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
Lattice Boltzmann method for bosons and fermions and the fourth-order Hermite polynomial expansion.
Coelho, Rodrigo C V; Ilha, Anderson; Doria, Mauro M; Pereira, R M; Aibe, Valter Yoshihiko
2014-04-01
The Boltzmann equation with the Bhatnagar-Gross-Krook collision operator is considered for the Bose-Einstein and Fermi-Dirac equilibrium distribution functions. We show that the expansion of the microscopic velocity in terms of Hermite polynomials must be carried to the fourth order to correctly describe the energy equation. The viscosity and thermal coefficients, previously obtained by Yang et al. [Shi and Yang, J. Comput. Phys. 227, 9389 (2008); Yang and Hung, Phys. Rev. E 79, 056708 (2009)] through the Uehling-Uhlenbeck approach, are also derived here. Thus the construction of a lattice Boltzmann method for the quantum fluid is possible provided that the Bose-Einstein and Fermi-Dirac equilibrium distribution functions are expanded to fourth order in the Hermite polynomials. PMID:24827360
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.
Affine and deformable registration based on polynomial expansion.
Farnebäck, Gunnar; Westin, Carl-Fredrik
2006-01-01
This paper presents a registration framework based on the polynomial expansion transform. The idea of polynomial expansion is that the image is locally approximated by polynomials at each pixel. Starting with observations of how the coefficients of ideal linear and quadratic polynomials change under translation and affine transformation, algorithms are developed to estimate translation and compute affine and deformable registration between a fixed and a moving image, from the polynomial expansion coefficients. All algorithms can be used for signals of any dimensionality. The algorithms are evaluated on medical data. PMID:17354971
Explicit energy expansion for general odd-degree polynomial potentials
NASA Astrophysics Data System (ADS)
Nanayakkara, Asiri; Mathanaranjan, Thilagarajah
2013-11-01
In this paper we derive an almost explicit analytic formula for asymptotic eigenenergy expansion of arbitrary odd-degree polynomial potentials of the form V (x) = (ix)2N+1 + β1x2N + β2x2N-1 + ··· + β2Nx, where β‧k are real or complex for 1 ⩽ k ⩽ 2N. The formula can be used to find semiclassical analytic expressions for eigenenergies up to any order, very efficiently. Each term of the expansion is given explicitly as a multinomial of the parameters β1,β2… and β2N of the potential. Unlike in the even-degree polynomial case, the highest-order term in the potential is pure imaginary and hence the system is non-Hermitian. Therefore all the integrations have been carried out along a contour enclosing two complex branch points, which lies within a wedge in the complex plane. With the help of some examples we demonstrate the accuracy of the method for both real and complex eigenspectra.
Optical homodyne tomography with polynomial series expansion
Benichi, Hugo; Furusawa, Akira
2011-09-15
We present and demonstrate a method for optical homodyne tomography based on the inverse Radon transform. Different from the usual filtered back-projection algorithm, this method uses an appropriate polynomial series to expand the Wigner function and the marginal distribution, and discretize Fourier space. We show that this technique solves most technical difficulties encountered with kernel deconvolution-based methods and reconstructs overall better and smoother Wigner functions. We also give estimators of the reconstruction errors for both methods and show improvement in noise handling properties and resilience to statistical errors.
Compressive sampling of polynomial chaos expansions: Convergence analysis and sampling strategies
Hampton, Jerrad; Doostan, Alireza
2015-01-01
Sampling orthogonal polynomial bases via Monte Carlo is of interest for uncertainty quantification of models with random inputs, using Polynomial Chaos (PC) expansions. It is known that bounding a probabilistic parameter, referred to as coherence, yields a bound on the number of samples necessary to identify coefficients in a sparse PC expansion via solution to an ℓ{sub 1}-minimization problem. Utilizing results for orthogonal polynomials, we bound the coherence parameter for polynomials of Hermite and Legendre type under their respective natural sampling distribution. In both polynomial bases we identify an importance sampling distribution which yields a bound with weaker dependence on the order of the approximation. For more general orthonormal bases, we propose the coherence-optimal sampling: a Markov Chain Monte Carlo sampling, which directly uses the basis functions under consideration to achieve a statistical optimality among all sampling schemes with identical support. We demonstrate these different sampling strategies numerically in both high-order and high-dimensional, manufactured PC expansions. In addition, the quality of each sampling method is compared in the identification of solutions to two differential equations, one with a high-dimensional random input and the other with a high-order PC expansion. In both cases, the coherence-optimal sampling scheme leads to similar or considerably improved accuracy.
Polynomial chaos expansion with random and fuzzy variables
NASA Astrophysics Data System (ADS)
Jacquelin, E.; Friswell, M. I.; Adhikari, S.; Dessombz, O.; Sinou, J.-J.
2016-06-01
A dynamical uncertain system is studied in this paper. Two kinds of uncertainties are addressed, where the uncertain parameters are described through random variables and/or fuzzy variables. A general framework is proposed to deal with both kinds of uncertainty using a polynomial chaos expansion (PCE). It is shown that fuzzy variables may be expanded in terms of polynomial chaos when Legendre polynomials are used. The components of the PCE are a solution of an equation that does not depend on the nature of uncertainty. Once this equation is solved, the post-processing of the data gives the moments of the random response when the uncertainties are random or gives the response interval when the variables are fuzzy. With the PCE approach, it is also possible to deal with mixed uncertainty, when some parameters are random and others are fuzzy. The results provide a fuzzy description of the response statistical moments.
Zhang, Yan; Sahinidis, Nikolaos V
2013-04-06
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.
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.
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. PMID:25791164
Higher order derivatives of R-Jacobi polynomials
NASA Astrophysics Data System (ADS)
Das, Sourav; Swaminathan, A.
2016-06-01
In this work, the R-Jacobi polynomials defined on the nonnegative real axis related to F-distribution are considered. Using their Sturm-Liouville system higher order derivatives are constructed. Orthogonality property of these higher ordered R-Jacobi polynomials are obtained besides their normal form, self-adjoint form and hypergeometric representation. Interesting results on the Interpolation formula and Gaussian quadrature formulae are obtained with numerical examples.
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
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
Rising, M. E.; Prinja, A. K.
2012-07-01
A critical neutron transport problem with random material properties is introduced. The total cross section and the average neutron multiplicity are assumed to be uncertain, characterized by the mean and variance with a log-normal distribution. The average neutron multiplicity and the total cross section are assumed to be uncorrected and the material properties for differing materials are also assumed to be uncorrected. The principal component analysis method is used to decompose the covariance matrix into eigenvalues and eigenvectors and then 'realizations' of the material properties can be computed. A simple Monte Carlo brute force sampling of the decomposed covariance matrix is employed to obtain a benchmark result for each test problem. In order to save computational time and to characterize the moments and probability density function of the multiplication factor the polynomial chaos expansion method is employed along with the stochastic collocation method. A Gauss-Hermite quadrature set is convolved into a multidimensional tensor product quadrature set and is successfully used to compute the polynomial chaos expansion coefficients of the multiplication factor. Finally, for a particular critical fuel pin assembly the appropriate number of random variables and polynomial expansion order are investigated. (authors)
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.
Simulation of stochastic systems via polynomial chaos expansions and convex optimization
NASA Astrophysics Data System (ADS)
Fagiano, Lorenzo; Khammash, Mustafa
2012-09-01
Polynomial chaos expansions represent a powerful tool to simulate stochastic models of dynamical systems. Yet, deriving the expansion's coefficients for complex systems might require a significant and nontrivial manipulation of the model, or the computation of large numbers of simulation runs, rendering the approach too time consuming and impracticable for applications with more than a handful of random variables. We introduce a computationally tractable technique for computing the coefficients of polynomial chaos expansions. The approach exploits a regularization technique with a particular choice of weighting matrices, which allows to take into account the specific features of polynomial chaos expansions. The method, completely based on convex optimization, can be applied to problems with a large number of random variables and uses a modest number of Monte Carlo simulations, while avoiding model manipulations. Additional information on the stochastic process, when available, can be also incorporated in the approach by means of convex constraints. We show the effectiveness of the proposed technique in three applications in diverse fields, including the analysis of a nonlinear electric circuit, a chaotic model of organizational behavior, and finally a chemical oscillator.
NASA Astrophysics Data System (ADS)
Rajabi, Mohammad Mahdi; Ataie-Ashtiani, Behzad; Simmons, Craig T.
2015-01-01
Real world models of seawater intrusion (SWI) require high computational efforts. This creates computational difficulties for the uncertainty propagation (UP) analysis of these models due the need for repeated numerical simulations in order to adequately capture the underlying statistics that describe the uncertainty in model outputs. Moreover, despite the obvious advantages of moment-independent global sensitivity analysis (SA) methods, these methods have rarely been employed for SWI and other complex groundwater models. The reason is that moment-independent global SA methods involve repeated UP analysis which further becomes computationally demanding. This study proposes the use of non-intrusive polynomial chaos expansions (PCEs) as a means to significantly accelerate UP analysis in SWI numerical modeling studies and shows that despite the highly non-linear and non-smooth input/output relationship that exists in SWI models, non-intrusive PCEs provide a reliable and yet computationally efficient surrogate of the original numerical model. The study illustrates that for the considered two and six dimensional UP problems, PCEs offer a more accurate estimation of the statistics describing the uncertainty in model outputs compared to Monte Carlo simulations based on the original numerical model. This study also shows that the use of non-intrusive PCEs in the estimation of the moment-independent sensitivity indices (i.e. delta indices) decreases the computational time by several orders of magnitude without causing significant loss of accuracy. The use of non-intrusive PCEs for the generation of SWI hazard maps is proposed to extend the practical applications of UP analysis in coastal aquifer management studies.
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. PMID:27227661
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.
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.
Polynomial Solutions of Nth Order Non-Homogeneous Differential Equations
ERIC Educational Resources Information Center
Levine, Lawrence E.; Maleh, Ray
2002-01-01
It was shown by Costa and Levine that the homogeneous differential equation (1-x[superscript N])y([superscript N]) + A[subscript N-1]x[superscript N-1)y([superscript N-1]) + A[subscript N-2]x[superscript N-2])y([superscript N-2]) + ... + A[subscript 1]xy[prime] + A[subscript 0]y = 0 has a finite polynomial solution if and only if [for…
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
NASA Astrophysics Data System (ADS)
Porta, G.; Tamellini, L.; Lever, V.; Riva, M.
2014-12-01
We present an inverse modeling procedure for the estimation of model parameters of sedimentary basins subject to compaction driven by mechanical and geochemical processes. We consider a sandstone basin whose dynamics are governed by a set of unknown key quantities. These include geophysical and geochemical system attributes as well as pressure and temperature boundary conditions. We derive a reduced (or surrogate) model of the system behavior based on generalized Polynomial Chaos Expansion (gPCE) approximations, which are directly linked to the variance-based Sobol indices associated with the selected uncertain model parameters. Parameter estimation is then performed within a Maximum Likelihood (ML) framework. We then study the way the ML inversion procedure can benefit from the adoption of anisotropic polynomial approximations (a-gPCE) in which the surrogate model is refined only with respect to selected parameters according to an analysis of the nonlinearity of the input-output mapping, as quantified through the Sobol sensitivity indices. Results are illustrated for a one-dimensional setting involving quartz cementation and mechanical compaction in sandstones. The reliability of gPCE and a-gPCE approximations in the context of the inverse modeling framework is assessed. The effects of (a) the strategy employed to build the surrogate model, leading either to a gPCE or a-gPCE representation, and (b) the type and quality of calibration data on the goodness of the parameter estimates is then explored.
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.
Effects of random stiffness variations in multistage rotors using the Polynomial Chaos Expansion
NASA Astrophysics Data System (ADS)
Seguí, B.; Faverjon, B.; Jacquet-Richardet, G.
2013-09-01
The paper presents a methodology that allows the investigation of the effects of random uncertainties on the global dynamics of multistage bladed discs systems. Uncertainties are accounted for as variations in the material properties of the blades. The multistage cyclic symmetry assumption is used to reduce the global problem. The random dynamics of the global system is obtained by applying the Polynomial Chaos Expansion. The methodology is applied to a two stage bladed disc assembly and the results of modal and forced response analysis are validated by comparisons with Monte-Carlo simulations. Possible interactions of multistage mode families in zones of high modal density due to uncertainties in the blades are discussed. Results obtained show that uncertainties may induce significant changes in the global dynamics of multistage assemblies and the proposed technique is shown to be efficient to capture those changes. The study classically evaluates the variations of frequencies and responses but also shows that the nature of mode shapes may be drastically affected by uncertainties.
On polynomial chaos expansion via gradient-enhanced ℓ1-minimization
NASA Astrophysics Data System (ADS)
Peng, Ji; Hampton, Jerrad; Doostan, Alireza
2016-04-01
Gradient-enhanced Uncertainty Quantification (UQ) has received recent attention, in which the derivatives of a Quantity of Interest (QoI) with respect to the uncertain parameters are utilized to improve the surrogate approximation. Polynomial chaos expansions (PCEs) are often employed in UQ, and when the QoI can be represented by a sparse PCE, ℓ1-minimization can identify the PCE coefficients with a relatively small number of samples. In this work, we investigate a gradient-enhanced ℓ1-minimization, where derivative information is computed to accelerate the identification of the PCE coefficients. For this approach, stability and convergence analysis are lacking, and thus we address these here with a probabilistic result. In particular, with an appropriate normalization, we show the inclusion of derivative information will almost-surely lead to improved conditions, e.g. related to the null-space and coherence of the measurement matrix, for a successful solution recovery. Further, we demonstrate our analysis empirically via three numerical examples: a manufactured PCE, an elliptic partial differential equation with random inputs, and a plane Poiseuille flow with random boundaries. These examples all suggest that including derivative information admits solution recovery at reduced computational cost.
NASA Astrophysics Data System (ADS)
Wang, Zhengzi
2015-08-01
The influence of ambient temperature is a big challenge to robust infrared face recognition. This paper proposes a new ambient temperature normalization algorithm to improve the performance of infrared face recognition under variable ambient temperatures. Based on statistical regression theory, a second order polynomial model is learned to describe the ambient temperature's impact on infrared face image. Then, infrared image was normalized to reference ambient temperature by the second order polynomial model. Finally, this normalization method is applied to infrared face recognition to verify its efficiency. The experiments demonstrate that the proposed temperature normalization method is feasible and can significantly improve the robustness of infrared face recognition.
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.
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.
On P -orderings, rings of integer-valued polynomials, and ultrametric analysis
NASA Astrophysics Data System (ADS)
Bhargava, Manjul
2009-10-01
We introduce two new notions of `` P -ordering'' and use them to define a three-parameter generalization of the usual factorial function. We then apply these notions of P -orderings and factorials to some classical problems in two distinct areas, namely: 1) the study of integer-valued polynomials and 2) P -adic analysis. Specifically, we first use these notions of P -orderings and factorials to construct explicit Polya-style regular bases for two natural families of rings of integer-valued polynomials defined on an arbitrary subset of a Dedekind domain. Second, we classify ``smooth'' functions on an arbitrary compact subset S of a local field, by constructing explicit interpolation series (i.e., orthonormal bases) for the Banach space of functions on S satisfying any desired conditions of continuous differentiability or local analyticity. Our constructions thus extend Mahler's Theorem (classifying the functions that are continuous on {Z}_p ) to a very general setting. In particular, our constructions prove that, for any epsilon>0 , the functions in any of the above Banach spaces can be epsilon -approximated by polynomials (with respect to their respective Banach norms). Thus we obtain the non-Archimedean analogues of the classical polynomial approximation theorems in real and complex analysis proven by Weierstrass, de la Vallee-Poussin, and Bernstein. Our proofs are effective.
Higher-order numerical methods derived from three-point polynomial interpolation
NASA Technical Reports Server (NTRS)
Rubin, S. G.; Khosla, P. K.
1976-01-01
Higher-order collocation procedures resulting in tridiagonal matrix systems are derived from polynomial spline interpolation and Hermitian finite-difference discretization. The equations generally apply for both uniform and variable meshes. Hybrid schemes resulting from different polynomial approximations for first and second derivatives lead to the nonuniform mesh extension of the so-called compact or Pade difference techniques. A variety of fourth-order methods are described and this concept is extended to sixth-order. Solutions with these procedures are presented for the similar and non-similar boundary layer equations with and without mass transfer, the Burgers equation, and the incompressible viscous flow in a driven cavity. Finally, the interpolation procedure is used to derive higher-order temporal integration schemes and results are shown for the diffusion equation.
A comparison of high-order polynomial and wave-based methods for Helmholtz problems
NASA Astrophysics Data System (ADS)
Lieu, Alice; Gabard, Gwénaël; Bériot, Hadrien
2016-09-01
The application of computational modelling to wave propagation problems is hindered by the dispersion error introduced by the discretisation. Two common strategies to address this issue are to use high-order polynomial shape functions (e.g. hp-FEM), or to use physics-based, or Trefftz, methods where the shape functions are local solutions of the problem (typically plane waves). Both strategies have been actively developed over the past decades and both have demonstrated their benefits compared to conventional finite-element methods, but they have yet to be compared. In this paper a high-order polynomial method (p-FEM with Lobatto polynomials) and the wave-based discontinuous Galerkin method are compared for two-dimensional Helmholtz problems. A number of different benchmark problems are used to perform a detailed and systematic assessment of the relative merits of these two methods in terms of interpolation properties, performance and conditioning. It is generally assumed that a wave-based method naturally provides better accuracy compared to polynomial methods since the plane waves or Bessel functions used in these methods are exact solutions of the Helmholtz equation. Results indicate that this expectation does not necessarily translate into a clear benefit, and that the differences in performance, accuracy and conditioning are more nuanced than generally assumed. The high-order polynomial method can in fact deliver comparable, and in some cases superior, performance compared to the wave-based DGM. In addition to benchmarking the intrinsic computational performance of these methods, a number of practical issues associated with realistic applications are also discussed.
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. PMID:26097104
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
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
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.
Drift kinetic equation exact through second order in gyroradius expansion
Simakov, Andrei N.; Catto, Peter J.
2005-01-01
The drift kinetic equation of Hazeltine [R. D. Hazeltine, Plasma Phys. 15, 77 (1973)] for a magnetized plasma of arbitrary collisionality is widely believed to be exact through the second order in the gyroradius expansion. It is demonstrated that this equation is only exact through the first order. The reason is that when evaluating the second-order gyrophase dependent distribution function, Hazeltine neglected contributions from the first-order gyrophase dependent distribution function, and then used this incomplete expression to derive the equation for the gyrophase independent distribution function. Consequently, the second-order distribution function and the stress tensor derived by this approach are incomplete. By relaxing slightly Hazeltine's orderings one is able to obtain a drift kinetic equation accurate through the second order in the gyroradius expansion. In addition, the gyroviscous stress tensor for plasmas of arbitrary collisionality is obtained.
NASA Astrophysics Data System (ADS)
Alvarez, G.; Şen, C.; Furukawa, N.; Motome, Y.; Dagotto, E.
2005-05-01
A software library is presented for the polynomial expansion method (PEM) of the density of states (DOS) introduced in [Y. Motome, N. Furukawa, J. Phys. Soc. Japan 68 (1999) 3853; N. Furukawa, Y. Motome, H. Nakata, Comput. Phys. Comm. 142 (2001) 410]. 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 [N. Furukawa, Y. Motome, J. Phys. Soc. Japan 73 (2004) 1482] which should be contrasted with the computational cost of the diagonalization technique that scales as O(N). The method is applied for the first time to a double exchange model with finite Hund coupling and also to diluted spin-fermion models. Program summaryTitle of library:TPEM Catalogue identifier: ADVK Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADVK Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland No. of lines in distributed program, including test data, etc.: 1707 No. of bytes in distributed program, including test data, etc.: 13 644 Distribution format:tar.gz Operating system:Linux, UNIX Number of files:4 plus 1 test program Programming language used:C Computer:PC Nature of the physical problem:The study of correlated electrons coupled to classical fields appears in the treatment of many materials of much current interest in condensed matter theory, e.g., manganites, diluted magnetic semiconductors and high temperature superconductors among others. Method of solution: Typically an exact diagonalization of the electronic sector is performed in this type of models for each configuration of classical fields, which are integrated using a classical Monte Carlo algorithm. A polynomial expansion of the density of states is able to replace the exact
Phantom Friedmann cosmologies and higher-order characteristics of expansion
Dabrowski, Mariusz P. . E-mail: mpdabfz@sus.univ.szczecin.pl; Stachowiak, Tomasz . E-mail: toms@oa.uj.edu.pl
2006-04-15
We discuss a more general class of phantom (p < -{rho}) cosmologies with various forms of both phantom (w < -1), and standard (w > -1) matter. We show that many types of evolution which include both Big-Bang and Big-Rip singularities are admitted and give explicit examples. Among some interesting models, there exist non-singular oscillating (or 'bounce') cosmologies, which appear due to a competition between positive and negative pressure of variety of matter content. From the point of view of the current observations the most interesting cosmologies are the ones which start with a Big-Bang and terminate at a Big-Rip. A related consequence of having a possibility of two types of singularities is that there exists an unstable static universe approached by the two asymptotic models-one of them reaches Big-Bang, and another reaches Big-Rip. We also give explicit relations between density parameters {omega} and the dynamical characteristics for these generalized phantom models, including higher-order observational characteristics such as jerk and 'kerk.' Finally, we discuss the observational quantities such as luminosity distance, angular diameter, and source counts, both in series expansion and explicitly, for phantom models. Our series expansion formulas for the luminosity distance and the apparent magnitude go as far as to the fourth-order in redshift z term, which includes explicitly not only the jerk, but also the 'kerk' (or 'snap') which may serve as an indicator of the curvature of the universe.
Rdzanek, Wojciech P
2016-06-01
This study deals with the classical problem of sound radiation of an excited clamped circular plate embedded into a flat rigid baffle. The system of the two coupled differential equations is solved, one for the excited and damped vibrations of the plate and the other one-the Helmholtz equation. An approach using the expansion into radial polynomials leads to results for the modal impedance coefficients useful for a comprehensive numerical analysis of sound radiation. The results obtained are accurate and efficient in a wide low frequency range and can easily be adopted for a simply supported circular plate. The fluid loading is included providing accurate results in resonance. PMID:27369144
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.
NASA Astrophysics Data System (ADS)
Triki, Houria; Porsezian, K.; Grelu, Philippe
2016-07-01
A generalized nonlinear Schrödinger equation with polynomial Kerr nonlinearity and non-Kerr terms of an arbitrarily higher order is investigated. This model can be applied to the femtosecond pulse propagation in highly-nonlinear optical media. We introduce a new chirping ansatz given as an expansion in powers of intensity of the light pulse and obtain both linear and nonlinear chirp contributions associated with propagating optical pulses. By taking the cubic-quintic-septic-nonic nonlinear Schrödinger (NLS) equation with seventh-order non-Kerr terms as an example for the generalized equation with Kerr and non-Kerr nonlinearity of arbitrary order, we derive families of chirped soliton solutions under certain parametric conditions. The solutions comprise bright, kink, anti-kink, and fractional-transform soliton solutions. In addition, we found the exact soliton solution for the model under consideration using a new ansatz. The parametric conditions for the existence of chirped solitons are also reported.
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
NASA Astrophysics Data System (ADS)
Chang, Phang; Isah, Abdulnasir
2016-02-01
In this paper we propose the wavelet operational method based on shifted Legendre polynomial to obtain the numerical solutions of nonlinear fractional-order chaotic system known by fractional-order Brusselator system. The operational matrices of fractional derivative and collocation method turn 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.
Conservation laws and a new expansion method for sixth order Boussinesq equation
NASA Astrophysics Data System (ADS)
Yokuş, Asıf; Kaya, Doǧan
2015-09-01
In this study, we analyze the conservation laws of a sixth order Boussinesq equation by using variational derivative. We get sixth order Boussinesq equation's traveling wave solutions with (1/G) -expansion method which we constitute newly by being inspired with (G/G) -expansion method which is suggested in [1]. We investigate conservation laws of the analytical solutions which we obtained by the new constructed method. The analytical solution's conductions which we get according to new expansion method are given graphically.
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.
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…
NASA Astrophysics Data System (ADS)
Laksâ, Arne
2015-11-01
B-splines are the de facto industrial standard for surface modelling in Computer Aided design. It is comparable to bend flexible rods of wood or metal. A flexible rod minimize the energy when bending, a third degree polynomial spline curve minimize the second derivatives. B-spline is a nice way of representing polynomial splines, it connect polynomial splines to corner cutting techniques, which induces many nice and useful properties. However, the B-spline representation can be expanded to something we can call general B-splines, i.e. both polynomial and non-polynomial splines. We will show how this expansion can be done, and the properties it induces, and examples of non-polynomial B-spline.
Some discrete multiple orthogonal polynomials
NASA Astrophysics Data System (ADS)
Arvesú, J.; Coussement, J.; van Assche, W.
2003-04-01
In this paper, we extend the theory of discrete orthogonal polynomials (on a linear lattice) to polynomials satisfying orthogonality conditions with respect to r positive discrete measures. First we recall the known results of the classical orthogonal polynomials of Charlier, Meixner, Kravchuk and Hahn (T.S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978; R. Koekoek and R.F. Swarttouw, Reports of the Faculty of Technical Mathematics and Informatics No. 98-17, Delft, 1998; A.F. Nikiforov et al., Classical Orthogonal Polynomials of a Discrete Variable, Springer, Berlin, 1991). These polynomials have a lowering and raising operator, which give rise to a Rodrigues formula, a second order difference equation, and an explicit expression from which the coefficients of the three-term recurrence relation can be obtained. Then we consider r positive discrete measures and define two types of multiple orthogonal polynomials. The continuous case (Jacobi, Laguerre, Hermite, etc.) was studied by Van Assche and Coussement (J. Comput. Appl. Math. 127 (2001) 317-347) and Aptekarev et al. (Multiple orthogonal polynomials for classical weights, manuscript). The families of multiple orthogonal polynomials (of type II) that we will study have a raising operator and hence a Rodrigues formula. This will give us an explicit formula for the polynomials. Finally, there also exists a recurrence relation of order r+1 for these multiple orthogonal polynomials of type II. We compute the coefficients of the recurrence relation explicitly when r=2.
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.
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.
NASA Astrophysics Data System (ADS)
Calogero, Francesco
2013-01-01
Some properties of a solvable N-body problem featuring several free parameters ("coupling constants") are investigated. Restrictions on its parameters are reported which guarantee that all its solutions are completely periodic with a fixed period independent of the initial data (isochrony). The restrictions on its parameters which guarantee the existence of equilibria are also identified. In this connection a remarkable second-order ODE—generally not of hypergeometric type, hence not reducible to those characterizing the classical polynomials—is studied: if its parameters satisfy a Diophantine condition, its general solution is a polynomial of degree N, the N zeros of which identify the equilibria of the N-body system.
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. 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…
NASA Astrophysics Data System (ADS)
Moraes, P. H. R. S.; Ribeiro, G.; Correa, R. A. C.
2016-07-01
In this work we present cosmological solutions from the simplest non-trivial polynomial function of T in f(R,T) theory of gravity, with R and T standing for the Ricci scalar and trace of the energy-momentum tensor, respectively. Although such an approach yields a highly non-linear differential equation for the scale factor, we show that it is possible to obtain analytical solutions for the cosmological parameters. For some values of the free parameters, the model is able to predict a transition from a decelerated to an accelerated expansion of the universe and the values of the deceleration parameter agree with observation.
The S-ordered Operator Expansions of One-mode and Two-mode Fresnel Operators and their Applications
NASA Astrophysics Data System (ADS)
Du, Jian-ming; Ren, Gang; Yu, Hai-jun; Zhang, Wen-hai
2016-08-01
By using the technique of integration within the s-ordered product of operators (IWSOP), we first deduce the s-ordered expansion of the one-mode and two-mode Fresnel operators. Employing the s-ordered operator expansion formula, the matrix elements of one-mode and two-mode Fresnel operator in the number state representation are also obtained, respectively.
Spreading lengths of Hermite polynomials
NASA Astrophysics Data System (ADS)
Sánchez-Moreno, P.; Dehesa, J. S.; Manzano, D.; Yáñez, R. J.
2010-03-01
The Renyi, Shannon and Fisher spreading lengths of the classical or hypergeometric orthogonal polynomials, which are quantifiers of their distribution all over the orthogonality interval, are defined and investigated. These information-theoretic measures of the associated Rakhmanov probability density, which are direct measures of the polynomial spreading in the sense of having the same units as the variable, share interesting properties: invariance under translations and reflections, linear scaling and vanishing in the limit that the variable tends towards a given definite value. The expressions of the Renyi and Fisher lengths for the Hermite polynomials are computed in terms of the polynomial degree. The combinatorial multivariable Bell polynomials, which are shown to characterize the finite power of an arbitrary polynomial, play a relevant role for the computation of these information-theoretic lengths. Indeed these polynomials allow us to design an error-free computing approach for the entropic moments (weighted Lq-norms) of Hermite polynomials and subsequently for the Renyi and Tsallis entropies, as well as for the Renyi spreading lengths. Sharp bounds for the Shannon length of these polynomials are also given by means of an information-theoretic-based optimization procedure. Moreover, the existence of a linear correlation between the Shannon length (as well as the second-order Renyi length) and the standard deviation is computationally proved. Finally, the application to the most popular quantum-mechanical prototype system, the harmonic oscillator, is discussed and some relevant asymptotical open issues related to the entropic moments, mentioned previously, are posed.
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 .
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.
Hermite base Bernoulli type polynomials on the umbral algebra
NASA Astrophysics Data System (ADS)
Dere, R.; Simsek, Y.
2015-01-01
The aim of this paper is to construct new generating functions for Hermite base Bernoulli type polynomials, which generalize not only the Milne-Thomson polynomials but also the two-variable Hermite polynomials. We also modify the Milne-Thomson polynomials, which are related to the Bernoulli polynomials and the Hermite polynomials. Moreover, by applying the umbral algebra to these generating functions, we derive new identities for the Bernoulli polynomials of higher order, the Hermite polynomials and numbers of higher order, and the Stirling numbers of the second kind.
NASA Astrophysics Data System (ADS)
Laloy, Eric; Rogiers, Bart; Vrugt, Jasper A.; Mallants, Dirk; Jacques, Diederik
2013-05-01
This study reports on two strategies for accelerating posterior inference of a highly parameterized and CPU-demanding groundwater flow model. Our method builds on previous stochastic collocation approaches, e.g., Marzouk and Xiu (2009) and Marzouk and Najm (2009), and uses generalized polynomial chaos (gPC) theory and dimensionality reduction to emulate the output of a large-scale groundwater flow model. The resulting surrogate model is CPU efficient and serves to explore the posterior distribution at a much lower computational cost using two-stage MCMC simulation. The case study reported in this paper demonstrates a two to five times speed-up in sampling efficiency.
NASA Astrophysics Data System (ADS)
Li, He; Gao, Yi-Tian; Liu, Li-Cai
2015-12-01
The Korteweg-de Vries (KdV)-type equations have been seen in fluid mechanics, plasma physics and lattice dynamics, etc. This paper will address the bilinearization problem for some higher-order KdV equations. Based on the relationship between the bilinear method and Bell-polynomial scheme, with introducing an auxiliary independent variable, we will present the general bilinear forms. By virtue of the symbolic computation, one- and two-soliton solutions are derived. Supported by the National Natural Science Foundation of China under Grant No. 11272023, the Open Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), and by the Fundamental Research Funds for the Central Universities of China under Grant No. 2011BUPTYB02
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.
Surya Mohan, P.; Tarvainen, Tanja; Schweiger, Martin; Pulkkinen, Aki; Arridge, Simon R.
2011-08-10
Highlights: {yields} We developed a variable order global basis scheme to solve light transport in 3D. {yields} Based on finite elements, the method can be applied to a wide class of geometries. {yields} It is computationally cheap when compared to the fixed order scheme. {yields} Comparisons with local basis method and other models demonstrate its accuracy. {yields} Addresses problems encountered n modeling of light transport in human brain. - Abstract: We propose the P{sub N} approximation based on a finite element framework for solving the radiative transport equation with optical tomography as the primary application area. The key idea is to employ a variable order spherical harmonic expansion for angular discretization based on the proximity to the source and the local scattering coefficient. The proposed scheme is shown to be computationally efficient compared to employing homogeneously high orders of expansion everywhere in the domain. In addition the numerical method is shown to accurately describe the void regions encountered in the forward modeling of real-life specimens such as infant brains. The accuracy of the method is demonstrated over three model problems where the P{sub N} approximation is compared against Monte Carlo simulations and other state-of-the-art methods.
NASA Astrophysics Data System (ADS)
Withers, Christopher S.; Nadarajah, Saralees
2016-07-01
A new class of polynomials pn(x) known as β-reciprocal polynomials is defined. Given a parameter ? that is not a root of -1, we show that the only β-reciprocal polynomials are pn(x) ≡ xn. When β is a root of -1, other polynomials are possible. For example, the Hermite polynomials are i-reciprocal, ?.
NASA Astrophysics Data System (ADS)
Park, Jun-Hyub; Shin, Myung-Soo; Kang, Dong-Joong; Lim, Sung-Jo; Ha, Jong-Eun
In this study, a system for non-contact in-situ measurement of strain during tensile test of thin films by using CCD camera with marking surface of specimen by black pen was implemented as a sensing device. To improve accuracy of measurement when CCD camera is used, this paper proposed a new method for measuring strain during tensile test of specimen with micrometer size. The size of pixel of CCD camera determines resolution of measurement, but the size of pixel can not satisfy the resolution required in tensile test of thin film because the extension of the specimen is very small during the tensile test. To increase resolution of measurement, the suggested method performs an accurate subpixel matching by applying 2nd order polynomial interpolation method to the conventional template matching. The algorithm was developed to calculate location of subpixel providing the best matching value by performing single dimensional polynomial interpolation from the results of pixel-based matching at a local region of image. The measurement resolution was less than 0.01 times of original pixel size. To verify the reliability of the system, the tensile test for the BeNi thin film was performed, which is widely used as a material in micro-probe tip. Tensile tests were performed and strains were measured using the proposed method and also the capacitance type displacement sensor for comparison. It is demonstrated that the new strain measurement system can effectively describe a behavior of materials after yield during the tensile test of the specimen at microscale with easy setup and better accuracy.
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)
Zhang, Xu
This paper introduces a class of polynomial maps in Euclidean spaces, investigates the conditions under which there exist Smale horseshoes and uniformly hyperbolic invariant sets, studies the chaotic dynamical behavior and strange attractors, and shows that some maps are chaotic in the sense of Li-Yorke or Devaney. This type of maps includes both the Logistic map and the Hénon map. For some diffeomorphisms with the expansion dimension equal to one or two in three-dimensional spaces, the conditions under which there exist Smale horseshoes and uniformly hyperbolic invariant sets on which the systems are topologically conjugate to the two-sided fullshift on finite alphabet are obtained; for some expanding maps, the chaotic region is analyzed by using the coupled-expansion theory and the Brouwer degree theory. For three types of higher-dimensional polynomial maps with degree two, the conditions under which there are Smale horseshoes and uniformly hyperbolic invariant sets are given, and the topological conjugacy between the maps on the invariant sets and the two-sided fullshift on finite alphabet is obtained. Some interesting maps with chaotic attractors and positive Lyapunov exponents in three-dimensional spaces are found by using computer simulations. In the end, two examples are provided to illustrate the theoretical results.
NASA Astrophysics Data System (ADS)
Laloy, Eric; Rogiers, Bart; Vrugt, Jasper; Mallants, Dirk; Jacques, Diederik
2013-04-01
This study presents a novel strategy for accelerating posterior exploration of highly parameterized and CPU-demanding hydrogeologic models. The method builds on the stochastic collocation approach of Marzouk and Xiu (2009) and uses the generalized polynomial chaos (gPC) framework to emulate the output of a groundwater flow model. The resulting surrogate model is CPU-efficient and allows for sampling the posterior parameter distribution at a much reduced computational cost. This surrogate distribution is subsequently employed to precondition a state-of-the-art two-stage Markov chain Monte Carlo (MCMC) simulation (Vrugt et al., 2009; Cui et al., 2011) of the original CPU-demanding flow model. Application of the proposed method to the hydrogeological characterization of a three-dimensional multi-layered aquifer shows a 2-5 times speed up in sampling efficiency.
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…
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.
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.
SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos
NASA Astrophysics Data System (ADS)
Ahlfeld, R.; Belkouchi, B.; Montomoli, F.
2016-09-01
A new arbitrary Polynomial Chaos (aPC) method is presented for moderately high-dimensional problems characterised by limited input data availability. The proposed methodology improves the algorithm of aPC and extends the method, that was previously only introduced as tensor product expansion, to moderately high-dimensional stochastic problems. The fundamental idea of aPC is to use the statistical moments of the input random variables to develop the polynomial chaos expansion. This approach provides the possibility to propagate continuous or discrete probability density functions and also histograms (data sets) as long as their moments exist, are finite and the determinant of the moment matrix is strictly positive. For cases with limited data availability, this approach avoids bias and fitting errors caused by wrong assumptions. In this work, an alternative way to calculate the aPC is suggested, which provides the optimal polynomials, Gaussian quadrature collocation points and weights from the moments using only a handful of matrix operations on the Hankel matrix of moments. It can therefore be implemented without requiring prior knowledge about statistical data analysis or a detailed understanding of the mathematics of polynomial chaos expansions. The extension to more input variables suggested in this work, is an anisotropic and adaptive version of Smolyak's algorithm that is solely based on the moments of the input probability distributions. It is referred to as SAMBA (PC), which is short for Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos. It is illustrated that for moderately high-dimensional problems (up to 20 different input variables or histograms) SAMBA can significantly simplify the calculation of sparse Gaussian quadrature rules. SAMBA's efficiency for multivariate functions with regard to data availability is further demonstrated by analysing higher order convergence and accuracy for a set of nonlinear test functions with 2, 5 and 10
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)
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. PMID:27190749
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.
NASA Astrophysics Data System (ADS)
Fridjine, S.; Amlouk, M.
In this study, we define a synthetic parameter: optothermal expansivity as a quantitative guide to evaluating and optimizing both the thermal and the optical performance of PV-T functional materials. The definition of this parameter, ψAB (Amlouk-Boubaker parameter), takes into account the thermal diffusivity and the optical effective absorptivity of the material. The values of this parameter, which seems to be a characteristic one, correspond to the total volume that contains a fixed amount of heat per unit time (m3 s-1) and can be considered as a 3D velocity of the transmitted heat inside the material. As the PV-T combined devices need to have simultaneous optical and thermal efficiency, we try to investigate some recently proposed materials (β-SnS2, In2S3, ZnS1-xSex|0 ≤x<0.5 and Zn-doped thioindate compounds) using the newly established ψAB/Eg abacus.
NASA Astrophysics Data System (ADS)
Mongeon, Michael C.
1996-03-01
This paper investigates the development of printer device profiles used in color document printing system environments when devices with intrinsically different gamut capabilities communicate with one another in a common (CIELAB) color space. While the main thrust of this activity focuses on the output printer, namely the Xerox 5760 printer, and its rendition of some device independent image description, characterizations are provided which investigate relative areas of photographic, monitor, and printer gamuts using a visual hue leaf comparison between devices. The printer is modeled using 4th-order polynomial regression which maps the device independent CIELAB image representation into device dependent printer CMYK. This technique results in 1.89 AEEavg over the training data set. Some key properties of the proposed calibration method are as follows: (1) Linearized CMYK tone reproduction curves with respect to AEEpaper to improve the distribution of calibration data in color space. (2) Application of GCR strategy and linearization to the calibration target prior to the regression on the measured CIELAB and original CMY values. Each strategy employs a K addition/No CMY removal method which maximizes printer gamut and relies on the regression to determine the appropriate CMY removal. The following GCR strategies are explored: CMY only (0% K addition), 50% K addition, 100% K addition, and non-linear K addition. A library of image processing algorithms is included, using LabView object oriented programming, which provides a modular approach for key color processing tasks. In the user interface, an image is selected with appropriate GCR strategy, and the program operates on the image. In general, the pictorial image quality is excellent for each GCR strategy with subtle differences between GCR approaches. Quantitative analysis of Q60 color matching performance is included.
NASA Astrophysics Data System (ADS)
Taghavi-Shahri, F.; Khanpour, Hamzeh; Atashbar Tehrani, S.; Alizadeh Yazdi, Z.
2016-06-01
We present a first QCD analysis of next-to-next-leading-order (NNLO) contributions of the spin-dependent parton distribution functions (PPDFs) in the nucleon and their uncertainties using the Jacobi polynomial approach. Having the NNLO contributions of the quark-quark and gluon-quark splitting functions in perturbative QCD [Nucl. Phys. B889, 351 (2014)], one can obtain the evolution of longitudinally polarized parton densities of hadrons up to NNLO accuracy of QCD. Very large sets of recent and up-to-date experimental data of spin structure functions of the proton g1p, neutron g1n, and deuteron g1d have been used in this analysis. The predictions for the NNLO calculations of the polarized parton distribution functions as well as the proton, neutron and deuteron polarized structure functions are compared with the corresponding results of the NLO approximation. We form a mutually consistent set of polarized PDFs due to the inclusion of the most available experimental data including the recently high-precision measurements from COMPASS16 experiments [Phys. Lett. B 753, 18 (2016)]. We have performed a careful estimation of the uncertainties using the most common and practical method, the Hessian method, for the polarized PDFs originating from the experimental errors. The proton, neutron and deuteron structure functions and also their first moments, Γp ,n ,d , are in good agreement with the experimental data at small and large momentum fractions of x . We will discuss how our knowledge of spin-dependence structure functions can improve at small and large values of x by the recent COMPASS16 measurements at CERN, the PHENIX and STAR measurements at RHIC, and at the future proposed colliders such as the Electron-Ion Collider.
Second-order many-body perturbation study on thermal expansion of solid carbon dioxide.
Li, Jinjin; Sode, Olaseni; Hirata, So
2015-01-13
An embedded-fragment ab initio second-order many-body perturbation (MP2) method is applied to an infinite three-dimensional crystal of carbon dioxide phase I (CO2-I), using the aug-cc-pVDZ and aug-cc-pVTZ basis sets, the latter in conjunction with a counterpoise correction for the basis-set superposition error. The equation of state, phonon frequencies, bulk modulus, heat capacity, Grüneisen parameter (including mode Grüneisen parameters for acoustic modes), thermal expansion coefficient (α), and thermal pressure coefficient (β) are computed. Of the factors that enter the expression of α, MP2 reproduces the experimental values of the heat capacity, Grüneisen parameter, and molar volume accurately. However, it proves to be exceedingly difficult to determine the remaining factor, the bulk modulus (B0), the computed value of which deviates from the observed value by 50-100%. As a result, α calculated by MP2 is systematically too low, while having the correct temperature dependence. The thermal pressure coefficient, β = αB0, which is independent of B0, is more accurately reproduced by theory up to 100 K. PMID:26574220
NASA Astrophysics Data System (ADS)
Guardia, M.; Kaloshin, V.; Zhang, J.
2016-07-01
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.
Investigating the Dimits Shift using the Second-order Cumulant Expansion Statistical Closure
NASA Astrophysics Data System (ADS)
St-Onge, D. A.; Krommes, J. A.
2015-11-01
The Dimits shift is the nonlinear upshift of the critical temperature gradient that signals the onset of collisionless ion-temperature-gradient-driven turbulence. This phenomenon is caused by the shearing away of turbulent streamers in the radial direction by poloidal zonal flows (ZFs). While the effect is witnessed in both gyrokinetic and gyrofluid simulations, there exists no analytical model that satisfactorily describes the mechanics through which it operates. In this work, a new model is developed by applying the second-order cumulant expansion closure to a simplified set of gyrofluid equations. In particular, we calculate the threshold for the zonostrophic instability of a two-field model, generalizing the work of Parker and Krommes on the modified Hasegawa-Mima equation, and assess whether the Reynolds-stress-generated ZFs can be destabilized in the model, thus indicating a Dimits shift. This work was supported by an NSERC PGS-D scholarship, as well as by U.S. DOE contract DE-AC02-09CH11466.
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, Shuai; Yang, Ping; Dong, Lizhi; Xu, Bing; Ao, Mingwu
2015-02-01
Walsh functions have been modified and utilized as binary-aberration-mode basis which are especially suitable for representing discrete wavefronts. However, when wavefront sensing techniques based on binary-aberration-mode detection trying to reconstruct common wavefronts with continuous forms, the Modified Walsh functions are incompetent. The limited space resolution of Modified Walsh functions will leave substantial residual wavefronts. In order to sidestep the space-resolution problem of binary-aberration modes, it's necessary to transform the Modified-Walsh-function expansion coefficients of wavefront to Zernike-polynomial coefficients and use Zernike polynomials to represent the wavefront to be reconstructed. For this reason, a transformation method for wavefront expansion coefficients of the two aberration modes is proposed. The principle of the transformation is the linear of wavefront expansion and the method of least squares. The numerical simulation demonstrates that the coefficient transformation with the transformation matrix is reliable and accurate.
Interval polynomial positivity
NASA Technical Reports Server (NTRS)
Bose, N. K.; Kim, K. D.
1989-01-01
It is shown that a univariate interval polynomial is globally positive if and only if two extreme polynomials are globally positive. It is shown that the global positivity property of a bivariate interval polynomial is completely determined by four extreme bivariate polynomials. The cardinality of the determining set for k-variate interval polynomials is 2k. One of many possible generalizations, where vertex implication for global positivity holds, is made by considering the parameter space to be the set dual of a boxed domain.
Federal Register 2010, 2011, 2012, 2013, 2014
2012-11-26
... From the Federal Register Online via the Government Publishing Office DEPARTMENT OF THE INTERIOR Public Land Order No. 7801; Withdrawal of Public Lands for Protection of Proposed Expansion of Twentynine Palms, CA Correction In notice document 2012-23479 beginning on page 58864 of the issue of Monday, September 24, 2012 make the...
Bispectrality of the Complementary Bannai-Ito Polynomials
NASA Astrophysics Data System (ADS)
Genest, Vincent X.; Vinet, Luc; Zhedanov, Alexei
2013-03-01
A one-parameter family of operators that have the complementary Bannai-Ito (CBI) polynomials as eigenfunctions is obtained. The CBI polynomials are the kernel partners of the Bannai-Ito polynomials and also correspond to a q→"1 limit of the Askey-Wilson polynomials. The eigenvalue equations for the CBI polynomials are found to involve second order Dunkl shift operators with reflections and exhibit quadratic spectra. The algebra associated to the CBI polynomials is given and seen to be a deformation of the Askey-Wilson algebra with an involution. The relation between the CBI polynomials and the recently discovered dual "1 Hahn and para-Krawtchouk polynomials, as well as their relation with the symmetric Hahn polynomials, is also discussed.
Linear 3 and 5-step methods using Taylor series expansion for solving special 3rd order ODEs
NASA Astrophysics Data System (ADS)
Rajabi, Marzieh; Ismail, Fudziah; Senu, Norazak
2016-06-01
Some new linear 3 and 5-step methods for solving special third order ordinary differential equations directly are constructed using Taylor's series expansion. A set of test problems are solved using the new method and the results are compared when the problem is reduced to a system of first order ordinary differential equations and then using the existing Runge-Kutta method. The numerical results have clearly shown the advantage and competency of the new methods.
Sage, M. H.; Blake, G. R.; Palstra, T. T. M.; Marquina, C.
2007-11-15
We report evidence for the phase coexistence of orbital orderings of different symmetry in RVO{sub 3} compounds with intermediate-size rare earths. Through a study by high-resolution x-ray powder diffraction and thermal expansion, we show that the competing orbital orderings are associated with the magnitude of the VO{sub 6} octahedral tilting and magnetic exchange striction in these compounds and that the phase-separated state is stabilized by lattice strains.
Properties of the one-dimensional Bose-Hubbard model from a high-order perturbative expansion
NASA Astrophysics Data System (ADS)
Damski, Bogdan; Zakrzewski, Jakub
2015-12-01
We employ a high-order perturbative expansion to characterize the ground state of the Mott phase of the one-dimensional Bose-Hubbard model. We compute for different integer filling factors the energy per lattice site, the two-point and density-density correlations, and expectation values of powers of the on-site number operator determining the local atom number fluctuations (variance, skewness, kurtosis). We compare these expansions to numerical simulations of the infinite-size system to determine their range of applicability. We also discuss a new sum rule for the density-density correlations that can be used in both equilibrium and non-equilibrium systems.
Blind phone segmentation based on spectral change detection using Legendre polynomial approximation.
Hoang, Dac-Thang; Wang, Hsiao-Chuan
2015-02-01
Phone segmentation involves partitioning a continuous speech signal into discrete phone units. In this paper, a method for automatic phone segmentation without prior knowledge of speech content is proposed. The signal spectrum was represented by band-energies. A segment of the band-energy curve was approximated using Legendre polynomial expansion, allowing Legendre polynomial coefficients to describe the properties of the segment. The spectral changes, which imply phone boundaries in the speech signal, were then detected by monitoring the variations of Legendre polynomial coefficients. A two-step algorithm for detecting phone boundaries was derived. The first step was to detect phone boundaries using first-order and second-order coefficients of the Legendre polynomial approximation. The second step was to locate slow spectral changes in the regions of concatenated voiced phones using zero-order coefficients of the Legendre polynomial approximation. This enabled the phone boundaries missed during the first step to be recovered. An evaluation using the TIMIT corpus indicated that the proposed method is comparable to or more accurate than previous methods. PMID:25698014
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.
Mayer expansion of the Nekrasov prepotential: The subleading ε2-order
NASA Astrophysics Data System (ADS)
Bourgine, Jean-Emile; Fioravanti, Davide
2016-05-01
The Mayer cluster expansion technique is applied to the Nekrasov instanton partition function of N = 2 SU (Nc) super Yang-Mills. The subleading small ε2-correction to the Nekrasov-Shatashvili limiting value of the prepotential is determined by a detailed analysis of all the one-loop diagrams. Indeed, several types of contributions can be distinguished according to their origin: long range interaction or potential expansion, clusters self-energy, internal structure, one-loop cyclic diagrams, etc. The field theory result derived more efficiently in [1], under some minor technical assumptions, receives here definite confirmation thanks to several remarkable cancellations: in this way, we may infer the validity of these assumptions for further computations in the field theoretical approach.
Heisenberg algebra, umbral calculus and orthogonal polynomials
Dattoli, G.; Levi, D.; Winternitz, P.
2008-05-15
Umbral calculus can be viewed as an abstract theory of the Heisenberg commutation relation [P,M]=1. In ordinary quantum mechanics, P is the derivative and M the coordinate operator. Here, we shall realize P as a second order differential operator and M as a first order integral one. We show that this makes it possible to solve large classes of differential and integrodifferential equations and to introduce new classes of orthogonal polynomials, related to Laguerre polynomials. These polynomials are particularly well suited for describing the so-called flatenned beams in laser theory.
NASA Astrophysics Data System (ADS)
Metaxas, Dimitrios
2008-09-01
I calculate the first corrections to the dynamical preexponential factor of the bubble nucleation rate for a relativistic first-order phase transition in an expanding cosmological background by estimating the effects of the Hubble expansion rate on the critical bubbles of Langer’s statistical theory of metastability. I also comment on possible applications and problems that arise when one considers the field theoretical extensions of these results (the Coleman De Luccia and Hawking-Moss instantons and decay rates).
Polynomial Graphs and Symmetry
ERIC Educational Resources Information Center
Goehle, Geoff; Kobayashi, Mitsuo
2013-01-01
Most quadratic functions are not even, but every parabola has symmetry with respect to some vertical line. Similarly, every cubic has rotational symmetry with respect to some point, though most cubics are not odd. We show that every polynomial has at most one point of symmetry and give conditions under which the polynomial has rotational or…
NASA Astrophysics Data System (ADS)
Bogner, Christian; Weinzierl, Stefan
The integrand of any multiloop integral is characterized after Feynman parametrization by two polynomials. In this review we summarize the properties of these polynomials. Topics covered in this paper include among others: spanning trees and spanning forests, the all-minors matrix-tree theorem, recursion relations due to contraction and deletion of edges, Dodgson's identity and matroids.
Zernike olivary polynomials for applications with olivary pupils.
Zheng, Yi; Sun, Shanshan; Li, Ying
2016-04-20
Orthonormal polynomials have been extensively applied in optical image systems. One important optical pupil, which is widely processed in lateral shearing interferometers (LSI) and subaperture stitch tests (SST), is the overlap region of two circular wavefronts that are displaced from each other. We call it an olivary pupil. In this paper, the normalized process of an olivary pupil in a unit circle is first presented. Then, using a nonrecursive matrix method, Zernike olivary polynomials (ZOPs) are obtained. Previously, Zernike elliptical polynomials (ZEPs) have been considered as an approximation over an olivary pupil. We compare ZOPs with their ZEPs counterparts. Results show that they share the same components but are in different proportions. For some low-order aberrations such as defocus, coma, and spherical, the differences are considerable and may lead to deviations. Using a least-squares method to fit coefficient curves, we present a power-series expansion form for the first 15 ZOPs, which can be used conveniently with less than 0.1% error. The applications of ZOP are demonstrated in wavefront decomposition, LSI interferogram reconstruction, and SST overlap domain evaluation. PMID:27140076
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
Pakhira, Anindya; Das, Saptarshi; Pan, Indranil; Das, Shantanu
2015-07-01
This paper uses the Continued Fraction Expansion (CFE) method for analog realization of fractional order differ-integrator and few special classes of fractional order (FO) controllers viz. Fractional Order Proportional-Integral-Derivative (FOPID) controller, FO[PD] controller and FO lead-lag compensator. Contemporary researchers have given several formulations for rational approximation of fractional order elements. However, approximation of the controllers studied in this paper, due to having fractional power of a rational transfer function, is not available in analog domain; although its digital realization already exists. This motivates us for applying CFE based analog realization technique for complicated FO controller structures to get equivalent rational transfer functions in terms of the controller tuning parameters. The symbolic expressions for rationalized transfer function in terms of the controller tuning parameters are especially important as ready references, without the need of running CFE algorithm every time and also helps in the synthesis of analog circuits for such FO controllers. PMID:25661163
More on rotations as spin matrix polynomials
Curtright, Thomas L.
2015-09-15
Any nonsingular function of spin j matrices always reduces to a matrix polynomial of order 2j. The challenge is to find a convenient form for the coefficients of the matrix polynomial. The theory of biorthogonal systems is a useful framework to meet this challenge. Central factorial numbers play a key role in the theoretical development. Explicit polynomial coefficients for rotations expressed either as exponentials or as rational Cayley transforms are considered here. Structural features of the results are discussed and compared, and large j limits of the coefficients are examined.
The Operator Product Expansion Beyond Leading Order for Spin-1/2 Fermions
NASA Astrophysics Data System (ADS)
Emmons, Samuel; Kang, Daekyoung; Platter, Lucas
2016-05-01
Strongly interacting systems of ultracold, two-component fermions have been studied using various techniques for many years. One technique that has been applied is a quantum field theoretical formulation of the zero-range model. In this framework, the Operator Product Expansion was used to derive universal relations for systems with a large scattering length. This corroborated and extended the work of Tan. We calculate finite range corrections to the momentum distribution using the OPE framework and demonstrate the utility of including the 1 /k6 tail from the OPE for the momentum distribution. Corrections to the universal relations for the system are calculated and expressed in terms of the S-wave effective range and an additional quantity D similar to Tan's contact which, in addition to the contact, relates various physical observables. We compare our results with quantum Monte Carlo calculations for the two-component Fermi gas with large scattering length. NSF Grant No. PHY-1516077; U.S. DOE Office of Science, Office of Nuclear Physics Contract Nos. DE-AC52-06NA25396, DE-AC05-00OR22725, an Early Career Research Award; LANL/LDRD Program.
NASA Astrophysics Data System (ADS)
Abbas, Gauhar; Ananthanarayan, B.; Caprini, Irinel; Fischer, Jan
2013-08-01
The moments of the hadronic spectral functions are of interest for the extraction of the strong coupling αs and other QCD parameters from the hadronic decays of the τ lepton. Motivated by the recent analyses of a large class of moments in the standard fixed-order and contour-improved perturbation theories, we consider the perturbative behavior of these moments in the framework of a QCD nonpower perturbation theory, defined by the technique of series acceleration by conformal mappings, which simultaneously implements renormalization-group summation and has a tame large-order behavior. Two recently proposed models of the Adler function are employed to generate the higher-order coefficients of the perturbation series and to predict the exact values of the moments, required for testing the properties of the perturbative expansions. We show that the contour-improved nonpower perturbation theories and the renormalization-group-summed nonpower perturbation theories have very good convergence properties for a large class of moments of the so-called “reference model,” including moments that are poorly described by the standard expansions. The results provide additional support for the plausibility of the description of the Adler function in terms of a small number of dominant renormalons.
Magnetic cluster expansion model for random and ordered magnetic face-centered cubic Fe-Ni-Cr alloys
NASA Astrophysics Data System (ADS)
Lavrentiev, M. Yu.; Wróbel, J. S.; Nguyen-Manh, D.; Dudarev, S. L.; Ganchenkova, M. G.
2016-07-01
A Magnetic Cluster Expansion model for ternary face-centered cubic Fe-Ni-Cr alloys has been developed, using DFT data spanning binary and ternary alloy configurations. Using this Magnetic Cluster Expansion model Hamiltonian, we perform Monte Carlo simulations and explore magnetic structures of alloys over the entire range of compositions, considering both random and ordered alloy structures. In random alloys, the removal of magnetic collinearity constraint reduces the total magnetic moment but does not affect the predicted range of compositions where the alloys adopt low-temperature ferromagnetic configurations. During alloying of ordered fcc Fe-Ni compounds with Cr, chromium atoms tend to replace nickel rather than iron atoms. Replacement of Ni by Cr in ordered alloys with high iron content increases the Curie temperature of the alloys. This can be explained by strong antiferromagnetic Fe-Cr coupling, similar to that found in bcc Fe-Cr solutions, where the Curie temperature increase, predicted by simulations as a function of Cr concentration, is confirmed by experimental observations. In random alloys, both magnetization and the Curie temperature decrease abruptly with increasing chromium content, in agreement with experiment.
NASA Astrophysics Data System (ADS)
Sahoo, S.; Saha Ray, S.
2016-04-01
In the present paper, we construct the analytical exact solutions of a nonlinear evolution equation in mathematical physics; namely time fractional modified KdV equation by using (G‧ / G)-expansion method and improved (G‧ / G)-expansion method. As a result, new types of exact analytical solutions are obtained.
NASA Astrophysics Data System (ADS)
Datta, Nilanjana; Hsieh, Min-Hsiu; Oppenheim, Jonathan
2016-05-01
State redistribution is the protocol in which given an arbitrary tripartite quantum state, with two of the subsystems initially being with Alice and one being with Bob, the goal is for Alice to send one of her subsystems to Bob, possibly with the help of prior shared entanglement. We derive an upper bound on the second order asymptotic expansion for the quantum communication cost of achieving state redistribution with a given finite accuracy. In proving our result, we also obtain an upper bound on the quantum communication cost of this protocol in the one-shot setting, by using the protocol of coherent state merging as a primitive.
NASA Astrophysics Data System (ADS)
Kéchichian, Jean A.
2011-09-01
A fourth order extension of the analytic form of the accelerations due to the luni-solar gravity perturbations along rotating axes is presented. These derivations are carried out in order to increase the accuracy of the dynamic modeling of perturbed optimal low-thrust transfers between general elliptic orbits, and enhance the fidelity of trajectory optimization software used in simulations and mission analyses. A set of rotating axes attached to the thrusting spacecraft is used such that both the thrust and perturbation accelerations due to Earth's geopotential and the luni-solar gravity are mathematically resolved along these axes prior to numerical integration of the actual trajectory. This Gaussian form of the state as well as the adjoint differential equations form a set of equations that are readily integrated and an iterative process is used to achieve convergence to a desired transfer. This analysis further reveals that further extensions to higher orders, say to the fifth order and beyond, are not needed to extract even more accuracy in the solutions because the minimum-time transfer solutions become fully stabilized in the sense that they do not exhibit any differences beyond a fraction of a second, or at most a few seconds even in the more extreme cases of very large orbits with apogee heights around 100,000 km with strong lunar influence.
NASA Astrophysics Data System (ADS)
Mironov, A.; Mkrtchyan, R.; Morozov, A.
2016-02-01
We present a universal knot polynomials for 2- and 3-strand torus knots in adjoint representation, by universalization of appropriate Rosso-Jones formula. According to universality, these polynomials coincide with adjoined colored HOMFLY and Kauffman polynomials at SL and SO/Sp lines on Vogel's plane, respectively and give their exceptional group's counterparts on exceptional line. We demonstrate that [m,n]=[n,m] topological invariance, when applicable, take place on the entire Vogel's plane. We also suggest the universal form of invariant of figure eight knot in adjoint representation, and suggest existence of such universalization for any knot in adjoint and its descendant representations. Properties of universal polynomials and applications of these results are discussed.
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.
Renormalized effective actions for the O(N) model at next-to-leading order of the 1/N expansion
Fejos, G.; Patkos, A.; Szep, Zs.
2009-07-15
A fully explicit renormalized quantum action functional is constructed for the O(N) model in the auxiliary field formulation at next-to-leading order (NLO) of the 1/N expansion. Counterterms are consistently and explicitly derived for arbitrary constant vacuum expectation value of the scalar and auxiliary fields. The renormalized NLO pion propagator is exact at this order and satisfies Goldstone's theorem. Elimination of the auxiliary field sector at the level of the functional provides with O(N{sup 0}) accuracy the renormalized effective action of the model in terms of the original variables. Alternative elimination of the pion and sigma propagators provides the renormalized NLO effective potential for the expectation values of the N vector and of the auxiliary field with the same accuracy.
2010-01-01
Background The complex data sets generated by higher-order polychromatic flow cytometry experiments are a challenge to analyze. Here we describe Exhaustive Expansion, a data analysis approach for deriving hundreds to thousands of cell phenotypes from raw data, and for interrogating these phenotypes to identify populations of biological interest given the experimental context. Methods We apply this approach to two studies, illustrating its broad applicability. The first examines the longitudinal changes in circulating human memory T cell populations within individual patients in response to a melanoma peptide (gp100209-2M) cancer vaccine, using 5 monoclonal antibodies (mAbs) to delineate subpopulations of viable, gp100-specific, CD8+ T cells. The second study measures the mobilization of stem cells in porcine bone marrow that may be associated with wound healing, and uses 5 different staining panels consisting of 8 mAbs each. Results In the first study, our analysis suggests that the cell surface markers CD45RA, CD27 and CD28, commonly used in historical lower order (2-4 color) flow cytometry analysis to distinguish memory from naïve and effector T cells, may not be obligate parameters in defining central memory T cells (TCM). In the second study, we identify novel phenotypes such as CD29+CD31+CD56+CXCR4+CD90+Sca1-CD44+, which may characterize progenitor cells that are significantly increased in wounded animals as compared to controls. Conclusions Taken together, these results demonstrate that Exhaustive Expansion supports thorough interrogation of complex higher-order flow cytometry data sets and aids in the identification of potentially clinically relevant findings. PMID:21034498
On the cardinality of twelfth degree polynomial
NASA Astrophysics Data System (ADS)
Lasaraiya, S.; Sapar, S. H.; Johari, M. A. Mohamat
2016-06-01
Let p be a prime and f (x, y) be a polynomial in Zp[x, y]. It is defined that the exponential sums associated with f modulo a prime pα is S (f :q )= ∑ e2/π i f (x ) q for α >1 , where f (x) is in Z[x] and the sum is taken over a complete set of residues x modulo positive integer q. Previous studies has shown that estimation of S (f; pα) is depends on the cardinality of the set of solutions to congruence equation associated with the polynomial. In order to estimate the cardinality, we need to have the value of p-adic sizes of common zeros of partial derivative polynomials associated with polynomial. Hence, p-adic method and newton polyhedron technique will be applied to this approach. After that, indicator diagram will be constructed and analyzed. The cardinality will in turn be used to estimate the exponential sums of the polynomials. This paper concentrates on the cardinality of the set of solutions to congruence equation associated with polynomial in the form of f (x, y) = ax12 + bx11y + cx10y2 + sx + ty + k.
Yao, Chenggui; Zou, Wei; Zhao, Qi
2012-06-01
The method of order parameter expansion is used to study the dynamical behavior in the globally delay-coupled nonidentical systems. Using the Landau-Stuart periodic system and Rössler chaotic oscillator to construct representative systems, the method can identify the boundary curves of amplitude death island analytically in the parameter space of the coupling and time delay. Furthermore, the parameter mismatch (diversity) effect on the size of island is investigated numerically. For the case of coupled chaotic Rössler systems with different timescales, the diversity increases the domain of death island monotonically. However, for the case of delay-coupled Landua-Stuart periodic systems with different frequencies, the average frequency turns out to be a critical role that determines change of size with the increase of diversity. PMID:22757556
NASA Astrophysics Data System (ADS)
Yao, Chenggui; Zou, Wei; Zhao, Qi
2012-06-01
The method of order parameter expansion is used to study the dynamical behavior in the globally delay-coupled nonidentical systems. Using the Landau-Stuart periodic system and Rössler chaotic oscillator to construct representative systems, the method can identify the boundary curves of amplitude death island analytically in the parameter space of the coupling and time delay. Furthermore, the parameter mismatch (diversity) effect on the size of island is investigated numerically. For the case of coupled chaotic Rössler systems with different timescales, the diversity increases the domain of death island monotonically. However, for the case of delay-coupled Landua-Stuart periodic systems with different frequencies, the average frequency turns out to be a critical role that determines change of size with the increase of diversity.
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.
Polynomials with small Mahler measure
NASA Astrophysics Data System (ADS)
Mossinghoff, M. J.
1998-10-01
We describe several searches for polynomials with integer coefficients and small Mahler measure. We describe the algorithm used to test Mahler measures. We determine all polynomials with degree at most 24 and Mahler measure less than 1.3, test all reciprocal and antireciprocal polynomials with height 1 and degree at most 40, and check certain sparse polynomials with height 1 and degree as large as 181. We find a new limit point of Mahler measures near 1.309, four new Salem numbers less than 1.3, and many new polynomials with small Mahler measure. None has measure smaller than that of Lehmer's degree 10 polynomial.
Calculators and Polynomial Evaluation.
ERIC Educational Resources Information Center
Weaver, J. F.
The intent of this paper is to suggest and illustrate how electronic hand-held calculators, especially non-programmable ones with limited data-storage capacity, can be used to advantage by students in one particular aspect of work with polynomial functions. The basic mathematical background upon which calculator application is built is summarized.…
ERIC Educational Resources Information Center
Gordon, Sheldon P.
1992-01-01
Demonstrates how the uniqueness and anonymity of a student's Social Security number can be utilized to create individualized polynomial equations that students can investigate using computers or graphing calculators. Students write reports of their efforts to find and classify all real roots of their equation. (MDH)
NASA Astrophysics Data System (ADS)
Roquet, F.; Madec, G.; McDougall, Trevor J.; Barker, Paul M.
2015-06-01
A new set of approximations to the standard TEOS-10 equation of state are presented. These follow a polynomial form, making it computationally efficient for use in numerical ocean models. Two versions are provided, the first being a fit of density for Boussinesq ocean models, and the second fitting specific volume which is more suitable for compressible models. Both versions are given as the sum of a vertical reference profile (6th-order polynomial) and an anomaly (52-term polynomial, cubic in pressure), with relative errors of ∼0.1% on the thermal expansion coefficients. A 75-term polynomial expression is also presented for computing specific volume, with a better accuracy than the existing TEOS-10 48-term rational approximation, especially regarding the sound speed, and it is suggested that this expression represents a valuable approximation of the TEOS-10 equation of state for hydrographic data analysis. In the last section, practical aspects about the implementation of TEOS-10 in ocean models are discussed.
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…
Determinants and Polynomial Root Structure
ERIC Educational Resources Information Center
De Pillis, L. G.
2005-01-01
A little known property of determinants is developed in a manner accessible to beginning undergraduates in linear algebra. Using the language of matrix theory, a classical result by Sylvester that describes when two polynomials have a common root is recaptured. Among results concerning the structure of polynomial roots, polynomials with pairs of…
Time-dependent generalized polynomial chaos
Gerritsma, Marc; Steen, Jan-Bart van der; Vos, Peter; Karniadakis, George
2010-11-01
Generalized polynomial chaos (gPC) has non-uniform convergence and tends to break down for long-time integration. The reason is that the probability density distribution (PDF) of the solution evolves as a function of time. The set of orthogonal polynomials associated with the initial distribution will therefore not be optimal at later times, thus causing the reduced efficiency of the method for long-time integration. Adaptation of the set of orthogonal polynomials with respect to the changing PDF removes the error with respect to long-time integration. In this method new stochastic variables and orthogonal polynomials are constructed as time progresses. In the new stochastic variable the solution can be represented exactly by linear functions. This allows the method to use only low order polynomial approximations with high accuracy. The method is illustrated with a simple decay model for which an analytic solution is available and subsequently applied to the three mode Kraichnan-Orszag problem with favorable results.
High degree interpolation polynomial in Newton form
NASA Technical Reports Server (NTRS)
Tal-Ezer, Hillel
1988-01-01
Polynomial interpolation is an essential subject in numerical analysis. Dealing with a real interval, it is well known that even if f(x) is an analytic function, interpolating at equally spaced points can diverge. On the other hand, interpolating at the zeroes of the corresponding Chebyshev polynomial will converge. Using the Newton formula, this result of convergence is true only on the theoretical level. It is shown that the algorithm which computes the divided differences is numerically stable only if: (1) the interpolating points are arranged in a different order, and (2) the size of the interval is 4.
Two-variable orthogonal polynomials of big q-Jacobi type
NASA Astrophysics Data System (ADS)
Lewanowicz, Stanislaw; Wozny, Pawel
2010-01-01
A four-parameter family of orthogonal polynomials in two discrete variables is defined for a weight function of basic hypergeometric type. The polynomials, which are expressed in terms of univariate big q-Jacobi polynomials, form an extension of Dunkl's bivariate (little) q-Jacobi polynomials [C.F. Dunkl, Orthogonal polynomials in two variables of q-Hahn and q-Jacobi type, SIAM J. Algebr. Discrete Methods 1 (1980) 137-151]. We prove orthogonality property of the new polynomials, and show that they satisfy a three-term relation in a vector-matrix notation, as well as a second-order partial q-difference equation.
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.
Polynomial harmonic GMDH learning networks for time series modeling.
Nikolaev, Nikolay Y; Iba, Hitoshi
2003-12-01
This paper presents a constructive approach to neural network modeling of polynomial harmonic functions. This is an approach to growing higher-order networks like these build by the multilayer GMDH algorithm using activation polynomials. Two contributions for enhancement of the neural network learning are offered: (1) extending the expressive power of the network representation with another compositional scheme for combining polynomial terms and harmonics obtained analytically from the data; (2) space improving the higher-order network performance with a backpropagation algorithm for further gradient descent learning of the weights, initialized by least squares fitting during the growing phase. Empirical results show that the polynomial harmonic version phGMDH outperforms the previous GMDH, a Neurofuzzy GMDH and traditional MLP neural networks on time series modeling tasks. Applying next backpropagation training helps to achieve superior polynomial network performances. PMID:14622880
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
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.
Orthogonal polynomials and tolerancing
NASA Astrophysics Data System (ADS)
Rogers, John R.
2011-10-01
Previous papers have established the inadvisability of applying tolerances directly to power-series aspheric coefficients. The basic reason is that the individual terms are far from orthogonal. Zernike surfaces and the new Forbes surface types have certain orthogonality properties over the circle described by the "normalization radius." However, at surfaces away from the stop, the optical beam is smaller than the surface, and the polynomials are not orthogonal over the area sampled by the beam. In this paper, we investigate the breakdown of orthogonality as the surface moves away from the aperture stop, and the implications of this to tolerancing.
A recursive algorithm for Zernike polynomials
NASA Technical Reports Server (NTRS)
Davenport, J. W.
1982-01-01
The analysis of a function defined on a rotationally symmetric system, with either a circular or annular pupil is discussed. In order to numerically analyze such systems it is typical to expand the given function in terms of a class of orthogonal polynomials. Because of their particular properties, the Zernike polynomials are especially suited for numerical calculations. Developed is a recursive algorithm that can be used to generate the Zernike polynomials up to a given order. The algorithm is recursively defined over J where R(J,N) is the Zernike polynomial of degree N obtained by orthogonalizing the sequence R(J), R(J+2), ..., R(J+2N) over (epsilon, 1). The terms in the preceding row - the (J-1) row - up to the N+1 term is needed for generating the (J,N)th term. Thus, the algorith generates an upper left-triangular table. This algorithm was placed in the computer with the necessary support program also included.
Polynomial approximation of functions in Sobolev spaces
NASA Technical Reports Server (NTRS)
Dupont, T.; Scott, R.
1980-01-01
Constructive proofs and several generalizations of approximation results of J. H. Bramble and S. R. Hilbert are presented. Using an averaged Taylor series, we represent a function as a polynomial plus a remainder. The remainder can be manipulated in many ways to give different types of bounds. Approximation of functions in fractional order Sobolev spaces is treated as well as the usual integer order spaces and several nonstandard Sobolev-like spaces.
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.
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.
Piecewise Polynomial Representations of Genomic Tracks
Tarabichi, Maxime; Detours, Vincent; Konopka, Tomasz
2012-01-01
Genomic data from micro-array and sequencing projects consist of associations of measured values to chromosomal coordinates. These associations can be thought of as functions in one dimension and can thus be stored, analyzed, and interpreted as piecewise-polynomial curves. We present a general framework for building piecewise polynomial representations of genome-scale signals and illustrate some of its applications via examples. We show that piecewise constant segmentation, a typical step in copy-number analyses, can be carried out within this framework for both array and (DNA) sequencing data offering advantages over existing methods in each case. Higher-order polynomial curves can be used, for example, to detect trends and/or discontinuities in transcription levels from RNA-seq data. We give a concrete application of piecewise linear functions to diagnose and quantify alignment quality at exon borders (splice sites). Our software (source and object code) for building piecewise polynomial models is available at http://sourceforge.net/projects/locsmoc/. PMID:23166601
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.
Piecewise polynomial representations of genomic tracks.
Tarabichi, Maxime; Detours, Vincent; Konopka, Tomasz
2012-01-01
Genomic data from micro-array and sequencing projects consist of associations of measured values to chromosomal coordinates. These associations can be thought of as functions in one dimension and can thus be stored, analyzed, and interpreted as piecewise-polynomial curves. We present a general framework for building piecewise polynomial representations of genome-scale signals and illustrate some of its applications via examples. We show that piecewise constant segmentation, a typical step in copy-number analyses, can be carried out within this framework for both array and (DNA) sequencing data offering advantages over existing methods in each case. Higher-order polynomial curves can be used, for example, to detect trends and/or discontinuities in transcription levels from RNA-seq data. We give a concrete application of piecewise linear functions to diagnose and quantify alignment quality at exon borders (splice sites). Our software (source and object code) for building piecewise polynomial models is available at http://sourceforge.net/projects/locsmoc/. PMID:23166601
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.
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…
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…
NASA Astrophysics Data System (ADS)
Sergeev, A.; Alharbi, F. H.; Jovanovic, R.; Kais, S.
2016-04-01
The gradient expansion of the kinetic energy density functional, when applied to atoms or finite systems, usually grossly overestimates the energy in the fourth order and generally diverges in the sixth order. We avoid the divergence of the integral by replacing the asymptotic series including the sixth order term in the integrand by a rational function. Padé approximants show moderate improvements in accuracy in comparison with partial sums of the series. The results are discussed for atoms and Hooke’s law model for two-electron atoms.
Wick polynomials and time-evolution of cumulants
NASA Astrophysics Data System (ADS)
Lukkarinen, Jani; Marcozzi, Matteo
2016-08-01
We show how Wick polynomials of random variables can be defined combinatorially as the unique choice, which removes all "internal contractions" from the related cumulant expansions, also in a non-Gaussian case. We discuss how an expansion in terms of the Wick polynomials can be used for derivation of a hierarchy of equations for the time-evolution of cumulants. These methods are then applied to simplify the formal derivation of the Boltzmann-Peierls equation in the kinetic scaling limit of the discrete nonlinear Schödinger equation (DNLS) with suitable random initial data. We also present a reformulation of the standard perturbation expansion using cumulants, which could simplify the problem of a rigorous derivation of the Boltzmann-Peierls equation by separating the analysis of the solutions to the Boltzmann-Peierls equation from the analysis of the corrections. This latter scheme is general and not tied to the DNLS evolution equations.
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.
On the field theory expansion of superstring five point amplitudes
NASA Astrophysics Data System (ADS)
Boels, Rutger H.
2013-11-01
A simple recursive expansion algorithm for the integrals of tree level superstring five point amplitudes in a flat background is given which reduces the expansion to simple symbol(ic) manipulations. This approach can be used for instance to prove the expansion is maximally transcendental to all orders and to verify several conjectures made in recent literature to high order. Closed string amplitudes follow from these open string results by the KLT relations. To obtain insight into these results in particular the maximal R-symmetry violating amplitudes (MRV) in type IIB superstring theory are studied. The obtained expansion of the open string amplitudes reduces the analysis for MRV amplitudes to the classification of completely symmetric polynomials of the external legs, up to momentum conservation. Using Molien's theorem as a counting tool this problem is solved by constructing an explicit nine element basis for this class. This theorem may be of wider interest: as is illustrated at higher points it can be used to calculate dimensions of polynomials of external momenta invariant under any finite group for in principle any number of legs, up to momentum conservation. In the closed (or mixed) case this follows after application of the Kawai-Lewellen-Tye [1] relations (or their analogons [2,3]).
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.
Hadamard Factorization of Stable Polynomials
NASA Astrophysics Data System (ADS)
Loredo-Villalobos, Carlos Arturo; Aguirre-Hernández, Baltazar
2011-11-01
The stable (Hurwitz) polynomials are important in the study of differential equations systems and control theory (see [7] and [19]). A property of these polynomials is related to Hadamard product. Consider two polynomials p,q ∈ R[x]:p(x) = anxn+an-1xn-1+...+a1x+a0q(x) = bmx m+bm-1xm-1+...+b1x+b0the Hadamard product (p × q) is defined as (p×q)(x) = akbkxk+ak-1bk-1xk-1+...+a1b1x+a0b0where k = min(m,n). Some results (see [16]) shows that if p,q ∈R[x] are stable polynomials then (p×q) is stable, also, i.e. the Hadamard product is closed; however, the reciprocal is not always true, that is, not all stable polynomial has a factorization into two stable polynomials the same degree n, if n> 4 (see [15]).In this work we will give some conditions to Hadamard factorization existence for stable polynomials.
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 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).
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. PMID:26547244
Roots of polynomials by ratio of successive derivatives
NASA Technical Reports Server (NTRS)
Crouse, J. E.; Putt, C. W.
1972-01-01
An order of magnitude study of the ratios of successive polynomial derivatives yields information about the number of roots at an approached root point and the approximate location of a root point from a nearby point. The location approximation improves as a root is approached, so a powerful convergence procedure becomes available. These principles are developed into a computer program which finds the roots of polynomials with real number coefficients.
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.
Polynomial Operators on Classes of Regular Languages
NASA Astrophysics Data System (ADS)
Klíma, Ondřej; Polák, Libor
We assign to each positive variety mathcal V and each natural number k the class of all (positive) Boolean combinations of the restricted polynomials, i.e. the languages of the form L_0a_1 L_1a_2dots a_ell L_ell, text{ where } ell≤ k, a 1,...,a ℓ are letters and L 0,...,L ℓ are languages from the variety mathcal V. For this polynomial operator we give a certain algebraic counterpart which works with identities satisfied by syntactic (ordered) monoids of languages considered. We also characterize the property that a variety of languages is generated by a finite number of languages. We apply our constructions to particular examples of varieties of languages which are crucial for a certain famous open problem concerning concatenation hierarchies.
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.
Mahler's Expansion and Boolean Functions
NASA Astrophysics Data System (ADS)
Michon, Jean-Francis; Valarcher, Pierre; YunÈs, Jean-Baptiste
2007-03-01
The substitution of X by X^2 in binomial polynomials generates sequences of integers by Mahler's expansion. We give some properties of these integers and a combinatorial interpretation with covers by projection. We also give applications to the classification of boolean functions. This sequence arose from our previous research on classification and complexity of Binary Decision Diagrams (BDD) associated with boolean functions.
NASA Astrophysics Data System (ADS)
Porter, Edward K.
2006-10-01
We introduce a new method for modelling the gravitational wave flux function of a test-mass particle inspiralling into an intermediate mass Schwarzschild black hole which is based on Chebyshev polynomials of the first kind. It is believed that these intermediate mass ratio inspiral events (IMRI) are expected to be seen in both the ground- and space-based detectors. Starting with the post-Newtonian expansion from black hole perturbation theory, we introduce a new Chebyshev approximation to the flux function, which due to a process called Chebyshev economization gives a model with faster convergence than either post-Newtonian- or Padé-based methods. As well as having excellent convergence properties, these polynomials are also very closely related to the elusive minimax polynomial. We find that at the last stable orbit, the error between the Chebyshev approximation and a numerically calculated flux is reduced, <1.8%, at all orders of approximation. We also find that the templates constructed using the Chebyshev approximation give better fitting factors, in general >0.99, and smaller errors, <1/10%, in the estimation of the chirp mass when compared to a fiducial exact waveform, constructed using the numerical flux and the exact expression for the orbital energy function, again at all orders of approximation. We also show that in the intermediate test-mass case, the new Chebyshev template is superior to both PN and Padé approximant templates, especially at lower orders of approximation.
NASA Astrophysics Data System (ADS)
Sánchez-Escobar, Juan Jaime; Barbosa Santillán, Liliana Ibeth
2015-09-01
This paper describes the use of a hybrid evolutionary optimization algorithm (HEOA) for computing the wavefront aberration from real interferometric data. By finding the near-optimal solution to an optimization problem, this algorithm calculates the Zernike polynomial expansion coefficients from a Fizeau interferogram, showing the validity for the reconstruction of the wavefront aberration. The proposed HEOA incorporates the advantages of both a multimember evolution strategy and locally weighted linear regression in order to minimize an objective function while avoiding premature convergence to a local minimum. The numerical results demonstrate that our HEOA is robust for analyzing real interferograms degraded by noise.
Discrete-time ℋ∞ control for nonlinear polynomial systems
NASA Astrophysics Data System (ADS)
Hernandez-Gonzalez, M.; Basin, M. V.
2015-02-01
This paper presents a solution of the suboptimal ? regulator problem for a class of discrete-time nonlinear polynomial systems. The solution is obtained by reducing the ? control problem to the corresponding ? one. A general solution has been obtained for a polynomial of an arbitrary order; then, finite-dimensional regulator equations are derived explicitly for a second-order polynomial. Numerical simulations have been carried out to show effectiveness of the proposed method.
Using Tutte polynomials to analyze the structure of the benzodiazepines
NASA Astrophysics Data System (ADS)
Cadavid Muñoz, Juan José
2014-05-01
Graph theory in general and Tutte polynomials in particular, are implemented for analyzing the chemical structure of the benzodiazepines. Similarity analysis are used with the Tutte polynomials for finding other molecules that are similar to the benzodiazepines and therefore that might show similar psycho-active actions for medical purpose, in order to evade the drawbacks associated to the benzodiazepines based medicine. For each type of benzodiazepines, Tutte polynomials are computed and some numeric characteristics are obtained, such as the number of spanning trees and the number of spanning forests. Computations are done using the computer algebra Maple's GraphTheory package. The obtained analytical results are of great importance in pharmaceutical engineering. As a future research line, the usage of the chemistry computational program named Spartan, will be used to extent and compare it with the obtained results from the Tutte polynomials of benzodiazepines.
Symmetric polynomials in information theory: Entropy and subentropy
Jozsa, Richard; Mitchison, Graeme
2015-06-15
Entropy and other fundamental quantities of information theory are customarily expressed and manipulated as functions of probabilities. Here we study the entropy H and subentropy Q as functions of the elementary symmetric polynomials in the probabilities and reveal a series of remarkable properties. Derivatives of all orders are shown to satisfy a complete monotonicity property. H and Q themselves become multivariate Bernstein functions and we derive the density functions of their Levy-Khintchine representations. We also show that H and Q are Pick functions in each symmetric polynomial variable separately. Furthermore, we see that H and the intrinsically quantum informational quantity Q become surprisingly closely related in functional form, suggesting a special significance for the symmetric polynomials in quantum information theory. Using the symmetric polynomials, we also derive a series of further properties of H and Q.
Polynomial Extensions of the Weyl C*-Algebra
NASA Astrophysics Data System (ADS)
Accardi, Luigi; Dhahri, Ameur
2015-09-01
We introduce higher order (polynomial) extensions of the unique (up to isomorphisms) nontrivial central extension of the Heisenberg algebra, which can be concretely realized as sub-Lie algebras of the polynomial algebra generated by the creation and annihilation operators in the Schrödinger representation. The simplest nontrivial of these extensions (the quadratic one) is isomorphic to the Galilei algebra, widely studied in quantum physics. By exponentiation of this representation we construct the corresponding polynomial analogue of the Weyl C*-algebra and compute the polynomial Weyl relations. From this we deduce the explicit form of the composition law of the associated nonlinear extensions of the 1-dimensional Heisenberg group. The above results are used to calculate a simple explicit form of the vacuum characteristic functions of the nonlinear field operators of the Galilei algebra, as well as of their moments. The corresponding measures turn out to be an interpolation family between Gaussian and Meixner, in particular Gamma.
Solving fuzzy polynomial equation and the dual fuzzy polynomial equation using the ranking method
NASA Astrophysics Data System (ADS)
Rahman, Nurhakimah Ab.; Abdullah, Lazim
2014-06-01
Fuzzy polynomials with trapezoidal and triangular fuzzy numbers have attracted interest among some researchers. Many studies have been done by researchers to obtain real roots of fuzzy polynomials. As a result, there are many numerical methods involved in obtaining the real roots of fuzzy polynomials. In this study, we will present the solution to the fuzzy polynomial equation and dual fuzzy polynomial equation using the ranking method of fuzzy numbers and subsequently transforming fuzzy polynomials to crisp polynomials. This transformation is performed using the ranking method based on three parameters, namely Value, Ambiguity and Fuzziness. Finally, we illustrate our approach with two numerical examples for fuzzy polynomial equation and dual fuzzy polynomial equation.
NASA Astrophysics Data System (ADS)
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.
NASA Astrophysics Data System (ADS)
Sakumichi, Naoyuki; Kawakami, Norio; Ueda, Masahito
2012-04-01
The quantum-statistical cluster expansion method of Lee and Yang is extended to investigate off-diagonal long-range order (ODLRO) in one-component and multicomponent mixtures of bosons or fermions. Our formulation is applicable to both a uniform system and a trapped system without local-density approximation and allows systematic expansions of one-particle and multiparticle reduced density matrices in terms of cluster functions, which are defined for the same system with Boltzmann statistics. Each term in this expansion can be associated with a Lee-Yang graph. We elucidate a physical meaning of each Lee-Yang graph; in particular, for a mixture of ultracold atoms and bound dimers, an infinite sum of the ladder-type Lee-Yang 0-graphs is shown to lead to Bose-Einstein condensation of dimers below the critical temperature. In the case of Bose statistics, an infinite series of Lee-Yang 1-graphs is shown to converge and gives the criteria of ODLRO at the one-particle level. Applications to a dilute Bose system of hard spheres are also made. In the case of Fermi statistics, an infinite series of Lee-Yang 2-graphs is shown to converge and gives the criteria of ODLRO at the two-particle level. Applications to a two-component Fermi gas in the tightly bound limit are also made.
Pan, Fengjuan; Li, Xiaohui; Lu, Fengqi; Wang, Xiaoming; Cao, Jiang; Kuang, Xiaojun; Véron, Emmanuel; Porcher, Florence; Suchomel, Matthew R; Wang, Jing; Allix, Mathieu
2015-09-21
Ordering of interpolated Ba(2+) chains and alternate Ta-O rows (TaO)(3+) in the pentagonal tunnels of tetragonal tungsten bronzes (TTB) is controlled by the nonstoichiometry in the highly nonstoichiometric Ba0.5-xTaO3-x system. In Ba0.22TaO2.72, the filling of Ba(2+) and (TaO)(3+) groups is partially ordered along the ab-plane of the simple TTB structure, resulting in a √2-type TTB superstructure (Pbmm), while in Ba0.175TaO2.675, the pentagonal tunnel filling is completely ordered along the b-axis of the simple TTB structure, leading to a triple TTB superstructure (P21212). Both superstructures show completely empty square tunnels favoring Ba(2+) conduction and feature unusual accommodation of Ta(5+) cations in the small triangular tunnels. In contrast with stoichiometric Ba6GaTa9O30, which shows linear thermal expansion of the cell parameters and monotonic decrease of permittivity with temperature within 100-800 K, these TTB superstructures and slightly nonstoichiometric simple TTB Ba0.4TaO2.9 display abnormally broad and frequency-dependent extrinsic dielectric relaxations in 10(3)-10(5) Hz above room temperature, a linear deviation of the c-axis thermal expansion around 600 K, and high dielectric permittivity ∼60-95 at 1 MHz at room temperature. PMID:26347025
Scalar Field Theories with Polynomial Shift Symmetries
NASA Astrophysics Data System (ADS)
Griffin, Tom; Grosvenor, Kevin T.; Hořava, Petr; Yan, Ziqi
2015-12-01
We continue our study of naturalness in nonrelativistic QFTs of the Lifshitz type, focusing on scalar fields that can play the role of Nambu-Goldstone (NG) modes associated with spontaneous symmetry breaking. Such systems allow for an extension of the constant shift symmetry to a shift by a polynomial of degree P in spatial coordinates. These "polynomial shift symmetries" in turn protect the technical naturalness of modes with a higher-order dispersion relation, and lead to a refinement of the proposed classification of infrared Gaussian fixed points available to describe NG modes in nonrelativistic theories. Generic interactions in such theories break the polynomial shift symmetry explicitly to the constant shift. It is thus natural to ask: Given a Gaussian fixed point with polynomial shift symmetry of degree P, what are the lowest-dimension operators that preserve this symmetry, and deform the theory into a self-interacting scalar field theory with the shift symmetry of degree P? To answer this (essentially cohomological) question, we develop a new graph-theoretical technique, and use it to prove several classification theorems. First, in the special case of P = 1 (essentially equivalent to Galileons), we reproduce the known Galileon N-point invariants, and find their novel interpretation in terms of graph theory, as an equal-weight sum over all labeled trees with N vertices. Then we extend the classification to P > 1 and find a whole host of new invariants, including those that represent the most relevant (or least irrelevant) deformations of the corresponding Gaussian fixed points, and we study their uniqueness.
NASA Astrophysics Data System (ADS)
Keat, Yap Hong; Atan, Kamel Ariffin Mohd; Sapar, Siti Hasana; Said, Mohamad Rushdan Md
2014-07-01
In this paper we apply Newton polyhedron technique in estimating the p-adic sizes of common zeros of partial derivative polynomial associated with a quartic polynomial. It is found that the p-adic sizes of a common zeros can be determined explicitly in terms of the p-adic orders of coefficients of dominant terms of polynomial.
Nishimoto, Yoshio
2015-09-01
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. PMID:26342360
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.
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.
NASA Astrophysics Data System (ADS)
Olson, Gordon L.
2012-04-01
When using polynomial expansions for the angular variables in the radiation transport equation, the usual procedure is to truncate the series by setting all higher order terms to zero. At low order, such simple closures may not give the optimum solution. This work tests alternate closures that scale either the time- or spatial-derivatives in the highest order equation. These scale factors can be chosen such that waves propagate at exactly the speed of light in optically thin media. Alternatively, they may be chosen to significantly improve the accuracy of low-order solutions with no additional computational cost. The same scaling procedure and scale factors work in one- and multi-dimensions. In multidimensions, reducing the order of a solution can save significant amounts of computer time.
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.
General complex polynomial root solver
NASA Astrophysics Data System (ADS)
Skowron, J.; Gould, A.
2012-12-01
This general complex polynomial root solver, implemented in Fortran and further optimized for binary microlenses, uses a new algorithm to solve polynomial equations and is 1.6-3 times faster than the ZROOTS subroutine that is commercially available from Numerical Recipes, depending on application. The largest improvement, when compared to naive solvers, comes from a fail-safe procedure that permits skipping the majority of the calculations in the great majority of cases, without risking catastrophic failure in the few cases that these are actually required.
Efficient modeling of photonic crystals with local Hermite polynomials
NASA Astrophysics Data System (ADS)
Boucher, C. R.; Li, Zehao; Albrecht, J. D.; Ram-Mohan, L. R.
2014-04-01
Developing compact algorithms for accurate electrodynamic calculations with minimal computational cost is an active area of research given the increasing complexity in the design of electromagnetic composite structures such as photonic crystals, metamaterials, optical interconnects, and on-chip routing. We show that electric and magnetic (EM) fields can be calculated using scalar Hermite interpolation polynomials as the numerical basis functions without having to invoke edge-based vector finite elements to suppress spurious solutions or to satisfy boundary conditions. This approach offers several fundamental advantages as evidenced through band structure solutions for periodic systems and through waveguide analysis. Compared with reciprocal space (plane wave expansion) methods for periodic systems, advantages are shown in computational costs, the ability to capture spatial complexity in the dielectric distributions, the demonstration of numerical convergence with scaling, and variational eigenfunctions free of numerical artifacts that arise from mixed-order real space basis sets or the inherent aberrations from transforming reciprocal space solutions of finite expansions. The photonic band structure of a simple crystal is used as a benchmark comparison and the ability to capture the effects of spatially complex dielectric distributions is treated using a complex pattern with highly irregular features that would stress spatial transform limits. This general method is applicable to a broad class of physical systems, e.g., to semiconducting lasers which require simultaneous modeling of transitions in quantum wells or dots together with EM cavity calculations, to modeling plasmonic structures in the presence of EM field emissions, and to on-chip propagation within monolithic integrated circuits.
Efficient modeling of photonic crystals with local Hermite polynomials
Boucher, C. R.; Li, Zehao; Albrecht, J. D.; Ram-Mohan, L. R.
2014-04-21
Developing compact algorithms for accurate electrodynamic calculations with minimal computational cost is an active area of research given the increasing complexity in the design of electromagnetic composite structures such as photonic crystals, metamaterials, optical interconnects, and on-chip routing. We show that electric and magnetic (EM) fields can be calculated using scalar Hermite interpolation polynomials as the numerical basis functions without having to invoke edge-based vector finite elements to suppress spurious solutions or to satisfy boundary conditions. This approach offers several fundamental advantages as evidenced through band structure solutions for periodic systems and through waveguide analysis. Compared with reciprocal space (plane wave expansion) methods for periodic systems, advantages are shown in computational costs, the ability to capture spatial complexity in the dielectric distributions, the demonstration of numerical convergence with scaling, and variational eigenfunctions free of numerical artifacts that arise from mixed-order real space basis sets or the inherent aberrations from transforming reciprocal space solutions of finite expansions. The photonic band structure of a simple crystal is used as a benchmark comparison and the ability to capture the effects of spatially complex dielectric distributions is treated using a complex pattern with highly irregular features that would stress spatial transform limits. This general method is applicable to a broad class of physical systems, e.g., to semiconducting lasers which require simultaneous modeling of transitions in quantum wells or dots together with EM cavity calculations, to modeling plasmonic structures in the presence of EM field emissions, and to on-chip propagation within monolithic integrated circuits.
On the minimum polynomial of supermatrices
NASA Astrophysics Data System (ADS)
Fellouris, Anargyros G.; Matiadou, Lina K.
2002-11-01
In this paper, a new selection of factors for the construction of the minimum polynomial of a supermatrix M is proposed, leading to null polynomials of M of lower degree than the degree of the corresponding polynomial obtained by using the method proposed in the work of Urrutia and Morales [1]. The case of (1 + 1) × (1 + 1) supermatrices has been completely discussed. Moreover, the main theorem concerning the construction of the minimum polynomial as a product of factors from the characteristic polynomial in the general case of (m + n) × (m + n) supermatrices is given. Finally, we prove that the minimum polynomial of a supermatrix M, in general, is not unique.
NASA Astrophysics Data System (ADS)
Liu, Shubin
1996-12-01
It has been shown previously that under certain circumstances the correlation energy density functional Ec[ρ] and its kinetic Tc[ρ] and potential Vc[ρ] components can be expanded in terms of homogeneous functionals An[ρ], with n=1,2,3,..., and where An[ρ] is homogeneous of degree (1-n) with respect to coordinate scaling. In this paper, we extend the analysis to expand similarly the pair distribution function gxc([ρ]r1,r2) and the second-order density matrix ρ2(r1,r2). It is found that both of them can be expanded under certain circumstances in terms of functionals an([ρ]r1,r2), with n=1,2,3,..., that are homogeneous of degree -n in coordinate scaling. The An[ρ] are explicitly obtained in terms of the an([ρ]r1,r2).
Polynomial Algorithms for Item Matching.
ERIC Educational Resources Information Center
Armstrong, Ronald D.; Jones, Douglas H.
1992-01-01
Polynomial algorithms are presented that are used to solve selected problems in test theory, and computational results from sample problems with several hundred decision variables are provided that demonstrate the benefits of these algorithms. The algorithms are based on optimization theory in networks (graphs). (SLD)
Entanglement conditions and polynomial identities
Shchukin, E.
2011-11-15
We develop a rather general approach to entanglement characterization based on convexity properties and polynomial identities. This approach is applied to obtain simple and efficient entanglement conditions that work equally well in both discrete as well as continuous-variable environments. Examples of violations of our conditions are presented.
Polynomial Beam Element Analysis Module
Energy Science and Technology Software Center (ESTSC)
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).
A New Functional Expansion for Polarization Coherence Tomography
NASA Astrophysics Data System (ADS)
Zhang, Hong; Ma, Peifeng; Wang, Chao; Zhang, Bo; Wu, Fan; Tang, Yixian
2011-03-01
In this paper we propose a different functional expansion for polarization coherence tomography (PCT) technique to reconstruct a vertical profile function. Assuming we have a priori knowledge of volume depth and ground topography, estimation of the profile coefficients is feasible. Instead of developing the profile function in a Fourier-Legendre series, we construct orthogonal family of function on [-1, 1] by the weight, deducing the first few orthogonal polynomials. And then we represent the vertical profile function using these orthogonal series, constructing the linear matrix by equation relations. Finally the coefficients are estimated by matrix inversion for the specific orthogonal polynomials. In this way the polynomials for approximation will be promoted up to four order using single-baseline data and up to six order using dual-baseline data. In terms of analysis of condition number of the linear matrix, we find that the CN in this way is smaller than the CN obtained in Fourier-Legendre series, indicating that the inversion in this way is more stable and less susceptible to noise. In the end this method is validated using simulated data.
Solutions of differential equations in a Bernstein polynomial basis
NASA Astrophysics Data System (ADS)
Idrees Bhatti, M.; Bracken, P.
2007-08-01
An algorithm for approximating solutions to differential equations in a modified new Bernstein polynomial basis is introduced. The algorithm expands the desired solution in terms of a set of continuous polynomials over a closed interval and then makes use of the Galerkin method to determine the expansion coefficients to construct a solution. Matrix formulation is used throughout the entire procedure. However, accuracy and efficiency are dependent on the size of the set of Bernstein polynomials and the procedure is much simpler compared to the piecewise B spline method for solving differential equations. A recursive definition of the Bernstein polynomials and their derivatives are also presented. The current procedure is implemented to solve three linear equations and one nonlinear equation, and excellent agreement is found between the exact and approximate solutions. In addition, the algorithm improves the accuracy and efficiency of the traditional methods for solving differential equations that rely on much more complicated numerical techniques. This procedure has great potential to be implemented in more complex systems where there are no exact solutions available except approximations.
Solving the interval type-2 fuzzy polynomial equation using the ranking method
NASA Astrophysics Data System (ADS)
Rahman, Nurhakimah Ab.; Abdullah, Lazim
2014-07-01
Polynomial equations with trapezoidal and triangular fuzzy numbers have attracted some interest among researchers in mathematics, engineering and social sciences. There are some methods that have been developed in order to solve these equations. In this study we are interested in introducing the interval type-2 fuzzy polynomial equation and solving it using the ranking method of fuzzy numbers. The ranking method concept was firstly proposed to find real roots of fuzzy polynomial equation. Therefore, the ranking method is applied to find real roots of the interval type-2 fuzzy polynomial equation. We transform the interval type-2 fuzzy polynomial equation to a system of crisp interval type-2 fuzzy polynomial equation. This transformation is performed using the ranking method of fuzzy numbers based on three parameters, namely value, ambiguity and fuzziness. Finally, we illustrate our approach by numerical example.
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.
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
Bifurcation of Kovalevskaya polynomial
El-Sabaa, F.M.
1995-10-01
The rotation of a rigid body about a fixed point in the Kovalevskaya case, where A = B = 2C, y{sub 0} = z{sub 0} = O (A, B, C are the principal moments of inertia; x{sub 0}, y{sub 0}, z{sub 0} represent the center of mass), has been reduced to quadrature, and the system can be integrated to a Riemann 0-function of two variables. The qualitative investigation of the motion of Kovalevskaya tops has been undertaken by many authors, starting with Applort and continuing with Kozlov. Kolossoff transformed the Kovalevskaya problem into plane motion under a certain potential force. By using elliptic coordinates, Kolossoff proved the inverse problem, i.e., he converted the plane motion system into a Kovalevskaya system. The qualitative investigation of the motion in the two-dimensional tori is given in order to obtain the bifurcation and the phase portrait of the problem.
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.
Quadratic-Like Dynamics of Cubic Polynomials
NASA Astrophysics Data System (ADS)
Blokh, Alexander; Oversteegen, Lex; Ptacek, Ross; Timorin, Vladlen
2016-02-01
A small perturbation of a quadratic polynomial f with a non-repelling fixed point gives a polynomial g with an attracting fixed point and a Jordan curve Julia set, on which g acts like angle doubling. However, there are cubic polynomials with a non-repelling fixed point, for which no perturbation results into a polynomial with Jordan curve Julia set. Motivated by the study of the closure of the Cubic Principal Hyperbolic Domain, we describe such polynomials in terms of their quadratic-like restrictions.
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.
Point estimation of simultaneous methods for solving polynomial equations
NASA Astrophysics Data System (ADS)
Petkovic, Miodrag S.; Petkovic, Ljiljana D.; Rancic, Lidija Z.
2007-08-01
The construction of computationally verifiable initial conditions which provide both the guaranteed and fast convergence of the numerical root-finding algorithm is one of the most important problems in solving nonlinear equations. Smale's "point estimation theory" from 1981 was a great advance in this topic; it treats convergence conditions and the domain of convergence in solving an equation f(z)=0 using only the information of f at the initial point z0. The study of a general problem of the construction of initial conditions of practical interest providing guaranteed convergence is very difficult, even in the case of algebraic polynomials. In the light of Smale's point estimation theory, an efficient approach based on some results concerning localization of polynomial zeros and convergent sequences is applied in this paper to iterative methods for the simultaneous determination of simple zeros of polynomials. We state new, improved initial conditions which provide the guaranteed convergence of frequently used simultaneous methods for solving algebraic equations: Ehrlich-Aberth's method, Ehrlich-Aberth's method with Newton's correction, Borsch-Supan's method with Weierstrass' correction and Halley-like (or Wang-Zheng) method. The introduced concept offers not only a clear insight into the convergence analysis of sequences generated by the considered methods, but also explicitly gives their order of convergence. The stated initial conditions are of significant practical importance since they are computationally verifiable; they depend only on the coefficients of a given polynomial, its degree n and initial approximations to polynomial zeros.
Properties of convergence for [omega],q-Bernstein polynomials
NASA Astrophysics Data System (ADS)
Wang, Heping
2008-04-01
In this paper, we discuss properties of the [omega],q-Bernstein polynomials introduced by S. Lewanowicz and P. Wozny in [S. Lewanowicz, P. Wozny, Generalized Bernstein polynomials, BIT 44 (1) (2004) 63-78], where f[set membership, variant]C[0,1], [omega],q>0, [omega][not equal to]1,q-1,...,q-n+1. When [omega]=0, we recover the q-Bernstein polynomials introduced by [G.M. Phillips, Bernstein polynomials based on the q-integers, Ann. Numer. Math. 4 (1997) 511-518]; when q=1, we recover the classical Bernstein polynomials. We compute the second moment of , and demonstrate that if f is convex and [omega],q[set membership, variant](0,1) or (1,[infinity]), then are monotonically decreasing in n for all x[set membership, variant][0,1]. We prove that for [omega][set membership, variant](0,1), qn[set membership, variant](0,1], the sequence converges to f uniformly on [0,1] for each f[set membership, variant]C[0,1] if and only if limn-->[infinity]qn=1. For fixed [omega],q[set membership, variant](0,1), we prove that the sequence converges for each f[set membership, variant]C[0,1] and obtain the estimates for the rate of convergence of by the modulus of continuity of f, and the estimates are sharp in the sense of order for Lipschitz continuous functions.
An error embedded method based on generalized Chebyshev polynomials
NASA Astrophysics Data System (ADS)
Kim, Philsu; Kim, Junghan; Jung, WonKyu; Bu, Sunyoung
2016-02-01
In this paper, we develop an error embedded method based on generalized Chebyshev polynomials for solving stiff initial value problems. The solution and the error at each integration step are calculated by generalized Chebyshev polynomials of two consecutive degrees having overlapping zeros, which enables us to minimize overall computational costs. Further the errors at each integration step are embedded in the algorithm itself. In terms of concrete convergence and stability analysis, the constructed algorithm turns out to have the 6th order convergence and an almost L-stability. We assess the proposed method with several numerical results, showing that it uses larger time step sizes and is numerically more efficient.
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
On adaptive weighted polynomial preconditioning for Hermitian positive definite matrices
NASA Technical Reports Server (NTRS)
Fischer, Bernd; Freund, Roland W.
1992-01-01
The conjugate gradient algorithm for solving Hermitian positive definite linear systems is usually combined with preconditioning in order to speed up convergence. In recent years, there has been a revival of polynomial preconditioning, motivated by the attractive features of the method on modern architectures. Standard techniques for choosing the preconditioning polynomial are based only on bounds for the extreme eigenvalues. Here a different approach is proposed, which aims at adapting the preconditioner to the eigenvalue distribution of the coefficient matrix. The technique is based on the observation that good estimates for the eigenvalue distribution can be derived after only a few steps of the Lanczos process. This information is then used to construct a weight function for a suitable Chebyshev approximation problem. The solution of this problem yields the polynomial preconditioner. In particular, we investigate the use of Bernstein-Szego weights.
NASA Astrophysics Data System (ADS)
Kalagov, G. A.; Kompaniets, M. V.; Nalimov, M. Yu.
2014-11-01
We use quantum-field renormalization group methods to study the phase transition in an equilibrium system of nonrelativistic Fermi particles with the "density-density" interaction in the formalism of temperature Green's functions. We especially attend to the case of particles with spins greater than 1/2 or fermionic fields with additional indices for some reason. In the vicinity of the phase transition point, we reduce this model to a ϕ 4 -type theory with a matrix complex skew-symmetric field. We define a family of instantons of this model and investigate the asymptotic behavior of quantum field expansions in this model. We calculate the β-functions of the renormalization group equation through the third order in the ( 4 ∈)-scheme. In the physical space dimensions D = 2, 3, we resum solutions of the renormalization group equation on trajectories of invariant charges. Our results confirm the previously proposed suggestion that in the system under consideration, there is a first-order phase transition into a superconducting state that occurs at a higher temperature than the classical theory predicts.
Borrel, Guillaume; Gaci, Nadia; Peyret, Pierre; O'Toole, Paul W.; Gribaldo, Simonetta
2014-01-01
Pyrrolysine (Pyl), the 22nd proteogenic amino acid, was restricted until recently to few organisms. Its translational use necessitates the presence of enzymes for synthesizing it from lysine, a dedicated amber stop codon suppressor tRNA, and a specific amino-acyl tRNA synthetase. The three genomes of the recently proposed Thermoplasmata-related 7th order of methanogens contain the complete genetic set for Pyl synthesis and its translational use. Here, we have analyzed the genomic features of the Pyl-coding system in these three genomes with those previously known from Bacteria and Archaea and analyzed the phylogeny of each component. This shows unique peculiarities, notably an amber tRNAPyl with an imperfect anticodon stem and a shortened tRNAPyl synthetase. Phylogenetic analysis indicates that a Pyl-coding system was present in the ancestor of the seventh order of methanogens and appears more closely related to Bacteria than to Methanosarcinaceae, suggesting the involvement of lateral gene transfer in the spreading of pyrrolysine between the two prokaryotic domains. We propose that the Pyl-coding system likely emerged once in Archaea, in a hydrogenotrophic and methanol-H2-dependent methylotrophic methanogen. The close relationship between methanogenesis and the Pyl system provides a possible example of expansion of a still evolving genetic code, shaped by metabolic requirements. PMID:24669202
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
NASA Astrophysics Data System (ADS)
Trigub, R. M.
2009-08-01
We prove a general direct theorem on the simultaneous pointwise approximation of smooth periodic functions and their derivatives by trigonometric polynomials and their derivatives with Hermitian interpolation. We study the order of approximation by polynomials whose graphs lie above or below the graph of the function on certain intervals. We prove several inequalities for Hermitian interpolation with absolute constants (for any system of nodes). For the first time we get a theorem on the best-order approximation of functions by polynomials with interpolation at a given system of nodes. We also provide a construction of Hermitian interpolating trigonometric polynomials for periodic functions (in the case of one node, these are trigonometric Taylor polynomials).
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.
From Jack polynomials to minimal model spectra
NASA Astrophysics Data System (ADS)
Ridout, David; Wood, Simon
2015-01-01
In this note, a deep connection between free field realizations of conformal field theories and symmetric polynomials is presented. We give a brief introduction into the necessary prerequisites of both free field realizations and symmetric polynomials, in particular Jack symmetric polynomials. Then we combine these two fields to classify the irreducible representations of the minimal model vertex operator algebras as an illuminating example of the power of these methods. While these results on the representation theory of the minimal models are all known, this note exploits the full power of Jack polynomials to present significant simplifications of the original proofs in the literature.
Spatial image polynomial decomposition with application to video classification
NASA Astrophysics Data System (ADS)
El Moubtahij, Redouane; Augereau, Bertrand; Tairi, Hamid; Fernandez-Maloigne, Christine
2015-11-01
This paper addresses the use of orthogonal polynomial basis transform in video classification due to its multiple advantages, especially for multiscale and multiresolution analysis similar to the wavelet transform. In our approach, we benefit from these advantages to reduce the resolution of the video by using a multiscale/multiresolution decomposition to define a new algorithm that decomposes a color image into geometry and texture component by projecting the image on a bivariate polynomial basis and considering the geometry component as the partial reconstruction and the texture component as the remaining part, and finally to model the features (like motion and texture) extracted from reduced image sequences by projecting them into a bivariate polynomial basis in order to construct a hybrid polynomial motion texture video descriptor. To evaluate our approach, we consider two visual recognition tasks, namely the classification of dynamic textures and recognition of human actions. The experimental section shows that the proposed approach achieves a perfect recognition rate in the Weizmann database and highest accuracy in the Dyntex++ database compared to existing methods.
Network meta-analysis of survival data with fractional polynomials
2011-01-01
Background Pairwise meta-analysis, indirect treatment comparisons and network meta-analysis for aggregate level survival data are often based on the reported hazard ratio, which relies on the proportional hazards assumption. This assumption is implausible when hazard functions intersect, and can have a huge impact on decisions based on comparisons of expected survival, such as cost-effectiveness analysis. Methods As an alternative to network meta-analysis of survival data in which the treatment effect is represented by the constant hazard ratio, a multi-dimensional treatment effect approach is presented. With fractional polynomials the hazard functions of interventions compared in a randomized controlled trial are modeled, and the difference between the parameters of these fractional polynomials within a trial are synthesized (and indirectly compared) across studies. Results The proposed models are illustrated with an analysis of survival data in non-small-cell lung cancer. Fixed and random effects first and second order fractional polynomials were evaluated. Conclusion (Network) meta-analysis of survival data with models where the treatment effect is represented with several parameters using fractional polynomials can be more closely fitted to the available data than meta-analysis based on the constant hazard ratio. PMID:21548941
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.
Polynomial chaotic inflation in supergravity
Nakayama, Kazunori; Takahashi, Fuminobu; Yanagida, Tsutomu T. E-mail: fumi@tuhep.phys.tohoku.ac.jp
2013-08-01
We present a general polynomial chaotic inflation model in supergravity, for which the predicted spectral index and tensor-to-scalar ratio can lie within the 1σ region allowed by the Planck results. Most importantly, the predicted tensor-to-scalar ratio is large enough to be probed in the on-going and future B-mode experiments. We study the inflaton dynamics and the subsequent reheating process in a couple of specific examples. The non-thermal gravitino production from the inflaton decay can be suppressed in a case with a discrete Z{sub 2} symmetry. We find that the reheating temperature can be naturally as high as O(10{sup 9−10}) GeV, sufficient for baryon asymmetry generation through (non-)thermal leptogenesis.
Fractal Trigonometric Polynomials for Restricted Range Approximation
NASA Astrophysics Data System (ADS)
Chand, A. K. B.; Navascués, M. A.; Viswanathan, P.; Katiyar, S. K.
2016-05-01
One-sided approximation tackles the problem of approximation of a prescribed function by simple traditional functions such as polynomials or trigonometric functions that lie completely above or below it. In this paper, we use the concept of fractal interpolation function (FIF), precisely of fractal trigonometric polynomials, to construct one-sided uniform approximants for some classes of continuous functions.
On the Waring problem for polynomial rings
Fröberg, Ralf; Ottaviani, Giorgio; Shapiro, Boris
2012-01-01
In this note we discuss an analog of the classical Waring problem for . Namely, we show that a general homogeneous polynomial of degree divisible by k≥2 can be represented as a sum of at most kn k-th powers of homogeneous polynomials in . Noticeably, kn coincides with the number obtained by naive dimension count. PMID:22460787
Point vortex equilibria related to Bessel polynomials
NASA Astrophysics Data System (ADS)
O'Neil, Kevin A.
2016-05-01
The method of polynomials is used to construct two families of stationary point vortex configurations. The vortices are placed at the reciprocals of the zeroes of Bessel polynomials. Configurations that translate uniformly, and configurations that are completely stationary, are obtained in this way.
Mangui Han
2004-12-19
Thermal expansion (TE) and magnetostriction (MS) measurements have been conducted for Gd{sub 5}(Si{sub x}Ge{sub 1-x}){sub 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{sub 5}(Si{sub x}Ge{sub 1-x}){sub 4} for x 0 {approx} 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{sub 5}(Si{sub 1.95}Ge{sub 2.05}), Gd{sub 5}(Si{sub 2}Ge{sub 2}), Gd{sub 5}(Si{sub 2.09}Ge{sub 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{sub 5}(Si{sub 0.15}Ge{sub 3.85}), it is partially reversible at some temperature range between the antiferromagnetic and the ferromagnetic state. For Gd{sub 5}(Si{sub 2.3}Ge{sub 1.7}) and Gd{sub 5}(Si{sub 3}Ge{sub 1}), it was a second order transformation between the paramagnetic and ferromagnetic state, because no {Delta}T have been found. Giant magnetostriction was only found on Gd{sub 5}(Si{sub 1.95}Ge{sub 2.05}), Gd{sub 5}(Si{sub 2}Ge{sub 2}), Gd{sub 5}(Si{sub 2.09}Ge{sub 1.91}) in their vicinity of first order transformation. MFM images have also been taken on polycrystal sample Gd{sub 5}(Si{sub 2.09}Ge{sub 1.91}) to investigate the transformation process. The results also indicates that the Curie temperature was lower and the thermally-induced strain higher in the sample made from lower purity level Gd starting materials compared with the sample made from high purity Gd metal. TE, MS, MFM and VSM measurements
Matrix product formula for Macdonald polynomials
NASA Astrophysics Data System (ADS)
Cantini, Luigi; de Gier, Jan; Wheeler, Michael
2015-09-01
We derive a matrix product formula for symmetric Macdonald polynomials. Our results are obtained by constructing polynomial solutions of deformed Knizhnik-Zamolodchikov equations, which arise by considering representations of the Zamolodchikov-Faddeev and Yang-Baxter algebras in terms of t-deformed bosonic operators. These solutions are generalized probabilities for particle configurations of the multi-species asymmetric exclusion process, and form a basis of the ring of polynomials in n variables whose elements are indexed by compositions. For weakly increasing compositions (anti-dominant weights), these basis elements coincide with non-symmetric Macdonald polynomials. Our formulas imply a natural combinatorial interpretation in terms of solvable lattice models. They also imply that normalizations of stationary states of multi-species exclusion processes are obtained as Macdonald polynomials at q = 1.
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.
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.
Series Expansion Method for Asymmetrical Percolation Models with Two Connection Probabilities
NASA Astrophysics Data System (ADS)
Inui, Norio; Komatsu, Genichi; Kameoka, Koichi
2000-01-01
In order to study the solvability of the percolation model based on Guttmann and Enting's conjecture, the power series for the percolation probability in the form of ∑nHn(q)pn is examined. Although the power series is given by calculating inverse of the transfer-matrix in principle, it is very hard to obtain the inverse matrix containing many complex polynomials as elements. We introduce a new series expansion technique which does not necessitate inverse operation for the transfer-matrix.By using the new procedure, we derive the series of the asymmetrical percolation probability including the isotropic percolation probability as a special case.
New families of superintegrable systems from Hermite and Laguerre exceptional orthogonal polynomials
Marquette, Ian; Quesne, Christiane
2013-04-15
In recent years, many exceptional orthogonal polynomials (EOP) were introduced and used to construct new families of 1D exactly solvable quantum potentials, some of which are shape invariant. In this paper, we construct from Hermite and Laguerre EOP and their related quantum systems new 2D superintegrable Hamiltonians with higher-order integrals of motion and the polynomial algebras generated by their integrals of motion. We obtain the finite-dimensional unitary representations of the polynomial algebras and the corresponding energy spectrum. We also point out a new type of degeneracies of the energy levels of these systems that is associated with holes in sequences of EOP.
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.
Efficient Multiplication of Polynomials on Graphics Hardware
NASA Astrophysics Data System (ADS)
Emeliyanenko, Pavel
We present the algorithm to multiply univariate polynomials with integer coefficients efficiently using the Number Theoretic transform (NTT) on Graphics Processing Units (GPU). The same approach can be used to multiply large integers encoded as polynomials. Our algorithm exploits fused multiply-add capabilities of the graphics hardware. NTT multiplications are executed in parallel for a set of distinct primes followed by reconstruction using the Chinese Remainder theorem (CRT) on the GPU. Our benchmarking experiences show the NTT multiplication performance up to 77 GMul/s. We compared our approach with CPU-based implementations of polynomial and large integer multiplication provided by NTL and GMP libraries.
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.
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.
NASA Astrophysics Data System (ADS)
saev, Yu N. I.; Kolchanova, V. A.; Tarasenko, S. S.; Tikhomirova, O. V.
2016-04-01
The paper proposes an original method of calculating the charge distribution on the surface of the conductive plate introduced into the external electrostatic field. The authors managed to obtain the polynomials which allow to solve the integral equation that establishes the relationship between charge distribution of the conductive plate and the potential distribution of the external field and the potential on the surface of the plate. The proposed algorithms solutions are valid in the presence of axial symmetry of the field and the plate. Examples of calculation of conductor charge distribution in the presence of external field by using a polynomial expansion have been presented. The comparisons of results calculated by the polynomial method and by known analytical solutions have been given
Zhao, Chunyu; Burge, James H
2013-12-16
Zernike polynomials are an orthonormal set of scalar functions over a circular domain, and are commonly used to represent wavefront phase or surface irregularity. In optical testing, slope or curvature of a surface or wavefront is sometimes measured instead, from which the surface or wavefront map is obtained. Previously we derived an orthonormal set of vector polynomials that fit to slope measurement data and yield the surface or wavefront map represented by Zernike polynomials. Here we define a 3-element curvature vector used to represent the second derivatives of a continuous surface, and derive a set of orthonormal curvature basis functions that are written in terms of Zernike polynomials. We call the new curvature functions the C polynomials. Closed form relations for the complete basis set are provided, and we show how to determine Zernike surface coefficients from the curvature data as represented by the C polynomials. PMID:24514717
Uncertainty quantification in simulations of epidemics using polynomial chaos.
Santonja, F; Chen-Charpentier, B
2012-01-01
Mathematical models based on ordinary differential equations are a useful tool to study the processes involved in epidemiology. Many models consider that the parameters are deterministic variables. But in practice, the transmission parameters present large variability and it is not possible to determine them exactly, and it is necessary to introduce randomness. In this paper, we present an application of the polynomial chaos approach to epidemiological mathematical models based on ordinary differential equations with random coefficients. Taking into account the variability of the transmission parameters of the model, this approach allows us to obtain an auxiliary system of differential equations, which is then integrated numerically to obtain the first-and the second-order moments of the output stochastic processes. A sensitivity analysis based on the polynomial chaos approach is also performed to determine which parameters have the greatest influence on the results. As an example, we will apply the approach to an obesity epidemic model. PMID:22927889
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.
The Rational Polynomial Coefficients Modification Using Digital Elevation Models
NASA Astrophysics Data System (ADS)
Alidoost, F.; Azizi, A.; Arefi, H.
2015-12-01
The high-resolution satellite imageries (HRSI) are as primary dataset for different applications such as DEM generation, 3D city mapping, change detection, monitoring, and deformation detection. The geo-location information of HRSI are stored in metadata called Rational Polynomial Coefficients (RPCs). There are many methods to improve and modify the RPCs in order to have a precise mapping. In this paper, an automatic approach is presented for the RPC modification using global Digital Elevation Models. The main steps of this approach are: relative digital elevation model generation, shift parameters calculation, sparse point cloud generation and shift correction, and rational polynomial fitting. Using some ground control points, the accuracy of the proposed method is evaluated based on statistical descriptors in which the results show that the geo-location accuracy of HRSI can be improved without using Ground Control Points (GCPs).
Orthogonal polynomial interpretation of Δ-Toda equations
NASA Astrophysics Data System (ADS)
Area, I.; Branquinho, A.; Foulquié Moreno, A.; Godoy, E.
2015-10-01
In this paper a discretization of Toda equations is analyzed. The correspondence between these Δ-Toda equations for the coefficients of the Jacobi operator and its resolvent function is established. It is shown that the spectral measure of these operators evolve in t like {(1+x)}1-t {{d}}μ (x) where {{d}}μ is a given positive Borel measure. The Lax pair for the Δ-Toda equations is derived and characterized in terms of linear functionals, where orthogonal polynomials which satisfy an Appell condition with respect to the forward difference operator Δ appear in a natural way. In order to illustrate the results of the paper we work out two examples of Δ-Toda equations related with Jacobi and Laguerre orthogonal polynomials.
Frameworks for Logically Classifying Polynomial-Time Optimisation Problems
NASA Astrophysics Data System (ADS)
Gate, James; Stewart, Iain A.
We show that a logical framework, based around a fragment of existential second-order logic formerly proposed by others so as to capture the class of polynomially-bounded P-optimisation problems, cannot hope to do so, under the assumption that P ≠ NP. We do this by exhibiting polynomially-bounded maximisation and minimisation problems that can be expressed in the framework but whose decision versions are NP-complete. We propose an alternative logical framework, based around inflationary fixed-point logic, and show that we can capture the above classes of optimisation problems. We use the inductive depth of an inflationary fixed-point as a means to describe the objective functions of the instances of our optimisation problems.
Symmetric multivariate polynomials as a basis for three-boson light-front wave functions.
Chabysheva, Sophia S; Elliott, Blair; Hiller, John R
2013-12-01
We develop a polynomial basis to be used in numerical calculations of light-front Fock-space wave functions. Such wave functions typically depend on longitudinal momentum fractions that sum to unity. For three particles, this constraint limits the two remaining independent momentum fractions to a triangle, for which the three momentum fractions act as barycentric coordinates. For three identical bosons, the wave function must be symmetric with respect to all three momentum fractions. Therefore, as a basis, we construct polynomials in two variables on a triangle that are symmetric with respect to the interchange of any two barycentric coordinates. We find that, through the fifth order, the polynomial is unique at each order, and, in general, these polynomials can be constructed from products of powers of the second- and third-order polynomials. The use of such a basis is illustrated in a calculation of a light-front wave function in two-dimensional ϕ(4) theory; the polynomial basis performs much better than the plane-wave basis used in discrete light-cone quantization. PMID:24483584
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
2014-09-01
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. PMID:25116333
Tutte Polynomial of Scale-Free Networks
NASA Astrophysics Data System (ADS)
Chen, Hanlin; Deng, Hanyuan
2016-05-01
The Tutte polynomial of a graph, or equivalently the q-state Potts model partition function, is a two-variable polynomial graph invariant of considerable importance in both statistical physics and combinatorics. The computation of this invariant for a graph is NP-hard in general. In this paper, we focus on two iteratively growing scale-free networks, which are ubiquitous in real-life systems. Based on their self-similar structures, we mainly obtain recursive formulas for the Tutte polynomials of two scale-free networks (lattices), one is fractal and "large world", while the other is non-fractal but possess the small-world property. Furthermore, we give some exact analytical expressions of the Tutte polynomial for several special points at ( x, y)-plane, such as, the number of spanning trees, the number of acyclic orientations, etc.
Hermite polynomials and quasi-classical asymptotics
Ali, S. Twareque; Engliš, Miroslav
2014-04-15
We study an unorthodox variant of the Berezin-Toeplitz type of quantization scheme, on a reproducing kernel Hilbert space generated by the real Hermite polynomials and work out the associated quasi-classical asymptotics.
Inequalities for a polynomial and its derivative
NASA Astrophysics Data System (ADS)
Chanam, Barchand; Dewan, K. K.
2007-12-01
Let , 1[less-than-or-equals, slant][mu][less-than-or-equals, slant]n, be a polynomial of degree n such that p(z)[not equal to]0 in z
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.)
Stochastic processes with orthogonal polynomial eigenfunctions
NASA Astrophysics Data System (ADS)
Griffiths, Bob
2009-12-01
Markov processes which are reversible with either Gamma, Normal, Poisson or Negative Binomial stationary distributions in the Meixner class and have orthogonal polynomial eigenfunctions are characterized as being processes subordinated to well-known diffusion processes for the Gamma and Normal, and birth and death processes for the Poisson and Negative Binomial. A characterization of Markov processes with Beta stationary distributions and Jacobi polynomial eigenvalues is also discussed.
Distortion theorems for polynomials on a circle
Dubinin, V N
2000-12-31
Inequalities for the derivatives with respect to {phi}=arg z the functions ReP(z), |P(z)|{sup 2} and arg P(z) are established for an algebraic polynomial P(z) at points on the circle |z|=1. These estimates depend, in particular, on the constant term and the leading coefficient of the polynomial P(z) and improve the classical Bernstein and Turan inequalities. The method of proof is based on the techniques of generalized reduced moduli.
Combinatorial and algorithm aspects of hyperbolic polynomials
Gurvits, Leonid I.
2004-01-01
Univariate polynomials with real roots appear quite often in modern combinatorics, especially in the context of integer polytopes. We discovered in this paper rather unexpected and very likely far-reaching connections between hyperbolic polynomials and many classical combinatorial and algorithmic problems. There are still several open problems. The most interesting is a hyperbolic generalization of the van der Waerden conjecture for permanents of doubly stochastic matrices.
Lin, J. C.; Tong, P. Lin, S.; Wang, B. S.; Song, W. H.; Tong, W.; Zou, Y. M.; Sun, Y. P.
2015-02-23
The thermal expansion and magnetic properties of antiperovskite manganese nitrides Ag{sub 1−x}NMn{sub 3+x} were reported. The substitution of Mn for Ag effectively broadens the temperature range of negative thermal expansion and drives it to cryogenic temperatures. As x increases, the paramagnetic (PM) to antiferromagnetic (AFM) phase transition temperature decreases. At x ∼ 0.2, the PM-AFM transition overlaps with the AFM to glass-like state transition. Above x = 0.2, two new distinct magnetic transitions were observed: One occurs above room temperature from PM to ferromagnetic (FM), and the other one evolves at a lower temperature (T{sup *}) below which both AFM and FM orderings are involved. Further, electron spin resonance measurement suggests that the broadened volume change near T{sup *} is closely related with the evolution of Γ{sup 5g} AFM ordering.
NASA Astrophysics Data System (ADS)
Lin, J. C.; Tong, P.; Tong, W.; Lin, S.; Wang, B. S.; Song, W. H.; Zou, Y. M.; Sun, Y. P.
2015-02-01
The thermal expansion and magnetic properties of antiperovskite manganese nitrides Ag1-xNMn3+x were reported. The substitution of Mn for Ag effectively broadens the temperature range of negative thermal expansion and drives it to cryogenic temperatures. As x increases, the paramagnetic (PM) to antiferromagnetic (AFM) phase transition temperature decreases. At x ˜ 0.2, the PM-AFM transition overlaps with the AFM to glass-like state transition. Above x = 0.2, two new distinct magnetic transitions were observed: One occurs above room temperature from PM to ferromagnetic (FM), and the other one evolves at a lower temperature (T*) below which both AFM and FM orderings are involved. Further, electron spin resonance measurement suggests that the broadened volume change near T* is closely related with the evolution of Γ5g AFM ordering.
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
On polynomial preconditioning for indefinite Hermitian matrices
NASA Technical Reports Server (NTRS)
Freund, Roland W.
1989-01-01
The minimal residual method is studied combined with polynomial preconditioning for solving large linear systems (Ax = b) with indefinite Hermitian coefficient matrices (A). The standard approach for choosing the polynomial preconditioners leads to preconditioned systems which are positive definite. Here, a different strategy is studied which leaves the preconditioned coefficient matrix indefinite. More precisely, the polynomial preconditioner is designed to cluster the positive, resp. negative eigenvalues of A around 1, resp. around some negative constant. In particular, it is shown that such indefinite polynomial preconditioners can be obtained as the optimal solutions of a certain two parameter family of Chebyshev approximation problems. Some basic results are established for these approximation problems and a Remez type algorithm is sketched for their numerical solution. The problem of selecting the parameters such that the resulting indefinite polynomial preconditioners speeds up the convergence of minimal residual method optimally is also addressed. An approach is proposed based on the concept of asymptotic convergence factors. Finally, some numerical examples of indefinite polynomial preconditioners are given.
An atlas of Rapp's 180-th order geopotential.
NASA Astrophysics Data System (ADS)
Melvin, P. J.
1986-08-01
Deprit's 1979 approach to the summation of the spherical harmonic expansion of the geopotential has been modified to spherical components and normalized Legendre polynomials. An algorithm has been developed which produces ten fields at the users option: the undulations of the geoid, three anomalous components of the gravity vector, or six components of the Hessian of the geopotential (gravity gradient). The algorithm is stable to high orders in single precision and does not treat the polar regions as a special case. Eleven contour maps of components of the anomalous geopotential on the surface of the ellipsoid are presented to validate the algorithm.
On the existence of polynomial time approximation schemes for OBDD minimization
NASA Astrophysics Data System (ADS)
Sieling, Detlef
The size of Ordered Binary Decision Diagrams (OBDDs) is determined by the chosen variable ordering. A poor choice may cause an OBDD to be too large to fit into the available memory. The decision variant of the variable ordering problem is known to be NP-complete. We strengthen this result by showing that there is no polynomial time approximation scheme for the variable ordering problem unless P = NP. We also prove a small lower bound on the performance ratio of a polynomial time approximation algorithm under the assumption P ≠ NP.
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.
Possible quantum algorithms for the Bollobas-Riordan-Tutte polynomial of a ribbon graph
NASA Astrophysics Data System (ADS)
Vélez, Mario; Ospina, Juan
2008-04-01
Three possible quantum algorithms, for the computation of the Bollobás-Riordan-Tutte polynomial of a given ribbon graph, are presented and discussed. The first possible algorithm is based on the spanning quasi-trees expansion for generalized Tutte polynomials of generalized graphs and on a quantum version of the Binary Decision Diagram (BDD) for quasi-trees . The second possible algorithm is based on the relation between the Kauffman bracket and the Tutte polynomial; and with an application of the recently introduced Aharonov-Arad-Eban-Landau quantum algorithm. The third possible algorithm is based on the relation between the HOMFLY polynomial and the Tutte polynomial; and with an application of the Wocjan-Yard quantum algorithm. It is claimed that these possible algorithms may be more efficient that the best known classical algorithms. These three algorithms may have interesting applications in computer science at general or in computational biology and bio-informatics in particular. A line for future research based on the categorification project is mentioned.
Online segmentation of time series based on polynomial least-squares approximations.
Fuchs, Erich; Gruber, Thiemo; Nitschke, Jiri; Sick, Bernhard
2010-12-01
The paper presents SwiftSeg, a novel technique for online time series segmentation and piecewise polynomial representation. The segmentation approach is based on a least-squares approximation of time series in sliding and/or growing time windows utilizing a basis of orthogonal polynomials. This allows the definition of fast update steps for the approximating polynomial, where the computational effort depends only on the degree of the approximating polynomial and not on the length of the time window. The coefficients of the orthogonal expansion of the approximating polynomial-obtained by means of the update steps-can be interpreted as optimal (in the least-squares sense) estimators for average, slope, curvature, change of curvature, etc., of the signal in the time window considered. These coefficients, as well as the approximation error, may be used in a very intuitive way to define segmentation criteria. The properties of SwiftSeg are evaluated by means of some artificial and real benchmark time series. It is compared to three different offline and online techniques to assess its accuracy and runtime. It is shown that SwiftSeg-which is suitable for many data streaming applications-offers high accuracy at very low computational costs. PMID:20975120
Constraints on SU(2) ⊗ SU(2) invariant polynomials for a pair of entangled qubits
NASA Astrophysics Data System (ADS)
Gerdt, V.; Khvedelidze, A.; Palii, Yu.
2011-06-01
We discuss the entanglement properties of two qubits in terms of polynomial invariants of the adjoint action of SU(2) ⊕ SU(2) group on the space of density matrices mathfrak{P}_ + . Since elements of mathfrak{P}_ + are Hermitian, non-negative fourth-order matrices with unit trace, the space of density matrices represents a semi-algebraic subset, mathfrak{P}_ + in mathbb{R}^{15} . We define mathfrak{P}_ + explicitly with the aid of polynomial inequalities in the Casimir operators of the enveloping algebra of SU(4) group. Using this result the optimal integrity basis for polynomial SU(2) ⊕ SU(2) invariants is proposed and the well-known Peres-Horodecki separability criterion for 2-qubit density matrices is given in the form of polynomial inequalities in three SU(4) Casimir invariants and two SU(2) ⊕ SU(2) scalars; namely, determinants of the so-called correlation and the Schlienz-Mahler entanglement matrices.
Kostant polynomials and the cohomology ring for G/B
Billey, Sara C.
1997-01-01
The Schubert calculus for G/B can be completely determined by a certain matrix related to the Kostant polynomials introduced in section 5 of Bernstein, Gelfand, and Gelfand [Bernstein, I., Gelfand, I. & Gelfand, S. (1973) Russ. Math. Surv. 28, 1–26]. The polynomials are defined by vanishing properties on the orbit of a regular point under the action of the Weyl group. For each element w in the Weyl group the polynomials also have nonzero values on the orbit points corresponding to elements which are larger in the Bruhat order than w. The main theorem given here is an explicit formula for these values. The matrix of orbit values can be used to determine the cup product for the cohomology ring for G/B, using only linear algebra or as described by Lascoux and Schützenberger [Lascoux, A. & Schützenberger, M.-P. (1982) C. R. Seances Acad. Sci. Ser. A 294, 447–450]. Complete proofs of all the theorems will appear in a forthcoming paper. PMID:11038536
Kostant polynomials and the cohomology ring for G/B.
Billey, S C
1997-01-01
The Schubert calculus for G/B can be completely determined by a certain matrix related to the Kostant polynomials introduced in section 5 of Bernstein, Gelfand, and Gelfand [Bernstein, I., Gelfand, I. & Gelfand, S. (1973) Russ. Math. Surv. 28, 1-26]. The polynomials are defined by vanishing properties on the orbit of a regular point under the action of the Weyl group. For each element w in the Weyl group the polynomials also have nonzero values on the orbit points corresponding to elements which are larger in the Bruhat order than w. The main theorem given here is an explicit formula for these values. The matrix of orbit values can be used to determine the cup product for the cohomology ring for G/B, using only linear algebra or as described by Lascoux and Schützenberger [Lascoux, A. & Schützenberger, M.-P. (1982) C. R. Seances Acad. Sci. Ser. A 294, 447-450]. Complete proofs of all the theorems will appear in a forthcoming paper. PMID:11038536