'differential' and (quantum) A-polynomial structures. We pay a special attention to identity between the A-polynomial equations for knots and Baxter equations for quantum relativistic integrable systems, related through Seiberg-Witten theory to 5d super-Yang-Mills models and through the AGT relation to the q-Virasoro algebra. This identity is an important ingredient of emerging a 3d- 5d generalization of the AGT relation. The shape of the Baxter equation (including the values of coefficients) depend on the choice of the knot/braid. Thus, like the case of KP integrability, where (some, so far torus) knots parameterize particular points of the Universal Grassmannian, in this relation they parameterize particular points in the moduli space of many-body integrable systems of relativistic type.
A Comparison of Approximation Modeling Techniques: Polynomial Versus Interpolating Models
NASA Technical Reports Server (NTRS)
Giunta, Anthony A.; Watson, Layne T.
1998-01-01
Two methods of creating approximation models are compared through the calculation of the modeling accuracy on test problems involving one, five, and ten independent variables. Here, the test problems are representative of the modeling challenges typically encountered in realistic engineering optimization problems. The first approximation model is a quadratic polynomial created using the method of least squares. This type of polynomial model has seen considerable use in recent engineering optimization studies due to its computational simplicity and ease of use. However, quadratic polynomial models may be of limited accuracy when the response data to be modeled have multiple local extrema. The second approximation model employs an interpolation scheme known as kriging developed in the fields of spatial statistics and geostatistics. This class of interpolating model has the flexibility to model response data with multiple local extrema. However, this flexibility is obtained at an increase in computational expense and a decrease in ease of use. The intent of this study is to provide an initial exploration of the accuracy and modeling capabilities of these two approximation methods.
Recognition of Arabic Sign Language Alphabet Using Polynomial Classifiers
NASA Astrophysics Data System (ADS)
Assaleh, Khaled; Al-Rousan, M.
2005-12-01
Building an accurate automatic sign language recognition system is of great importance in facilitating efficient communication with deaf people. In this paper, we propose the use of polynomial classifiers as a classification engine for the recognition of Arabic sign language (ArSL) alphabet. Polynomial classifiers have several advantages over other classifiers in that they do not require iterative training, and that they are highly computationally scalable with the number of classes. Based on polynomial classifiers, we have built an ArSL system and measured its performance using real ArSL data collected from deaf people. We show that the proposed system provides superior recognition results when compared with previously published results using ANFIS-based classification on the same dataset and feature extraction methodology. The comparison is shown in terms of the number of misclassified test patterns. The reduction in the rate of misclassified patterns was very significant. In particular, we have achieved a 36% reduction of misclassifications on the training data and 57% on the test data.
Cusp Catastrophe Polynomial Model: Power and Sample Size Estimation
Chen, Ding-Geng(Din); Chen, Xinguang(Jim); Lin, Feng; Tang, Wan; Lio, Y. L.; Guo, (Tammy) Yuanyuan
2016-01-01
Guastello’s polynomial regression method for solving cusp catastrophe model has been widely applied to analyze nonlinear behavior outcomes. However, no statistical power analysis for this modeling approach has been reported probably due to the complex nature of the cusp catastrophe model. Since statistical power analysis is essential for research design, we propose a novel method in this paper to fill in the gap. The method is simulation-based and can be used to calculate statistical power and sample size when Guastello’s polynomial regression method is used to cusp catastrophe modeling analysis. With this novel approach, a power curve is produced first to depict the relationship between statistical power and samples size under different model specifications. This power curve is then used to determine sample size required for specified statistical power. We verify the method first through four scenarios generated through Monte Carlo simulations, and followed by an application of the method with real published data in modeling early sexual initiation among young adolescents. Findings of our study suggest that this simulation-based power analysis method can be used to estimate sample size and statistical power for Guastello’s polynomial regression method in cusp catastrophe modeling. PMID:27158562
Tensor calculus in polar coordinates using Jacobi polynomials
NASA Astrophysics Data System (ADS)
Vasil, Geoffrey M.; Burns, Keaton J.; Lecoanet, Daniel; Olver, Sheehan; Brown, Benjamin P.; Oishi, Jeffrey S.
2016-11-01
Spectral methods are an efficient way to solve partial differential equations on domains possessing certain symmetries. The utility of a method depends strongly on the choice of spectral basis. In this paper we describe a set of bases built out of Jacobi polynomials, and associated operators for solving scalar, vector, and tensor partial differential equations in polar coordinates on a unit disk. By construction, the bases satisfy regularity conditions at r = 0 for any tensorial field. The coordinate singularity in a disk is a prototypical case for many coordinate singularities. The work presented here extends to other geometries. The operators represent covariant derivatives, multiplication by azimuthally symmetric functions, and the tensorial relationship between fields. These arise naturally from relations between classical orthogonal polynomials, and form a Heisenberg algebra. Other past work uses more specific polynomial bases for solving equations in polar coordinates. The main innovation in this paper is to use a larger set of possible bases to achieve maximum bandedness of linear operations. We provide a series of applications of the methods, illustrating their ease-of-use and accuracy.
NASA Astrophysics Data System (ADS)
Kaporin, I. E.
2012-02-01
In order to precondition a sparse symmetric positive definite matrix, its approximate inverse is examined, which is represented as the product of two sparse mutually adjoint triangular matrices. In this way, the solution of the corresponding system of linear algebraic equations (SLAE) by applying the preconditioned conjugate gradient method (CGM) is reduced to performing only elementary vector operations and calculating sparse matrix-vector products. A method for constructing the above preconditioner is described and analyzed. The triangular factor has a fixed sparsity pattern and is optimal in the sense that the preconditioned matrix has a minimum K-condition number. The use of polynomial preconditioning based on Chebyshev polynomials makes it possible to considerably reduce the amount of scalar product operations (at the cost of an insignificant increase in the total number of arithmetic operations). The possibility of an efficient massively parallel implementation of the resulting method for solving SLAEs is discussed. For a sequential version of this method, the results obtained by solving 56 test problems from the Florida sparse matrix collection (which are large-scale and ill-conditioned) are presented. These results show that the method is highly reliable and has low computational costs.
Regression-based adaptive sparse polynomial dimensional decomposition for sensitivity analysis
NASA Astrophysics Data System (ADS)
Tang, Kunkun; Congedo, Pietro; Abgrall, Remi
2014-11-01
Polynomial dimensional decomposition (PDD) is employed in this work for global sensitivity analysis and uncertainty quantification of stochastic systems subject to a large number of random input variables. Due to the intimate structure between PDD and Analysis-of-Variance, PDD is able to provide simpler and more direct evaluation of the Sobol' sensitivity indices, when compared to polynomial chaos (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 the standard method unaffordable for real engineering applications. In order to address this problem of curse of dimensionality, this work proposes a variance-based adaptive strategy aiming to build a cheap meta-model by sparse-PDD with PDD coefficients computed by regression. During this adaptive procedure, the model representation by PDD only contains few terms, so that the cost to resolve repeatedly the linear system of the least-square regression problem is negligible. The size of the final sparse-PDD representation is much smaller than the full PDD, since only significant terms are eventually retained. Consequently, a much less number of calls to the deterministic model is required to compute the final PDD coefficients.
A categorification of the quantum sl(N)-link polynomials using foams
NASA Astrophysics Data System (ADS)
Vaz, Pedro
2008-07-01
In this thesis we define and study a categorification of the sl(N)-link polynomial using foams, for Ngeq 3. For N=3 we define the universal sl(3)-link homology, using foams, which depends on three parameters and show that it is functorial, up to scalars, with respect to link cobordisms. Our theory is integral. We show that tensoring it with Q yields a theory which is equivalent to the rational universal Khovanov-Rozansky sl(3)-link homology. For Ngeq 4 we construct a rational theory categorifying the sl(N)-link polynomial using foams. Our theory is functorial, up to scalars, with respect to link cobordisms. To evaluate closed foams we use the Kapustin-Li formula. We show that for any link our homology is isomorphic to the Khovanov-Rozansky homology. We conjecture that the theory is integral and we compute the conjectured integral sl(N)-link homology for the (2,m)-torus links and show that it has torsion of order N.
Polynomial chaos for the computation of annual energy production in wind farm layout optimization
NASA Astrophysics Data System (ADS)
Padrón, A. S.; Stanley, A. P. J.; Thomas, J. J.; Alonso, J. J.; Ning, A.
2016-09-01
Careful management of wake interference is essential to further improve Annual Energy Production (AEP) of wind farms. Wake effects can be minimized through optimization of turbine layout, wind farm control, and turbine design. Realistic wind farm optimization is challenging because it has numerous design degrees of freedom and must account for the stochastic nature of wind. In this paper we provide a framework for calculating AEP for any relevant uncertain (stochastic) variable of interest. We use Polynomial Chaos (PC) to efficiently quantify the effect of the stochastic variables—wind direction and wind speed—on the statistical outputs of interest (AEP) for wind farm layout optimization. When the stochastic variable includes the wind direction, polynomial chaos is one order of magnitude more accurate in computing the AEP when compared to commonly used simplistic integration techniques (rectangle rule), especially for non grid-like wind farm layouts. Furthermore, PC requires less simulations for the same accuracy. This allows for more efficient optimization and uncertainty quantification of wind farm energy production.
NASA Astrophysics Data System (ADS)
Muller, Melanie J. I.; Korolev, Kirill S.; Murray, Andrew W.; Nelson, David R.
2012-02-01
The expansion of a species into new territory is often strongly influenced by the presence of other species. This effect is particularly striking for the case of mutualistic species that enhance each other's proliferation. Examples range from major events in evolutionary history, such as the spread and diversification of flowering plants due to their mutualism with pollen-dispersing insects, to modern examples like the surface colonisation of multi-species microbial biofilms. Here, we investigate the spread of cross-feeding strains of the budding yeast Saccharomyces cerevisiae on an agar surface as a model system for expanding mutualists. Depending on the degree of mutualism, the two strains form distinctive spatial patterns during their range expansion. This change in spatial patterns can be understood as a phase transition within a stepping stone model generalized to two mutualistic species.
Evolutionary polynomial regression applied to rainfall triggered landslide reactivation alert
NASA Astrophysics Data System (ADS)
Simeone, Vincenzo; Doglioni, Angelo; Fiorillo, Francesco
2010-05-01
Evolutionary Polynomial Regression (EPR) is a hybrid evolutionary modelling paradigm, which allows for the construction of explicit model equations, starting from measured data. It was successfully applied to multiple cases study as well as to different model aims, i.e. dynamic natural systems, pipe burst analysis, geotechnical soil characterization, etc. A landslide located on a slope on the Adriatic coast of south Italy, close to the small town of Petacciato, is here investigated. In particular, starting from the rainfall data, which are available for the last 110 years as daily records, and from 11 activation episodes which range between 1932 and 2009, a data-driven model aimed at describing the reactivation as function of cumulative rainfall values was identified. Petacciato landslide is a deep large landslide; it lays on a slope characterized by outcropping Pleistocenic blue clays, which are somewhere spaced out of thin loamy-sandy layers. Moving towards the upper part of the slope, which is closer to the town, blue clays are progressively replaced by sand and conglomerates. The slope is also characterized by frequent terraces which show slope inversions. The landslide is quite wide and the slope is characterized by a low steepness. In addition, the bottom of the sea, close to the shore, is very gently deepening, thus excluding an effect of water on the slope stability. For these reasons, the dynamic of Petacciato landslide is quite difficult to be interpreted. The particular mechanism of the landslide as well as the exceptional data availability, in particular in terms of reactivations, make it possible to cope with this system according to a data mining approach. It was observed that the landslide reactivated after long rainy periods, and then rainfall was assumed as a triggering factor. Therefore, cumulative rainfall values were constructed in order to account with long periods, up to 500 days. These were used as candidate inputs for the construction of a
Order-N Electronic Structure Calculation of n-TYPE GaAs Quantum Dots
NASA Astrophysics Data System (ADS)
Nomura, S.; Iitaka, T.
2008-10-01
A linear scale method for calculating electronic properties of large and complex systems is introduced within a local density approximation. The method is based on the Chebyshev polynomial expansion and the time-dependent method, which is tested in calculating the electronic structure of a model n-type GaAs quantum dot.
Thermal expansion of glassy polymers.
Davy, K W; Braden, M
1992-01-01
The thermal expansion of a number of glassy polymers of interest in dentistry has been studied using a quartz dilatometer. In some cases, the expansion was linear and therefore the coefficient of thermal expansion readily determined. Other polymers exhibited non-linear behaviour and values appropriate to different temperature ranges are quoted. The linear coefficient of thermal expansion was, to a first approximation, a function of both the molar volume and van der Waal's volume of the repeating unit.
Tutty, O.
2015-01-01
With the goal of providing the first example of application of a recently proposed method, thus demonstrating its ability to give results in principle, global stability of a version of the rotating Couette flow is examined. The flow depends on the Reynolds number and a parameter characterizing the magnitude of the Coriolis force. By converting the original Navier–Stokes equations to a finite-dimensional uncertain dynamical system using a partial Galerkin expansion, high-degree polynomial Lyapunov functionals were found by sum-of-squares of polynomials optimization. It is demonstrated that the proposed method allows obtaining the exact global stability limit for this flow in a range of values of the parameter characterizing the Coriolis force. Outside this range a lower bound for the global stability limit was obtained, which is still better than the energy stability limit. In the course of the study, several results meaningful in the context of the method used were also obtained. Overall, the results obtained demonstrate the applicability of the recently proposed approach to global stability of the fluid flows. To the best of our knowledge, it is the first case in which global stability of a fluid flow has been proved by a generic method for the value of a Reynolds number greater than that which could be achieved with the energy stability approach. PMID:26730219
Homentcovschi, Dorel; Miles, Ronald N.
2012-01-01
The paper applies the re-expansion method for analyzing planar discontinuities at the junction of two axi-symmetrical circular waveguides. The normal modes in the two waveguides are expanded at the junction plane into a system of functions accounting for velocity singularities at the corner points. As the new expansion has a high convergence order, only a few terms have to be considered for obtaining the solution of most practical problems. This paper gives the equivalent impedance accounting for nonplanar waves into a plane-wave analysis and also the scattering matrix describing the coupling of arbitrary modes at each side of the discontinuity valid in the case of many propagating modes in both sides of the duct. The last section applies the re-expansion technique to some concentric expansion chambers providing an explicit formula for the transmission loss coefficient. PMID:22352491
Higher-order spectra for identification of nonlinear modal coupling
NASA Astrophysics Data System (ADS)
Hickey, Daryl; Worden, Keith; Platten, Michael F.; Wright, Jan R.; Cooper, Jonathan E.
2009-05-01
Over the past four decades considerable work has been done in the area of power spectrum estimation. The information contained within the power spectrum relates to a signal's autocorrelation or 'second-order statistics'. The power spectrum provides a complete statistical description of a Gaussian process; however, a problem with this information is that it is phase blind. This problem is addressed if one turns to a system's frequency response function (FRF). The FRF graphs the magnitude and phase of the frequency response of a system; in order to do this it requires information regarding the frequency content of the input and output signals. Situations arise in science and engineering whereby signal analysts are required to look beyond second-order statistics and analyse a signal's higher-order statistics (HOS). HOS or spectra give information on a signal's deviation from Gaussianity and consequently are a good indicator function for the presence of nonlinearity within a system. One of the main problems in nonlinear system identification is that of high modal density. Many modelling schemes involve making some expansion of the nonlinear restoring force in terms of polynomial or other basis terms. If more than one degree-of-freedom is involved this becomes a multivariate problem and the number of candidate terms in the expansion grows explosively with the order of nonlinearity and the number of degrees-of-freedom. This paper attempts to use HOS to detect and qualify nonlinear behaviour for a number of symmetrical and asymmetrical systems over a range of degrees-of-freedom. In doing so the paper also attempts to show that HOS are a more sensitive tool than the FRF in detecting nonlinearity. Furthermore, the object of this paper is to try and identify which modes couple in a nonlinear manner in order to reduce the number of candidate coupling terms, for a model, as much as possible. The bispectrum method has previously been applied to simple low-DOF systems with high
Expansion: A Plan for Success.
ERIC Educational Resources Information Center
Callahan, A.P.
This report provides selling brokers' guidelines for the successful expansion of their operations outlining a basic method of preparing an expansion plan. Topic headings are: The Pitfalls of Expansion (The Language of Business, Timely Financial Reporting, Regulatory Agencies of Government, Preoccupation with the Facade of Business, A Business Is a…
Path-integral approach to the Wigner-Kirkwood expansion.
Jizba, Petr; Zatloukal, Václav
2014-01-01
We study the high-temperature behavior of quantum-mechanical path integrals. Starting from the Feynman-Kac formula, we derive a functional representation of the Wigner-Kirkwood perturbation expansion for quantum Boltzmann densities. As shown by its applications to different potentials, the presented expansion turns out to be quite efficient in generating analytic form of the higher-order expansion coefficients. To put some flesh on the bare bones, we apply the expansion to obtain basic thermodynamic functions of the one-dimensional anharmonic oscillator. Further salient issues, such as generalization to the Bloch density matrix and comparison with the more customary world-line formulation, are discussed.
High-Temperature Expansions for Frenkel-Kontorova Model
NASA Astrophysics Data System (ADS)
Takahashi, K.; Mannari, I.; Ishii, T.
1995-02-01
Two high-temperature series expansions of the Frenkel-Kontorova (FK) model are investigated: the high-temperature approximation of Schneider-Stoll is extended to the FK model having the density ρ ≠ 1, and an alternative series expansion in terms of the modified Bessel function is examined. The first six-order terms for both expansions in free energy are explicitly obtained and compared with Ishii's approximation of the transfer-integral method. The specific heat based on the expansions is discussed by comparing with those of the transfer-integral method and Monte Carlo simulation.
Shearing expansion-free spherical anisotropic fluid evolution
Herrera, L.; Santos, N. O.; Wang Anzhong
2008-10-15
Spherically symmetric expansion-free distributions are systematically studied. The entire set of field equations and junction conditions are presented for a general distribution of dissipative anisotropic fluid (principal stresses unequal), and the expansion-free condition is integrated. In order to understand the physical meaning of expansion-free motion, two different definitions for the radial velocity of a fluid element are discussed. It is shown that the appearance of a cavity is inevitable in the expansion-free evolution. The nondissipative case is considered in detail, and the Skripkin model is recovered.
NASA Astrophysics Data System (ADS)
Carvalho, Luis A.
2005-06-01
Zernike Polynomials have been successfully used for many years in optics. Nevertheless there are some recent discussions regarding their accuracy when applied to surfaces such as the human cornea. A set of synthetic surfaces resembling several common corneal anomalies was sampled and was also used to compute the optical path difference using a simple ray-tracing procedure. The Root Mean Square Error between the Zernike Polynomials fit and the theoretical elevation and WF error surface was computed for both surfaces and for all number of Zernike terms. We have found that RMSE for the simplest, most symmetric corneal surface (spherical shape) and for the most complex shape (post-radial keratotomy) both the optical path difference and surface elevation, for 1 through 36 Zernike terms, range from: 421.4 to 0.8 microns, and 421.4 to 8.2 microns, respectively; mean RMSE for maximum Zernike terms for both surfaces were 4.5 microns. Computations in this work suggest that, for surfaces such as post-RK, keratoconus or post-keratoplasty, even more than 36 terms may be necessary in order to obtain minimum precision requirements. We suggest that the number of Zernike Polynomial should not be a global fixed conventional value but rather based on specific surface properties.
NASA Astrophysics Data System (ADS)
Cieplak, Agnieszka; Slosar, Anze
2017-01-01
The Lyman-alpha forest has become a powerful cosmological probe of the underlying matter distribution at high redshift. It is a highly non-linear field with much information present beyond the two-point statistics of the power spectrum. The flux probability distribution function (PDF) in particular has been used as a successful probe of small-scale physics. In addition to the cosmological evolution however, it is also sensitive to pixel noise, spectrum resolution, and continuum fitting, all of which lead to possible biased estimators. Here we argue that measuring coefficients of the Legendre polynomial expansion of the PDF offers several advantages over the binned PDF as is commonly done. Since the n-th coefficient can be expressed as a linear combination of the first n moments of the field, this allows for the coefficients to be measured in the presence of noise and allows for a clear route towards marginalization over the mean flux. In addition, we use hydrodynamic cosmological simulations to demonstrate that in the presence of noise, a finite number of these coefficients are well measured with a very sharp transition into noise dominance. This compresses the information into a finite small number of well-measured quantities.
Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems
Marzouk, Youssef M. Najm, Habib N.
2009-04-01
We consider a Bayesian approach to nonlinear inverse problems in which the unknown quantity is a spatial or temporal field, endowed with a hierarchical Gaussian process prior. Computational challenges in this construction arise from the need for repeated evaluations of the forward model (e.g., in the context of Markov chain Monte Carlo) and are compounded by high dimensionality of the posterior. We address these challenges by introducing truncated Karhunen-Loeve expansions, based on the prior distribution, to efficiently parameterize the unknown field and to specify a stochastic forward problem whose solution captures that of the deterministic forward model over the support of the prior. We seek a solution of this problem using Galerkin projection on a polynomial chaos basis, and use the solution to construct a reduced-dimensionality surrogate posterior density that is inexpensive to evaluate. We demonstrate the formulation on a transient diffusion equation with prescribed source terms, inferring the spatially-varying diffusivity of the medium from limited and noisy data.
Multidimensional Gravitational Models: Fluxbrane and S-Brane Solutions with Polynomials
NASA Astrophysics Data System (ADS)
Ivashchuk, V. D.; Melnikov, V. N.
2007-06-01
Main results in obtaining exact solutions for multidimensional models and their application to solving main problems of modern cosmology and black hole physics are described. Some new results on composite fluxbrane and S-brane solutions for a wide class of intersection rules are presented. These solutions are defined on a product manifold R* × M1 × ... × Mn which contains n Ricci-flat spaces M1,...,Mn with 1-dimensional R* and M1. They are defined up to a set of functions obeying non-linear differential equations equivalent to Toda-type equations with certain boundary conditions imposed. Exact solutions corresponding to configurations with two branes and intersections related to simple Lie algebras C2 and G2 are obtained. In these cases the functions Hs(z), s = 1, 2, are polynomials of degrees: (3, 4) and (6, 10), respectively, in agreement with a conjecture suggested earlier. Examples of simple S-brane solutions describing an accelerated expansion of a certain factor-space are given explicitely.
From one-dimensional fields to Vlasov equilibria: theory and application of Hermite polynomials
NASA Astrophysics Data System (ADS)
Allanson, O.; Neukirch, T.; Troscheit, S.; Wilson, F.
2016-06-01
We consider the theory and application of a solution method for the inverse problem in collisionless equilibria, namely that of calculating a Vlasov-Maxwell equilibrium for a given macroscopic (fluid) equilibrium. Using Jeans' theorem, the equilibrium distribution functions are expressed as functions of the constants of motion, in the form of a Maxwellian multiplied by an unknown function of the canonical momenta. In this case it is possible to reduce the inverse problem to inverting Weierstrass transforms, which we achieve by using expansions over Hermite polynomials. A sufficient condition on the pressure tensor is found which guarantees the convergence and the boundedness of the candidate solution, when satisfied. This condition is obtained by elementary means, and it is clear how to put it into practice. We also argue that for a given pressure tensor for which our method applies, there always exists a positive distribution function solution for a sufficiently magnetised plasma. Illustrative examples of the use of this method with both force-free and non-force-free macroscopic equilibria are presented, including the full verification of a recently derived distribution function for the force-free Harris sheet (Allanson et al., Phys. Plasmas, vol. 22 (10), 2015, 102116). In the effort to model equilibria with lower values of the plasma β, solutions for the same macroscopic equilibrium in a new gauge are calculated, with numerical results presented for βpl=0.05.
Bou Matar, Olivier; Gasmi, Noura; Zhou, Huan; Goueygou, Marc; Talbi, Abdelkrim
2013-03-01
A numerical method to compute propagation constants and mode shapes of elastic waves in layered piezoelectric-piezomagnetic composites, potentially deposited on a substrate, is described. The basic feature of the method is the expansion of particle displacement, stress fields, electric and magnetic potentials in each layer on different polynomial bases: Legendre for a layer of finite thickness and Laguerre for the semi-infinite substrate. The exponential convergence rate of the method for propagation of Love waves is numerically verified. The main advantage of the method is to directly determine complex wave numbers for a given frequency via a matricial eigenvalue problem, in a way that no transcendental equation has to be solved. Results are presented and the method is discussed.
Load regulating expansion fixture
Wagner, Lawrence M.; Strum, Michael J.
1998-01-01
A free standing self contained device for bonding ultra thin metallic films, such as 0.001 inch beryllium foils. The device will regulate to a predetermined load for solid state bonding when heated to a bonding temperature. The device includes a load regulating feature, whereby the expansion stresses generated for bonding are regulated and self adjusting. The load regulator comprises a pair of friction isolators with a plurality of annealed copper members located therebetween. The device, with the load regulator, will adjust to and maintain a stress level needed to successfully and economically complete a leak tight bond without damaging thin foils or other delicate components.
Load regulating expansion fixture
Wagner, L.M.; Strum, M.J.
1998-12-15
A free standing self contained device for bonding ultra thin metallic films, such as 0.001 inch beryllium foils is disclosed. The device will regulate to a predetermined load for solid state bonding when heated to a bonding temperature. The device includes a load regulating feature, whereby the expansion stresses generated for bonding are regulated and self adjusting. The load regulator comprises a pair of friction isolators with a plurality of annealed copper members located therebetween. The device, with the load regulator, will adjust to and maintain a stress level needed to successfully and economically complete a leak tight bond without damaging thin foils or other delicate components. 1 fig.
NASA Astrophysics Data System (ADS)
Gelman, L.; Petrunin, I.; Komoda, J.
2010-02-01
The new chirp-Wigner higher order spectra (CWHOS) are proposed for transient signals with any known nonlinear polynomial variation of instantaneous frequency. The proposed technique is effective for nonlinearity detection for transient signals with nonlinear polynomial time variation of the instantaneous frequency.
Expansible quantum secret sharing network
NASA Astrophysics Data System (ADS)
Sun, Ying; Xu, Sheng-Wei; Chen, Xiu-Bo; Niu, Xin-Xin; Yang, Yi-Xian
2013-08-01
In the practical applications, member expansion is a usual demand during the development of a secret sharing network. However, there are few consideration and discussion on network expansibility in the existing quantum secret sharing schemes. We propose an expansible quantum secret sharing scheme with relatively simple and economical quantum resources and show how to split and reconstruct the quantum secret among an expansible user group in our scheme. Its trait, no requirement of any agent's assistant during the process of member expansion, can help to prevent potential menaces of insider cheating. We also give a discussion on the security of this scheme from three aspects.
The Corolla Polynomial for Spontaneously Broken Gauge Theories
NASA Astrophysics Data System (ADS)
Prinz, David
2016-09-01
In Kreimer and Yeats (Electr. J. Comb. 41-41, 2013), Kreimer et al. (Annals Phys. 336, 180-222, 2013) and Sars (2015) the Corolla Polynomial C ({Γ }) in C [a_{h1}, ldots , a_{h_{ \\vert {Γ }^{[1/2]} \\vert }}] was introduced as a graph polynomial in half-edge variables {ah}_{h in {Γ }^{[1/2]}} over a 3-regular scalar quantum field theory (QFT) Feynman graph Γ. It allows for a covariant quantization of pure Yang-Mills theory without the need for introducing ghost fields, clarifies the relation between quantum gauge theory and scalar QFT with cubic interaction and translates back the problem of renormalizing quantum gauge theory to the problem of renormalizing scalar QFT with cubic interaction (which is super renormalizable in 4 dimensions of spacetime). Furthermore, it is, as we believe, useful for computer calculations. In Prinz (2015) on which this paper is based the formulation of Kreimer and Yeats (Electr. J. Comb. 41-41, 2013), Kreimer et al. (Annals Phys. 336, 180-222, 2013) and Sars (2015) gets slightly altered in a fashion specialized in the case of the Feynman gauge. It is then formulated as a graph polynomial C ({Γ } ) in C [a_{h_{1 ± }}, ldots , a_{h_{ \\vert {Γ }^{[1/2]} \\vert } {h}_{± }}, b_{h1}, ldots , b_{h_{ \\vert {Γ }^{[1/2]} \\vert }}] in three different types of half-edge variables {a_{h+} , a_{h-} , bh}_{h in {Γ }^{[1/2]}} . This formulation is also suitable for the generalization to the case of spontaneously broken gauge theories (in particular all bosons from the Standard Model), as was first worked out in Prinz (2015) and gets reviewed here.
Polynomial search and global modeling: Two algorithms for modeling chaos.
Mangiarotti, S; Coudret, R; Drapeau, L; Jarlan, L
2012-10-01
Global modeling aims to build mathematical models of concise description. Polynomial Model Search (PoMoS) and Global Modeling (GloMo) are two complementary algorithms (freely downloadable at the following address: http://www.cesbio.ups-tlse.fr/us/pomos_et_glomo.html) designed for the modeling of observed dynamical systems based on a small set of time series. Models considered in these algorithms are based on ordinary differential equations built on a polynomial formulation. More specifically, PoMoS aims at finding polynomial formulations from a given set of 1 to N time series, whereas GloMo is designed for single time series and aims to identify the parameters for a selected structure. GloMo also provides basic features to visualize integrated trajectories and to characterize their structure when it is simple enough: One allows for drawing the first return map for a chosen Poincaré section in the reconstructed space; another one computes the Lyapunov exponent along the trajectory. In the present paper, global modeling from single time series is considered. A description of the algorithms is given and three examples are provided. The first example is based on the three variables of the Rössler attractor. The second one comes from an experimental analysis of the copper electrodissolution in phosphoric acid for which a less parsimonious global model was obtained in a previous study. The third example is an exploratory case and concerns the cycle of rainfed wheat under semiarid climatic conditions as observed through a vegetation index derived from a spatial sensor.
Ibañez, Diego Rodríguez; Gómez-Pedrero, José A; Alonso, Jose; Quiroga, Juan A
2016-03-21
A new method for fitting a series of Zernike polynomials to point clouds defined over connected domains of arbitrary shape defined within the unit circle is presented in this work. The method is based on the application of machine learning fitting techniques by constructing an extended training set in order to ensure the smooth variation of local curvature over the whole domain. Therefore this technique is best suited for fitting points corresponding to ophthalmic lenses surfaces, particularly progressive power ones, in non-regular domains. We have tested our method by fitting numerical and real surfaces reaching an accuracy of 1 micron in elevation and 0.1 D in local curvature in agreement with the customary tolerances in the ophthalmic manufacturing industry.
A ROM-Less Direct Digital Frequency Synthesizer Based on Hybrid Polynomial Approximation
Omran, Qahtan Khalaf; Islam, Mohammad Tariqul; Misran, Norbahiah; Faruque, Mohammad Rashed Iqbal
2014-01-01
In this paper, a novel design approach for a phase to sinusoid amplitude converter (PSAC) has been investigated. Two segments have been used to approximate the first sine quadrant. A first linear segment is used to fit the region near the zero point, while a second fourth-order parabolic segment is used to approximate the rest of the sine curve. The phase sample, where the polynomial changed, was chosen in such a way as to achieve the maximum spurious free dynamic range (SFDR). The invented direct digital frequency synthesizer (DDFS) has been encoded in VHDL and post simulation was carried out. The synthesized architecture exhibits a promising result of 90 dBc SFDR. The targeted structure is expected to show advantages for perceptible reduction of hardware resources and power consumption as well as high clock speeds. PMID:24892092
A ROM-less direct digital frequency synthesizer based on hybrid polynomial approximation.
Omran, Qahtan Khalaf; Islam, Mohammad Tariqul; Misran, Norbahiah; Faruque, Mohammad Rashed Iqbal
2014-01-01
In this paper, a novel design approach for a phase to sinusoid amplitude converter (PSAC) has been investigated. Two segments have been used to approximate the first sine quadrant. A first linear segment is used to fit the region near the zero point, while a second fourth-order parabolic segment is used to approximate the rest of the sine curve. The phase sample, where the polynomial changed, was chosen in such a way as to achieve the maximum spurious free dynamic range (SFDR). The invented direct digital frequency synthesizer (DDFS) has been encoded in VHDL and post simulation was carried out. The synthesized architecture exhibits a promising result of 90 dBc SFDR. The targeted structure is expected to show advantages for perceptible reduction of hardware resources and power consumption as well as high clock speeds.
Generalized MICZ-Kepler system, duality, polynomial, and deformed oscillator algebras
Marquette, Ian
2010-10-15
We present the quadratic algebra of the generalized MICZ-Kepler system in three-dimensional Euclidean space E{sub 3} and its dual, the four-dimensional singular oscillator, in four-dimensional Euclidean space E{sub 4}. We present their realization in terms of a deformed oscillator algebra using the Daskaloyannis construction. The structure constants are, in these cases, functions not only of the Hamiltonian but also of other integrals commuting with all generators of the quadratic algebra. We also present a new algebraic derivation of the energy spectrum of the MICZ-Kepler system on the three sphere S{sup 3} using a quadratic algebra. These results point out also that results and explicit formula for structure functions obtained for quadratic, cubic, and higher order polynomial algebras in the context of two-dimensional superintegrable systems may be applied to superintegrable systems in higher dimensions with and without monopoles.
Ayouz, Mehdi; Babikov, Dmitri
2012-01-01
New global potential energy surface for the ground electronic state of ozone is constructed at the complete basis set level of the multireference configuration interaction theory. A method of fitting the data points by analytical permutationally invariant polynomial function is adopted. A small set of 500 points is preoptimized using the old surface of ozone. In this procedure the positions of points in the configuration space are chosen such that the RMS deviation of the fit is minimized. New ab initio calculations are carried out at these points and are used to build new surface. Additional points are addedmore » to the vicinity of the minimum energy path in order to improve accuracy of the fit, particularly in the region where the surface of ozone exhibits a shallow van der Waals well. New surface can be used to study formation of ozone at thermal energies and its spectroscopy near the dissociation threshold.« less
NASA Technical Reports Server (NTRS)
Yevdokimov, V. P.; Natenzon, M. Y.
1980-01-01
The effectiveness of the algorithm of polynomial striction of the data is examined, and the selection of an algorithm is carried out, according to the results of measurements of the components of the permanent magnetic field in communication sequences from the Venera-9 space vehicle, where the shock waves were recorded. It is shown that the most effective algorithm, according to the coefficient of striction, is the algorithm of the zero order interpolator. Also examined is the increase in effectiveness with the introduction of a variable threshold of comparison. The root-mean-square deviation of the error of regeneration of the data with various thresholds is calculated, and qualitative analysis of the distortions is carried out, having shown a considerable change in the form of the small discontinuities, which may be interpreted as potential interplanetary shock waves.
Generalized MICZ-Kepler system, duality, polynomial, and deformed oscillator algebras
NASA Astrophysics Data System (ADS)
Marquette, Ian
2010-10-01
We present the quadratic algebra of the generalized MICZ-Kepler system in three-dimensional Euclidean space E3 and its dual, the four-dimensional singular oscillator, in four-dimensional Euclidean space E4. We present their realization in terms of a deformed oscillator algebra using the Daskaloyannis construction. The structure constants are, in these cases, functions not only of the Hamiltonian but also of other integrals commuting with all generators of the quadratic algebra. We also present a new algebraic derivation of the energy spectrum of the MICZ-Kepler system on the three sphere S3 using a quadratic algebra. These results point out also that results and explicit formula for structure functions obtained for quadratic, cubic, and higher order polynomial algebras in the context of two-dimensional superintegrable systems may be applied to superintegrable systems in higher dimensions with and without monopoles.
Charge-based MOSFET model based on the Hermite interpolation polynomial
NASA Astrophysics Data System (ADS)
Colalongo, Luigi; Richelli, Anna; Kovacs, Zsolt
2017-04-01
An accurate charge-based compact MOSFET model is developed using the third order Hermite interpolation polynomial to approximate the relation between surface potential and inversion charge in the channel. This new formulation of the drain current retains the same simplicity of the most advanced charge-based compact MOSFET models such as BSIM, ACM and EKV, but it is developed without requiring the crude linearization of the inversion charge. Hence, the asymmetry and the non-linearity in the channel are accurately accounted for. Nevertheless, the expression of the drain current can be worked out to be analytically equivalent to BSIM, ACM and EKV. Furthermore, thanks to this new mathematical approach the slope factor is rigorously defined in all regions of operation and no empirical assumption is required.