Mason, JC
2002-01-01
Chebyshev polynomials crop up in virtually every area of numerical analysis, and they hold particular importance in recent advances in subjects such as orthogonal polynomials, polynomial approximation, numerical integration, and spectral methods. Yet no book dedicated to Chebyshev polynomials has been published since 1990, and even that work focused primarily on the theoretical aspects. A broad, up-to-date treatment is long overdue.Providing highly readable exposition on the subject''s state of the art, Chebyshev Polynomials is just such a treatment. It includes rigorous yet down-to-earth coverage of the theory along with an in-depth look at the properties of all four kinds of Chebyshev polynomials-properties that lead to a range of results in areas such as approximation, series expansions, interpolation, quadrature, and integral equations. Problems in each chapter, ranging in difficulty from elementary to quite advanced, reinforce the concepts and methods presented.Far from being an esoteric subject, Chebysh...
IIR approximations to the fractional differentiator/integrator using Chebyshev polynomials theory.
Romero, M; de Madrid, A P; Mañoso, C; Vinagre, B M
2013-07-01
This paper deals with the use of Chebyshev polynomials theory to achieve accurate discrete-time approximations to the fractional-order differentiator/integrator in terms of IIR filters. These filters are obtained using the Chebyshev-Padé and the Rational Chebyshev approximations, two highly accurate numerical methods that can be computed with ease using available software. They are compared against other highly accurate approximations proposed in the literature. It is also shown how the frequency response of the fractional-order integrator approximations can be easily improved at low frequencies. PMID:23507506
Application of Chebyshev Polynomial to simulated modeling
Institute of Scientific and Technical Information of China (English)
CHI Hai-hong; LI Dian-pu
2006-01-01
Chebyshev polynomial is widely used in many fields, and used usually as function approximation in numerical calculation. In this paper, Chebyshev polynomial expression of the propeller properties across four quadrants is given at first, then the expression of Chebyshev polynomial is transformed to ordinary polynomial for the need of simulation of propeller dynamics. On the basis of it,the dynamical models of propeller across four quadrants are given. The simulation results show the efficiency of mathematical model.
Kaporin, I. E.
2012-02-01
In order to precondition a sparse symmetric positive definite matrix, its approximate inverse is examined, which is represented as the product of two sparse mutually adjoint triangular matrices. In this way, the solution of the corresponding system of linear algebraic equations (SLAE) by applying the preconditioned conjugate gradient method (CGM) is reduced to performing only elementary vector operations and calculating sparse matrix-vector products. A method for constructing the above preconditioner is described and analyzed. The triangular factor has a fixed sparsity pattern and is optimal in the sense that the preconditioned matrix has a minimum K-condition number. The use of polynomial preconditioning based on Chebyshev polynomials makes it possible to considerably reduce the amount of scalar product operations (at the cost of an insignificant increase in the total number of arithmetic operations). The possibility of an efficient massively parallel implementation of the resulting method for solving SLAEs is discussed. For a sequential version of this method, the results obtained by solving 56 test problems from the Florida sparse matrix collection (which are large-scale and ill-conditioned) are presented. These results show that the method is highly reliable and has low computational costs.
Blind Signature Scheme Based on Chebyshev Polynomials
Directory of Open Access Journals (Sweden)
Maheswara Rao Valluri
2011-12-01
Full Text Available A blind signature scheme is a cryptographic protocol to obtain a valid signature for a message from a signer such that signer’s view of the protocol can’t be linked to the resulting message signature pair. This paper presents blind signature scheme using Chebyshev polynomials. The security of the given scheme depends upon the intractability of the integer factorization problem and discrete logarithms ofChebyshev polynomials.
Digital terrain modeling with the Chebyshev polynomials
Florinsky, I V
2015-01-01
Mathematical problems of digital terrain analysis include interpolation of digital elevation models (DEMs), DEM generalization and denoising, and computation of morphometric variables by calculation of partial derivatives of elevation. Traditionally, these procedures are based on numerical treatments of two-variable discrete functions of elevation. We developed a spectral analytical method and algorithm based on high-order orthogonal expansions using the Chebyshev polynomials of the first kind with the subsequent Fejer summation. The method and algorithm are intended for DEM analytical treatment, such as, DEM global approximation, denoising, and generalization as well as computation of morphometric variables by analytical calculation of partial derivatives. To test the method and algorithm, we used a DEM of the Northern Andes including 230,880 points (the elevation matrix 480 $\\times$ 481). DEMs were reconstructed with 480, 240, 120, 60, and 30 expansion coefficients. The first and second partial derivatives ...
On the Connection Coefficients of the Chebyshev-Boubaker Polynomials
Directory of Open Access Journals (Sweden)
Paul Barry
2013-01-01
Full Text Available The Chebyshev-Boubaker polynomials are the orthogonal polynomials whose coefficient arrays are defined by ordinary Riordan arrays. Examples include the Chebyshev polynomials of the second kind and the Boubaker polynomials. We study the connection coefficients of this class of orthogonal polynomials, indicating how Riordan array techniques can lead to closed-form expressions for these connection coefficients as well as recurrence relations that define them.
Discrete Fourier Analysis and Chebyshev Polynomials with G2 Group
Directory of Open Access Journals (Sweden)
Huiyuan Li
2012-10-01
Full Text Available The discrete Fourier analysis on the 30°-60°-90° triangle is deduced from the corresponding results on the regular hexagon by considering functions invariant under the group G2, which leads to the definition of four families generalized Chebyshev polynomials. The study of these polynomials leads to a Sturm-Liouville eigenvalue problem that contains two parameters, whose solutions are analogues of the Jacobi polynomials. Under a concept of m-degree and by introducing a new ordering among monomials, these polynomials are shown to share properties of the ordinary orthogonal polynomials. In particular, their common zeros generate cubature rules of Gauss type.
Deprit, A.
1975-01-01
A theory for generating segmented ephemerides is discussed as a means for fast generation and simple retrieval of nominal orbit data. Over a succession of finite intervals of time, the orbit is represented by a best approximation expressed by Chebyshev polynomials. Storage of coefficients tables for Chebyshev polynomials is seen as a method to reduce data and decrease transmission costs. A general algorithm was constructed and computer programs were designed. The possibility of storing an ephemeris for a few days in the on-board computer, or in microprocessors attached to the data collectors is suggested.
Cryptanalysis of Multiplicative Coupled Cryptosystems Based on the Chebyshev Polynomials
Shakiba, Ali; Hooshmandasl, Mohammad Reza; Meybodi, Mohsen Alambardar
2016-06-01
In this work, we propose a class of public-key cryptosystems called multiplicative coupled cryptosystem, or MCC for short, as well as discuss its security within three different models. Moreover, we discuss a chaotic instance of MCC based on the first and the second types of Chebyshev polynomials over real numbers for these three security models. To avoid round-off errors in floating point arithmetic as well as to enhance the security of the chaotic instance discussed, the Chebyshev polynomials of the first and the second types over a finite field are employed. We also consider the efficiency of the proposed MCCs. The discussions throughout the paper are supported by practical examples.
Higher-order Chebyshev rational approximation method (CRAM)
International Nuclear Information System (INIS)
The burnup equations can in principle be solved by computing the exponential of the burnup matrix. However, due to the difficult numerical characteristics of burnup matrices, the problem is extremely stiff, and the matrix exponential solution was long considered infeasible for an entire burnup system containing over a thousand nuclides. After discovering that the eigenvalues of burnup matrices are generally confined to a region near the negative real axis, the Chebyshev rational approximation method (CRAM) was introduced as a novel method to solve the burnup equations. It can be characterized as the best rational function on the negative real axis and it has been shown to be capable of simultaneously solving an entire burnup system both accurately and efficiently. The main difficulty in using CRAM for computing the matrix exponential is determining the coefficients of the rational function for a given approximation order. Some polynomial CRAM coefficients have been published in 1984, and based on these literature values, CRAM approximations up to the order 16 have been thus far applied in burnup calculations. The topic of this paper is the computation of CRAM approximations and their application to burnup equations. A Remez-type method utilizing the equioscillation property of best approximations is used to construct the CRAM approximants for approximation orders 1,. . . , 50. Numerical results are presented for a large burnup system and for a decay system. It is demonstrated that higher-order CRAM can be used to accurately solve the burnup equations even with time steps of the order of millions of years. (author)
Generalized Chebyshev-like Approximation for Low-pass Filter
Directory of Open Access Journals (Sweden)
Hisham L. Swady
2011-06-01
Full Text Available Analog filters constitute indispensible component of analog circuits and still playing an important part in interface with analog real world. realizing filters with odd order is preferred because of its time response . Therefore, this paper is conducted to introduce a new generalized Chebyshev – like approximation for analog filters. The analyses presented to realize the filters with odd order. This proposed novel approach offer good results in terms of flat delay and time domain response. Also, the achieved results are validated by comparison to normal Chebyshev filter via investigation several examples.
Energy Technology Data Exchange (ETDEWEB)
Benasser Algehawi, Mohammed, E-mail: malgehawi@yahoo.co [School of Computer Sciences, Universiti Sains Malaysia, Penang, 11800 (Malaysia); Samsudin, Azman, E-mail: azman@cs.usm.m [School of Computer Sciences, Universiti Sains Malaysia, Penang, 11800 (Malaysia)
2010-10-18
We present a method to extract key pairs needed for the Identity Based Encryption (IBE) scheme from extended Chebyshev polynomial over finite fields Z{sub p}. Our proposed scheme relies on the hard problem and the bilinear property of the extended Chebyshev polynomial over Z{sub p}. The proposed system is applicable, secure, and reliable.
APPLICATION OF NEWTON'S AND CHEBYSHEV'S METHODS TO PARALLEL FACTORIZATION OF POLYNOMIALS
Institute of Scientific and Technical Information of China (English)
Shi-ming Zheng
2001-01-01
In this paper it is shown in two different ways that one of the family of parallel iterations to determine all real quadratic factors of polynomials presented in [12] is Newton's method applied to the special equation (1.7) below. Furthermore, we apply Chebyshev's method to (1.7) and obtain a new parallel iteration for factorization of polynomials. Finally, some properties of the parallel iterations are discussed.
Applying Semigroup Property of Enhanced Chebyshev Polynomials to Anonymous Authentication Protocol
Hong Lai; Jinghua Xiao; Lixiang Li; Yixian Yang
2012-01-01
We apply semigroup property of enhanced Chebyshev polynomials to present an anonymous authentication protocol. This paper aims at improving security and reducing computational and storage overhead. The proposed scheme not only has much lower computational complexity and cost in the initialization phase but also allows the users to choose their passwords freely. Moreover, it can provide revocation of lost or stolen smart card, which can resist man-in-the-middle attack and off-line dictionary a...
Bulbs of Period Two in the Family of Chebyshev-Halley Iterative Methods on Quadratic Polynomials
Directory of Open Access Journals (Sweden)
Alicia Cordero
2013-01-01
Full Text Available The parameter space associated to the parametric family of Chebyshev-Halley on quadratic polynomials shows a dynamical richness worthy of study. This analysis has been initiated by the authors in previous works. Every value of the parameter belonging to the same connected component of the parameter space gives rise to similar dynamical behavior. In this paper, we focus on the search of regions in the parameter space that gives rise to the appearance of attractive orbits of period two.
Kopeliovich, Vladimir B
2016-01-01
The angular dependence of the cumulative particles production off nuclei near the kinematical boundary for multistep process is defined by characteristic polynomials in angular variables $J_N^2(z_N^\\theta)$, where $\\theta$ is the polar angle defining the momentum of the final (cumulative) particle, $z_N^\\theta = cos (\\theta/N)$, the integer $N$ being the multiplicity of the process (the number of interactions). Physical argumentation, exploring the small phase space method, leads to the appearance of equations for these polynomials $J_N^2[cos(\\pi/N)]=0$. The recurrent relations between polynomials with different $N$ are obtained, and their connection with known in mathematics Chebyshev polynomials of 2-d kind is established. As a result of this equality, differential cross section of the cumulative particle production has characteristic behaviour $d\\sigma \\sim 1/ \\sqrt {\\pi - \\theta}$ at $\\theta \\sim \\pi$ (the backward focusing effect). Such behaviour takes place for any multiplicity of the interaction, begin...
Weighted discrete least-squares polynomial approximation using randomized quadratures
Zhou, Tao; Narayan, Akil; Xiu, Dongbin
2015-10-01
We discuss the problem of polynomial approximation of multivariate functions using discrete least squares collocation. The problem stems from uncertainty quantification (UQ), where the independent variables of the functions are random variables with specified probability measure. We propose to construct the least squares approximation on points randomly and uniformly sampled from tensor product Gaussian quadrature points. We analyze the stability properties of this method and prove that the method is asymptotically stable, provided that the number of points scales linearly (up to a logarithmic factor) with the cardinality of the polynomial space. Specific results in both bounded and unbounded domains are obtained, along with a convergence result for Chebyshev measure. Numerical examples are provided to verify the theoretical results.
Novel Image Encryption Scheme Based on Chebyshev Polynomial and Duffing Map
2014-01-01
We present a novel image encryption algorithm using Chebyshev polynomial based on permutation and substitution and Duffing map based on substitution. Comprehensive security analysis has been performed on the designed scheme using key space analysis, visual testing, histogram analysis, information entropy calculation, correlation coefficient analysis, differential analysis, key sensitivity test, and speed test. The study demonstrates that the proposed image encryption algorithm shows advantages of more than 10113 key space and desirable level of security based on the good statistical results and theoretical arguments. PMID:25143970
Solutions for the Klein-Gordon and Dirac equations on the lattice based on Chebyshev polynomials
Faustino, Nelson
2016-01-01
The main goal of this paper is to adopt a multivector calculus scheme to study finite difference discretizations of Klein-Gordon and Dirac equations for which Chebyshev polynomials of the first kind may be used to represent a set of solutions. The development of a well-adapted discrete Clifford calculus framework based on spinor fields allows us to represent, using solely projection based arguments, the solutions for the discretized Dirac equations from the knowledge of the solutions of the discretized Klein-Gordon equation. Implications of those findings on the interpretation of the lattice fermion doubling problem is briefly discussed.
Contourlet Filter Design Based on Chebyshev Best Uniform Approximation
Yang, Guoan; Fang, Xiaofeng; Jing, Mingli; Zhang, Songjun; Hou, Ming
2010-12-01
The contourlet transform can deal effectively with images which have directional information such as contour and texture. In contrast to wavelets for which there exists many good filters, the contourlet filter design for image processing applications is still an ongoing work. Therefore, this paper presents an approach for designing the contourlet filter based on the Chebyshev best uniform approximation for achieving an efficient image denoising applications using hidden Markov tree models in the contourlet domain. Here, we design both the optimal 9/7 wavelet filter banks with rational coefficients and new pkva 12 filter. In this paper, the Laplacian pyramid followed by the direction filter banks decomposition in the contourlet transform using the two filter banks above and the image denoising applications in the contourlet hidden Markov tree model are implemented, respectively. The experimental results show that the denoising performance of the test image Zelda in terms of peak signal-to-noise ratio is improved by 0.33 dB than using CDF 9/7 filter banks with irrational coefficients on the JPEG2000 standard and standard pkva 12 filter, and visual effects are as good as compared with the research results of Duncan D.-Y. Po and Minh N. Do.
Discrete least squares approximation with polynomial vectors
Van Barel, Marc; Bultheel, Adhemar
1993-01-01
We give a solution of a discrete least squares approximation problem in terms of orthogonal polynomial vectors. The degrees of the polynomial elements of these vectors can be different. An algorithm is constructed computing the coefficients of recurrence relations for the orthogonal polynomial vectors. In case the function values are prescribed in points on the real line or on the unit circle variants of the original algorithm can be designed which are an order of magnitude more efficient. Al...
Energy Technology Data Exchange (ETDEWEB)
Lorber, A.A.; Carey, G.F.; Bova, S.W.; Harle, C.H. [Univ. of Texas, Austin, TX (United States)
1996-12-31
The connection between the solution of linear systems of equations by iterative methods and explicit time stepping techniques is used to accelerate to steady state the solution of ODE systems arising from discretized PDEs which may involve either physical or artificial transient terms. Specifically, a class of Runge-Kutta (RK) time integration schemes with extended stability domains has been used to develop recursion formulas which lead to accelerated iterative performance. The coefficients for the RK schemes are chosen based on the theory of Chebyshev iteration polynomials in conjunction with a local linear stability analysis. We refer to these schemes as Chebyshev Parameterized Runge Kutta (CPRK) methods. CPRK methods of one to four stages are derived as functions of the parameters which describe an ellipse {Epsilon} which the stability domain of the methods is known to contain. Of particular interest are two-stage, first-order CPRK and four-stage, first-order methods. It is found that the former method can be identified with any two-stage RK method through the correct choice of parameters. The latter method is found to have a wide range of stability domains, with a maximum extension of 32 along the real axis. Recursion performance results are presented below for a model linear convection-diffusion problem as well as non-linear fluid flow problems discretized by both finite-difference and finite-element methods.
Exponential Polynomial Approximation with Unrestricted Upper Density
Institute of Scientific and Technical Information of China (English)
Xiang Dong YANG
2011-01-01
We take a new approach to obtaining necessary and sufficient conditions for the incompleteness of exponential polynomials in Lp/α, where Lp/α is the weighted Banach space of complex continuous functions f defined on the real axis (R)satisfying (∫+∞/-∞|f(t)|pe-α(t)dt)1/p, 1 < p < ∞, and α(t) is a nonnegative continuous function defined on the real axis (R). In this paper, the upper density of the sequence which forms the exponential polynomials is not required to be finite. In the study of weighted polynomial approximation, consideration of the case is new.
Smith Simon J
1999-01-01
For a fixed integer and , let denote the th fundamental polynomial for Hermite–Fejér interpolation on the Chebyshev nodes . (So is the unique polynomial of degree at most which satisfies , and whose first derivatives vanish at each .) In this paper it is established that It is also shown that is an increasing function of , and the best possible bound so that for all , and is obtained. The results generalise those for Lagrange interpolation, obtained by P. Erdős an...
Error Minimization of Polynomial Approximation of Delta
Indian Academy of Sciences (India)
Islam Sana; Sadiq Muhammad; Qureshi Muhammad Shahid
2008-09-01
The difference between Universal time (UT) and Dynamical time (TD), known as Delta ( ) is tabulated for the first day of each year in the Astronomical Almanac. During the last four centuries it is found that there are large differences between its values for two consecutive years. Polynomial approximations have been developed to obtain the values of for any time of a year for the period AD 1620 to AD 2000 (Meeu 2000) as no dynamical theories describe the variations in . In this work, a new set of polynomials for is obtained for the period AD 1620 to AD 2007 that is found to produce better results compared to previous attempts.
A New Six-Parameter Model Based on Chebyshev Polynomials for Solar Cells
Directory of Open Access Journals (Sweden)
Shu-xian Lun
2015-01-01
Full Text Available This paper presents a new current-voltage (I-V model for solar cells. It has been proved that series resistance of a solar cell is related to temperature. However, the existing five-parameter model ignores the temperature dependence of series resistance and then only accurately predicts the performance of monocrystalline silicon solar cells. Therefore, this paper uses Chebyshev polynomials to describe the relationship between series resistance and temperature. This makes a new parameter called temperature coefficient for series resistance introduced into the single-diode model. Then, a new six-parameter model for solar cells is established in this paper. This new model can improve the accuracy of the traditional single-diode model and reflect the temperature dependence of series resistance. To validate the accuracy of the six-parameter model in this paper, five kinds of silicon solar cells with different technology types, that is, monocrystalline silicon, polycrystalline silicon, thin film silicon, and tripe-junction amorphous silicon, are tested at different irradiance and temperature conditions. Experiment results show that the six-parameter model proposed in this paper is an I-V model with moderate computational complexity and high precision.
Polynomial approximation of functions in Sobolev spaces
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.
Institute of Scientific and Technical Information of China (English)
呼家源; 白慧; 王慧
2015-01-01
根据 Legendre多项式的正交性，利用一个组合恒等式，得到由 Legendre 多项式表示x2n和x2n+1的具体形式，进而建立了 Legendre多项式和Chebyshev多项式的关系式。%Based on the orthogonality of the Legendre polynomials and combinational method,x2n and x2n+1 expressed by the combinatorial sums of the Legendre polynomials in this paper.Then according to these identities,Chebyshev polynomials and Legendre polynomials are connected with each other in definite forms.
International Nuclear Information System (INIS)
In this study, we use the spectral collocation method using Chebyshev polynomials for spatial derivatives and fourth order Runge-Kutta method for time integration to solve the generalized Burger's-Huxley equation (GBHE). To reduce round-off error in spectral collocation (pseudospectral) method we use preconditioning. Firstly, theory of application of Chebyshev spectral collocation method with preconditioning (CSCMP) and domain decomposition on the generalized Burger's-Huxley equation presented. This method yields a system of ordinary differential algebric equations (DAEs). Secondly, we use fourth order Runge-Kutta formula for the numerical integration of the system of DAEs. The numerical results obtained by this way have been compared with the exact solution to show the efficiency of the method.
Approximate protein structural alignment in polynomial time.
Kolodny, Rachel; Linial, Nathan
2004-08-17
Alignment of protein structures is a fundamental task in computational molecular biology. Good structural alignments can help detect distant evolutionary relationships that are hard or impossible to discern from protein sequences alone. Here, we study the structural alignment problem as a family of optimization problems and develop an approximate polynomial-time algorithm to solve them. For a commonly used scoring function, the algorithm runs in O(n(10)/epsilon(6)) time, for globular protein of length n, and it detects alignments that score within an additive error of epsilon from all optima. Thus, we prove that this task is computationally feasible, although the method that we introduce is too slow to be a useful everyday tool. We argue that such approximate solutions are, in fact, of greater interest than exact ones because of the noisy nature of experimentally determined protein coordinates. The measurement of similarity between a pair of protein structures used by our algorithm involves the Euclidean distance between the structures (appropriately rigidly transformed). We show that an alternative approach, which relies on internal distance matrices, must incorporate sophisticated geometric ingredients if it is to guarantee optimality and run in polynomial time. We use these observations to visualize the scoring function for several real instances of the problem. Our investigations yield insights on the computational complexity of protein alignment under various scoring functions. These insights can be used in the design of scoring functions for which the optimum can be approximated efficiently and perhaps in the development of efficient algorithms for the multiple structural alignment problem. PMID:15304646
Chebyshev type lattice path weight polynomials by a constant term method
Brak, R
2009-01-01
We prove a constant term theorem which is useful for finding weight polynomials for Ballot/Motzkin paths in a strip with a fixed number of arbitrary `decorated' weights as well as an arbitrary `background' weight. Our CT theorem, like Viennot's lattice path theorem from which it is derived primarily by a change of variable lemma, is expressed in terms of orthogonal polynomials which in our applications of interest often turn out to be non-classical. Hence we also present an efficient method for finding explicit closed form polynomial expressions for these non-classical orthogonal polynomials. Our method for finding the closed form polynomial expressions relies on simple combinatorial manipulations of Viennot's diagrammatic representation for orthogonal polynomials. In the course of the paper we also provide a new proof of Viennot's original orthogonal polynomial lattice path theorem. The new proof is of interest because it uses diagonalization of the transfer matrix, but gets around difficulties that have ari...
Banerjee, Amartya S; Hu, Wei; Yang, Chao; Pask, John E
2016-01-01
The Discontinuous Galerkin (DG) electronic structure method employs an adaptive local basis set to solve the equations of density functional theory in a discontinuous Galerkin framework. The methodology is implemented in the Discontinuous Galerkin Density Functional Theory (DGDFT) code for large-scale parallel electronic structure calculations. In DGDFT, the basis is generated on-the-fly to capture the local material physics, and can systematically attain chemical accuracy with only a few tens of degrees of freedom per atom. Hence, DGDFT combines the key advantage of planewave basis sets in terms of systematic improvability with that of localized basis sets in reducing basis size. A central issue for large-scale calculations, however, is the computation of the electron density from the discretized Hamiltonian in an efficient and scalable manner. We show in this work how Chebyshev polynomial filtered subspace iteration (CheFSI) can be used to address this issue and push the envelope in large-scale materials si...
Conditional Density Approximations with Mixtures of Polynomials
DEFF Research Database (Denmark)
Varando, Gherardo; López-Cruz, Pedro L.; Nielsen, Thomas Dyhre;
2015-01-01
Mixtures of polynomials (MoPs) are a non-parametric density estimation technique especially designed for hybrid Bayesian networks with continuous and discrete variables. Algorithms to learn one- and multi-dimensional (marginal) MoPs from data have recently been proposed. In this paper we introduce...... basis functions (MoTBFs). The empirical results show that the proposed methods generally yield models that are comparable to or significantly better than those found using the MoTBF-based method....
Animating Nested Taylor Polynomials to Approximate a Function
Mazzone, Eric F.; Piper, Bruce R.
2010-01-01
The way that Taylor polynomials approximate functions can be demonstrated by moving the center point while keeping the degree fixed. These animations are particularly nice when the Taylor polynomials do not intersect and form a nested family. We prove a result that shows when this nesting occurs. The animations can be shown in class or…
Pointwise Approximation Theorems for Combinations and Derivatives of Bernstein Polynomials
Institute of Scientific and Technical Information of China (English)
Lin Sen XIE
2005-01-01
We establish the pointwise approximation theorems for the combinations of Bernstein polynomials by the rth Ditzian-Totik modulus of smoothness ωγφ(f, t) where φ is an admissible step-weight function. An equivalence relation between the derivatives of these polynomials and the smoothness of functions is also obtained.
Inner approximations for polynomial matrix inequalities and robust stability regions
Henrion, Didier
2011-01-01
Following a polynomial approach, many robust fixed-order controller design problems can be formulated as optimization problems whose set of feasible solutions is modelled by parametrized polynomial matrix inequalities (PMI). These feasibility sets are typically nonconvex. Given a parametrized PMI set, we provide a hierarchy of linear matrix inequality (LMI) problems whose optimal solutions generate inner approximations modelled by a single polynomial sublevel set. Those inner approximations converge in a strong analytic sense to the nonconvex original feasible set, with asymptotically vanishing conservatism. One may also impose the hierarchy of inner approximations to be nested or convex. In the latter case they do not converge any more to the feasible set, but they can be used in a convex optimization framework at the price of some conservatism. Finally, we show that the specific geometry of nonconvex polynomial stability regions can be exploited to improve convergence of the hierarchy of inner approximation...
APPROXIMATION BY GENERALIZED MKZ-OPERATORS IN POLYNOMIAL WEIGHTED SPACES
Institute of Scientific and Technical Information of China (English)
Lucyna Rempulska; Mariola Skorupka
2007-01-01
We prove some approximation properties of generalized Meyer-K(o)nig and Zeller operators for differentiable functions in polynomial weighted spaces. The results extend some results proved in [ 1-3,7-16].
An overview on polynomial approximation of NP-hard problems
Directory of Open Access Journals (Sweden)
Paschos Vangelis Th.
2009-01-01
Full Text Available The fact that polynomial time algorithm is very unlikely to be devised for an optimal solving of the NP-hard problems strongly motivates both the researchers and the practitioners to try to solve such problems heuristically, by making a trade-off between computational time and solution's quality. In other words, heuristic computation consists of trying to find not the best solution but one solution which is 'close to' the optimal one in reasonable time. Among the classes of heuristic methods for NP-hard problems, the polynomial approximation algorithms aim at solving a given NP-hard problem in poly-nomial time by computing feasible solutions that are, under some predefined criterion, as near to the optimal ones as possible. The polynomial approximation theory deals with the study of such algorithms. This survey first presents and analyzes time approximation algorithms for some classical examples of NP-hard problems. Secondly, it shows how classical notions and tools of complexity theory, such as polynomial reductions, can be matched with polynomial approximation in order to devise structural results for NP-hard optimization problems. Finally, it presents a quick description of what is commonly called inapproximability results. Such results provide limits on the approximability of the problems tackled.
Structured matrix based methods for approximate polynomial GCD
Boito, Paola
2011-01-01
Defining and computing a greatest common divisor of two polynomials with inexact coefficients is a classical problem in symbolic-numeric computation. The first part of this book reviews the main results that have been proposed so far in the literature. As usual with polynomial computations, the polynomial GCD problem can be expressed in matrix form: the second part of the book focuses on this point of view and analyses the structure of the relevant matrices, such as Toeplitz, Toepliz-block and displacement structures. New algorithms for the computation of approximate polynomial GCD are presented, along with extensive numerical tests. The use of matrix structure allows, in particular, to lower the asymptotic computational cost from cubic to quadratic order with respect to polynomial degree. .
CHEBYSHEV APPROXIMATION OF THE SECOND KIND OF MODIFIED BESSEL FUNCTION OF ORDER ZERO
Institute of Scientific and Technical Information of China (English)
张璟; 周哲玮
2004-01-01
The second kind of modified Bessel function of order zero is the solutions of many problems in engineering. Modified Bessel equation is transformed by exponential transformation and expanded by J. P. Boyd' s rational Chebyshev basis.
Approximation and polynomial convexity in several complex variables
Ölçücüoğlu, Büke; Olcucuoglu, Buke
2009-01-01
This thesis is a survey on selected topics in approximation theory. The topics use either the techniques from the theory of several complex variables or those that arise in the study of the subject. We also go through elementary theory of polynomially convex sets in complex analysis.
The Laplace transform and polynomial approximation in L2
DEFF Research Database (Denmark)
Labouriau, Rodrigo
2016-01-01
This short note gives a sufficient condition for having the class of polynomials dense in the space of square integrable functions with respect to a finite measure dominated by the Lebesgue measure in the real line, here denoted by L2. It is shown that if the Laplace transform of the measure...... in play is bounded in a neighbourhood of the origin, then the moments of all order are finite and the class of polynomials is dense in L2. The existence of the moments of all orders is well known for the case where the measure is concentrated in the positive real line (see Feller, 1966), but the result...... concerning the polynomial approximation is original, even thought the proof is relatively simple. Additionally, an alternative stronger condition (easier to be verified) not involving the calculation of the Laplace transform is given. The condition essentially says that the density of the measure should have...
Polynomial approximation and cubature at approximate Fekete and Leja points of the cylinder
De Marchi, Stefano
2011-01-01
The paper deals with polynomial interpolation, least-square approximation and cubature of functions defined on the rectangular cylinder, $K=D\\times [-1,1]$, with $D$ the unit disk. The nodes used for these processes are the {\\it Approximate Fekete Points} (AFP) and the {\\it Discrete Leja Points} (DLP) extracted from suitable {\\it Weakly Admissible Meshes} (WAMs) of the cylinder. From the analysis of the growth of the Lebesgue constants, approximation and cubature errors, we show that the AFP and the DLP extracted from WAM are good points for polynomial approximation and numerical integration of functions defined on the cylinder.
Application of a Local Polynomial Approximation Chaotic Time Series Prediction
Orzeszko, Witold
2004-01-01
Chaos theory has become a new approach to financial processes analysis. Due to complicated dynamics, chaotic time series seem to be random and, in consequence, unpredictable. In fact, unlike truly random processes, chaotic dynamics can be forecasted very precisely in a short run. In this paper, a local polynomial approximation is presented. Its efficiency, as a method of building short-term predictors of chaotic time series, has been examined. The presented method has been applied to forecast...
Time-delay polynomial networks and rates of approximation
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Irwin W. Sandberg
1998-01-01
Full Text Available We consider a large family of finite memory causal time-invariant maps G from an input set S to a set of ℝ-valued functions, with the members of both sets of functions defined on the nonnegative integers, and we give an upper bound on the error in approximating a G using a two-stage structure consisting of a tapped delay line and a static polynomial network N . This upper bound depends on the degree of the multivariable polynomial that characterizes N. Also given is a lower bound on the worst-case error in approximating a G using polynomials of a fixed maximum degree. These upper and lower bounds differ only by a multiplicative constant. We also give a corresponding result for the approximation of not-necessarily-causal input–output maps with inputs and outputs that may depend on more than one variable. This result is of interest, for example, in connection with image processing.
SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos
Ahlfeld, R.; Belkouchi, B.; Montomoli, F.
2016-09-01
A new arbitrary Polynomial Chaos (aPC) method is presented for moderately high-dimensional problems characterised by limited input data availability. The proposed methodology improves the algorithm of aPC and extends the method, that was previously only introduced as tensor product expansion, to moderately high-dimensional stochastic problems. The fundamental idea of aPC is to use the statistical moments of the input random variables to develop the polynomial chaos expansion. This approach provides the possibility to propagate continuous or discrete probability density functions and also histograms (data sets) as long as their moments exist, are finite and the determinant of the moment matrix is strictly positive. For cases with limited data availability, this approach avoids bias and fitting errors caused by wrong assumptions. In this work, an alternative way to calculate the aPC is suggested, which provides the optimal polynomials, Gaussian quadrature collocation points and weights from the moments using only a handful of matrix operations on the Hankel matrix of moments. It can therefore be implemented without requiring prior knowledge about statistical data analysis or a detailed understanding of the mathematics of polynomial chaos expansions. The extension to more input variables suggested in this work, is an anisotropic and adaptive version of Smolyak's algorithm that is solely based on the moments of the input probability distributions. It is referred to as SAMBA (PC), which is short for Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos. It is illustrated that for moderately high-dimensional problems (up to 20 different input variables or histograms) SAMBA can significantly simplify the calculation of sparse Gaussian quadrature rules. SAMBA's efficiency for multivariate functions with regard to data availability is further demonstrated by analysing higher order convergence and accuracy for a set of nonlinear test functions with 2, 5 and 10
Boyd, John P.; Yu, Fu
2011-02-01
We compare seven different strategies for computing spectrally-accurate approximations or differential equation solutions in a disk. Separation of variables for the Laplace operator yields an analytic solution as a Fourier-Bessel series, but this usually converges at an algebraic (sub-spectral) rate. The cylindrical Robert functions converge geometrically but are horribly ill-conditioned. The Zernike and Logan-Shepp polynomials span the same space, that of Cartesian polynomials of a given total degree, but the former allows partial factorization whereas the latter basis facilitates an efficient algorithm for solving the Poisson equation. The Zernike polynomials were independently rediscovered several times as the product of one-sided Jacobi polynomials in radius with a Fourier series in θ. Generically, the Zernike basis requires only half as many degrees of freedom to represent a complicated function on the disk as does a Chebyshev-Fourier basis, but the latter has the great advantage of being summed and interpolated entirely by the Fast Fourier Transform instead of the slower matrix multiplication transforms needed in radius by the Zernike basis. Conformally mapping a square to the disk and employing a bivariate Chebyshev expansion on the square is spectrally accurate, but clustering of grid points near the four singularities of the mapping makes this method less efficient than the rest, meritorious only as a quick-and-dirty way to adapt a solver-for-the-square to the disk. Radial basis functions can match the best other spectral methods in accuracy, but require slow non-tensor interpolation and summation methods. There is no single “best” basis for the disk, but we have laid out the merits and flaws of each spectral option.
Optimal approximation of harmonic growth clusters by orthogonal polynomials
Balogh, Ferenc
2008-01-01
Interface dynamics in two-dimensional systems with a maximal number of conservation laws gives an accurate theoretical model for many physical processes, from the hydrodynamics of immiscible, viscous flows (zero surface-tension limit of Hele-Shaw flows, [1]), to the granular dynamics of hard spheres [2], and even diffusion-limited aggregation [3]. Although a complete solution for the continuum case exists [4, 5], efficient approximations of the boundary evolution are very useful due to their practical applications [6]. In this article, the approximation scheme based on orthogonal polynomials with a deformed Gaussian kernel [7] is discussed, as well as relations to potential theory.
Optimal approximation of harmonic growth clusters by orthogonal polynomials
Energy Technology Data Exchange (ETDEWEB)
Teodorescu, Razvan [Los Alamos National Laboratory
2008-01-01
Interface dynamics in two-dimensional systems with a maximal number of conservation laws gives an accurate theoreticaI model for many physical processes, from the hydrodynamics of immiscible, viscous flows (zero surface-tension limit of Hele-Shaw flows), to the granular dynamics of hard spheres, and even diffusion-limited aggregation. Although a complete solution for the continuum case exists, efficient approximations of the boundary evolution are very useful due to their practical applications. In this article, the approximation scheme based on orthogonal polynomials with a deformed Gaussian kernel is discussed, as well as relations to potential theory.
Uniform and Pointwise Shape Preserving Approximation by Algebraic Polynomials
Kopotun, K A; Prymak, A; Shevchuk, I A
2011-01-01
We survey developments, over the last thirty years, in the theory of Shape Preserving Approximation (SPA) by algebraic polynomials on a finite interval. In this article, "shape" refers to (finitely many changes of) monotonicity, convexity, or q-monotonicity of a function (for definition, see Section 4). It is rather well known that it is possible to approximate a function by algebraic polynomials that preserve its shape (i.e., the Weierstrass approximation theorem is valid for SPA). At the same time, the degree of SPA is much worse than the degree of best unconstrained approximation in some cases, and it is "about the same" in others. Numerous results quantifying this difference in degrees of SPA and unconstrained approximation have been obtained in recent years, and the main purpose of this article is to provide a "bird's-eye view" on this area, and discuss various approaches used. In particular, we present results on the validity and invalidity of uniform and pointwise estimates in terms of various moduli o...
The BQP-hardness of approximating the Jones polynomial
Aharonov, Dorit; Arad, Itai
2011-03-01
A celebrated important result due to Freedman et al (2002 Commun. Math. Phys. 227 605-22) states that providing additive approximations of the Jones polynomial at the kth root of unity, for constant k=5 and k>=7, is BQP-hard. Together with the algorithmic results of Aharonov et al (2005) and Freedman et al (2002 Commun. Math. Phys. 227 587-603), this gives perhaps the most natural BQP-complete problem known today and motivates further study of the topic. In this paper, we focus on the universality proof; we extend the result of Freedman et al (2002) to ks that grow polynomially with the number of strands and crossings in the link, thus extending the BQP-hardness of Jones polynomial approximations to all values to which the AJL algorithm applies (Aharonov et al 2005), proving that for all those values, the problems are BQP-complete. As a side benefit, we derive a fairly elementary proof of the Freedman et al density result, without referring to advanced results from Lie algebra representation theory, making this important result accessible to a wider audience in the computer science research community. We make use of two general lemmas we prove, the bridge lemma and the decoupling lemma, which provide tools for establishing the density of subgroups in SU(n). Those tools seem to be of independent interest in more general contexts of proving the quantum universality. Our result also implies a completely classical statement, that the multiplicative approximations of the Jones polynomial, at exactly the same values, are #P-hard, via a recent result due to Kuperberg (2009 arXiv:0908.0512). Since the first publication of those results in their preliminary form (Aharonov and Arad 2006 arXiv:quant-ph/0605181), the methods we present here have been used in several other contexts (Aharonov and Arad 2007 arXiv:quant-ph/0702008; Peter and Stephen 2008 Quantum Inf. Comput. 8 681). The present paper is an improved and extended version of the results presented by Aharonov and Arad
High Resolution of the ECG Signal by Polynomial Approximation
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G. Rozinaj
2006-04-01
Full Text Available Averaging techniques as temporal averaging and space averaging have been successfully used in many applications for attenuating interference [6], [7], [8], [9], [10]. In this paper we introduce interference removing of the ECG signal by polynomial approximation, with smoothing discrete dependencies, to make up for averaging methods. The method is suitable for low-level signals of the electrical activity of the heart often less than 10 m V. Most low-level signals arising from PR, ST and TP segments which can be detected eventually and their physiologic meaning can be appreciated. Of special importance for the diagnostic of the electrical activity of the heart is the activity bundle of His between P and R waveforms. We have established an artificial sine wave to ECG signal between P and R wave. The aim focus is to verify the smoothing method by polynomial approximation if the SNR (signal-to-noise ratio is negative (i.e. a signal is lower than noise.
Canonical representation for approximating solution of fuzzy polynomial equations
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M. Salehnegad
2010-06-01
Full Text Available In this paper, the concept of canonical representation is proposed to find fuzzy roots of fuzzy polynomial equations. We transform fuzzy polynomial equations to system of crisp polynomial equations, this transformation is perform by using canonical representation based on three parameters Value, Ambiguity and Fuzziness.
ON COEFFICIENT POLYNOMIALS OF CUBIC HERMITE-PAD(E) APPROXIMATIONS TO THE EXPONENTIAL FUNCTION
Institute of Scientific and Technical Information of China (English)
Cheng-de Zheng; Guo-can Wang; Zhi-bin Li
2005-01-01
The polynomials related with cubic Hermite-Pade approximation to the exponential function are investigated which have degrees at most n, m, s respectively. A connection is given between the coefficients of each of the polynomials and certain hypergeometric functions, which leads to a simple expression for a polynomial in a special case. Contour integral representations of the polynomials are given. By using of the saddle point method the exact asymptotics of the polynomials are derived as n, m, s tend to infinity through certain ray sequence. Some further uniform asymptotic aspects of the polynomials are also discussed.
Approximation to Continuous Functions by a Kind of Interpolation Polynomials
Institute of Scientific and Technical Information of China (English)
Yuan Xue-gang; Wang De-hui
2001-01-01
In this paper, an interpolation polynomial operator Fn (f; l, x ) is constructed based on the zeros of a kind of Jacobi polynomials as the interpolation nodes. For any continuous function f(x)∈ Cb[1,1] (0≤b≤l) Fn(f; l,x) converges to f(x) uniformly, where l is an odd number.
Polynomial approximations of a class of stochastic multiscale elasticity problems
Hoang, Viet Ha; Nguyen, Thanh Chung; Xia, Bingxing
2016-06-01
We consider a class of elasticity equations in {mathbb{R}^d} whose elastic moduli depend on n separated microscopic scales. The moduli are random and expressed as a linear expansion of a countable sequence of random variables which are independently and identically uniformly distributed in a compact interval. The multiscale Hellinger-Reissner mixed problem that allows for computing the stress directly and the multiscale mixed problem with a penalty term for nearly incompressible isotropic materials are considered. The stochastic problems are studied via deterministic problems that depend on a countable number of real parameters which represent the probabilistic law of the stochastic equations. We study the multiscale homogenized problems that contain all the macroscopic and microscopic information. The solutions of these multiscale homogenized problems are written as generalized polynomial chaos (gpc) expansions. We approximate these solutions by semidiscrete Galerkin approximating problems that project into the spaces of functions with only a finite number of N gpc modes. Assuming summability properties for the coefficients of the elastic moduli's expansion, we deduce bounds and summability properties for the solutions' gpc expansion coefficients. These bounds imply explicit rates of convergence in terms of N when the gpc modes used for the Galerkin approximation are chosen to correspond to the best N terms in the gpc expansion. For the mixed problem with a penalty term for nearly incompressible materials, we show that the rate of convergence for the best N term approximation is independent of the Lamé constants' ratio when it goes to {infty}. Correctors for the homogenization problem are deduced. From these we establish correctors for the solutions of the parametric multiscale problems in terms of the semidiscrete Galerkin approximations. For two-scale problems, an explicit homogenization error which is uniform with respect to the parameters is deduced. Together
Institute of Scientific and Technical Information of China (English)
陈宇; 韦鹏程
2011-01-01
将Chebyshev多项式与模运算相结合,对其定义在实数域上进行了扩展,经过理论验证和数据分析,总结出实数域多项式应用于公钥密码的一些性质.利用RSA公钥算法和EIGamal公钥算法的算法结构,提出基于有限域离散Chebyshev多项式的公钥密码算法.该算法结构类似于RSA算法,其安全性基于大数因式分解的难度或者与El-Gamal的离散对数难度相当,能够抵抗对于RSA的选择密文攻击,并且易于软件实现.%By combining Chebyshev polynomials with modulus compute,extending Chebyshev polynomials' definition domain to real number, some conclusions were drawn by theoretic verification and data analysis. Making use of the framework of the traditional public-key algorithm RSA and ElGamal, proposed a chaotic public-key encryption algorithm based on extending discrete Chebyshev polynomials' definition domain to Real number. Its security is based on the intractability of the integer factorization problem as RSA,and it is able to resist the chosen cipher-text attack against RSA and easy to be implemented.
Directory of Open Access Journals (Sweden)
Mohsen Razzaghi
2000-01-01
Full Text Available A direct method for finding the solution of variational problems using a hybrid function is discussed. The hybrid functions which consist of block-pulse functions plus Chebyshev polynomials are introduced. An operational matrix of integration and the integration of the cross product of two hybrid function vectors are presented and are utilized to reduce a variational problem to the solution of an algebraic equation. Illustrative examples are included to demonstrate the validity and applicability of the technique.
Polynomial time approximation schemes for the traveling repairman and other minimum latency problems
Sitters, R.A.
2013-01-01
We give a polynomial time, (1 + \\epsilon)-approximation algorithm for the traveling repairman problem (TRP) in the Euclidean plane, on weighted planar graphs, and on weighted trees. This improves on the known quasi-polynomial time approximation schemes for these problems. The algorithm is based on a
Polynomial time approximation schemes for the traveling repairman and other minimum latency problems
Sitters, R.A.; Chekuri, C.
2014-01-01
We give a polynomial time, (1 + ∊)-approximation algorithm for the traveling repairman problem (TRP) in the Euclidean plane, on weighted planar graphs, and on weighted trees. This improves on the known quasi-polynomial time approximation schemes for these problems. The algorithm is based on a simple
Polynomial birth-death distribution approximation in Wasserstein distance
Xia, Aihua; Zhang, Fuxi
2008-01-01
The polynomial birth-death distribution (abbr. as PBD) on $\\ci=\\{0,1,2, >...\\}$ or $\\ci=\\{0,1,2, ..., m\\}$ for some finite $m$ introduced in Brown & Xia (2001) is the equilibrium distribution of the birth-death process with birth rates $\\{\\alpha_i\\}$ and death rates $\\{\\beta_i\\}$, where $\\a_i\\ge0$ and $\\b_i\\ge0$ are polynomial functions of $i\\in\\ci$. The family includes Poisson, negative binomial, binomial and hypergeometric distributions. In this paper, we give probabilistic proofs of variou...
A novel computational approach to approximate fuzzy interpolation polynomials.
Jafarian, Ahmad; Jafari, Raheleh; Mohamed Al Qurashi, Maysaa; Baleanu, Dumitru
2016-01-01
This paper build a structure of fuzzy neural network, which is well sufficient to gain a fuzzy interpolation polynomial of the form [Formula: see text] where [Formula: see text] is crisp number (for [Formula: see text], which interpolates the fuzzy data [Formula: see text]. Thus, a gradient descent algorithm is constructed to train the neural network in such a way that the unknown coefficients of fuzzy polynomial are estimated by the neural network. The numeral experimentations portray that the present interpolation methodology is reliable and efficient. PMID:27625982
Terui, Akira
2010-01-01
We present an extension of our GPGCD method, an iterative method for calculating approximate greatest common divisor (GCD) of univariate polynomials, to polynomials with the complex coefficients. For a given pair of polynomials and a degree, our algorithm finds a pair of polynomials which has a GCD of the given degree and whose coefficients are perturbed from those in the original inputs, making the perturbations as small as possible, along with the GCD. In our GPGCD method, the problem of approximate GCD is transfered to a constrained minimization problem, then solved with a so-called modified Newton method, which is a generalization of the gradient-projection method, by searching the solution iteratively. While our original method is designed for polynomials with the real coefficients, we extend it to accept polynomials with the complex coefficients in this paper.
Institute of Scientific and Technical Information of China (English)
CHENG Min; WANG Guojin
2004-01-01
NURBS curve is one of the most commonly used tools in CAD systems and geometric modeling for its various specialties, which means that its shape is locally adjustable as well as its continuity order, and it can represent a conic curve precisely. But how to do degree reduction of NURBS curves in a fast and efficient way still remains a puzzling problem. By applying the theory of the best uniform approximation of Chebyshev polynomials and the explicit matrix representation of NURBS curves, this paper gives the necessary and sufficient condition for degree reducible NURBS curves in an explicit form.And a new way of doing degree reduction of NURBS curves is also presented, including the multi-degree reduction of a NURBS curve on each knot span and the multi-degree reduction of a whole NURBS curve. This method is easy to carry out, and only involves simple calculations. It provides a new way of doing degree reduction of NURBS curves,which can be widely used in computer graphics and industrial design.
Approximation properties of SzÃ¡sz type operators based on Charlier polynomials
KAJLA, ARUN; AGRAWAL, Purshottam Narain
2015-01-01
In the present paper, we study some approximation properties of the Sz\\'{a}sz type operators involving Charlier polynomials introduced by Varma and Ta\\c{s}delen in 2012. First, we establish approximation in a Lipschitz type space and weighted approximation theorems for these operators. Then we obtain the error in the approximation of functions having derivatives of bounded variation.
GPGCD, an Iterative Method for Calculating Approximate GCD, for Multiple Univariate Polynomials
Terui, Akira
2010-01-01
We present an extension of our GPGCD method, an iterative method for calculating approximate greatest common divisor (GCD) of univariate polynomials, to multiple polynomial inputs. For a given pair of polynomials and a degree, our algorithm finds a pair of polynomials which has a GCD of the given degree and whose coefficients are perturbed from those in the original inputs, making the perturbations as small as possible, along with the GCD. In our GPGCD method, the problem of approximate GCD is transferred to a constrained minimization problem, then solved with the so-called modified Newton method, which is a generalization of the gradient-projection method, by searching the solution iteratively. In this paper, we extend our method to accept more than two polynomials with the real coefficients as an input.
GPGCD, an Iterative Method for Calculating Approximate GCD, for Multiple Univariate Polynomials
Terui, Akira
We present an extension of our GPGCD method, an iterative method for calculating approximate greatest common divisor (GCD) of univariate polynomials, to multiple polynomial inputs. For a given pair of polynomials and a degree, our algorithm finds a pair of polynomials which has a GCD of the given degree and whose coefficients are perturbed from those in the original inputs, making the perturbations as small as possible, along with the GCD. In our GPGCD method, the problem of approximate GCD is transferred to a constrained minimization problem, then solved with the so-called modified Newton method, which is a generalization of the gradient-projection method, by searching the solution iteratively. In this paper, we extend our method to accept more than two polynomials with the real coefficients as an input.
Delta T: Polynomial Approximation of Time Period 1620–2013
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M. Khalid
2014-01-01
Full Text Available The difference between the Uniform Dynamical Time and Universal Time is referred to as ΔT (delta T. Delta T is used in numerous astronomical calculations, that is, eclipses,and length of day. It is additionally required to reduce quantified positions of minor planets to a uniform timescale for the purpose of orbital determination. Since Universal Time is established on the basis of the variable rotation of planet Earth, the quantity ΔT mirrors the unevenness of that rotation, and so it changes slowly, but rather irregularly, as time passes. We have worked on empirical formulae for estimating ΔT and have discovered a set of polynomials of the 4th order with nine intervals which is accurate within the range of ±0.6 seconds for the duration of years 1620–2013.
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
Semi-infinite optimization with sums of exponentials via polynomial approximation
Dumitrescu, Bogdan; Sicleru, Bogdan C.; Avram, Florin
2014-01-01
We propose a general method for optimization with semi-infinite constraints that involve a linear combination of functions, focusing on the case of the exponential function. Each function is lower and upper bounded on sub-intervals by low-degree polynomials. Thus, the constraints can be approximated with polynomial inequalities that can be implemented with linear matrix inequalities. Convexity is preserved, but the problem has now a finite number of constraints. We show how to take advantage ...
Polynomial Approximation of Functions: Historical Perspective and New Tools
Kidron, Ivy
2003-01-01
This paper examines the effect of applying symbolic computation and graphics to enhance students' ability to move from a visual interpretation of mathematical concepts to formal reasoning. The mathematics topics involved, Approximation and Interpolation, were taught according to their historical development, and the students tried to follow the…
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
Derivation of reduced model for control system design using Chebyshev techniques
International Nuclear Information System (INIS)
New methods are developed for reduced-order modelling of high-order, linear, time-invariant systems characterized by a transfer function. The first method is based on manipulating two Chebyshev polynomial series, one representing the frequency characteristics of the high-order system and the other representing the approximating low-order model. The proposed method can be viewed as generalizing the classical Pade approximation problem, with Chebyshev polynomial series being over a desired frequency interval instead of a power series about a single frequency point. The second method is based on approximating the high-order transfer function in terms of best Chebyshev approximation on a desired domain in the complex plane. An algorithm to find for a complex function best Chebyshev rational approximations in the complex plane is suggested and its theoretical basis confirmed. The algorithm is based on a complex version of Lawson algorithm that is applied to a complex version of a rational least square approximation program. (author)
Real Scalar Field Scattering with Polynomial Approximation around Schwarzschild-de Sitter Black-hole
Liu, Molin; Zhang, Jingfei; Yu, Fei
2008-01-01
As one of the fitting methods, the polynomial approximation is effective to process sophisticated problem. In this paper, we employ this approach to handle the scattering of scalar field around the Schwarzschild-de Sitter black-hole. The complex relationship between tortoise coordinate and radial coordinate is replaced by the approximate polynomial. The Schr$\\ddot{o}$dinger-like equation, the real boundary conditions and the polynomial approximation construct a full Sturm-Liouville type problem. Then this boundary value problem can be solved numerically according to two limiting cases: the first one is the Nariai black-hole whose horizons are close to each other, the second one is when the horizons are widely separated. Compared with previous results (Brevik and Tian), the field near the event horizon and cosmological horizon can have a better description.
Local Random Quantum Circuits are Approximate Polynomial-Designs
Brandão, Fernando G. S. L.; Harrow, Aram W.; Horodecki, Michał
2016-09-01
We prove that local random quantum circuits acting on n qubits composed of O( t 10 n 2) many nearest neighbor two-qubit gates form an approximate unitary t-design. Previously it was unknown whether random quantum circuits were a t-design for any t > 3. The proof is based on an interplay of techniques from quantum many-body theory, representation theory, and the theory of Markov chains. In particular we employ a result of Nachtergaele for lower bounding the spectral gap of frustration-free quantum local Hamiltonians; a quasi-orthogonality property of permutation matrices; a result of Oliveira which extends to the unitary group the path-coupling method for bounding the mixing time of random walks; and a result of Bourgain and Gamburd showing that dense subgroups of the special unitary group, composed of elements with algebraic entries, are ∞-copy tensor-product expanders. We also consider pseudo-randomness properties of local random quantum circuits of small depth and prove that circuits of depth O( t 10 n) constitute a quantum t-copy tensor-product expander. The proof also rests on techniques from quantum many-body theory, in particular on the detectability lemma of Aharonov, Arad, Landau, and Vazirani. We give applications of the results to cryptography, equilibration of closed quantum dynamics, and the generation of topological order. In particular we show the following pseudo-randomness property of generic quantum circuits: Almost every circuit U of size O( n k ) on n qubits cannot be distinguished from a Haar uniform unitary by circuits of size O( n ( k-9)/11) that are given oracle access to U.
Pieper, Andreas; Kreutzer, Moritz; Alvermann, Andreas; Galgon, Martin; Fehske, Holger; Hager, Georg; Lang, Bruno; Wellein, Gerhard
2016-11-01
We study Chebyshev filter diagonalization as a tool for the computation of many interior eigenvalues of very large sparse symmetric matrices. In this technique the subspace projection onto the target space of wanted eigenvectors is approximated with filter polynomials obtained from Chebyshev expansions of window functions. After the discussion of the conceptual foundations of Chebyshev filter diagonalization we analyze the impact of the choice of the damping kernel, search space size, and filter polynomial degree on the computational accuracy and effort, before we describe the necessary steps towards a parallel high-performance implementation. Because Chebyshev filter diagonalization avoids the need for matrix inversion it can deal with matrices and problem sizes that are presently not accessible with rational function methods based on direct or iterative linear solvers. To demonstrate the potential of Chebyshev filter diagonalization for large-scale problems of this kind we include as an example the computation of the 102 innermost eigenpairs of a topological insulator matrix with dimension 109 derived from quantum physics applications.
λ-PDF AND GEGENBAUER POLYNOMIAL APPROXIMATION FOR DYNAMIC RESPONSE PROBLEMS OF RANDOM STRUCTURES
Institute of Scientific and Technical Information of China (English)
FANG Tong; LENG Xiaolei; MA Xiaoping; MENG Guang
2004-01-01
A bounded, mono-peak, and symmetrically distributed probability density function,called λ-PDF, together with the Gegenbauer polynomial approximation, is used in dynamic response problems of random structures. The λ-PDF can reasonably model a variety of random parameters in engineering random structures. The Gegenbauer polynomial approximation can be viewed as a new extension of the weighted residual method into the random space. Both of them can be easily used by scientists and engineers, and applied to a variety of response problems of random structures. The numerical example shows the effectiveness of the proposed method to study dynamic phenomena in random structures.
Chkifa, Abdellah
2015-04-08
Motivated by the numerical treatment of parametric and stochastic PDEs, we analyze the least-squares method for polynomial approximation of multivariate functions based on random sampling according to a given probability measure. Recent work has shown that in the univariate case, the least-squares method is quasi-optimal in expectation in [A. Cohen, M A. Davenport and D. Leviatan. Found. Comput. Math. 13 (2013) 819–834] and in probability in [G. Migliorati, F. Nobile, E. von Schwerin, R. Tempone, Found. Comput. Math. 14 (2014) 419–456], under suitable conditions that relate the number of samples with respect to the dimension of the polynomial space. Here “quasi-optimal” means that the accuracy of the least-squares approximation is comparable with that of the best approximation in the given polynomial space. In this paper, we discuss the quasi-optimality of the polynomial least-squares method in arbitrary dimension. Our analysis applies to any arbitrary multivariate polynomial space (including tensor product, total degree or hyperbolic crosses), under the minimal requirement that its associated index set is downward closed. The optimality criterion only involves the relation between the number of samples and the dimension of the polynomial space, independently of the anisotropic shape and of the number of variables. We extend our results to the approximation of Hilbert space-valued functions in order to apply them to the approximation of parametric and stochastic elliptic PDEs. As a particular case, we discuss “inclusion type” elliptic PDE models, and derive an exponential convergence estimate for the least-squares method. Numerical results confirm our estimate, yet pointing out a gap between the condition necessary to achieve optimality in the theory, and the condition that in practice yields the optimal convergence rate.
Chebyshev Finite Difference Method for Fractional Boundary Value Problems
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Boundary
2015-09-01
Full Text Available This paper presents a numerical method for fractional differential equations using Chebyshev finite difference method. The fractional derivatives are described in the Caputo sense. Numerical results show that this method is of high accuracy and is more convenient and efficient for solving boundary value problems involving fractional ordinary differential equations. AMS Subject Classification: 34A08 Keywords and Phrases: Chebyshev polynomials, Gauss-Lobatto points, fractional differential equation, finite difference 1. Introduction The idea of a derivative which interpolates between the familiar integer order derivatives was introduced many years ago and has gained increasing importance only in recent years due to the development of mathematical models of a certain situations in engineering, materials science, control theory, polymer modelling etc. For example see [20, 22, 25, 26]. Most fractional order differential equations describing real life situations, in general do not have exact analytical solutions. Several numerical and approximate analytical methods for ordinary differential equation Received: December 2014; Accepted: March 2015 57 Journal of Mathematical Extension Vol. 9, No. 3, (2015, 57-71 ISSN: 1735-8299 URL: http://www.ijmex.com Chebyshev Finite Difference Method for Fractional Boundary Value Problems H. Azizi Taft Branch, Islamic Azad University Abstract. This paper presents a numerical method for fractional differential equations using Chebyshev finite difference method. The fractional derivative
Generating the patterns of variation with GeoGebra: the case of polynomial approximations
Attorps, Iiris; Björk, Kjell; Radic, Mirko
2016-01-01
In this paper, we report a teaching experiment regarding the theory of polynomial approximations at the university mathematics teaching in Sweden. The experiment was designed by applying Variation theory and by using the free dynamic mathematics software GeoGebra. The aim of this study was to investigate if the technology-assisted teaching of Taylor polynomials compared with traditional way of work at the university level can support the teaching and learning of mathematical concepts and ideas. An engineering student group (n = 19) was taught Taylor polynomials with the assistance of GeoGebra while a control group (n = 18) was taught in a traditional way. The data were gathered by video recording of the lectures, by doing a post-test concerning Taylor polynomials in both groups and by giving one question regarding Taylor polynomials at the final exam for the course in Real Analysis in one variable. In the analysis of the lectures, we found Variation theory combined with GeoGebra to be a potentially powerful tool for revealing some critical aspects of Taylor Polynomials. Furthermore, the research results indicated that applying Variation theory, when planning the technology-assisted teaching, supported and enriched students' learning opportunities in the study group compared with the control group.
Institute of Scientific and Technical Information of China (English)
GU Chuan-qing; PAN Bao-zhen; WU Bei-bei
2006-01-01
To solve Fredholm integral equations of the second kind, a generalized linear functional is introduced and a new function-valued Padé-type approximation is defined.By means of the power series expansion of the solution, this method can construct an approximate solution to solve the given integral equation. On the basis of the orthogonal polynomials, two useful determinant expressions of the numerator polynomial and the denominator polynomial for padé-type approximation are explicitly given.
Approximating the Value of a Concurrent Reachability Game in the Polynomial Time Hierarchy
DEFF Research Database (Denmark)
Frederiksen, Søren Kristoffer Stiil; Miltersen, Peter Bro
2013-01-01
We show that the value of a finite-state concurrent reachability game can be approximated to arbitrary precision in TFNP[NP], that is, in the polynomial time hierarchy. Previously, no better bound than PSPACE was known for this problem. The proof is based on formulating a variant of the state...
Bota, C.; Cǎruntu, B.; Bundǎu, O.
2013-10-01
In this paper we applied the Squared Remainder Minimization Method (SRMM) to find analytic approximate polynomial solutions for Riccati differential equations. Two examples are included to demonstrated the validity and applicability of the method. The results are compared to those obtained by other methods.
Generating the Patterns of Variation with GeoGebra: The Case of Polynomial Approximations
Attorps, Iiris; Björk, Kjell; Radic, Mirko
2016-01-01
In this paper, we report a teaching experiment regarding the theory of polynomial approximations at the university mathematics teaching in Sweden. The experiment was designed by applying Variation theory and by using the free dynamic mathematics software GeoGebra. The aim of this study was to investigate if the technology-assisted teaching of…
Approximation by polynomials and Blaschke products having all zeros on a circle
Farmer, David W
2010-01-01
We show that a nonvanishing analytic function on a domain in the unit disc can be approximated by (a scalar multiple of) a Blaschke product whose zeros lie on a prescribed circle enclosing the domain. We also give a new proof of the analogous classical result for polynomials. A connection is made to universality results for the Riemann zeta function.
Approximation of a switched linear system by a nonswitched homogeneous polynomial system
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
It is shown that given a switched linear control system which satisfy the controllability rank condition, it is possible to construct a non-switched homogenous polynomial control system in such a way that the trajectories of the latter may be arbitrarily well approximated norm on finite time intervals.
On the role of polynomials in RBF-FD approximations: I. Interpolation and accuracy
Flyer, Natasha; Fornberg, Bengt; Bayona, Victor; Barnett, Gregory A.
2016-09-01
Radial basis function-generated finite difference (RBF-FD) approximations generalize classical grid-based finite differences (FD) from lattice-based to scattered node layouts. This greatly increases the geometric flexibility of the discretizations and makes it easier to carry out local refinement in critical areas. Many different types of radial functions have been considered in this RBF-FD context. In this study, we find that (i) polyharmonic splines (PHS) in conjunction with supplementary polynomials provide a very simple way to defeat stagnation (also known as saturation) error and (ii) give particularly good accuracy for the tasks of interpolation and derivative approximations without the hassle of determining a shape parameter. In follow-up studies, we will focus on how to best use these hybrid RBF polynomial bases for FD approximations in the contexts of solving elliptic and hyperbolic type PDEs.
Nth-order flat approximation of the signum function by a polynomial
Hosenthien, H. H.
1972-01-01
In the interval studied, the signum function, sgn x, was demonstrated to be uniquely approximated by an odd polynomial f sub n (x) of order 2n-1, for which the approximation is nth order flat with respect to the points (1,1) and (-1,-1). A theorem was proved which states that for even integers n or = 2, the approximating polynomial has a pair of nonzero real roots + or - x sub n such that the x sub n form a monotonically decreasing sequence which converges to the root of 2 as n approaches infinity. For odd n i, f sub n (x) represents a strictly increasing monotonic function for all real x. As n tends to infinity, f sub n (x) converges to sgn x uniformly in two interval ranges.
On the optimal polynomial approximation of stochastic PDEs by galerkin and collocation methods
Beck, Joakim
2012-09-01
In this work we focus on the numerical approximation of the solution u of a linear elliptic PDE with stochastic coefficients. The problem is rewritten as a parametric PDE and the functional dependence of the solution on the parameters is approximated by multivariate polynomials. We first consider the stochastic Galerkin method, and rely on sharp estimates for the decay of the Fourier coefficients of the spectral expansion of u on an orthogonal polynomial basis to build a sequence of polynomial subspaces that features better convergence properties, in terms of error versus number of degrees of freedom, than standard choices such as Total Degree or Tensor Product subspaces. We consider then the Stochastic Collocation method, and use the previous estimates to introduce a new class of Sparse Grids, based on the idea of selecting a priori the most profitable hierarchical surpluses, that, again, features better convergence properties compared to standard Smolyak or tensor product grids. Numerical results show the effectiveness of the newly introduced polynomial spaces and sparse grids. © 2012 World Scientific Publishing Company.
On Chebyshev-Markov rational functions over several intervals
Lukashov, AL
1998-01-01
Chebyshev-Markov rational functions are the solutions of the following extremal problem [GRAPHICS] with K being a compact subset of R and omega(n)(x) being a fixed real polynomial of degree less than n, positive on K. A parametric representation of Chebyshev-Markov rational functions is found for K
OPTIMAL ERROR ESTIMATES OF THE PARTITION OF UNITY METHOD WITH LOCAL POLYNOMIAL APPROXIMATION SPACES
Institute of Scientific and Technical Information of China (English)
Yun-qing Huang; Wei Li; Fang Su
2006-01-01
In this paper, we provide a theoretical analysis of the partition of unity finite element method(PUFEM), which belongs to the family of meshfree methods. The usual error analysis only shows the order of error estimate to the same as the local approximations[12].Using standard linear finite element base functions as partition of unity and polynomials as local approximation space, in 1-d case, we derive optimal order error estimates for PUFEM interpolants. Our analysis show that the error estimate is of one order higher than the local approximations. The interpolation error estimates yield optimal error estimates for PUFEM solutions of elliptic boundary value problems.
Institute of Scientific and Technical Information of China (English)
Fran(c)ois Chaplais
2006-01-01
In applications it is useful to compute the local average of a function f(u) of an input u from empirical statistics on u. A very simple relation exists when the local averages are given by a Haar approximation. The question is to know if it holds for higher order approximation methods. To do so,it is necessary to use approximate product operators defined over linear approximation spaces. These products are characterized by a Strang and Fix like condition. An explicit construction of these product operators is exhibited for piecewise polynomial functions, using Hermite interpolation. The averaging relation which holds for the Haar approximation is then recovered when the product is defined by a two point Hermite interpolation.
Modified Chebyshev Collocation Method for Solving Differential Equations
Directory of Open Access Journals (Sweden)
M Ziaul Arif
2015-05-01
Full Text Available This paper presents derivation of alternative numerical scheme for solving differential equations, which is modified Chebyshev (Vieta-Lucas Polynomial collocation differentiation matrices. The Scheme of modified Chebyshev (Vieta-Lucas Polynomial collocation method is applied to both Ordinary Differential Equations (ODEs and Partial Differential Equations (PDEs cases. Finally, the performance of the proposed method is compared with finite difference method and the exact solution of the example. It is shown that modified Chebyshev collocation method more effective and accurate than FDM for some example given.
Fedorova, A; Parsa, Z
1998-01-01
In this paper we present applications of methods from wavelet analysis to polynomial approximations for a number of accelerator physics problems. According to variational approach in the general case we have the solution as a multiresolution (multiscales) expansion in the base of compactly supported wavelet basis. We give extension of our results to the cases of periodic orbital particle motion and arbitrary variable coefficients. Then we consider more flexible variational method which is based on biorthogonal wavelet approach. Also we consider different variational approach, which is applied to each scale.
White matter structure assessment from reduced HARDI data using low-rank polynomial approximations.
Gur, Yaniv; Jiao, Fangxiang; Zhu, Stella Xinghua; Johnson, Chris R
2012-10-01
Assessing white matter fiber orientations directly from DWI measurements in single-shell HARDI has many advantages. One of these advantages is the ability to model multiple fibers using fewer parameters than are required to describe an ODF and, thus, reduce the number of DW samples needed for the reconstruction. However, fitting a model directly to the data using Gaussian mixture, for instance, is known as an initialization-dependent unstable process. This paper presents a novel direct fitting technique for single-shell HARDI that enjoys the advantages of direct fitting without sacrificing the accuracy and stability even when the number of gradient directions is relatively low. This technique is based on a spherical deconvolution technique and decomposition of a homogeneous polynomial into a sum of powers of linear forms, known as a symmetric tensor decomposition. The fiber-ODF (fODF), which is described by a homogeneous polynomial, is approximated here by a discrete sum of even-order linear-forms that are directly related to rank-1 tensors and represent single-fibers. This polynomial approximation is convolved to a single-fiber response function, and the result is optimized against the DWI measurements to assess the fiber orientations and the volume fractions directly. This formulation is accompanied by a robust iterative alternating numerical scheme which is based on the Levenberg-Marquardt technique. Using simulated data and in vivo, human brain data we show that the proposed algorithm is stable, accurate and can model complex fiber structures using only 12 gradient directions. PMID:24818174
ON NEWMAN-TYPE RATIONAL INTERPOLATION TO |x| AT THE CHEBYSHEV NODES OF THE SECOND KIND
Institute of Scientific and Technical Information of China (English)
Laiyi Zhu; Zhaolin Dong
2006-01-01
Recently Brutman and Passow considered Newman-type rational interpolation to |x| induced by arbitrary set of symmetric nodes in [-1, 1] and gave the general estimation of the approximation error.By their methods one could establish the exact order of approximation for some special nodes. In the present paper we consider the special case where the interpolation nodes are the zeros of the Chebyshev polynomial of the second kind and prove that in this case the exact order of approximation is O (1/nlnn).
Ait-Haddou, Rachid
2015-06-04
We show that the best degree reduction of a given polynomial P from degree n to m with respect to the discrete (Formula presented.)-norm is equivalent to the best Euclidean distance of the vector of h-Bézier coefficients of P from the vector of degree raised h-Bézier coefficients of polynomials of degree m. Moreover, we demonstrate the adequacy of h-Bézier curves for approaching the problem of weighted discrete least squares approximation. Applications to discrete orthogonal polynomials are also presented. © 2015 Springer Science+Business Media Dordrecht
Algebraic multilevel iteration methods and the best approximation to $1/x$ in the uniform norm
Kraus, Johannes; Zikatanov, Ludmil
2010-01-01
In this note, we provide simple convergence analysis for the algebraic multilevel iteration methods. We consider two examples of AMLI methods with different polynomial acceleration. The first one is based on shifted and scaled Chebyshev polynomial and the other on the polynomial of best approximation to $x^{-1}$ on a finite interval with positive endpoints in the uniform norm. The construction of the latter polynomial is of interest by itself, and we have included a derivation of a 3 term recurrence relation for computing this polynomial. We have also derived several inequalities related to the error of best approximation, which we applied in the AMLI analysis.
An approximation polynomial-time algorithm for a sequence bi-clustering problem
Kel'manov, A. V.; Khamidullin, S. A.
2015-06-01
We consider a strongly NP-hard problem of partitioning a finite sequence of vectors in Euclidean space into two clusters using the criterion of the minimal sum of the squared distances from the elements of the clusters to the centers of the clusters. The center of one of the clusters is to be optimized and is determined as the mean value over all vectors in this cluster. The center of the other cluster is fixed at the origin. Moreover, the partition is such that the difference between the indices of two successive vectors in the first cluster is bounded above and below by prescribed constants. A 2-approximation polynomial-time algorithm is proposed for this problem.
Narimani, Mohammand; Lam, H K; Dilmaghani, R; Wolfe, Charles
2011-06-01
Relaxed linear-matrix-inequality-based stability conditions for fuzzy-model-based control systems with imperfect premise matching are proposed. First, the derivative of the Lyapunov function, containing the product terms of the fuzzy model and fuzzy controller membership functions, is derived. Then, in the partitioned operating domain of the membership functions, the relations between the state variables and the mentioned product terms are represented by approximated polynomials in each subregion. Next, the stability conditions containing the information of all subsystems and the approximated polynomials are derived. In addition, the concept of the S-procedure is utilized to release the conservativeness caused by considering the whole operating region for approximated polynomials. It is shown that the well-known stability conditions can be special cases of the proposed stability conditions. Simulation examples are given to illustrate the validity of the proposed approach.
The algebra of two dimensional generalized Chebyshev-Koornwinder oscillator
Energy Technology Data Exchange (ETDEWEB)
Borzov, V. V., E-mail: borzov.vadim@yandex.ru [Department of Mathematics, St. Petersburg State University of Telecommunications, 191186, Moika 61, St. Petersburg (Russian Federation); Damaskinsky, E. V., E-mail: evd@pdmi.ras.ru [Department of Natural Sciences, Institute of Defense Technical Engineering (VITI), 191123, Zacharievskaya 22, St. Petersburg (Russian Federation)
2014-10-15
In the previous works of Borzov and Damaskinsky [“Chebyshev-Koornwinder oscillator,” Theor. Math. Phys. 175(3), 765–772 (2013)] and [“Ladder operators for Chebyshev-Koornwinder oscillator,” in Proceedings of the Days on Diffraction, 2013], the authors have defined the oscillator-like system that is associated with the two variable Chebyshev-Koornwinder polynomials. We call this system the generalized Chebyshev-Koornwinder oscillator. In this paper, we study the properties of infinite-dimensional Lie algebra that is analogous to the Heisenberg algebra for the Chebyshev-Koornwinder oscillator. We construct the exact irreducible representation of this algebra in a Hilbert space H of functions that are defined on a region which is bounded by the Steiner hypocycloid. The functions are square-integrable with respect to the orthogonality measure for the Chebyshev-Koornwinder polynomials and these polynomials form an orthonormalized basis in the space H. The generalized oscillator which is studied in the work can be considered as the simplest nontrivial example of multiboson quantum system that is composed of three interacting oscillators.
Van Raemdonck, Mario; Alcoba, Diego R; Poelmans, Ward; De Baerdemacker, Stijn; Torre, Alicia; Lain, Luis; Massaccesi, Gustavo E; Van Neck, Dimitri; Bultinck, Patrick
2015-09-14
A class of polynomial scaling methods that approximate Doubly Occupied Configuration Interaction (DOCI) wave functions and improve the description of dynamic correlation is introduced. The accuracy of the resulting wave functions is analysed by comparing energies and studying the overlap between the newly developed methods and full configuration interaction wave functions, showing that a low energy does not necessarily entail a good approximation of the exact wave function. Due to the dependence of DOCI wave functions on the single-particle basis chosen, several orbital optimisation algorithms are introduced. An energy-based algorithm using the simulated annealing method is used as a benchmark. As a computationally more affordable alternative, a seniority number minimising algorithm is developed and compared to the energy based one revealing that the seniority minimising orbital set performs well. Given a well-chosen orbital basis, it is shown that the newly developed DOCI based wave functions are especially suitable for the computationally efficient description of static correlation and to lesser extent dynamic correlation.
Directory of Open Access Journals (Sweden)
Madeira Sara C
2009-06-01
Full Text Available Abstract Background The ability to monitor the change in expression patterns over time, and to observe the emergence of coherent temporal responses using gene expression time series, obtained from microarray experiments, is critical to advance our understanding of complex biological processes. In this context, biclustering algorithms have been recognized as an important tool for the discovery of local expression patterns, which are crucial to unravel potential regulatory mechanisms. Although most formulations of the biclustering problem are NP-hard, when working with time series expression data the interesting biclusters can be restricted to those with contiguous columns. This restriction leads to a tractable problem and enables the design of efficient biclustering algorithms able to identify all maximal contiguous column coherent biclusters. Methods In this work, we propose e-CCC-Biclustering, a biclustering algorithm that finds and reports all maximal contiguous column coherent biclusters with approximate expression patterns in time polynomial in the size of the time series gene expression matrix. This polynomial time complexity is achieved by manipulating a discretized version of the original matrix using efficient string processing techniques. We also propose extensions to deal with missing values, discover anticorrelated and scaled expression patterns, and different ways to compute the errors allowed in the expression patterns. We propose a scoring criterion combining the statistical significance of expression patterns with a similarity measure between overlapping biclusters. Results We present results in real data showing the effectiveness of e-CCC-Biclustering and its relevance in the discovery of regulatory modules describing the transcriptomic expression patterns occurring in Saccharomyces cerevisiae in response to heat stress. In particular, the results show the advantage of considering approximate patterns when compared to state of
Polynomial Approximation Algorithms for the TSP and the QAP with a Factorial Domination Number
DEFF Research Database (Denmark)
Gutin, Gregory; Yeo, Anders
2002-01-01
Glover and Punnen (J. Oper. Res. Soc. 48 (1997) 502) asked whether there exists a polynomial time algorithm that always produces a tour which is not worse than at least n!/p(n) tours for some polynomial p(n) for every TSP instance on n cities. They conjectured that, unless P = NP, the answer to t...
Carpenter, William C.
1991-01-01
Engineering optimization problems involve minimizing some function subject to constraints. In areas such as aircraft optimization, the constraint equations may be from numerous disciplines such as transfer of information between these disciplines and the optimization algorithm. They are also suited to problems which may require numerous re-optimizations such as in multi-objective function optimization or to problems where the design space contains numerous local minima, thus requiring repeated optimizations from different initial designs. Their use has been limited, however, by the fact that development of response surfaces randomly selected or preselected points in the design space. Thus, they have been thought to be inefficient compared to algorithms to the optimum solution. A development has taken place in the last several years which may effect the desirability of using response surfaces. It may be possible that artificial neural nets are more efficient in developing response surfaces than polynomial approximations which have been used in the past. This development is the concern of the work.
Halman, Nir; Klabjan, Diego; Mostagir, Mohamed; Orlin, Jim; Simchi-Levi, David
2009-01-01
The single-item stochastic inventory control problem is to find an inventory replenishment policy in the presence of independent discrete stochastic demands under periodic review and finite time horizon. In this paper, we prove that this problem is intractable and design for it a fully polynomial-time approximation scheme.
Simple polynomial approximation to modified Bethe formula low-energy electron stopping powers data
Energy Technology Data Exchange (ETDEWEB)
Taborda, A., E-mail: ana.taborda@irsn.fr [Institut de Radioprotection et de Sûreté Nucléaire (IRSN), PRP-HOM/SDI/LEDI, BP-17, 92262 Fontenay-aux-Roses (France); Desbrée, A. [Institut de Radioprotection et de Sûreté Nucléaire (IRSN), PRP-HOM/SDI/LEDI, BP-17, 92262 Fontenay-aux-Roses (France); Reis, M.A. [C" 2TN, Campus Tecnológico e Nuclear, Instituto Superior Técnico, Universidade de Lisboa, EN10 km139.7, 2685-066 Bobadela LRS (Portugal)
2015-08-01
A recently published detailed and exhaustive paper on cross-sections for ionisation induced by keV electrons clearly shows that electron phenomena occurring in parallel with X-ray processes may have been dramatically overlooked for many years, mainly when low atomic number species are involved since, in these cases, the fluorescence coefficient is smaller than the Auger yield. An immediate problem is encountered while attempting to tackle the issue. Accounting for electron phenomena requires the knowledge of the stopping power of electrons within, at least, a reasonably small error. Still, the Bethe formula for stopping powers is known to not be valid for electron energies below 30 keV, and its use leads to values far off experimental ones. Recently, a few authors have addressed this problem and both detailed tables of electron stopping powers for various atomic species and attempts to simplify the calculations, have emerged. Nevertheless, its implementation in software routines to efficiently calculate keV electron effects in materials quickly becomes a bit cumbersome. Following a procedure already used to establish efficient methods to calculate ionisation cross-sections by protons and alpha particles, it became clear that a simple polynomial approximation could be set, which allows retrieving the electronic stopping powers with errors of less than 20% for energies above 500 eV and less than 50% for energies between 50 eV and 500 eV. In this work, we present this approximation which, based on just six parameters, allows to recover electron stopping power values that are less than 20% different from recently published experimentally validated tabulated data.
Simple polynomial approximation to modified Bethe formula low-energy electron stopping powers data
International Nuclear Information System (INIS)
A recently published detailed and exhaustive paper on cross-sections for ionisation induced by keV electrons clearly shows that electron phenomena occurring in parallel with X-ray processes may have been dramatically overlooked for many years, mainly when low atomic number species are involved since, in these cases, the fluorescence coefficient is smaller than the Auger yield. An immediate problem is encountered while attempting to tackle the issue. Accounting for electron phenomena requires the knowledge of the stopping power of electrons within, at least, a reasonably small error. Still, the Bethe formula for stopping powers is known to not be valid for electron energies below 30 keV, and its use leads to values far off experimental ones. Recently, a few authors have addressed this problem and both detailed tables of electron stopping powers for various atomic species and attempts to simplify the calculations, have emerged. Nevertheless, its implementation in software routines to efficiently calculate keV electron effects in materials quickly becomes a bit cumbersome. Following a procedure already used to establish efficient methods to calculate ionisation cross-sections by protons and alpha particles, it became clear that a simple polynomial approximation could be set, which allows retrieving the electronic stopping powers with errors of less than 20% for energies above 500 eV and less than 50% for energies between 50 eV and 500 eV. In this work, we present this approximation which, based on just six parameters, allows to recover electron stopping power values that are less than 20% different from recently published experimentally validated tabulated data
A Randomized Fully Polynomial-time Approximation Scheme for Weighted Perfect Matching in the Plane
Directory of Open Access Journals (Sweden)
Yasser M. Abd El-Latif
2012-12-01
Full Text Available — In the approximate Euclidean min-weighted perfect matching problem, a set V of 2n points in the plane and a real number 0 are given. Usually, a solution of this problem is a partition of points of V into n pairs such that the sum of the distances between the paired points is at most (1 times the optimal solution.In this paper, the authors give a randomized algorithm which follows a Monte-Carlo method. This algorithm is a randomized fully polynomial-time approximation scheme for the given problem. Fortunately, the suggested algorithm is a one tackled the matching problem in both Euclidean nonbipartite and bipartite cases.The presented algorithm outlines as follows: With repeating 1/ times, we choose a point from V to build the suitable pair satisfying the suggested condition on the distance. If this condition is achieved, then remove the points of the constructed pair from V and put this pair in M (the output set of the solution. Then, choose a point and the nearest point of it from the remaining points in V to construct a pair and put it inM . Remove the two points of the constructed pair from V and repeat this process until V becomes an empty set. Obviously, this method is very simple. Furthermore, our algorithm can be applied without any modification on complete weighted graphs K mand complete weighted bipartite graphs Kn,n, where n,m 1and m is an even.
Karassiov, V. P.; A. A. Gusev; Vinitsky, S. I.
2001-01-01
We compare exact and SU(2)-cluster approximate calculation schemes to determine dynamics of the second-harmonic generation model using its reformulation in terms of a polynomial Lie algebra $su_{pd}(2)$ and related spectral representations of the model evolution operator realized in algorithmic forms. It enabled us to implement computer experiments exhibiting a satisfactory accuracy of the cluster approximations in a large range of characteristic model parameters.
Tal-Ezer, Hillel
1987-01-01
During the process of solving a mathematical model numerically, there is often a need to operate on a vector v by an operator which can be expressed as f(A) while A is NxN matrix (ex: exp(A), sin(A), A sup -1). Except for very simple matrices, it is impractical to construct the matrix f(A) explicitly. Usually an approximation to it is used. In the present research, an algorithm is developed which uses a polynomial approximation to f(A). It is reduced to a problem of approximating f(z) by a polynomial in z while z belongs to the domain D in the complex plane which includes all the eigenvalues of A. This problem of approximation is approached by interpolating the function f(z) in a certain set of points which is known to have some maximal properties. The approximation thus achieved is almost best. Implementing the algorithm to some practical problem is described. Since a solution to a linear system Ax = b is x= A sup -1 b, an iterative solution to it can be regarded as a polynomial approximation to f(A) = A sup -1. Implementing the algorithm in this case is also described.
Institute of Scientific and Technical Information of China (English)
刘哲; 宋余庆; 宋旼珊
2013-01-01
针对原有一元正交多项式混合模型只能根据灰度特征分割图像的问题,提出一种基于多元Chebyshev正交多项式混合模型的多维特征的医学图像分割方法.首先,根据Fourier分析方法与张量积理论推导出图像的多元Chebyshev正交多项式,并构建多元正交多项式的非参数混合模型,用最小均方差(MISE)估计每一个模型的平滑参数；然后,用EM算法求解正交多项式系数和模型的混合比.此方法不需要对模型作任何假设,可以有效克服“模型失配”问题.通过实验,表明了该分割方法的有效性.%To solve the problem of over-reliance on priori assumptions of the parameter methods for finite mixture models and the problem that monic Chebyshev orthogonal polynomials can only process the gray images, a segmentation method of mixture models of multivariate Chebyshev orthogonal polynomials for color image was proposed in this paper. First,the multivariate Chebyshev orthogonal polynomials was derived by the Fourier analysis and the tensor product theory, and the nonparametric mixture model of multivariate orthogonal polynomials was proposed. And the mean integrated squared error(MISE) was used to estimate the smoothing parameter for each model. Second, the expectation maximum(EM) algorithm was used to estimate the orthogonal polynomial coefficients and the model of the weight. This method does not require any prior assumptions on the model, and it can effectively overcome the "model mismatch" problem. The experimental results with the images show that this method can achieve better segmentation results than the mean-shift method.
Geddes, K. O.
1977-01-01
If a linear ordinary differential equation with polynomial coefficients is converted into integrated form then the formal substitution of a Chebyshev series leads to recurrence equations defining the Chebyshev coefficients of the solution function. An explicit formula is presented for the polynomial coefficients of the integrated form in terms of the polynomial coefficients of the differential form. The symmetries arising from multiplication and integration of Chebyshev polynomials are exploited in deriving a general recurrence equation from which can be derived all of the linear equations defining the Chebyshev coefficients. Procedures for deriving the general recurrence equation are specified in a precise algorithmic notation suitable for translation into any of the languages for symbolic computation. The method is algebraic and it can therefore be applied to differential equations containing indeterminates.
Approximation of Iteration Number for Gauss-Seidel Using Redlich-Kister Polynomial
Directory of Open Access Journals (Sweden)
M. K. Hasan
2010-01-01
Full Text Available Problem statement: Development of mathematical models based on set of observed data plays a crucial role to describe and predict any phenomena in science, engineering and economics. Therefore, the main purpose of this study was to compare the efficiency of Arithmetic Mean (AM, Geometric Mean (GM and Explicit Group (EG iterative methods to solve system of linear equations via estimation of unknown parameters in linear models. Approach: The system of linear equations for linear models generated by using least square method based on (m+1 set of observed data for number of Gauss-Seidel iteration from various grid sizes. Actually there were two types of linear models considered such as piece-wise linear polynomial and piece-wise Redlich-Kister polynomial. All unknown parameters of these models estimated and calculated by using three proposed iterative methods. Results: Thorough several implementations of numerical experiments, the accuracy for formulations of two proposed models had shown that the use of the third-order Redlich-Kister polynomial has high accuracy compared to linear polynomial case. Conclusion: The efficiency of AM and GM iterative methods based on the Redlich-Kister polynomial is superior as compared to EG iterative method.
Migliorati, G.
2013-05-30
In this work we consider the random discrete L^2 projection on polynomial spaces (hereafter RDP) for the approximation of scalar quantities of interest (QOIs) related to the solution of a partial differential equation model with random input parameters. In the RDP technique the QOI is first computed for independent samples of the random input parameters, as in a standard Monte Carlo approach, and then the QOI is approximated by a multivariate polynomial function of the input parameters using a discrete least squares approach. We consider several examples including the Darcy equations with random permeability, the linear elasticity equations with random elastic coefficient, and the Navier--Stokes equations in random geometries and with random fluid viscosity. We show that the RDP technique is well suited to QOIs that depend smoothly on a moderate number of random parameters. Our numerical tests confirm the theoretical findings in [G. Migliorati, F. Nobile, E. von Schwerin, and R. Tempone, Analysis of the Discrete $L^2$ Projection on Polynomial Spaces with Random Evaluations, MOX report 46-2011, Politecnico di Milano, Milano, Italy, submitted], which have shown that, in the case of a single uniformly distributed random parameter, the RDP technique is stable and optimally convergent if the number of sampling points is proportional to the square of the dimension of the polynomial space. Here optimality means that the weighted $L^2$ norm of the RDP error is bounded from above by the best $L^\\\\infty$ error achievable in the given polynomial space, up to logarithmic factors. In the case of several random input parameters, the numerical evidence indicates that the condition on quadratic growth of the number of sampling points could be relaxed to a linear growth and still achieve stable and optimal convergence. This makes the RDP technique very promising for moderately high dimensional uncertainty quantification.
Energy Technology Data Exchange (ETDEWEB)
Blackett, S.A. [Univ. of Auckland (New Zealand). Dept of Engineering Science
1996-02-01
Numerical analysis is an important part of Engineering. Frequently relationships are not adequately understood, or too complicated to be represented by theoretical formulae. Instead, empirical approximations based on observed relationships can be used for simple fast and accurate evaluations. Historically, storage of data has been a large constraint on approximately methods. So the challenge is to find a sufficiently accurate representation of data which is valid over as large a range as possible while requiring the storage of only a few numerical values. Polynomials, popular as approximation functions because of their simplicity, can be used to represent simple data. Equation 1.1 shows a simple 3rd order polynomial approximation. However, just increasing the order and number of terms included in a polynomial approximation does not improve the overall result. Although the function may fit exactly to observed data, between these points it is likely that the approximation is increasingly less smooth and probably inadequate. An alternative to adding further terms to the approximation is to make the approximation rational. Equation 1.2 shows a rational polynomial, 3rd order in the numerator and denominator. A rational polynomial approximation allows poles and this can greatly enhance an approximation. In Sections 2 and 3 two different methods for fitting rational polynomials to a given data set are detailed. In Section 4, consideration is given to different rational polynomials used on adjacent regions. Section 5 shows the performance of the rational polynomial algorithms. Conclusions are presented in Section 6.
A Fast, Simple, and Stable Chebyshev--Legendre Transform Using an Asymptotic Formula
Hale, Nicholas
2014-02-06
A fast, simple, and numerically stable transform for converting between Legendre and Chebyshev coefficients of a degree N polynomial in O(N(log N)2/ log log N) operations is derived. The fundamental idea of the algorithm is to rewrite a well-known asymptotic formula for Legendre polynomials of large degree as a weighted linear combination of Chebyshev polynomials, which can then be evaluated by using the discrete cosine transform. Numerical results are provided to demonstrate the efficiency and numerical stability. Since the algorithm evaluates a Legendre expansion at an N +1 Chebyshev grid as an intermediate step, it also provides a fast transform between Legendre coefficients and values on a Chebyshev grid. © 2014 Society for Industrial and Applied Mathematics.
Institute of Scientific and Technical Information of China (English)
汪首坤; 彭建敏; 刘洋
2013-01-01
On the basis of working principle and its characteristics of linear variable differential transformer (LVDT) displacement sensor, this paper put forward a method to deal with the nonlinear issue of output signals at both ends of the sensor using Chebyshev best approximation principle. The sensor's effective range is adaptively divided into linear and nonlinear regions, linear processing would be made for signals of two regions, respectively. The linear working range and the corresponding linear approximable straight-line function of the sensor are determined according to Chebyshev best approximation principle, while signals in nonlinear regions are linearized using rational B-spline function. A signal processor was designed based on MSP430 microcontroller and a test platform composed of a stepper motor straight-line units and a standard laser sensor was established. Experiments have been implemented to verify the feasibility of proposed method with 85 mm range of LVDT displacement sensor. The results show that this method can effectively improve the linearity and precision of the sensor and extend the working range of displacement sensor.%针对LVDT位移传感器两端输出信号的非线性问题,提出了一种基于切比雪夫最佳逼近原理的信号处理方法.该方法将传感器有效量程自适应地分为线性和非线性区域.线性工作范围和对应直线逼近函数利用切比雪夫一次最佳逼近自适应确定,非线性区域信号采用有理B样条函数进行线性化处理.设计了基于MSP430单片机的信号处理器,搭建了基于步进电机直线台和标准激光传感器的试验平台,对该算法进行实验验证.实验选用量程为85 mm的LVDT位移传感器,实验结果表明,该方法将传感器的非线性误差从2.47％降至0.30％,测量平均误差绝对值从0.64 mm降至0.12 mm,有效改善了传感器的线性度和精度,延展了其工作范围.
Chebyshev Finite Difference Method for Solving Constrained Quadratic Optimal Control Problems
M Maleki; M. Dadkhah Tirani
2011-01-01
. In this paper the Chebyshev finite difference method is employed for finding the approximate solution of time varying constrained optimal control problems. This approach consists of reducing the optimal control problem to a nonlinear mathematical programming problem. To this end, the collocation points (Chebyshev Gauss-Lobatto nodes) are introduced then the state and control variables are approximated using special Chebyshev series with unknown parameters. The performan...
Institute of Scientific and Technical Information of China (English)
刘哲; 宋余庆; 陈健美; 谢从华; 宋旼珊
2011-01-01
To solve the problem of over-reliance on priori assumptions of the parameter methods for finite mixture models, a nonparametric mixture model of Chebyshev orthogonal polynomials of the second kind for image segmentation method is proposed in this paper. Firstly, an image nonparametric misture model based on Chebyshev orthogonal polynomials of the second kind is designed. The mixture identification step based on the maximisation of the likelihood can be realised without hypothesis on the distribution of the conditional probability density function(PDF). In this paper, we intend to give some simulation results for the determination of the smoothing parameter, and use mean integrated squared error (MISE) estimation of the smoothing parameter for each model. Secondly, the stochastic expectation maximum (SEM) algorithm is used to estimate the Chebyshev orthogonal polynomial coefficients and the model of the weight. This method does not require any priori assumptions on the model, and it can effectively overcome the "model mismatch" problem. The algorithm finds the most likely number of classes and their associated model parameters and generates a segmentation of the image by classifying the pixels into these classes. Compared with the segmentation methods of other orthogonal polynomials, this new method is much more fast in speed and better segmentation quality. The experimental results about the image segmentation show that this method is better than the Gaussian mixture model segmentation results.%有参混合模型需要假设模型为某种已知的参数模型,而实际数据往往很难假设出这种参数模型的分布.为此,提出一种二类切比雪夫正交多项式的非参数图像混合模型分割方法.首先,设计出一种基于二类切比雪夫正交多项式的图像非参数混合模型,每一个模型的平滑参数根据误差方法和最小的准则进行计算.然后,利用随机期望最大(SEM)算法求解正交多项式系数和每
Reduction of Linear Programming to Linear Approximation
Vaserstein, Leonid N.
2006-01-01
It is well known that every Chebyshev linear approximation problem can be reduced to a linear program. In this paper we show that conversely every linear program can be reduced to a Chebyshev linear approximation problem.
Fully Polynomial Approximation Schemes for Single-Item Capacitated Economic Lot-Sizing Problems
C.P.M. van Hoesel; A.P.M. Wagelmans (Albert)
1997-01-01
textabstractNP-hard cases of the single-item capacitated lot-sizing problem have been the topic of extensive research and continue to receive considerable attention. However, surprisingly few theoretical results have been published on approximation methods for these problems. To the best of our kno
An inequality for polynomials with elliptic majorant
Nikolov Geno
1999-01-01
Let be the transformed Chebyshev polynomial of the first kind, where . We show here that has the greatest uniform norm in of its -th derivative among all algebraic polynomials of degree not exceeding , which vanish at and satisfy the inequality at the points .
Application of polynomial preconditioners to conservation laws
Geurts, Bernard J.; Buuren, van René; Lu, Hao
2000-01-01
Polynomial preconditioners which are suitable in implicit time-stepping methods for conservation laws are reviewed and analyzed. The preconditioners considered are either based on a truncation of a Neumann series or on Chebyshev polynomials for the inverse of the system-matrix. The latter class of p
Pade approximants for functions with branch points - strong asymptotics of Nuttall-Stahl polynomials
Aptekarev, Alexander I
2011-01-01
Let f be a germ of an analytic function at infinity that can be analytically continued along any path in the complex plane deprived of a finite set of points, f \\in\\mathcal{A}(\\bar{\\C} \\setminus A), \\sharp A <\\infty. J. Nuttall has put forward the important relation between the maximal domain of f where the function has a single-valued branch and the domain of convergence of the diagonal Pade approximants for f. The Pade approximants, which are rational functions and thus single-valued, approximate a holomorphic branch of f in the domain of their convergence. At the same time most of their poles tend to the boundary of the domain of convergence and the support of their limiting distribution models the system of cuts that makes the function f single-valued. Nuttall has conjectured (and proved for many important special cases) that this system of cuts has minimal logarithmic capacity among all other systems converting the function f to a single-valued branch. Thus the domain of convergence corresponds to the...
Sagnol, Guillaume
2010-01-01
The theory of "optimal experimental design" explains how to best select experiments in order to estimate a set of parameters. The quality of the estimation can be measured by the confidence ellipsoids of a certain estimator. This leads to concave maximization problems in which the objective function is nondecreasing with respect to the L\\"owner ordering of symmetric matrices, and is applied to the "information matrix" describing the structure of these confidence ellipsoids. In a number of real-world applications, the variables controlling the experimental design are discrete, or binary. This paper provides approximability bounds for this NP-hard problem. In particular, we establish a matrix inequality which shows that the objective function is submodular, from which it follows that the greedy approach, which has often been used for this problem, always gives a design within $1-1/e$ of the optimum. We next study the design found by rounding the solution of the continuous relaxed problem, an approach which has ...
NAPX: A Polynomial Time Approximation Scheme for the Noah's Ark Problem
Hickey, G; Maheshwari, A; Zeh, N
2008-01-01
The Noah's Ark Problem (NAP) is an NP-Hard optimization problem with relevance to ecological conservation management. It asks to maximize the phylogenetic diversity (PD) of a set of taxa given a fixed budget, where each taxon is associated with a cost of conservation and a probability of extinction. NAP has received renewed interest with the rise in availability of genetic sequence data, allowing PD to be used as a practical measure of biodiversity. However, only simplified instances of the problem, where one or more parameters are fixed as constants, have as of yet been addressed in the literature. We present NAPX, the first algorithm for the general version of NAP that returns a $1 - \\epsilon$ approximation of the optimal solution. It runs in $O(\\frac{n B^2 h^2 \\log^2n}{\\log^2(1 - \\epsilon)})$ time where $n$ is the number of species, and $B$ is the total budget and $h$ is the height of the input tree. We also provide improved bounds for its expected running time.
Energy Technology Data Exchange (ETDEWEB)
Ceolin, C., E-mail: celina.ceolin@gmail.com [Universidade Federal de Santa Maria (UFSM), Frederico Westphalen, RS (Brazil). Centro de Educacao Superior Norte; Schramm, M.; Bodmann, B.E.J.; Vilhena, M.T., E-mail: celina.ceolin@gmail.com [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Programa de Pos-Graduacao em Engenharia Mecanica
2015-07-01
Recently the stationary neutron diffusion equation in heterogeneous rectangular geometry was solved by the expansion of the scalar fluxes in polynomials in terms of the spatial variables (x; y), considering the two-group energy model. The focus of the present discussion consists in the study of an error analysis of the aforementioned solution. More specifically we show how the spatial subdomain segmentation is related to the degree of the polynomial and the Lipschitz constant. This relation allows to solve the 2-D neutron diffusion problem for second degree polynomials in each subdomain. This solution is exact at the knots where the Lipschitz cone is centered. Moreover, the solution has an analytical representation in each subdomain with supremum and infimum functions that shows the convergence of the solution. We illustrate the analysis with a selection of numerical case studies. (author)
A Note on The Convexity of Chebyshev Sets
Directory of Open Access Journals (Sweden)
Sangeeta
2009-07-01
Full Text Available Perhaps one of the major unsolved problem in Approximation Theoryis: Whether or not every Chebyshev subset of a Hilbert space must be convex. Many partial answers to this problem are available in the literature. R.R. Phelps[Proc. Amer. Math. Soc. 8 (1957, 790-797] showed that a Chebyshev set in an inner product space (or in a strictly convex normed linear space is convex if the associated metric projection is non-expansive. We extend this result to metricspaces.
Institute of Scientific and Technical Information of China (English)
王雷
2008-01-01
<正>Polynomial functions are among the sim- plest expressions in algebra.They are easy to evaluate:only addition and repeated multipli- cation are required.Because of this,they are often used to approximate other more compli-
Chebyshev Finite Difference Method for Solving Constrained Quadratic Optimal Control Problems
Directory of Open Access Journals (Sweden)
M. Maleki*
2011-06-01
Full Text Available . In this paper the Chebyshev finite difference method is employed for finding the approximate solution of time varying constrained optimal control problems. This approach consists of reducing the optimal control problem to a nonlinear mathematical programming problem. To this end, the collocation points (Chebyshev Gauss-Lobatto nodes are introduced then the state and control variables are approximated using special Chebyshev series with unknown parameters. The performance index is parameterized and the system dynamics and constraints are then replaced with a set of algebraic equations. Numerical examples are included to demonstrate the validity and applicability of the technique.
Wang, Chunxiao; Liu, Hongya
2008-01-01
As one exact candidate of the higher dimensional black hole, the 5D Ricci-flat Schwarzschild-de Sitter black string space presents something interesting. In this paper, we give a numerical solution to the real scalar field around the Nariai black hole by the polynomial approximation. Unlike the previous tangent approximation, this fitting function makes a perfect match in the leading intermediate region and gives a good description near both the event and the cosmological horizons. We can read from our results that the wave is close to a harmonic one with the tortoise coordinate. Furthermore, with the actual radial coordinate the waves pile up almost equally near the both horizons.
Short-time Chebyshev wave packet method for molecular photoionization
Sun, Zhaopeng; Zheng, Yujun
2016-08-01
In this letter we present the extended usage of short-time Chebyshev wave packet method in the laser induced molecular photoionization dynamics. In our extension, the polynomial expansion of the exponential in the time evolution operator, the Hamiltonian operator can act on the wave packet directly which neatly avoids the matrix diagonalization. This propagation scheme is of obvious advantages when the dynamical system has large Hamiltonian matrix. Computational simulations are performed for the calculation of photoelectronic distributions from intense short pulse ionization of K2 and NaI which represent the Born-Oppenheimer (BO) model and Non-BO one, respectively.
Chebyshev and Fourier spectral methods
Boyd, John P
2001-01-01
Completely revised text focuses on use of spectral methods to solve boundary value, eigenvalue, and time-dependent problems, but also covers Hermite, Laguerre, rational Chebyshev, sinc, and spherical harmonic functions, as well as cardinal functions, linear eigenvalue problems, matrix-solving methods, coordinate transformations, methods for unbounded intervals, spherical and cylindrical geometry, and much more. 7 Appendices. Glossary. Bibliography. Index. Over 160 text figures.
Taylor多项式逼近函数的计算机模拟%Computer Simulation of Approximation by Taylor Polynomials
Institute of Scientific and Technical Information of China (English)
张和平; 王凯
2011-01-01
To help better understanding Taylor expansion, we select a typical teaching case which uses the drawing power of MATLAB to demonstrate visually the approximation of functions by their Taylor polynomials.%针对典型教学案例,借助MATLAB的绘图与动画功能,通过编制程序并绘制图像,探讨Taylor多项式逼近函数的直观解释,以求帮助学生深入理解函数的Taylor展开式.
Mello, M. M.; Ventura, L.
2015-03-01
A method using different light sources and sensors have already been used to approximate weighting functions to calculate light transmittance in sunglasses. Although it made possible a low cost equipment that inform the user about its sunglasses, each transmittance test is still dependent of its components. We tested two methods, using polynomial approximation and artificial neural network, that would open the possibility for the use of a fixed light source and sensor for all light transmittance tests from the standard. Spectrophotometry, visible transmittance and traffic light transmittance was calculated in 45 lenses of sunglasses, used as samples for testing the methodologies. The tests included a white LED, a RGB sensor, and electronic for control and signal acquisition. Bland - Altman analysis tool was used to calculate the agreement between the method and the transmittances calculated in the spectrophotometer. Both methods, had an approximation within the deviation limit required by NBR15111. The system with the polynomial regression showed lower deviations than artificial neural networks. A larger number of samples can improve the methods in order to obtain an optimal calibration that includes all sunglasses. No meter in the market can calculate accurately all light transmittances measurements required for the sunglasses. The methodology was applied only for the visible light, while UV and infrared spectrum remains to be tested. The methodology tested presented a way for simple low-cost equipment for all light transmittance tests in sunglasses.
UNCOUPLING LAMINAR CONJUGATE HEAT TRANSFER THROUGH CHEBYSHEV POLYNOMIAL
Directory of Open Access Journals (Sweden)
ANTONIO J. BULA
2010-01-01
verificados con la solución obtenida por medio de software CFD comercial, FIDAP ®. La solución ncluyo el cálculo del coeficiente de transferencia de calor, el número de Nusselt, el número de Biot, todos tanto local como promedio. La distribución de temperatura en la interface también fue obtenida.
Energy Technology Data Exchange (ETDEWEB)
Yuste, Santos Bravo; Abad, Enrique, E-mail: santos@unex.es [Departamento de Fisica, Universidad de Extremadura, E-06071 Badajoz (Spain)
2011-02-18
We present an iterative method to obtain approximations to Bessel functions of the first kind J{sub p}(x) (p > -1) via the repeated application of an integral operator to an initial seed function f{sub 0}(x). The class of seed functions f{sub 0}(x) leading to sets of increasingly accurate approximations f{sub n}(x) is considerably large and includes any polynomial. When the operator is applied once to a polynomial of degree s, it yields a polynomial of degree s + 2, and so the iteration of this operator generates sets of increasingly better polynomial approximations of increasing degree. We focus on the set of polynomial approximations generated from the seed function f{sub 0}(x) = 1. This set of polynomials is useful not only for the computation of J{sub p}(x) but also from a physical point of view, as it describes the long-time decay modes of certain fractional diffusion and diffusion-wave problems.
Direct method for variational problems via hybrid of block-pulse and chebyshev functions
Directory of Open Access Journals (Sweden)
Razzaghi Mohsen
2000-01-01
Full Text Available A direct method for finding the solution of variational problems using a hybrid function is discussed. The hybrid functions which consist of block-pulse functions plus Chebyshev polynomials are introduced. An operational matrix of integration and the integration of the cross product of two hybrid function vectors are presented and are utilized to reduce a variational problem to the solution of an algebraic equation. Illustrative examples are included to demonstrate the validity and applicability of the technique.
Chebyshev collocation spectral lattice Boltzmann method for simulation of low-speed flows.
Hejranfar, Kazem; Hajihassanpour, Mahya
2015-01-01
In this study, the Chebyshev collocation spectral lattice Boltzmann method (CCSLBM) is developed and assessed for the computation of low-speed flows. Both steady and unsteady flows are considered here. The discrete Boltzmann equation with the Bhatnagar-Gross-Krook approximation based on the pressure distribution function is considered and the space discretization is performed by the Chebyshev collocation spectral method to achieve a highly accurate flow solver. To provide accurate unsteady solutions, the time integration of the temporal term in the lattice Boltzmann equation is made by the fourth-order Runge-Kutta scheme. To achieve numerical stability and accuracy, physical boundary conditions based on the spectral solution of the governing equations implemented on the boundaries are used. An iterative procedure is applied to provide consistent initial conditions for the distribution function and the pressure field for the simulation of unsteady flows. The main advantage of using the CCSLBM over other high-order accurate lattice Boltzmann method (LBM)-based flow solvers is the decay of the error at exponential rather than at polynomial rates. Note also that the CCSLBM applied does not need any numerical dissipation or filtering for the solution to be stable, leading to highly accurate solutions. Three two-dimensional (2D) test cases are simulated herein that are a regularized cavity, the Taylor vortex problem, and doubly periodic shear layers. The results obtained for these test cases are thoroughly compared with the analytical and available numerical results and show excellent agreement. The computational efficiency of the proposed solution methodology based on the CCSLBM is also examined by comparison with those of the standard streaming-collision (classical) LBM and two finite-difference LBM solvers. The study indicates that the CCSLBM provides more accurate and efficient solutions than these LBM solvers in terms of CPU and memory usage and an exponential
Hedayatrasa, Saeid; Bui, Tinh Quoc; Zhang, Chuanzeng; Lim, Chee Wah
2014-02-01
Numerical modeling of the Lamb wave propagation in functionally graded materials (FGMs) by a two-dimensional time-domain spectral finite element method (SpFEM) is presented. The high-order Chebyshev polynomials as approximation functions are used in the present formulation, which provides the capability to take into account the through thickness variation of the material properties. The efficiency and accuracy of the present model with one and two layers of 5th order spectral elements in modeling wave propagation in FGM plates are analyzed. Different excitation frequencies in a wide range of 28-350 kHz are investigated, and the dispersion properties obtained by the present model are verified by reference results. The through thickness wave structure of two principal Lamb modes are extracted and analyzed by the symmetry and relative amplitude of the vertical and horizontal oscillations. The differences with respect to Lamb modes generated in homogeneous plates are explained. Zero-crossing and wavelet signal processing-spectrum decomposition procedures are implemented to obtain phase and group velocities and their dispersion properties. So it is attested how this approach can be practically employed for simulation, calibration and optimization of Lamb wave based nondestructive evaluation techniques for the FGMs. The capability of modeling stress wave propagation through the thickness of an FGM specimen subjected to impact load is also investigated, which shows that the present method is highly accurate as compared with other existing reference data.
Institute of Scientific and Technical Information of China (English)
E. H. Doha; S. I. El-Soubhy
2001-01-01
Numerical Analysis,Clarendon Press,Oxford,1972.[10]Doha,E.H. and Helal,M. A. ,An Accurate Double Chebyshev Spectral Approximation for Parabolic Partial Differential Equations,J. Egypt. Math. Soc.,5 (1997),No. 1,83-101.[11]Gottlieb,D. and Orszag,S.A. ,Numerical Analysis of Spectral Methods: Theory and Applications,CBMS-NSF Regional Conf. Series in Applied Mathematics,Vol.[2]6,Society for Industrial and Applied Mathamatics,Philadelphia,PA,1977.[12]Karageorghis,A. ,Chebyshev Spectral Methods for Solving Two-Point Boundary Value Problems Arising in Heat Transfer,Comput. Methods Appl. Mech. Eng. ,70(1988),103-121.[13]Karageorghis,A. ,A Note on the Chebyshev Coefficients of the General-Order Derivative of an Infinitely Differentiable Function,J. Comput. Appl. Math.,21(1988),129-132.[14]Karageorghis,A. ,A Note on the Chebyshev Coefficients of the Moments of the General Order Derivative of an Infinitely Differentiable Function,J. Comput. Appl. Math. ,21(1988),383-386.[15]Karageorghis,A. and Phillips,T.N. ,On the Coefficients of Differentiated Expansions of UItraspherical Polynomials,ICASE Report No. 89- 65,NASA Langley Research Center,Hampton,VA,1989 and Appl. Num. Math.,9(1992),133-141.[16]Luke,Y. ,The Special Functions and Their Approximations,Vol. 1,Academic Press,New York,1969.[17]Phillips,T.N. ,On the Legendre Coefficients of a General Order Derivative of an Inifintely Differentiable Function,IMA J. Numer. Anal. ,8(1988),455-459.[18]Phillps,T.N. and Karageorghis,A. ,On the Coefficients of Integrated Exapansions of Ultraspherical Polynomials,SIAM J. Numer. Anal. ,27(1990),823-830.Manuscript Received：2000年4月27日Manuscript Revised：2001年5月15日Published：2001年9月1日
Institute of Scientific and Technical Information of China (English)
范文玎; 孙光耀; 张彬航; 陈锐; 郝丽娟
2016-01-01
燃耗计算在反应堆设计、分析研究中起着重要作用.相比于传统点燃耗算法,切比雪夫有理逼近方法(Chebyshev rational approximation method,CRAM)具有计算速度快、精度高的优点.基于超级蒙特卡罗核计算仿真软件系统SuperMC(Super Monte Carlo Simulation Program for Nuclear and Radiation Process),采用切比雪夫有理逼近方法和桶排序能量查找方法,进行了蒙特卡罗燃耗计算的初步研究与验证.通过燃料棒燃耗例题以及IAEA-ADS(International Atomic Energy Agency-Accelerator Driven Systems)国际基准题,初步验证了该燃耗计算方法的正确性,且IAEA-ADS基准题测试表明,与统一能量网格方法相比,桶排序能量查找方法在保证了计算效率的同时减少了内存开销.%Background:Burnup calculation is the key point of reactor design and analysis. It's significant to calculate the burnup situation and isotopic atom density accurately while a reactor is being designed.Purpose:Based on the Monte Carlo particle simulation code SuperMC (Super Monte Carlo Simulation Program for Nuclear and Radiation Process), this paper aimed to conduct preliminary study and verification on Monte Carlo burnup calculations. Methods:For the characteristics of accuracy, this paper adopted Chebyshev rational approximation method (CRAM) as the point-burnup algorithm. Moreover, instead of the union energy grids method, this paper adopted an energy searching method based on bucket sort algorithm, which reduced the memory overhead on the condition that the calculation efficiency is ensured.Results:By calculating the fuel rod burnup problem and the IAEA-ADS (International Atomic Energy Agency - Accelerator Driven Systems) international benchmark, the simulation results were basically consistent with Serpent and other counties' results, respectively. In addition, the bucket sort energy searching method reduced about 95% storage space compared with union energy grids method for IAEA
3-D vibration analysis of annular sector plates using the Chebyshev-Ritz method
Zhou, D.; Lo, S. H.; Cheung, Y. K.
2009-02-01
The three-dimensional free vibration of annular sector plates with various boundary conditions is studied by means of the Chebyshev-Ritz method. The analysis is based on the three-dimensional small strain linear elasticity theory. The product of Chebyshev polynomials satisfying the necessary boundary conditions is selected as admissible functions in such a way that the governing eigenvalue equation can be conveniently derived through an optimization process by the Ritz method. The boundary functions guarantee the satisfaction of the geometric boundary conditions of the plates and the Chebyshev polynomials provide the robustness for numerical calculation. The present study provides a full vibration spectrum for the thick annular sector plates, which cannot be given by the two-dimensional (2-D) theories such as the Mindlin theory. Comprehensive numerical results with high accuracy are systematically produced, which can be used as benchmark to evaluate other numerical methods. The effect of radius ratio, thickness ratio and sector angle on natural frequencies of the plates with a sector angle from 120° to 360° is discussed in detail. The three-dimensional vibration solutions for plates with a re-entrant sector angle (larger than 180°) and shallow helicoidal shells (sector angle larger than 360°) with a small helix angle are presented for the first time.
Freud, Géza
1971-01-01
Orthogonal Polynomials contains an up-to-date survey of the general theory of orthogonal polynomials. It deals with the problem of polynomials and reveals that the sequence of these polynomials forms an orthogonal system with respect to a non-negative m-distribution defined on the real numerical axis. Comprised of five chapters, the book begins with the fundamental properties of orthogonal polynomials. After discussing the momentum problem, it then explains the quadrature procedure, the convergence theory, and G. Szegő's theory. This book is useful for those who intend to use it as referenc
Wang, Zhiheng
2015-01-01
A simple multidomain Chebyshev pseudo-spectral method is developed for two-dimensional fluid flow and heat transfer over square cylinders. The incompressible Navier-Stokes equations with primitive variables are discretized in several subdomains of the computational domain. The velocities and pressure are discretized with the same order of Chebyshev polynomials, i.e., the PN-PN method. The Projection method is applied in coupling the pressure with the velocity. The present method is first validated by benchmark problems of natural convection in a square cavity. Then the method based on multidomains is applied to simulate fluid flow and heat transfer from square cylinders. The numerical results agree well with the existing results. © Taylor & Francis Group, LLC.
Xie, Jiaquan; Huang, Qingxue; Yang, Xia
2016-01-01
In this paper, we are concerned with nonlinear one-dimensional fractional convection diffusion equations. An effective approach based on Chebyshev operational matrix is constructed to obtain the numerical solution of fractional convection diffusion equations with variable coefficients. The principal characteristic of the approach is the new orthogonal functions based on Chebyshev polynomials to the fractional calculus. The corresponding fractional differential operational matrix is derived. Then the matrix with the Tau method is utilized to transform the solution of this problem into the solution of a system of linear algebraic equations. By solving the linear algebraic equations, the numerical solution is obtained. The approach is tested via examples. It is shown that the proposed algorithm yields better results. Finally, error analysis shows that the algorithm is convergent. PMID:27504247
Almost Chebyshev set with respect to bounded subsets
Institute of Scientific and Technical Information of China (English)
李冲; 王兴华
1997-01-01
The uniqueness and existence of restricted Chebyshev center with respect to arbitrary subset are investigated. The concept of almost Chebyshev sets with respect to bounded subsets is introduced. It is proved that each closed subset in a reflexive locally uniformly convex (uniformly convex, respectively) Banach space is an almost Chebyshev subset with respect to compact convex subsets (bounded convex subsets and bounded subsets, respectively).
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…
Image contrast enhancement using Chebyshev wavelet moments
Uchaev, Dm. V.; Uchaev, D. V.; Malinnikov, V. A.
2015-12-01
A new algorithm for image contrast enhancement in the Chebyshev moment transform (CMT) domain is introduced. This algorithm is based on a contrast measure that is defined as the ratio of high-frequency to zero-frequency content in the bands of CMT matrix. Our algorithm enables to enhance a large number of high-spatial-frequency coefficients, that are responsible for image details, without severely degrading low-frequency contributions. To enhance high-frequency Chebyshev coefficients we use a multifractal spectrum of scaling exponents (SEs) for Chebyshev wavelet moment (CWM) magnitudes, where CWMs are multiscale realization of Chebyshev moments (CMs). This multifractal spectrum is very well suited to extract meaningful structures on images of natural scenes, because these images have a multifractal character. Experiments with test images show some advantages of the proposed algorithm as compared to other widely used image enhancement algorithms. The main advantage of our algorithm is the following: the algorithm very well highlights image details during image contrast enhancement.
基于Chebyshev的概率公钥密码体制%Probabilistic public-key cryptosystem based on Chebyshev
Institute of Scientific and Technical Information of China (English)
程学海; 徐江峰
2013-01-01
This paper introduced the definition and the properties of Chebyshev polynomial.According to the deterministic public-key cryptosystem of Chebyshev polynomial,found that it couldn't resist chosen cipher-text attacks.Combining with the security model against chosen cipher-text attacks,this paper proposed the probabilistic public-key cryptosystem of Chebyshev polynomial.The analysis show that the proposed cryptosystem is correct.Through the result of the reduction proof,the proposed cryptosystem can resist the adaptive chosen cipher-text attacks and has the IND-CCA2 security.%介绍了Chebyshev多项式的定义和相关性质,针对确定性Chebyshev多项式公钥密码体制进行了研究,发现其不能抵抗选择密文攻击.结合抵抗选择密文攻击的安全模型,提出了基于有限域的Chebyshev多项式的概率公钥密码体制,分析结果表明该密码体制是正确的.通过归约证明,该密码体制能够抵挡适应性选择密文攻击,具有抵抗选择密文攻击的IND-CCA2安全性.
Efficient Prime Counting and the Chebyshev Primes
Directory of Open Access Journals (Sweden)
Michel Planat
2013-01-01
Full Text Available The function where is the logarithm integral and the number of primes up to is well known to be positive up to the (very large Skewes' number. Likewise, according to Robin's work, the functions and , where and are Chebyshev summatory functions, are positive if and only if Riemann hypothesis (RH holds. One introduces the jump function at primes and one investigates , , and . In particular, , and for . Besides, for any odd , an infinite set of the so-called Chebyshev primes. In the context of RH, we introduce the so-called Riemann primes as champions of the function (or of the function . Finally, we find a good prime counting function , that is found to be much better than the standard Riemann prime counting function.
Laptev V. N.; Sergeev A. E.; Sergeev E. A.
2015-01-01
The article presents the theorem of Chebyshev on the distribution of primes, considering functions that approximated prime numbers. We have also considered a new function, which is quite good for approximation of prime numbers. A review of the known results on distribution of prime numbers is given as well
Polynomially Bounded Sequences and Polynomial Sequences
Directory of Open Access Journals (Sweden)
Okazaki Hiroyuki
2015-09-01
Full Text Available In this article, we formalize polynomially bounded sequences that plays an important role in computational complexity theory. Class P is a fundamental computational complexity class that contains all polynomial-time decision problems [11], [12]. It takes polynomially bounded amount of computation time to solve polynomial-time decision problems by the deterministic Turing machine. Moreover we formalize polynomial sequences [5].
s-Numbers sequences for homogeneous polynomials
Caliskan, Erhan; Rueda, Pilar
2015-01-01
We extend the well known theory of $s$-numbers of linear operators to homogeneous polynomials defined between Banach spaces. Approximation, Kolmogorov and Gelfand numbers of polynomials are introduced and some well-known results of the linear and multilinear settings are obtained for homogeneous polynomials.
On the stability and accuracy of least squares approximations
Cohen, Albert; Leviatan, Dany
2011-01-01
We consider the problem of reconstructing an unknown function $f$ on a domain $X$ from samples of $f$ at $n$ randomly chosen points with respect to a given measure $\\rho_X$. Given a sequence of linear spaces $(V_m)_{m>0}$ with ${\\rm dim}(V_m)=m\\leq n$, we study the least squares approximations from the spaces $V_m$. It is well known that such approximations can be inaccurate when $m$ is too close to $n$, even when the samples are noiseless. Our main result provides a criterion on $m$ that describes the needed amount of regularization to ensure that the least squares method is stable and that its accuracy, measured in $L^2(X,\\rho_X)$, is comparable to the best approximation error of $f$ by elements from $V_m$. We illustrate this criterion for various approximation schemes, such as trigonometric polynomials, with $\\rho_X$ being the uniform measure, and algebraic polynomials, with $\\rho_X$ being either the uniform or Chebyshev measure. For such examples we also prove similar stability results using deterministic...
Directory of Open Access Journals (Sweden)
M. Heydari
2013-05-01
Full Text Available A new and effective direct method to determine the numerical solution of linear and nonlinear differential-algebraic equations (DAEs is proposed. The method consists of expanding the required approximate solution as the elements of Chebyshev cardinal functions. The operational matrices for the integration and product of the Chebyshev cardinal functions are presented. A general procedure for forming these matrices is given. These matrices play an important role in modelling of problems. By using these operational matrices together, a differentialalgebraic equation can be transformed to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique
A new class of three-variable orthogonal polynomials and their recurrences relations
Institute of Scientific and Technical Information of China (English)
2008-01-01
A new class of three-variable orthogonai polynomials,defined as eigenfunctions of a second order PDE operator,is studied.These polynomials are orthogonal over a curved tetrahedron region, which can be seen as a mapping from a traditional tetrahedron,and can be taken as an extension of the 2-D Steiner domain.The polynomials can be viewed as Jacobi polynomials on such a domain.Three- term relations are derived explicitly.The number of the individual terms,involved in the recurrences relations,are shown to be independent on the total degree of the polynomials.The numbers now are determined to be five and seven,with respect to two conjugate variables z,(?) and a real variable r, respectively.Three examples are discussed in details,which can be regarded as the analogues of the Chebyshev polynomials of the first and the second kinds,and Legendre polynomials.
A new class of three-variable orthogonal polynomials and their recurrences relations
Institute of Scientific and Technical Information of China (English)
SUN JiaChang
2008-01-01
A new class of three-variable orthogonal polynomials, defined as eigenfunctions of a second order PDE operator, is studied. These polynomials are orthogonal over a curved tetrahedron region,which can be seen as a mapping from a traditional tetrahedron, and can be taken as an extension of the 2-D Steiner domain. The polynomials can be viewed as Jacobi polynomials on such a domain. Threeterm relations are derived explicitly. The number of the individual terms, involved in the recurrences relations, are shown to be independent on the total degree of the polynomials. The numbers now are determined to be five and seven, with respect to two conjugate variables z, (z) and a real variable r,respectively. Three examples are discussed in details, which can be regarded as the analogues of the Chebyshev polynomials of the first and the second kinds, and Legendre polynomials.
Performance comparison of polynomial representations for optimizing optical freeform systems
Brömel, A.; Gross, H.; Ochse, D.; Lippmann, U.; Ma, C.; Zhong, Y.; Oleszko, M.
2015-09-01
Optical systems can benefit strongly from freeform surfaces, however the choice of the right representation isn`t an easy one. Classical representations like X-Y-polynomials, as well as Zernike-polynomials are often used for such systems, but should have some disadvantage regarding their orthogonality, resulting in worse convergence and reduced quality in final results compared to newer representations like the Q-polynomials by Forbes. Additionally the supported aperture is a circle, which can be a huge drawback in case of optical systems with rectangular aperture. In this case other representations like Chebyshev-or Legendre-polynomials come into focus. There are a larger number of possibilities; however the experience with these newer representations is rather limited. Therefore in this work the focus is on investigating the performance of four widely used representations in optimizing two ambitious systems with very different properties: Three-Mirror-Anastigmat and an anamorphic System. The chosen surface descriptions offer support for circular or rectangular aperture, as well as different grades of departure from rotational symmetry. The basic shapes are for example a conic or best-fit-sphere and the polynomial set is non-, spatial or slope-orthogonal. These surface representations were chosen to evaluate the impact of these aspects on the performance optimization of the two example systems. Freeform descriptions investigated here were XY-polynomials, Zernike in Fringe representation, Q-polynomials by Forbes, as well as 2-dimensional Chebyshev-polynomials. As a result recommendations for the right choice of freeform surface representations for practical issues in the optimization of optical systems can be given.
Expansions of one density via polynomials orthogonal with respect to the other
Szabłowski, Paweł J
2010-01-01
We expand Chebyshev polynomials and some of its linear combination in linear combinations of q-Hermite, Rogers and Al Salam-Chihara polynomials and vice versa. We use these expansions to obtain expansions of the some densities, including q-Normal and some related to it, in infinite series of orthogonal polynomials allowing deeper analysis, discovering new properties. On the way we find an easy proof of expansion of of Poisson-Mehler kernels for q-Hermite polynomials and also its inverse. We also formulate simple rule relating one set of orthogonal polynomials to the other given the properties of the ratio of the respective densities of measures orthogonalizing these polynomials sets.
Polynomial chaotic inflation in supergravity revisited
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Kazunori Nakayama
2014-10-01
Full Text Available We revisit a polynomial chaotic inflation model in supergravity which we proposed soon after the Planck first data release. Recently some issues have been raised in Ref. [12], concerning the validity of our polynomial chaotic inflation model. We study the inflaton dynamics in detail, and confirm that the inflaton potential is very well approximated by a polynomial potential for the parameters of our interest in any practical sense, and in particular, the spectral index and the tensor-to-scalar ratio can be estimated by single-field approximation. This justifies our analysis of the polynomial chaotic inflation in supergravity.
ON COPOSITIVE APPROXIMATION IN SOME CLASSICAL SPACES OF SEQUENCES
Institute of Scientific and Technical Information of China (English)
Aref Kamal
2003-01-01
In this paper the author writes a simple characterization for the best copositive approximation in c; the space of convergent sequences, by elements of finite dimensional Chebyshev subspaces, and shows that it is unique.
Institute of Scientific and Technical Information of China (English)
尤文坚; 梁兵; 李荫军
2013-01-01
The interpolation of sensor characteristic parameters need be processed rapidly in the field of test and control to meet the real-time requirements. A sensor characteristic parameters interpolation system based on Chebyshev algorithm is designed. The system uses function approximation with Chebyshev polynomial, and characteristic parameters curves of different accuracy can be fitted on request. The main performance indicators is calculated, and interpolation curves is printed. Moreover, the system can combine with intelligent sensor or embedded system easily. After long time application, the reliability and validity of the system has been proved. The designed system achieves good results.%在测试及控制领域,需要对传感器特性参数的拟合进行快速处理,满足系统实时性的要求.设计了一个基于切比雪夫算法的传感器特性参数拟合系统,使用切比雪夫多项式进行函数逼近,能够根据要求拟合出不同精度的特性参数曲线,并计算出主要的性能指标,打印出拟合曲线,并且能方便地与智能传感系统或嵌入式控制系统相融合.系统经过长时间的应用,证明了其可靠性及有效性,取得了良好的效果.
Factoring multivariate integral polynomials.
Lenstra, A.K.
1983-01-01
An algorithm is presented to factorize polynomials in several variables with integral coefficients that is polynomial-time in the degrees of the polynomial to be factored, for any fixed number of variables. The algorithm generalizes the algorithm presented by A. K. Lenstra et al. to factorize integral polynomials in one variable.
Directory of Open Access Journals (Sweden)
Jiri Hrivnak
2016-08-01
Full Text Available The aim of this paper is to make an explicit link between the Weyl-orbit functions and the corresponding polynomials, on the one hand, and to several other families of special functions and orthogonal polynomials on the other. The cornerstone is the connection that is made between the one-variable orbit functions of A1 and the four kinds of Chebyshev polynomials. It is shown that there exists a similar connection for the two-variable orbit functions of A2 and a specific version of two variable Jacobi polynomials. The connection with recently studied G2-polynomials is established. Formulas for connection between the four types of orbit functions of Bn or Cn and the (antisymmetric multivariate cosine and sine functions are explicitly derived.
New classes of test polynomials of polynomial algebras
Institute of Scientific and Technical Information of China (English)
冯克勤; 余解台
1999-01-01
A polynomial p in a polynomial algebra over a field is called a test polynomial if any endomorphism of the polynomial algebra that fixes p is an automorphism. some classes of new test polynomials recognizing nonlinear automorphisms of polynomial algebras are given. In the odd prime characteristic case, test polynomials recognizing non-semisimple automorphisms are also constructed.
Orthogonal Polynomials and their Applications
Dehesa, Jesús; Marcellan, Francisco; Francia, José; Vinuesa, Jaime
1988-01-01
The Segovia meeting set out to stimulate an intensive exchange of ideas between experts in the area of orthogonal polynomials and its applications, to present recent research results and to reinforce the scientific and human relations among the increasingly international community working in orthogonal polynomials. This volume contains original research papers as well as survey papers about fundamental questions in the field (Nevai, Rakhmanov & López) and its relationship with other fields such as group theory (Koornwinder), Padé approximation (Brezinski), differential equations (Krall, Littlejohn) and numerical methods (Rivlin).
BEST APPROXIMATION BY DOWNWARD SETS WITH APPLICATIONS
Institute of Scientific and Technical Information of China (English)
H.Mohebi; A. M. Rubinov
2006-01-01
We develop a theory of downward sets for a class of normed ordered spaces. We study best approximation in a normed ordered space X by elements of downward sets, and give necessary and sufficient conditions for any element of best approximation by a closed downward subset of X. We also characterize strictly downward subsets of X, and prove that a downward subset of X is strictly downward if and only if each its boundary point is Chebyshev. The results obtained are used for examination of some Chebyshev pairs (W,x), where x ∈ X and W is a closed downward subset of X.
Piecewise Extended Chebyshev Spaces: a numerical test for design
Beccari, Carolina Vittoria; Casciola, Giulio; Mazure, Marie-Laurence
2016-01-01
Given a number of Extended Chebyshev (EC) spaces on adjacent intervals, all of the same dimension, we join them via convenient connection matrices without increasing the dimension. The global space is called a Piecewise Extended Chebyshev (PEC) Space. In such a space one can count the total number of zeroes of any non-zero element, exactly as in each EC-section-space. When this number is bounded above in the global space the same way as in its section-spaces, we say that it is an Extended Che...
Polynomial selections and separation by polynomials
Wasowicz, Szymon
2008-01-01
Necessary and sufficient conditions under which two real functions defined on the real interval can be separated by a polynomial are given. An immediate consequence of the main result is the existence of the polynomial separation of convex functions of higher order. Another application is some Hyers-Ulam-stability-type result.
Cen, Xiuli; Zhao, Yulin; Liang, Haihua
2014-01-01
In this paper, we study the number of limit cycles which bifurcate from the periodic orbits of cubic polynomial vector fields of Lotka-Volterra type having a rational first integral of degree 2, under polynomial perturbations of degree $n$. The analysis is carried out by estimating the number of zeros of the corresponding Abelian integrals. Moreover, using \\emph{Chebyshev criterion}, we show that the sharp upper bound for the number of zeros of the Abelian integrals defined on each period ann...
Factoring Polynomials and Fibonacci.
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)
Branched polynomial covering maps
DEFF Research Database (Denmark)
Hansen, Vagn Lundsgaard
2002-01-01
A Weierstrass polynomial with multiple roots in certain points leads to a branched covering map. With this as the guiding example, we formally define and study the notion of a branched polynomial covering map. We shall prove that many finite covering maps are polynomial outside a discrete branch...... set. Particular studies are made of branched polynomial covering maps arising from Riemann surfaces and from knots in the 3-sphere. (C) 2001 Elsevier Science B.V. All rights reserved....
Generalized Fibonacci-Lucas Polynomials
Directory of Open Access Journals (Sweden)
Mamta Singh
2013-12-01
Full Text Available Various sequences of polynomials by the names of Fibonacci and Lucas polynomials occur in the literature over a century. The Fibonacci polynomials and Lucas polynomials are famous for possessing wonderful and amazing properties and identities. In this paper, Generalized Fibonacci-Lucas Polynomials are introduced and defined by the recurrence relation with and . Some basic identities of Generalized Fibonacci-Lucas Polynomials are obtained by method of generating function. Keywords: Fibonacci polynomials, Lucas polynomials, Generalized Fibonacci polynomials, Generalized Fibonacci-Lucas polynomials.
Thermodynamic characterization of networks using graph polynomials
Ye, Cheng; Peron, Thomas K DM; Silva, Filipi N; Rodrigues, Francisco A; Costa, Luciano da F; Torsello, Andrea; Hancock, Edwin R
2015-01-01
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...
CHEBYSHEV ACCELERATION TECHNIQUE FOR SOLVING FUZZY LINEAR SYSTEM
Directory of Open Access Journals (Sweden)
S.H. Nasseri
2009-10-01
Full Text Available In this paper, Chebyshev acceleration technique is used to solve the fuzzy linear system (FLS. This method is discussed in details and followed by summary of some other acceleration techniques. Moreover, we show that in some situations that the methods such as Jacobi, Gauss-Sidel, SOR and conjugate gradient is divergent, our proposed method is applicable and the acquired results are illustrated by some numerical examples.
Chebyshev's bias for composite numbers with restricted prime divisors
Pieter Moree
2001-01-01
Let P(x,d,a) denote the number of primes p<=x with p=a(mod d). Chebyshev's bias is the phenomenon that `more often' P(x;d,n)>P(x;d,r), than the other way around, where n is a quadratic non-residue mod d and r is a quadratic residue mod d. If P(x;d,n)>=P(x;d,r) for every x up to some large number, th
Chebyshev matrix product state approach for spectral functions
Holzner, Andreas; Weichselbaum, Andreas; McCulloch, Ian P.; Schollwöck, Ulrich; von Delft, Jan
2011-05-01
We show that recursively generated Chebyshev expansions offer numerically efficient representations for calculating zero-temperature spectral functions of one-dimensional lattice models using matrix product state (MPS) methods. The main features of this Chebyshev matrix product state (CheMPS) approach are as follows: (i) it achieves uniform resolution over the spectral function’s entire spectral width; (ii) it can exploit the fact that the latter can be much smaller than the model’s many-body bandwidth; (iii) it offers a well-controlled broadening scheme that allows finite-size effects to be either resolved or smeared out, as desired; (iv) it is based on using MPS tools to recursively calculate a succession of Chebyshev vectors |tn>, (v) the entanglement entropies of which were found to remain bounded with increasing recursion order n for all cases analyzed here; and (vi) it distributes the total entanglement entropy that accumulates with increasing n over the set of Chebyshev vectors |tn>, which need not be combined into a single vector. In this way, the growth in entanglement entropy that usually limits density matrix renormalization group (DMRG) approaches is packaged into conveniently manageable units. We present zero-temperature CheMPS results for the structure factor of spin-(1)/(2) antiferromagnetic Heisenberg chains and perform a detailed finite-size analysis. Making comparisons to three benchmark methods, we find that CheMPS (a) yields results comparable in quality to those of correction-vector DMRG, at dramatically reduced numerical cost; (b) agrees well with Bethe ansatz results for an infinite system, within the limitations expected for numerics on finite systems; and (c) can also be applied in the time domain, where it has potential to serve as a viable alternative to time-dependent DMRG (in particular, at finite temperatures). Finally, we present a detailed error analysis of CheMPS for the case of the noninteracting resonant level model.
Energy Technology Data Exchange (ETDEWEB)
Spata, Michael [Old Dominion Univ., Norfolk, VA (United States)
2012-08-01
An experiment was conducted at Jefferson Lab's Continuous Electron Beam Accelerator Facility to develop a beam-based technique for characterizing the extent of the nonlinearity of the magnetic fields of a beam transport system. Horizontally and vertically oriented pairs of air-core kicker magnets were simultaneously driven at two different frequencies to provide a time-dependent transverse modulation of the beam orbit relative to the unperturbed reference orbit. Fourier decomposition of the position data at eight different points along the beamline was then used to measure the amplitude of these frequencies. For a purely linear transport system one expects to find solely the frequencies that were applied to the kickers with amplitudes that depend on the phase advance of the lattice. In the presence of nonlinear fields one expects to also find harmonics of the driving frequencies that depend on the order of the nonlinearity. Chebyshev polynomials and their unique properties allow one to directly quantify the magnitude of the nonlinearity with the minimum error. A calibration standard was developed using one of the sextupole magnets in a CEBAF beamline. The technique was then applied to a pair of Arc 1 dipoles and then to the magnets in the Transport Recombiner beamline to measure their multipole content as a function of transverse position within the magnets.
$BKW$-Operators for Chebyshev Systems
Ishii, Takashi; Izuchi, Keiji
1999-01-01
This paper is concerned with Korovkin type approximation theorems. We characterize $BKW$-operators on the Banach space of real valued continuous functions on the unit interval for the test functions $\\{1,t,t^2,t^3\\}$. It is also investigated when subtraction of composition operators are $BKW$-operators for $\\{1,t,t^2,t^3,t^4\\}$.
Some Undecidable Problems on Approximability of NP Optimization Problems
Institute of Scientific and Technical Information of China (English)
黄雄
1996-01-01
In this paper some undecidable problems on approximability of NP optimization problems are investigated.In particular,the following problems are all undecidable:(1) Given an NP optimization problem,is it approximable in polynomial time?(2)For any polynomial-time computable function r(n),given a polynomial time approximable NP optimization problem,has it a polynomial-time approximation algorithm with approximation performance ratio r(n) (r(n)-approximable)?(3)For any polynomial-time computable functions r(n),r'(n),where r'(n)
International Nuclear Information System (INIS)
The criticality problem is studied based on one-speed time-dependent neutron transport theory, for a uniform and finite slab, using the Marshak boundary condition. The time-dependent neutron transport equation is reduced to a stationary equation. The variation of the critical thickness of the time-dependent system is investigated by using the linear anisotropic scattering kernel together with the combination of forward and backward scattering. Numerical calculations for various combinations of the scattering parameters and selected values of the time decay constant and the reflection coefficient are performed by using the Chebyshev polynomials approximation method. The results are compared with those previously obtained by other methods which are available in the literature.
Quantum Grothendieck polynomials
Kirillov, Anatol N.
1996-01-01
We study the algebraic aspects of (small) quantum equivariant $K$-theory of flag manifold. Lascoux-Sch\\"utzenberger's type formula for quantum double and quantum double dual Grothendieck polynomials and the quantum Cauchy identity for quantum Grothendieck polynomials are obtained.
Polynomial Graphs and Symmetry
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…
Jack polynomials in superspace
Desrosiers, P; Mathieu, P
2003-01-01
This work initiates the study of {\\it orthogonal} symmetric polynomials in superspace. Here we present two approaches leading to a family of orthogonal polynomials in superspace that generalize the Jack polynomials. The first approach relies on previous work by the authors in which eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland Hamiltonian were constructed. Orthogonal eigenfunctions are now obtained by diagonalizing the first nontrivial element of a bosonic tower of commuting conserved charges not containing this Hamiltonian. Quite remarkably, the expansion coefficients of these orthogonal eigenfunctions in the supermonomial basis are stable with respect to the number of variables. The second and more direct approach amounts to symmetrize products of non-symmetric Jack polynomials with monomials in the fermionic variables. This time, the orthogonality is inherited from the orthogonality of the non-symmetric Jack polynomials, and the value of the norm is given exp...
Nonnegativity of uncertain polynomials
Directory of Open Access Journals (Sweden)
iljak Dragoslav D.
1998-01-01
Full Text Available The purpose of this paper is to derive tests for robust nonnegativity of scalar and matrix polynomials, which are algebraic, recursive, and can be completed in finite number of steps. Polytopic families of polynomials are considered with various characterizations of parameter uncertainty including affine, multilinear, and polynomic structures. The zero exclusion condition for polynomial positivity is also proposed for general parameter dependencies. By reformulating the robust stability problem of complex polynomials as positivity of real polynomials, we obtain new sufficient conditions for robust stability involving multilinear structures, which can be tested using only real arithmetic. The obtained results are applied to robust matrix factorization, strict positive realness, and absolute stability of multivariable systems involving parameter dependent transfer function matrices.
An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method
Energy Technology Data Exchange (ETDEWEB)
Belendez, A., E-mail: a.belendez@ua.e [Departamento de Fisica, Ingenieria de Sistemas y Teoria de la Senal, Universidad de Alicante, Apartado 99, E-03080 Alicante (Spain); Mendez, D.I. [Departamento de Fisica, Ingenieria de Sistemas y Teoria de la Senal, Universidad de Alicante, Apartado 99, E-03080 Alicante (Spain); Fernandez, E. [Departamento de Optica, Farmacologia y Anatomia, Universidad de Alicante, Apartado 99, E-03080 Alicante (Spain); Marini, S. [Departamento de Fisica, Ingenieria de Sistemas y Teoria de la Senal, Universidad de Alicante, Apartado 99, E-03080 Alicante (Spain); Pascual, I. [Departamento de Optica, Farmacologia y Anatomia, Universidad de Alicante, Apartado 99, E-03080 Alicante (Spain)
2009-08-03
The nonlinear oscillations of a Duffing-harmonic oscillator are investigated by an approximated method based on the 'cubication' of the initial nonlinear differential equation. In this cubication method the restoring force is expanded in Chebyshev polynomials and the original nonlinear differential equation is approximated by a Duffing equation in which the coefficients for the linear and cubic terms depend on the initial amplitude, A. The replacement of the original nonlinear equation by an approximate Duffing equation allows us to obtain explicit approximate formulas for the frequency and the solution as a function of the complete elliptic integral of the first kind and the Jacobi elliptic function, respectively. These explicit formulas are valid for all values of the initial amplitude and we conclude this cubication method works very well for the whole range of initial amplitudes. Excellent agreement of the approximate frequencies and periodic solutions with the exact ones is demonstrated and discussed and the relative error for the approximate frequency is as low as 0.071%. Unlike other approximate methods applied to this oscillator, which are not capable to reproduce exactly the behaviour of the approximate frequency when A tends to zero, the cubication method used in this Letter predicts exactly the behaviour of the approximate frequency not only when A tends to infinity, but also when A tends to zero. Finally, a closed-form expression for the approximate frequency is obtained in terms of elementary functions. To do this, the relationship between the complete elliptic integral of the first kind and the arithmetic-geometric mean as well as Legendre's formula to approximately obtain this mean are used.
Simulation of electrically driven jet using Chebyshev collocation method
Institute of Scientific and Technical Information of China (English)
无
2011-01-01
The model of electrically driven jet is governed by a series of quasi 1D dimensionless partial differential equations(PDEs).Following the method of lines,the Chebyshev collocation method is employed to discretize the PDEs and obtain a system of differential-algebraic equations(DAEs).By differentiating constrains in DAEs twice,the system is transformed into a set of ordinary differential equations(ODEs) with invariants.Then the implicit differential equations solver "ddaskr" is used to solve the ODEs and ...
Best polynomial degree reduction on q-lattices with applications to q-orthogonal polynomials
Ait-Haddou, Rachid
2015-06-07
We show that a weighted least squares approximation of q-Bézier coefficients provides the best polynomial degree reduction in the q-L
Mironov, A; Morozov, A
2015-01-01
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, 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 representation. Properties of universal polynomials and applications of these results are discussed.
Yu, Jiun-Hung
2012-01-01
Polynomial remainder codes are a large class of codes derived from the Chinese remainder theorem that includes Reed-Solomon codes as a special case. In this paper, we revisit these codes and study them more carefully than in previous work. We explicitly allow the code symbols to be polynomials of different degrees, which leads to two different notions of weight and distance. Algebraic decoding is studied in detail. If the moduli are not irreducible, the notion of an error locator polynomial is replaced by an error factor polynomial. We then obtain a collection of gcd-based decoding algorithms, some of which are not quite standard even when specialized to Reed-Solomon codes.
Weighted approximation with varying weight
Totik, Vilmos
1994-01-01
A new construction is given for approximating a logarithmic potential by a discrete one. This yields a new approach to approximation with weighted polynomials of the form w"n"(" "= uppercase)P"n"(" "= uppercase). The new technique settles several open problems, and it leads to a simple proof for the strong asymptotics on some L p(uppercase) extremal problems on the real line with exponential weights, which, for the case p=2, are equivalent to power- type asymptotics for the leading coefficients of the corresponding orthogonal polynomials. The method is also modified toyield (in a sense) uniformly good approximation on the whole support. This allows one to deduce strong asymptotics in some L p(uppercase) extremal problems with varying weights. Applications are given, relating to fast decreasing polynomials, asymptotic behavior of orthogonal polynomials and multipoint Pade approximation. The approach is potential-theoretic, but the text is self-contained.
Additive and polynomial representations
Krantz, David H; Suppes, Patrick
1971-01-01
Additive and Polynomial Representations deals with major representation theorems in which the qualitative structure is reflected as some polynomial function of one or more numerical functions defined on the basic entities. Examples are additive expressions of a single measure (such as the probability of disjoint events being the sum of their probabilities), and additive expressions of two measures (such as the logarithm of momentum being the sum of log mass and log velocity terms). The book describes the three basic procedures of fundamental measurement as the mathematical pivot, as the utiliz
Approximate calculation of the conditional Wiener integral in quantum mechanics problem
International Nuclear Information System (INIS)
Application of the functional integration method to obtain some characteristics of quantum mechanics system in the Euclidean formulation of theory is considered. The conditional Wiener integrals are calculated using our approximate formulas, which are exact for the functional polynomials of certain degree. The use of the method is demonstrated taking the anharmonic oscillator with Hsub(g)=1/2(psup(2)+xsup(2))+gxsup(4) and Hsup(f)=1/2(psup(2)+xsup(2))+1/2(xsup(2)-fsup(2))sup(2) as an example. The E0, E1 energies of the ground and first excited states of this system, propagator G(r)= and wave function squared of the ground state |phi0(x)|2 are calculated. The evaluation of the integrals is performed using the Gauss and Chebyshev quadrature formulas. The comparison of our numerical results with the values obtained by other authors using both Monte Carlo method on the lattice and approximation of paths in the Feynman integral is presented. This comparison demonstrates a higher efficiency of the method used
STABILITY OF SWITCHED POLYNOMIAL SYSTEMS
Institute of Scientific and Technical Information of China (English)
Zhiqiang LI; Yupeng QIAO; Hongsheng QI; Daizhan CHENG
2008-01-01
This paper investigates the stability of (switched) polynomial systems. Using semi-tensor product of matrices, the paper develops two tools for testing the stability of a (switched) polynomial system. One is to convert a product of multi-variable polynomials into a canonical form, and the other is an easily verifiable sufficient condition to justify whether a multi-variable polynomial is positive definite. Using these two tools, the authors construct a polynomial function as a candidate Lyapunov function and via testing its derivative the authors provide some sufficient conditions for the global stability of polynomial systems.
Design and Use of a Learning Object for Finding Complex Polynomial Roots
Benitez, Julio; Gimenez, Marcos H.; Hueso, Jose L.; Martinez, Eulalia; Riera, Jaime
2013-01-01
Complex numbers are essential in many fields of engineering, but students often fail to have a natural insight of them. We present a learning object for the study of complex polynomials that graphically shows that any complex polynomials has a root and, furthermore, is useful to find the approximate roots of a complex polynomial. Moreover, we…
Kreso, Dijana; Tichy, Robert F.
2015-01-01
Starting from Ritt's classical theorems, we give a survey of results in functional decomposition of polynomials and of applications in Diophantine equations. This includes sufficient conditions for the indecomposability of polynomials, the study of decompositions of lacunary polynomials and the finiteness criterion for the equations of type f(x) = g(y).
Calculators and Polynomial Evaluation.
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.…
Hetyei, Gábor
2010-01-01
We introduce the short toric polynomial associated to a graded Eulerian poset. This polynomial contains the same information as the two toric polynomials introduced by Stanley, but allows different algebraic manipulations. The intertwined recurrence defining Stanley's toric polynomials may be replaced by a single recurrence, in which the degree of the discarded terms is independent of the rank. A short toric variant of the formula by Bayer and Ehrenborg, expressing the toric $h$-vector in terms of the $cd$-index, may be stated in a rank-independent form, and it may be shown using weighted lattice path enumeration and the reflection principle. We use our techniques to derive a formula expressing the toric $h$-vector of a dual simplicial Eulerian poset in terms of its $f$-vector. This formula implies Gessel's formula for the toric $h$-vector of a cube, and may be used to prove that the nonnegativity of the toric $h$-vector of a simple polytope is a consequence of the Generalized Lower Bound Theorem holding for ...
JACOBI POLYNOMIALS USED TO INVERT THE LAPLACE TRANSFORM
Institute of Scientific and Technical Information of China (English)
A.Al-Shuaibi; F.Al-Rawjih
2004-01-01
Given the Laplace transform F(s) of a function f(t), we develop a new algorithm to find an approximation to f(t) by the use of the classical Jacobi polynomials. The main contribution of our work is the development of a new and very effective method to determine the coefficients in the finite series expansion that approximation f(t) in terms of Jacobi polynomials. Some numerical examples are illustrated.
Computing the Alexander Polynomial Numerically
DEFF Research Database (Denmark)
Hansen, Mikael Sonne
2006-01-01
Explains how to construct the Alexander Matrix and how this can be used to compute the Alexander polynomial numerically.......Explains how to construct the Alexander Matrix and how this can be used to compute the Alexander polynomial numerically....
ON PROPERTIES OF DIFFERENCE POLYNOMIALS
Institute of Scientific and Technical Information of China (English)
Chen Zongxuan; Huang Zhibo; Zheng Xiumin
2011-01-01
We study the value distribution of difference polynomials of meromorphic functions, and extend classical theorems of Tumura-Clunie type to difference polynomials. We also consider the value distribution of f(z)f(z+c).
Chromatic polynomials for simplicial complexes
DEFF Research Database (Denmark)
Møller, Jesper Michael; Nord, Gesche
2016-01-01
In this note we consider s s -chromatic polynomials for finite simplicial complexes. When s=1 s=1 , the 1 1 -chromatic polynomial is just the usual graph chromatic polynomial of the 1 1 -skeleton. In general, the s s -chromatic polynomial depends on the s s -skeleton and its value at r r is the n...
Mapped Chebyshev pseudospectral method to study multiple scale phenomena
alexandrescu, Adrian; Salgueiro, Jose R; Perez-Garcia, Victor M
2007-01-01
In the framework of mapped pseudospectral methods, we introduce a new polynomial-type mapping function in order to describe accurately the dynamics of systems developing almost singular structures. Using error criteria related to the spectral interpolation error, the new polynomial-type mapping is compared against previously proposed mappings for the study of collapse and shock wave phenomena. As a physical application, we study the dynamics of two coupled beams, described by coupled nonlinear Schr\\"odinger equations and modeling beam propagation in an atomic coherent media, whose spatial sizes differs up to several orders of magnitude. It is demonstrated, also by numerical simulations, that the accuracy properties of the new polynomial-type mapping outperforms in orders of magnitude the ones of the other studied mapping functions.
Fast and High-Quality Bilateral Filtering Using Gauss-Chebyshev Approximation
Ghosh, Sanjay; Chaudhury, Kunal N.
2016-01-01
The bilateral filter is an edge-preserving smoother that has diverse applications in image processing, computer vision, computer graphics, and computational photography. The filter uses a spatial kernel along with a range kernel to perform edge-preserving smoothing. In this paper, we consider the Gaussian bilateral filter where both the kernels are Gaussian. A direct implementation of the Gaussian bilateral filter requires $O(\\sigma_s^2)$ operations per pixel, where $\\sigma_s$ is the standard...
MHD Falkner-Skan flow of Maxwell fluid by rational Chebyshev collocation method
Institute of Scientific and Technical Information of China (English)
S. ABBASBANDY; T. HAYAT; H. R. GHEHSAREH; A. ALSAEDI
2013-01-01
The magnetohydrodynamics (MHD) Falkner-Skan flow of the Maxwell fluid is studied. Suitable transform reduces the partial differential equation into a nonlinear three order boundary value problem over a semi-infinite interval. An eﬃcient approach based on the rational Chebyshev collocation method is performed to find the solution to the proposed boundary value problem. The rational Chebyshev collocation method is equipped with the orthogonal rational Chebyshev function which solves the problem on the semi-infinite domain without truncating it to a finite domain. The obtained results are presented through the illustrative graphs and tables which demonstrate the affectivity, stability, and convergence of the rational Chebyshev collocation method. To check the accuracy of the obtained results, a numerical method is applied for solving the problem. The variations of various embedded parameters into the problem are examined.
Hubbell rectangular source integral calculation using a fast Chebyshev wavelets method.
Manai, K; Belkadhi, K
2016-07-01
An integration method based on Chebyshev wavelets is presented and used to calculate the Hubbell rectangular source integral. A study of the convergence and the accuracy of the method was carried out by comparing it to previous studies. PMID:27152913
AN ACCURATE SOLUTION OF THE POISSON EQUATION BY THE FINITE DIFFERENCE-CHEBYSHEV-TAU METHOD
Institute of Scientific and Technical Information of China (English)
Hani I. Siyyam
2001-01-01
A new finite difference-Chebyshev-Tau method for the solution of the twodimensional Poisson equation is presented. Some of the numerical results are also presented which indicate that the method is satisfactory and compatible to other methods.
Kuipers, J.
2012-06-01
New features of the symbolic algebra package Form 4 are discussed. Most importantly, these features include polynomial factorization and polynomial gcd computation. Examples of their use are shown. One of them is an exact version of Mincer which gives answers in terms of rational polynomials and 5 master integrals.
R.J. Stroeker (Roel)
2002-01-01
textabstractA Q-derived polynomial is a univariate polynomial, defined over the rationals, with the property that its zeros, and those of all its derivatives are rational numbers. There is a conjecture that says that Q-derived polynomials of degree 4 with distinct roots for themselves and all their
Thermodynamic characterization of networks using graph polynomials
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.
On the Relation between Composite Right-/Left-Handed Transmission Lines and Chebyshev Filters
Directory of Open Access Journals (Sweden)
Changjun Liu
2009-01-01
Full Text Available Composite right-/left-handed (CRLH transmission lines have gained great interest in the microwave community. In practical applications, such CRLH sections realized by series and shunt resonators have a finite length. Starting from the observation that a high-order Chebyshev filter also exhibits a periodic central section of very similar structure, the relations between finite length CRHL transmission lines and Chebyshev filters are discussed in this paper. It is shown that a finite length CRLH transmission line in the balanced case is equivalent to the central part of a low-ripple high-order Chebyshev band-pass filter, and a dual-CRLH transmission line in the balanced case is equivalent to a low-ripple high-order Chebyshev band-stop filter. The nonperiodic end sections of a Chebyshev filter can be regarded as matching sections, thus leading to an even better amplitude and phase response. It is also shown that, equally to a CRHL transmission line, a Chebyshev filter exhibits negative phase velocity in part of its passband. As a consequence, an improved behavior of finite length CRLH transmission lines may be achieved adding matching sections based on filter theory; this is demonstrated by a simulation example.
Complex Polynomial Vector Fields
DEFF Research Database (Denmark)
Dias, Kealey
The two branches of dynamical systems, continuous and discrete, correspond to the study of differential equations (vector fields) and iteration of mappings respectively. In holomorphic dynamics, the systems studied are restricted to those described by holomorphic (complex analytic) functions...... vector fields. Since the class of complex polynomial vector fields in the plane is natural to consider, it is remarkable that its study has only begun very recently. There are numerous fundamental questions that are still open, both in the general classification of these vector fields, the decomposition...... of parameter spaces into structurally stable domains, and a description of the bifurcations. For this reason, the talk will focus on these questions for complex polynomial vector fields....
A Characterization of Polynomials
DEFF Research Database (Denmark)
Andersen, Kurt Munk
1996-01-01
Given the problem:which functions f(x) are characterized by a relation of the form:f[x1,x2,...,xn]=h(x1+x2+...+xn), where n>1 and h(x) is a given function? Here f[x1,x2,...,xn] denotes the divided difference on n points x1,x2,...,xn of the function f(x).The answer is: f(x) is a polynomial of degree...
Some discrete multiple orthogonal polynomials
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.
Oblivious Polynomial Evaluation
Institute of Scientific and Technical Information of China (English)
Hong-Da Li; Dong-Yao Ji; Deng-Guo Feng; Bao Li
2004-01-01
The problem of two-party oblivious polynomial evaluation(OPE)is studied,where one party(Alice)has a polynomial P(x)and the other party(Bob)with an input x wants to learn P(x)in such an oblivious way that Bob obtains P(x)without learning any additional information about P except what is implied by P(x)and Alice does not know Bob's input x.The former OPE protocols are based on an intractability assumption except for OT protocols.In fact,evaluating P(x)is equivalent to computing the product of the coefficient vectors(a0,...,an)and(1,...,xn).Using this idea,an efficient scale product protocol of two vectors is proposed first and then two OPE protocols are presented which do not need any other cryptographic assumption except for OT protocol.Compared with the existing OPE protocol,another characteristic of the proposed protocols is the degree of the polynomial is private.Another OPE protocol works in case of existence of untrusted third party.
Polynomial Learning of Distribution Families
Belkin, Mikhail
2010-01-01
The question of polynomial learnability of probability distributions, particularly Gaussian mixture distributions, has recently received significant attention in theoretical computer science and machine learning. However, despite major progress, the general question of polynomial learnability of Gaussian mixture distributions still remained open. The current work resolves the question of polynomial learnability for Gaussian mixtures in high dimension with an arbitrary fixed number of components. The result on learning Gaussian mixtures relies on an analysis of distributions belonging to what we call "polynomial families" in low dimension. These families are characterized by their moments being polynomial in parameters and include almost all common probability distributions as well as their mixtures and products. Using tools from real algebraic geometry, we show that parameters of any distribution belonging to such a family can be learned in polynomial time and using a polynomial number of sample points. The r...
The Medusa Algorithm for Polynomial Matings
DEFF Research Database (Denmark)
Boyd, Suzanne Hruska; Henriksen, Christian
2012-01-01
The Medusa algorithm takes as input two postcritically finite quadratic polynomials and outputs the quadratic rational map which is the mating of the two polynomials (if it exists). Specifically, the output is a sequence of approximations for the parameters of the rational map, as well as an image...... of its Julia set. Whether these approximations converge is answered using Thurston's topological characterization of rational maps. This algorithm was designed by John Hamal Hubbard, and implemented in 1998 by Christian Henriksen and REU students David Farris and Kuon Ju Liu. In this paper we describe...... the algorithm and its implementation, discuss some output from the program (including many pictures) and related questions. Specifically, we include images and a discussion for some shared matings, Lattès examples, and tuning sequences of matings....
Approximate Implicitization Using Linear Algebra
Directory of Open Access Journals (Sweden)
Oliver J. D. Barrowclough
2012-01-01
Full Text Available We consider a family of algorithms for approximate implicitization of rational parametric curves and surfaces. The main approximation tool in all of the approaches is the singular value decomposition, and they are therefore well suited to floating-point implementation in computer-aided geometric design (CAGD systems. We unify the approaches under the names of commonly known polynomial basis functions and consider various theoretical and practical aspects of the algorithms. We offer new methods for a least squares approach to approximate implicitization using orthogonal polynomials, which tend to be faster and more numerically stable than some existing algorithms. We propose several simple propositions relating the properties of the polynomial bases to their implicit approximation properties.
A Deterministic and Polynomial Modified Perceptron Algorithm
Directory of Open Access Journals (Sweden)
Olof Barr
2006-01-01
Full Text Available We construct a modified perceptron algorithm that is deterministic, polynomial and also as fast as previous known algorithms. The algorithm runs in time O(mn3lognlog(1/ρ, where m is the number of examples, n the number of dimensions and ρ is approximately the size of the margin. We also construct a non-deterministic modified perceptron algorithm running in timeO(mn2lognlog(1/ρ.
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.
Symmetric functions and Hall polynomials
MacDonald, Ian Grant
1998-01-01
This reissued classic text is the acclaimed second edition of Professor Ian Macdonald's groundbreaking monograph on symmetric functions and Hall polynomials. The first edition was published in 1979, before being significantly expanded into the present edition in 1995. This text is widely regarded as the best source of information on Hall polynomials and what have come to be known as Macdonald polynomials, central to a number of key developments in mathematics and mathematical physics in the 21st century Macdonald polynomials gave rise to the subject of double affine Hecke algebras (or Cherednik algebras) important in representation theory. String theorists use Macdonald polynomials to attack the so-called AGT conjectures. Macdonald polynomials have been recently used to construct knot invariants. They are also a central tool for a theory of integrable stochastic models that have found a number of applications in probability, such as random matrices, directed polymers in random media, driven lattice gases, and...
The Tutte polynomial of some self-similar graphs
Donno, Alfredo
2010-01-01
We study the Tutte polynomial of two infinite families of finite graphs: the Sierpi\\'{n}ski graphs, which are finite approximations of the well-known Sierpi\\'{n}ski gasket, and the Schreier graphs of the Hanoi Towers group $H^{(3)}$ acting on the rooted ternary tree. For both of them, we recursively describe the Tutte polynomial and we compute several special evaluations of it, giving interesting results about the combinatorial structure of these graphs.
Modeling Microwave Structures in Time Domain Using Laguerre Polynomials
Z. Raida; Lacik, J.
2006-01-01
The paper is focused on time domain modeling of microwave structures by the method of moments. Two alternative schemes with weighted Laguerre polynomials are presented. Thanks to their properties, these schemes are free of late time oscillations. Further, the paper is aimed to effective and accurate evaluation of Green's functions integrals within these schemes. For this evaluation, a first- and second-order polynomial approximation is developed. The last part of the paper deals with mode...
Witt Rings and Permutation Polynomials
Institute of Scientific and Technical Information of China (English)
Qifan Zhang
2005-01-01
Let p be a prime number. In this paper, the author sets up a canonical correspondence between polynomial functions over Z/p2Z and 3-tuples of polynomial functions over Z/pZ. Based on this correspondence, he proves and reproves some fundamental results on permutation polynomials mod pl. The main new result is the characterization of strong orthogonal systems over Z/p1Z.
Deformed Mittag-Leffler Polynomials
Miomir S. Stankovic; Marinkovic, Sladjana D.; Rajkovic, Predrag M.
2010-01-01
The starting point of this paper are the Mittag-Leffler polynomials introduced by H. Bateman [1]. Based on generalized integer powers of real numbers and deformed exponential function, we introduce deformed Mittag-Leffler polynomials defined by appropriate generating function. We investigate their recurrence relations, differential properties and orthogonality. Since they have all zeros on imaginary axes, we also consider real polynomials with real zeros associated to them.
Polynomial Regression on Riemannian Manifolds
Hinkle, Jacob; Fletcher, P Thomas; Joshi, Sarang
2012-01-01
In this paper we develop the theory of parametric polynomial regression in Riemannian manifolds and Lie groups. We show application of Riemannian polynomial regression to shape analysis in Kendall shape space. Results are presented, showing the power of polynomial regression on the classic rat skull growth data of Bookstein as well as the analysis of the shape changes associated with aging of the corpus callosum from the OASIS Alzheimer's study.
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.
Derivations and identities for Kravchuk polynomials
Bedratyuk, Leonid
2012-01-01
We introduce the notion of Kravchuk derivations of the polynomial algebra. We prove that any element of the kernel of the derivation gives a polynomial identity satisfied by the Kravchuk polynomials. Also, we prove that any kernel element of the basic Weitzenb\\"ok derivations yields a polynomial identity satisfied by the Kravchuk polynomials. We describe the corresponding intertwining maps.
Chebyshev blossoming in Müntz spaces: Toward shaping with Young diagrams
Ait-Haddou, Rachid
2013-08-01
The notion of a blossom in extended Chebyshev spaces offers adequate generalizations and extra-utilities to the tools for free-form design schemes. Unfortunately, such advantages are often overshadowed by the complexity of the resulting algorithms. In this work, we show that for the case of Müntz spaces with integer exponents, the notion of a Chebyshev blossom leads to elegant algorithms whose complexities are embedded in the combinatorics of Schur functions. We express the blossom and the pseudo-affinity property in Müntz spaces in terms of Schur functions. We derive an explicit expression for the Chebyshev-Bernstein basis via an inductive argument on nested Müntz spaces. We also reveal a simple algorithm for dimension elevation. Free-form design schemes in Müntz spaces with Young diagrams as shape parameters are discussed. © 2013 Elsevier Ltd. All rights reserved.
Lai, Hong; Orgun, Mehmet A.; Pieprzyk, Josef; Li, Jing; Luo, Mingxing; Xiao, Jinghua; Xiao, Fuyuan
2016-08-01
We propose an approach that achieves high-capacity quantum key distribution using Chebyshev-map values corresponding to Lucas numbers coding. In particular, we encode a key with the Chebyshev-map values corresponding to Lucas numbers and then use k-Chebyshev maps to achieve consecutive and flexible key expansion and apply the pre-shared classical information between Alice and Bob and fountain codes for privacy amplification to solve the security of the exchange of classical information via the classical channel. Consequently, our high-capacity protocol does not have the limitations imposed by orbital angular momentum and down-conversion bandwidths, and it meets the requirements for longer distances and lower error rates simultaneously.
Design of Non-Uniform Linear Antenna Arrays Using DolphChebyshev and Binomial Methods
Directory of Open Access Journals (Sweden)
Jean-François D. Essiben
2015-08-01
Full Text Available This paper explores the analytical methods of synthesizing linear antenna arrays. The synthesis employed is based on non-uniform methods. In particular, the Dolph-Chebyshev and binomial methods are used, so as to improve the directivity of the array and to reduce the level of the secondary lobes by adjusting the geometrical and electric parameters of the array. The radiation patterns, the directivity, and the array factors of the uniform and the non-uniform methods are presented. It is shown that the Chebyshev arrays have better directivity than binomial arrays for the same number of elements and separation distance, while binomial arrays have very low side lobes compared with Chebyshev and uniform excitation arrays. Finally, numerical results of both methods are analyzed and compared.
Parallel Construction of Irreducible Polynomials
DEFF Research Database (Denmark)
Frandsen, Gudmund Skovbjerg
Let arithmetic pseudo-NC^k denote the problems that can be solved by log space uniform arithmetic circuits over the finite prime field GF(p) of depth O(log^k (n + p)) and size polynomial in (n + p). We show that the problem of constructing an irreducible polynomial of specified degree over GF(p) ...
Polynomial Regressions and Nonsense Inference
DEFF Research Database (Denmark)
Ventosa-Santaulària, Daniel; Rodríguez-Caballero, Carlos Vladimir
behavior. We extend Phillips’ (1986) results by proving an inference drawn from polynomial specifications, under stochastic nonstationarity, is misleading unless the variables cointegrate. We use a generalized polynomial specification as a vehicle to study its asymptotic and finite-sample properties. Our...
Complex Polynomial Vector Fields
DEFF Research Database (Denmark)
The two branches of dynamical systems, continuous and discrete, correspond to the study of differential equations (vector fields) and iteration of mappings respectively. In holomorphic dynamics, the systems studied are restricted to those described by holomorphic (complex analytic) functions...... or meromorphic (allowing poles as singularities) functions. There already exists a well-developed theory for iterative holomorphic dynamical systems, and successful relations found between iteration theory and flows of vector fields have been one of the main motivations for the recent interest in holomorphic...... vector fields. Since the class of complex polynomial vector fields in the plane is natural to consider, it is remarkable that its study has only begun very recently. There are numerous fundamental questions that are still open, both in the general classification of these vector fields, the decomposition...
CHEBYSHEV WEIGHTED NORM LEAST-SQUARES SPECTRAL METHODS FOR THE ELLIPTIC PROBLEM
Institute of Scientific and Technical Information of China (English)
Sang Dong Kim; Byeong Chun Shin
2006-01-01
We develop and analyze a first-order system least-squares spectral method for the second-order elliptic boundary value problem with variable coefficients. We first analyze the Chebyshev weighted norm least-squares functional defined by the sum of the L2w-and H-1w,- norm of the residual equations and then we replace the negative norm by the discrete negative norm and analyze the discrete Chebyshev weighted least-squares method. The spectral convergence is derived for the proposed method. We also present various numerical experiments. The Legendre weighted least-squares method can be easily developed by following this paper.
Certified counting of roots of random univariate polynomials
Cleveland, Joseph; Hauenstein, Jonathan D; Haywood, Ian; Mehta, Dhagash; Morse, Anthony; Robol, Leonardo; Schlenk, Taylor
2014-01-01
A challenging problem in computational mathematics is to compute roots of a high-degree univariate random polynomial. We combine an efficient multiprecision implementation for solving high-degree random polynomials with two certification methods, namely Smale's $\\alpha$-theory and one based on Gerschgorin's theorem, for showing that a given numerical approximation is in the quadratic convergence region of Newton's method of some exact solution. With this combination, we can certifiably count the number of real roots of random polynomials. We quantify the difference between the two certification procedures and list the salient features of both of them. After benchmarking on random polynomials where the coefficients are drawn from the Gaussian distribution, we obtain novel experimental results for the Cauchy distribution case.
Convergence of the variants of a Chebyshev-Halley iteration family%一类变异型 Chebyshev-Halley迭代法的收敛性
Institute of Scientific and Technical Information of China (English)
徐秀斌; 包振威; 何濛
2015-01-01
研究了一类变异型Chebyshev-Halley迭代法的收敛性。给出了在满足条件‖F"( x)‖≤ω(‖x‖)时的迭代法收敛性判据及半局部收敛性的证明，最后分析了参数α的变化对收敛半径的影响，以期为某种参数的选择提供依据。%Under condition ‖F"( x)‖≤ω(‖x‖) , the convergence of the variants of a Chebyshev-Halley iteration family was discussed. The convergence criterion and semi-local convergence were obtained. The impact of the change of parameter α on the convergence radius was analyzed, so a kind of choosing criterion for parameters was provided.
Befriending Askey-Wilson polynomials
Szabłowski, Paweł J
2011-01-01
Although our main interest is with the Askey-Wilson (AW) polynomials we recall and review four other families of the so-called Askey-Wilson scheme of polynomials. We do this for completeness as well as for better exposition of AW properties. Our main results concentrate on the complex parameters case, revealing new fascinating symmetries between the variables and some of the parameters. In particular we express Askey-Wilson polynomials as linear combinations of Al-Salam--Chihara (ASC) polynomials which together with the obtained earlier expansion of the Askey-Wilson density forms complete generalization of the situation met in the case of Al-Salam--Chihara and q-Hermite polynomials and the Poisson-Mehler expansion formula. As a by-product we get useful identities involving ASC polynomials. Finally by certain re-scaling of variables and parameters we arrive to AW polynomials and AW densities that have clear probabilistic interpretation. We recall some known and present some believed to be unknown identities an...
Generalized companion matrix for approximate GCD
Boito, Paola
2011-01-01
We study a variant of the univariate approximate GCD problem, where the coe?- cients of one polynomial f(x)are known exactly, whereas the coe?cients of the second polynomial g(x)may be perturbed. Our approach relies on the properties of the matrix which describes the operator of multiplication by gin the quotient ring C[x]=(f). In particular, the structure of the null space of the multiplication matrix contains all the essential information about GCD(f; g). Moreover, the multiplication matrix exhibits a displacement structure that allows us to design a fast algorithm for approximate GCD computation with quadratic complexity w.r.t. polynomial degrees.
Institute of Scientific and Technical Information of China (English)
田亚芳; 周素静; 王霞
2012-01-01
利用权函数法,给出非线性方程求根的Chebyshev-Halley方法的几类改进方法,证明方法六阶收敛到单根.Chebyshev-Halley方法的效率指数为1.442,改进后的两步方法的效率指数为1.565.最后给出数值试验,且与牛顿法,Chebyshev-Halley 方法及其它已知的方程求根方法做了比较.结果表明方法具有一定的优越性.%Applying the method of weight functions, new families of improvements of Chebyshev-Halley methods are given. The convergence property is proved and the present methods converge to the simple roots with sixth-order. The efficiency index of Chebyshev-Halley methods is 1.442 and the modified two-step Chebyshev-Halley methods' is 1.565. In the end , numerical tests are given and the present methods are compared with Newton's method, Chebyshev-Halley methods and other known root-finding methods. The results show that the proposed methods have some more advantages than other methods.
Rational offset approximation of rational Bézier curves
Institute of Scientific and Technical Information of China (English)
CHENG Min; WANG Guo-jin
2006-01-01
The problem of parametric speed approximation of a rational curve is raised in this paper. Offset curves are widely used in various applications. As for the reason that in most cases the offset curves do not preserve the same polynomial or rational polynomial representations, it arouses difficulty in applications. Thus approximation methods have been introduced to solve this problem. In this paper, it has been pointed out that the crux of offset curve approximation lies in the approximation of parametric speed. Based on the Jacobi polynomial approximation theory with endpoints interpolation, an algebraic rational approximation algorithm of offset curve, which preserves the direction of normal, is presented.
Learning rates of least-square regularized regression with polynomial kernels
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
This paper presents learning rates for the least-square regularized regression algorithms with polynomial kernels. The target is the error analysis for the regression problem in learning theory. A regularization scheme is given, which yields sharp learning rates. The rates depend on the dimension of polynomial space and polynomial reproducing kernel Hilbert space measured by covering numbers. Meanwhile, we also establish the direct approximation theorem by Bernstein-Durrmeyer operators in Lρ2X with Borel probability measure.
Learning rates of least-square regularized regression with polynomial kernels
Institute of Scientific and Technical Information of China (English)
LI BingZheng; WANG GuoMao
2009-01-01
This paper presents learning rates for the least-square regularized regression algorithms with polynomial kernels. The target is the error analysis for the regression problem in learning theory. A regularization scheme is given, which yields sharp learning rates. The rates depend on the dimension of polynomial space and polynomial reproducing kernel Hilbert space measured by covering numbers. Meanwhile, we also establish the direct approximation theorem by Bernstein-Durrmeyer operators in Lpx2 with Borel probability measure.
MATLAB Solution of microstrip chebyshev Low pass filter system parameters by insertion loss method
Amiya Dey; Avijit Paul; Tanajit Manna
2012-01-01
We propose the practical solution for overcoming the tedious job of detail calculation and corresponding rigorous analysis of several system parameter values associated with the design, testing and troubleshooting of Microwave Microstrip Chebyshev Lowpass Filter (LPF) by Insertion Loss Method. The entire above spoken Microwave Lowpass filter system modeling and performance analysis are implemented using MATLAB.
Orthogonal polynomials and deformed oscillators
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.
Exact constants in approximation theory
Korneichuk, N
1991-01-01
This book is intended as a self-contained introduction for non-specialists, or as a reference work for experts, to the particular area of approximation theory that is concerned with exact constants. The results apply mainly to extremal problems in approximation theory, which in turn are closely related to numerical analysis and optimization. The book encompasses a wide range of questions and problems: best approximation by polynomials and splines; linear approximation methods, such as spline-approximation; optimal reconstruction of functions and linear functionals. Many of the results are base
An introduction to orthogonal polynomials
Chihara, Theodore S
2011-01-01
Assuming no further prerequisites than a first undergraduate course in real analysis, this concise introduction covers general elementary theory related to orthogonal polynomials. It includes necessary background material of the type not usually found in the standard mathematics curriculum. Suitable for advanced undergraduate and graduate courses, it is also appropriate for independent study. Topics include the representation theorem and distribution functions, continued fractions and chain sequences, the recurrence formula and properties of orthogonal polynomials, special functions, and some
Characterizing Polynomial Time Computability of Rational and Real Functions
Directory of Open Access Journals (Sweden)
Walid Gomaa
2009-11-01
Full Text Available Recursive analysis was introduced by A. Turing [1936], A. Grzegorczyk [1955], and D. Lacombe [1955]. It is based on a discrete mechanical framework that can be used to model computation over the real numbers. In this context the computational complexity of real functions defined over compact domains has been extensively studied. However, much less have been done for other kinds of real functions. This article is divided into two main parts. The first part investigates polynomial time computability of rational functions and the role of continuity in such computation. On the one hand this is interesting for its own sake. On the other hand it provides insights into polynomial time computability of real functions for the latter, in the sense of recursive analysis, is modeled as approximations of rational computations. The main conclusion of this part is that continuity does not play any role in the efficiency of computing rational functions. The second part defines polynomial time computability of arbitrary real functions, characterizes it, and compares it with the corresponding notion over rational functions. Assuming continuity, the main conclusion is that there is a conceptual difference between polynomial time computation over the rationals and the reals manifested by the fact that there are polynomial time computable rational functions whose extensions to the reals are not polynomial time computable and vice versa.
Evaluations of topological Tutte polynomials
Ellis-Monaghan, Joanna A
2011-01-01
We find a number of new combinatorial identities for, and interpretations of evaluations of, the topological Tutte polynomials of Las Vergnas, $L(G)$, and of and Bollob\\'as and Riordan, $R(G)$, as well as for the classical Tutte polynomial $T(G)$. For example, we express $R(G)$ and $T(G)$ as a sum of chromatic polynomials, show that $R(G)$ counts non-crossing graph states and $k$-valuations, and reformulate the Four Colour Theorem in terms of $R(G)$. Our main approach is to apply identities for the topological transition polynomial, one involving twisted duals, and one involving doubling the edges of a graph. These identities for the transition polynomial allow us to show that the Penrose polynomial $P(G)$ can be recovered from $R(G)$, a fact that we use to obtain identities and interpretations for $R(G)$. We also consider enumeration of circuits in medial graphs and use this to relate $R(G)$ and $L(G)$ for graphs embedded in low genus surfaces.
Bannai-Ito polynomials and dressing chains
Derevyagin, Maxim; Tsujimoto, Satoshi; Vinet, Luc; Zhedanov, Alexei
2012-01-01
Schur-Delsarte-Genin (SDG) maps and Bannai-Ito polynomials are studied. SDG maps are related to dressing chains determined by quadratic algebras. The Bannai-Ito polynomials and their kernel polynomials -- the complementary Bannai-Ito polynomials -- are shown to arise in the framework of the SDG maps.
Uniqueness and Zeros of -Shift Difference Polynomials
Indian Academy of Sciences (India)
Kai Liu; Xin-Ling Liu; Ting-Bin Cao
2011-08-01
In this paper, we consider the zero distributions of -shift difference polynomials of meromorphic functions with zero order, and obtain two theorems that extend the classical Hayman results on the zeros of differential polynomials to -shift difference polynomials. We also investigate the uniqueness problem of -shift difference polynomials that share a common value.
Birth-death processes and associated polynomials
Doorn, van Erik A.
2003-01-01
We consider birth-death processes on the nonnegative integers and the corresponding sequences of orthogonal polynomials called birth-death polynomials. The sequence of associated polynomials linked with a sequence of birth-death polynomials and its orthogonalizing measure can be used in the analysis
On the Irreducibility of Some Composite Polynomials
Directory of Open Access Journals (Sweden)
M. Alizadeh
2012-06-01
Full Text Available . In this paper we study the irreducibility of some composite polynomials, constructed by a polynomial composition method over finite fields. Finally, a recurrent method for constructing families of irreducible polynomials of higher degree from given irreducible polynomials over finite fields is given
A Kantorovich-Stancu Type Generalization of Szasz Operators including Brenke Type Polynomials
Directory of Open Access Journals (Sweden)
Rabia Aktaş
2013-01-01
Full Text Available We introduce a Kantorovich-Stancu type modification of a generalization of Szasz operators defined by means of the Brenke type polynomials and obtain approximation properties of these operators. Also, we give a Voronovskaya type theorem for Kantorovich-Stancu type operators including Gould-Hopper polynomials.
Continuous and discrete best polynomial degree reduction with Jacobi and Hahn weights
Ait-Haddou, Rachid
2016-03-02
We show that the weighted least squares approximation of Bézier coefficients with Hahn weights provides the best polynomial degree reduction in the Jacobi L2L2-norm. A discrete analogue of this result is also provided. Applications to Jacobi and Hahn orthogonal polynomials are presented.
Differentiation by integration using orthogonal polynomials, a survey
E. Diekema; T.H. Koornwinder
2012-01-01
This survey paper discusses the history of approximation formulas for n-th order derivatives by integrals involving orthogonal polynomials. There is a large but rather disconnected corpus of literature on such formulas. We give some results in greater generality than in the literature. Notably we un
A dynamic inequality generation scheme for polynomial programming
Ghaddar, B.; Vera Lizcano, J.C.; Anjos, M.F.
2016-01-01
Hierarchies of semidefinite programs have been used to approximate or even solve polynomial programs. This approach rapidly becomes computationally expensive and is often tractable only for problems of small size. In this paper, we propose a dynamic inequality generation scheme to generate valid pol
Learning Mixtures of Polynomials of Conditional Densities from Data
DEFF Research Database (Denmark)
L. López-Cruz, Pedro; Nielsen, Thomas Dyhre; Bielza, Concha;
2013-01-01
Mixtures of polynomials (MoPs) are a non-parametric density estimation technique for hybrid Bayesian networks with continuous and discrete variables. We propose two methods for learning MoP ap- proximations of conditional densities from data. Both approaches are based on learning MoP approximations......- ods with the approach for learning mixtures of truncated basis functions from data....
Eigenvalues and eigenfunctions for the ground state of polynomial potentials
International Nuclear Information System (INIS)
Analytic approximations for the ground state eigenvalues and eigenfunctions of polynomial potentials are found using an extended two-point quasi-rational approximation technique. In this procedure, the approximants are obtained through the power series and asymptotic expansion of the logarithmic derivative of the ground state eigenfunction, leaving the energy eigenvalue as a free parameter. A first approximation to the energy is obtained by imposing the condition that the rational approximating function must not have defects. Later, an iteration procedure leads to very precise energy eigenvalues. The method is described in detail using several explicit potentials as examples
Institute of Scientific and Technical Information of China (English)
王德明; 刘停战
2009-01-01
考虑求解非线性方程的迭代解法,研究带一个参数的三阶Chebyshev-HMley型迭代族方法.在文中对该迭代族进行修正,把二阶导数进行近似代替,得到了一族新的双参数三阶迭代族,并且当参数取特殊值的时候,又可以得到一个具有四阶收敛的新迭代算法.
Modeling Microwave Structures in Time Domain Using Laguerre Polynomials
Directory of Open Access Journals (Sweden)
Z. Raida
2006-09-01
Full Text Available The paper is focused on time domain modeling of microwave structures by the method of moments. Two alternative schemes with weighted Laguerre polynomials are presented. Thanks to their properties, these schemes are free of late time oscillations. Further, the paper is aimed to effective and accurate evaluation of Green's functions integrals within these schemes. For this evaluation, a first- and second-order polynomial approximation is developed. The last part of the paper deals with modeling microstrip structures in the time domain. Conditions of impedance matching are derived, and the proposed approach is verified by modeling a microstrip filter.
An orthogonal basis for the hyperbolic hybrid polynomial space
Institute of Scientific and Technical Information of China (English)
HUANG Yu; WANG GuoZhao
2007-01-01
Motivated by the wide usage of the Tchebyshev basis and Legendre basis in the algebra polynomial space, we construct an orthogonal basis with the properties of the H-Bézier basis in the hyperbolic hybrid polynomial space, which is similar to the Legendre basis and holds remarkable properties. Moreover, we derive the transformation matrices that map the H-Bézier basis and the orthogonal basis forms into each other. An example for approximating the degree reduction of the H- Bézier curves is sketched to illustrate the utility of the orthogonal basis.
Orthogonal Polynomials and Special Functions
Assche, Walter
2003-01-01
The set of lectures from the Summer School held in Leuven in 2002 provide an up-to-date account of recent developments in orthogonal polynomials and special functions, in particular for algorithms for computer algebra packages, 3nj-symbols in representation theory of Lie groups, enumeration, multivariable special functions and Dunkl operators, asymptotics via the Riemann-Hilbert method, exponential asymptotics and the Stokes phenomenon. The volume aims at graduate students and post-docs working in the field of orthogonal polynomials and special functions, and in related fields interacting with orthogonal polynomials, such as combinatorics, computer algebra, asymptotics, representation theory, harmonic analysis, differential equations, physics. The lectures are self-contained requiring only a basic knowledge of analysis and algebra, and each includes many exercises.
Plain Polynomial Arithmetic on GPU
Anisul Haque, Sardar; Moreno Maza, Marc
2012-10-01
As for serial code on CPUs, parallel code on GPUs for dense polynomial arithmetic relies on a combination of asymptotically fast and plain algorithms. Those are employed for data of large and small size, respectively. Parallelizing both types of algorithms is required in order to achieve peak performances. In this paper, we show that the plain dense polynomial multiplication can be efficiently parallelized on GPUs. Remarkably, it outperforms (highly optimized) FFT-based multiplication up to degree 212 while on CPU the same threshold is usually at 26. We also report on a GPU implementation of the Euclidean Algorithm which is both work-efficient and runs in linear time for input polynomials up to degree 218 thus showing the performance of the GCD algorithm based on systolic arrays.
Rational Chebyshev Spectral Transform for the dynamics of high-power laser diodes
Javaloyes, J
2014-01-01
This manuscript details the use of the rational Chebyshev transform for describing the transverse dynamics of high-power laser diodes, either broad area lasers, index guided lasers or monolithic master oscillator power amplifier devices. This spectral method can be used in combination with the delay algebraic equation approach developed in \\cite{JB-OE-12}, which allows to substantially reduce the computation time. The theory is presented in such a way that it encompasses the case of the Fourier spectral transform presented in \\cite{PJB-JSTQE-13} as a particular case. It is also extended to the consideration of index guiding with an arbitrary profile. Because their domain of definition is infinite, the convergence properties of the Chebyshev Rational functions allow handling the boundary conditions with higher accuracy than with the previously studied Fourier method. As practical examples, we solve the beam propagation problem with and without index guiding: we obtain excellent results and an improvement of th...
Operation analysis of a Chebyshev-Pantograph leg mechanism for a single DOF biped robot
Liang, Conghui; Ceccarelli, Marco; Takeda, Yukio
2012-12-01
In this paper, operation analysis of a Chebyshev-Pantograph leg mechanism is presented for a single degree of freedom (DOF) biped robot. The proposed leg mechanism is composed of a Chebyshev four-bar linkage and a pantograph mechanism. In contrast to general fully actuated anthropomorphic leg mechanisms, the proposed leg mechanism has peculiar features like compactness, low-cost, and easy-operation. Kinematic equations of the proposed leg mechanism are formulated for a computer oriented simulation. Simulation results show the operation performance of the proposed leg mechanism with suitable characteristics. A parametric study has been carried out to evaluate the operation performance as function of design parameters. A prototype of a single DOF biped robot equipped with two proposed leg mechanisms has been built at LARM (Laboratory of Robotics and Mechatronics). Experimental test shows practical feasible walking ability of the prototype, as well as drawbacks are discussed for the mechanical design.
Rational Chebyshev spectral transform for the dynamics of broad-area laser diodes
International Nuclear Information System (INIS)
This manuscript details the use of the rational Chebyshev transform for describing the transverse dynamics of broad-area laser diodes and amplifiers. This spectral method can be used in combination with the delay algebraic equations approach developed in [1], which substantially reduces the computation time. The theory is presented in such a way that it encompasses the case of the Fourier spectral transform presented in [2] as a particular case. It is also extended to the consideration of index guiding with an arbitrary transverse profile. Because their domain of definition is infinite, the convergence properties of the Chebyshev rational functions allow handling the boundary conditions with higher accuracy than with the previously studied Fourier transform method. As practical examples, we solve the beam propagation problem with and without index guiding: we obtain excellent results and an improvement of the integration time between one and two orders of magnitude as compared with a fully distributed two dimensional model
On Polynomial Solutions of Linear Differential Equations with Polynomial Coefficients
Si, Do Tan
1977-01-01
Demonstrates a method for solving linear differential equations with polynomial coefficients based on the fact that the operators z and D + d/dz are known to be Hermitian conjugates with respect to the Bargman and Louck-Galbraith scalar products. (MLH)
Efficient generation of correlated random numbers using Chebyshev-optimal magnitude-only IIR filters
Rodriguez, Alejandro; Johnson, Steven G.
2007-01-01
We compare several methods for the efficient generation of correlated random sequences (colored noise) by filtering white noise to achieve a desired correlation spectrum. We argue that a class of IIR filter-design techniques developed in the 1970s, which obtain the global Chebyshev-optimum minimum-phase filter with a desired magnitude and arbitrary phase, are uniquely suited for this problem but have seldom been used. The short filters that result from such techniques are crucial for applicat...
Directory of Open Access Journals (Sweden)
M. H. Heydari
2013-01-01
Full Text Available An efficient Chebyshev wavelets method for solving a class of nonlinear fractional integrodifferential equations in a large interval is developed, and a new technique for computing nonlinear terms in such equations is proposed. Existence of a unique solution for such equations is proved. Convergence and error analysis of the proposed method are investigated. Moreover in order to show efficiency of the proposed method, the new approach is compared with some numerical methods.
Szatmári, Daniel
2015-12-01
Disadvantages of the currently used Křovák's map projection in the Slovak Republic, such as large scale distortion, became evident after the division of Czechoslovakia. The aim of this paper is to show the results of the optimization of cartographic projections using Chebyshev's theorem for conformal projections and its application to the territory of the Slovak Republic. The calculus used, the scale distortions achieved and their comparison with the scale distortions of currently used map projections will be demonstrated.
Symbolic computation of Appell polynomials using Maple
Directory of Open Access Journals (Sweden)
H. Alkahby
2001-07-01
Full Text Available This work focuses on the symbolic computation of Appell polynomials using the computer algebra system Maple. After describing the traditional approach of constructing Appell polynomials, the paper examines the operator method of constructing the same Appell polynomials. The operator approach enables us to express the Appell polynomial as Bessel function whose coefficients are Euler and Bernuolli numbers. We have also constructed algorithms using Maple to compute Appell polynomials based on the methods we have described. The achievement is the construction of Appell polynomials for any function of bounded variation.
Solutions of differential equations in a Bernstein polynomial basis
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.
Characterizing Polynomial Time Computability of Rational and Real Functions
Gomaa, Walid
2009-01-01
Recursive analysis was introduced by A. Turing [1936], A. Grzegorczyk [1955], and D. Lacombe [1955]. It is based on a discrete mechanical framework that can be used to model computation over the real numbers. In this context the computational complexity of real functions defined over compact domains has been extensively studied. However, much less have been done for other kinds of real functions. This article is divided into two main parts. The first part investigates polynomial time computability of rational functions and the role of continuity in such computation. On the one hand this is interesting for its own sake. On the other hand it provides insights into polynomial time computability of real functions for the latter, in the sense of recursive analysis, is modeled as approximations of rational computations. The main conclusion of this part is that continuity does not play any role in the efficiency of computing rational functions. The second part defines polynomial time computability of arbitrary real ...
Local polynomial Whittle estimation of perturbed fractional processes
DEFF Research Database (Denmark)
Frederiksen, Per; Nielsen, Frank; Nielsen, Morten Ørregaard
We propose a semiparametric local polynomial Whittle with noise (LPWN) estimator of the memory parameter in long memory time series perturbed by a noise term which may be serially correlated. The estimator approximates the spectrum of the perturbation as well as that of the short-memory component...... of the signal by two separate polynomials. Including these polynomials we obtain a reduction in the order of magnitude of the bias, but also in‡ate the asymptotic variance of the long memory estimate by a multiplicative constant. We show that the estimator is consistent for d 2 (0; 1), asymptotically normal...... for d ε (0, 3/4), and if the spectral density is infinitely smooth near frequency zero, the rate of convergence can become arbitrarily close to the parametric rate, pn. A Monte Carlo study reveals that the LPWN estimator performs well in the presence of a serially correlated perturbation term...
Applying polynomial filtering to mass preconditioned Hybrid Monte Carlo
Haar, Taylor; Zanotti, James; Nakamura, Yoshifumi
2016-01-01
The use of mass preconditioning or Hasenbusch filtering in modern Hybrid Monte Carlo simulations is common. At light quark masses, multiple filters (three or more) are typically used to reduce the cost of generating dynamical gauge fields; however, the task of tuning a large number of Hasenbusch mass terms is non-trivial. The use of short polynomial approximations to the inverse has been shown to provide an effective UV filter for HMC simulations. In this work we investigate the application of polynomial filtering to the mass preconditioned Hybrid Monte Carlo algorithm as a means of introducing many time scales into the molecular dynamics integration with a simplified parameter tuning process. A generalized multi-scale integration scheme that permits arbitrary step- sizes and can be applied to Omelyan-style integrators is also introduced. We find that polynomial-filtered mass-preconditioning (PF-MP) performs as well as or better than standard mass preconditioning, with significantly less fine tuning required.
Etessami, Kousha; Yannakakis, Mihalis
2012-01-01
We show that one can approximate the least fixed point solution for a multivariate system of monotone probabilistic max(min) polynomial equations, referred to as maxPPSs (and minPPSs, respectively), in time polynomial in both the encoding size of the system of equations and in log(1/epsilon), where epsilon > 0 is the desired additive error bound of the solution. (The model of computation is the standard Turing machine model.) We establish this result using a generalization of Newton's method which applies to maxPPSs and minPPSs, even though the underlying functions are only piecewise-differentiable. This generalizes our recent work which provided a P-time algorithm for purely probabilistic PPSs. These equations form the Bellman optimality equations for several important classes of infinite-state Markov Decision Processes (MDPs). Thus, as a corollary, we obtain the first polynomial time algorithms for computing to within arbitrary desired precision the optimal value vector for several classes of infinite-state...
Prestack traveltime approximations
Alkhalifah, Tariq Ali
2011-01-01
Most prestack traveltime relations we tend work with are based on homogeneous (or semi-homogenous, possibly effective) media approximations. This includes the multi-focusing or double square-root (DSR) and the common reflection stack (CRS) equations. Using the DSR equation, I analyze the associated eikonal form in the general source-receiver domain. Like its wave-equation counterpart, it suffers from a critical singularity for horizontally traveling waves. As a result, I derive expansion based solutions of this eikonal based on polynomial expansions in terms of the reflection and dip angles in a generally inhomogenous background medium. These approximate solutions are free of singularities and can be used to estimate travetimes for small to moderate offsets (or reflection angles) in a generally inhomogeneous medium. A Marmousi example demonstrates the usefulness of the approach. © 2011 Society of Exploration Geophysicists.
On Modular Counting with Polynomials
DEFF Research Database (Denmark)
Hansen, Kristoffer Arnsfelt
2006-01-01
For any integers m and l, where m has r sufficiently large (depending on l) factors, that are powers of r distinct primes, we give a construction of a (symmetric) polynomial over Z_m of degree O(\\sqrt n) that is a generalized representation (commonly also called weak representation) of the MODl...
Two polynomial division inequalities in
Directory of Open Access Journals (Sweden)
Goetgheluck P
1998-01-01
Full Text Available This paper is a first attempt to give numerical values for constants and , in classical estimates and where is an algebraic polynomial of degree at most and denotes the -metric on . The basic tools are Markov and Bernstein inequalities.
Global Polynomial Kernel Hazard Estimation
DEFF Research Database (Denmark)
Hiabu, Munir; Miranda, Maria Dolores Martínez; Nielsen, Jens Perch;
2015-01-01
This paper introduces a new bias reducing method for kernel hazard estimation. The method is called global polynomial adjustment (GPA). It is a global correction which is applicable to any kernel hazard estimator. The estimator works well from a theoretical point of view as it asymptotically redu...
The Ridge Function Representation of Polynomials and an Application to Neural Networks
Institute of Scientific and Technical Information of China (English)
Ting Fan XIE; Fei Long CAO
2011-01-01
The first goal of this paper is to establish some properties of the ridge function representation for multivariate polynomials,and the second one is to apply these results to the problem of approximation by neural networks.We find that for continuous functions,the rate of approximation obtained by a neural network with one hidden layer is no slower than that of an algebraic polynomial.
Preeti*; Dr. Amandeep Singh Sappal; Dr. Hardeep Singh Ryait,
2012-01-01
The concept of decimation plays very important role in the modern digital communication systems. Digital decimators or decimation filters are used to decrease the sampling rate of a sampled signal in digital domain. The Farrow Structure provides an efficient way to implement the decimation filter using polynomial approximation method for arbitrary sample rate change which offers the option of continuously adjustable resample ratio. In this paper, Lagrange polynomial approximation method has ...
Directory of Open Access Journals (Sweden)
Preeti,
2012-06-01
Full Text Available The concept of decimation plays very important role in the modern digital communication systems. Digital decimators or decimation filters are used to decrease the sampling rate of a sampled signal in digital domain. The Farrow Structure provides an efficient way to implement the decimation filter using polynomial approximation method for arbitrary sample rate change which offers the option of continuously adjustable resample ratio. In this paper, Lagrange polynomial approximation method has been used to implement Farrow structure based decimator. The optimum filter coefficients have been calculated using Lagrange Polynomials. Simulation results have been presented.
Solutions of interval type-2 fuzzy polynomials using a new ranking method
Rahman, Nurhakimah Ab.; Abdullah, Lazim; Ghani, Ahmad Termimi Ab.; Ahmad, Noor'Ani
2015-10-01
A few years ago, a ranking method have been introduced in the fuzzy polynomial equations. Concept of the ranking method is proposed to find actual roots of fuzzy polynomials (if exists). Fuzzy polynomials are transformed to system of crisp polynomials, performed by using ranking method based on three parameters namely, Value, Ambiguity and Fuzziness. However, it was found that solutions based on these three parameters are quite inefficient to produce answers. Therefore in this study a new ranking method have been developed with the aim to overcome the inherent weakness. The new ranking method which have four parameters are then applied in the interval type-2 fuzzy polynomials, covering the interval type-2 of fuzzy polynomial equation, dual fuzzy polynomial equations and system of fuzzy polynomials. The efficiency of the new ranking method then numerically considered in the triangular fuzzy numbers and the trapezoidal fuzzy numbers. Finally, the approximate solutions produced from the numerical examples indicate that the new ranking method successfully produced actual roots for the interval type-2 fuzzy polynomials.
On the Computational Complexity of L_{1}-Approximation
DEFF Research Database (Denmark)
Oliva, Paulo Borges
2002-01-01
t is well known that for a given continuous function f : [0, 1] and a number n there exists a unique polynomial pn Pn (polynomials of degree n) which best L1-approximates f. We establish the first upper bound on the complexity of the sequence (pn)n , assuming f is polynomial-time computable. Our ...
Sheffer and Non-Sheffer Polynomial Families
Directory of Open Access Journals (Sweden)
G. Dattoli
2012-01-01
Full Text Available By using the integral transform method, we introduce some non-Sheffer polynomial sets. Furthermore, we show how to compute the connection coefficients for particular expressions of Appell polynomials.
Improved Approximation for the Directed Spanner Problem
Bhattacharyya, Arnab; Makarychev, Konstantin
2010-01-01
We prove that the size of the sparsest directed k-spanner of a graph can be approximated in polynomial time to within a factor of $\\tilde{O}(\\sqrt{n})$, for all k >= 3. This improves the $\\tilde{O}(n^{2/3})$-approximation recently shown by Dinitz and Krauthgamer.
Exceptional Polynomials and SUSY Quantum Mechanics
Chaitanya, K V S Shiv; Panigrahi, Prasanta K; Radhakrishnan, R; Srinivasan, V
2011-01-01
We show that the existence of exceptional polynomials leads to the presence of non-trivial supersymmetry. The existence of these polynomials leads to two distinct potentials in the Schr\\"odinger equation having the same eigenvalues. We describe two ways of finding new exactly solvable (ES) potentials. All Schr\\"odinger equations having the Laguerre and the Jacobi polynomials as solutions, have non-trivial supersymmetric partners with corresponding exceptional polynomials as solutions.
An Improved Volumetric Estimation Using Polynomial Regression
Noraini Abdullah; Amran Ahmed; Zainodin Hj. Jubok
2011-01-01
The polynomial regression (PR) technique is used to estimate the parameters of the dependent variable having a polynomial relationship with the independent variable. Normality and nonlinearity exhibit polynomial characterization of power terms greater than 2. Polynomial Regression models (PRM) with the auxiliary variables are considered up to their third order interactions. Preliminary, multicollinearity between the independent variables is minimized and statistical tests involving the Global...
Positive trigonometric polynomials and signal processing applications
Dumitrescu, Bogdan
2007-01-01
Presents the results on positive trigonometric polynomials within a unitary framework; the theoretical results obtained partly from the general theory of real polynomials, partly from self-sustained developments. This book provides information on the theory of sum-of-squares trigonometric polynomials in two parts: theory and applications.
Lattice Platonic Solids and their Ehrhart polynomial
Directory of Open Access Journals (Sweden)
E. J. Ionascu
2013-01-01
Full Text Available First, we calculate the Ehrhart polynomial associated to an arbitrary cube with integer coordinates for its vertices. Then, we use this result to derive relationships between the Ehrhart polynomials for regular lattice tetrahedra and those for regular lattice octahedra. These relations allow one to reduce the calculation of these polynomials to only one coefficient.
General Eulerian Numbers and Eulerian Polynomials
Directory of Open Access Journals (Sweden)
Tingyao Xiong
2013-01-01
Full Text Available We will generalize the definitions of Eulerian numbers and Eulerian polynomials to general arithmetic progressions. Under the new definitions, we have been successful in extending several well-known properties of traditional Eulerian numbers and polynomials to the general Eulerian polynomials and numbers.
Lattice Platonic Solids and their Ehrhart polynomial
Ionascu, Eugen J
2011-01-01
First, we calculate the Ehrhart polynomial associated to an arbitrary cube with integer coordinates for its vertices. Then, we use this result to derive relationships between the Ehrhart polynomials for regular lattice tetrahedrons and those for regular lattice octahedrons. These relations allow one to reduce the calculation of these polynomials to only one coefficient.
Energy Technology Data Exchange (ETDEWEB)
Chalasani, P.; Saias, I. [Los Alamos National Lab., NM (United States); Jha, S. [Carnegie Mellon Univ., Pittsburgh, PA (United States)
1996-04-08
As increasingly large volumes of sophisticated options (called derivative securities) are traded in world financial markets, determining a fair price for these options has become an important and difficult computational problem. Many valuation codes use the binomial pricing model, in which the stock price is driven by a random walk. In this model, the value of an n-period option on a stock is the expected time-discounted value of the future cash flow on an n-period stock price path. Path-dependent options are particularly difficult to value since the future cash flow depends on the entire stock price path rather than on just the final stock price. Currently such options are approximately priced by Monte carlo methods with error bounds that hold only with high probability and which are reduced by increasing the number of simulation runs. In this paper the authors show that pricing an arbitrary path-dependent option is {number_sign}-P hard. They show that certain types f path-dependent options can be valued exactly in polynomial time. Asian options are path-dependent options that are particularly hard to price, and for these they design deterministic polynomial-time approximate algorithms. They show that the value of a perpetual American put option (which can be computed in constant time) is in many cases a good approximation to the value of an otherwise identical n-period American put option. In contrast to Monte Carlo methods, the algorithms have guaranteed error bounds that are polynormally small (and in some cases exponentially small) in the maturity n. For the error analysis they derive large-deviation results for random walks that may be of independent interest.
Normal BGG solutions and polynomials
Cap, A; Hammerl, M
2012-01-01
First BGG operators are a large class of overdetermined linear differential operators intrinsically associated to a parabolic geometry on a manifold. The corresponding equations include those controlling infinitesimal automorphisms, higher symmetries, and many other widely studied PDE of geometric origin. The machinery of BGG sequences also singles out a subclass of solutions called normal solutions. These correspond to parallel tractor fields and hence to (certain) holonomy reductions of the canonical normal Cartan connection. Using the normal Cartan connection, we define a special class of local frames for any natural vector bundle associated to a parabolic geometry. We then prove that the coefficient functions of any normal solution of a first BGG operator with respect to such a frame are polynomials in the normal coordinates of the parabolic geometry. A bound on the degree of these polynomials in terms of representation theory data is derived. For geometries locally isomorphic to the homogeneous model of ...
BSDEs with polynomial growth generators
Directory of Open Access Journals (Sweden)
Philippe Briand
2000-01-01
Full Text Available In this paper, we give existence and uniqueness results for backward stochastic differential equations when the generator has a polynomial growth in the state variable. We deal with the case of a fixed terminal time, as well as the case of random terminal time. The need for this type of extension of the classical existence and uniqueness results comes from the desire to provide a probabilistic representation of the solutions of semilinear partial differential equations in the spirit of a nonlinear Feynman-Kac formula. Indeed, in many applications of interest, the nonlinearity is polynomial, e.g, the Allen-Cahn equation or the standard nonlinear heat and Schrödinger equations.
Directory of Open Access Journals (Sweden)
S. China Venkateswarlu
2013-07-01
Full Text Available This paper investigates the effect of Dolph-Chebyshev window frequency response Side lobe Attenuation on the improvement of Speech quality in terms of six objective quality measures. In Speech Enhancement process, signal corrupted by noise is segmented into frames and each segment is Windowed using Dolph-Chebyshev Window with variation in the side lobe attenuation parameter α. The Windowed Speech segments are applied to the Weiner Filter Speech Enhancement algorithm and the Enhanced Speech signal is reconstructed in its time domain. The focus is to study the effect of Dolph-Chebyshev Window spectral side lobe attenuation on the Speech Enhancement process. For different side lobe attenuations of the Dolph-Chebyshev Window frequency response, speech quality objective measures have been computed. From this study, it is observed that the Side lobe Attenuation parameter α plays an important role on the Speech enhancement process in terms of six objective quality measures. The results are compared with the measures of Hanning window and an optimum side lobe attenuation parameter in dB for the Dolph-Chebyshev Window is proposed for better speech quality
Quantitative equidistribution for the solutions of a system of sparse polynomial equations
D'Andrea, Carlos; Sombra, Martín
2012-01-01
For a sequence of n Laurent polynomials in n variables with complex coefficients such that the size of the coefficients is not too big with respect to the facet resultants of the input sequence, we show that the solutions in the algebraic torus of the system of equations defined by this sequence, are approximately equidistributed near the unit polycircle. This generalizes to the multivariate case, up to a exponent, a classical result due to Erd\\"os and Tur\\'an on the distribution of the arguments of the roots of a univariate polynomial. We apply this result to bound the number of real roots of a system of Laurent polynomials, and to study the asymptotic distribution of the roots of systems of Laurent polynomials over the integers, and of random systems of Laurent polynomials over the field of complex numbers.
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.
Stable piecewise polynomial vector fields
Directory of Open Access Journals (Sweden)
Claudio Pessoa
2012-09-01
Full Text Available Let $N={y>0}$ and $S={y<0}$ be the semi-planes of $mathbb{R}^2$ having as common boundary the line $D={y=0}$. Let $X$ and $Y$ be polynomial vector fields defined in $N$ and $S$, respectively, leading to a discontinuous piecewise polynomial vector field $Z=(X,Y$. This work pursues the stability and the transition analysis of solutions of $Z$ between $N$ and $S$, started by Filippov (1988 and Kozlova (1984 and reformulated by Sotomayor-Teixeira (1995 in terms of the regularization method. This method consists in analyzing a one parameter family of continuous vector fields $Z_{epsilon}$, defined by averaging $X$ and $Y$. This family approaches $Z$ when the parameter goes to zero. The results of Sotomayor-Teixeira and Sotomayor-Machado (2002 providing conditions on $(X,Y$ for the regularized vector fields to be structurally stable on planar compact connected regions are extended to discontinuous piecewise polynomial vector fields on $mathbb{R}^2$. Pertinent genericity results for vector fields satisfying the above stability conditions are also extended to the present case. A procedure for the study of discontinuous piecewise vector fields at infinity through a compactification is proposed here.
Pattern Matching under Polynomial Transformation
Butman, Ayelet; Clifford, Raphael; Jalsenius, Markus; Lewenstein, Noa; Porat, Benny; Porat, Ely; Sach, Benjamin
2011-01-01
We consider a class of pattern matching problems where a polynomial transformation can be applied to the pattern at every alignment. Given a pattern of length m and a longer text of length n where both are assumed to contain integer values only, we show O(n log m) algorithms for pattern matching under linear transformations even when wildcard symbols can occur in the input. We then show how to extend the technique to polynomial transformations of arbitrary degree. Next we consider the problem of finding the minimum Hamming distance under polynomial transformation. We show that, for any epsilon > 0, there cannot exist an O(nm^(1-epsilon)) algorithm for additive and linear transformations conditional on the hardness of the classic 3SUM problem. Finally, we consider a version of the Hamming distance problem under additive transformations with a bound k on the maximum distance that need be reported. We give a deterministic O(nk log k) time solution which we then improve by careful use of randomisation to O(n sqrt...
On Ternary Inclusion-Exclusion Polynomials
Bachman, Gennady
2010-01-01
Taking a combinatorial point of view on cyclotomic polynomials leads to a larger class of polynomials we shall call the inclusion-exclusion polynomials. This gives a more appropriate setting for certain types of questions about the coefficients of these polynomials. After establishing some basic properties of inclusion-exclusion polynomials we turn to a detailed study of the structure of ternary inclusion-exclusion polynomials. The latter subclass is exemplified by cyclotomic polynomials $\\Phi_{pqr}$, where $p
Computing Nash Equilibria: Approximation and Smoothed Complexity
Chen, Xi; Deng, Xiaotie; Teng, Shang-Hua
2006-01-01
We show that the BIMATRIX game does not have a fully polynomial-time approximation scheme, unless PPAD is in P. In other words, no algorithm with time polynomial in n and 1/\\epsilon can compute an \\epsilon-approximate Nash equilibrium of an n by nbimatrix game, unless PPAD is in P. Instrumental to our proof, we introduce a new discrete fixed-point problem on a high-dimensional cube with a constant side-length, such as on an n-dimensional cube with side-length 7, and show that they are PPAD-co...
A New Parallel Domain-Decomposed Chebyshev Collocation Method for Atmospheric and Oceanic Modeling
Directory of Open Access Journals (Sweden)
Yu-Ming Tsai
2012-01-01
Full Text Available Spectral methods seek the solution to a differential equation in terms of series of known smooth function. The Chebyshev series possesses the exponential-convergence property regardless of the imposed boundary condition, and therefore is suited for the regional modeling. We propose a new domain-decomposed Chebyshev collocation method which facilitates an efficient parallel implementation. The boundary conditions for the individual sub-domains are exchanged through one grid interval overlapping. This approach is validated using the one dimensional advection equation and the inviscid _ equation. We further tested the vortex formation and propagation problems using two-dimensional nonlinear shallow water equations. The domain decomposition approach in general gave more accurate solutions compared to that of the single domain calculation. Moreover, our approach retains the exponential error convergence and conservation of mass and the quadratic quantities such as kinetic energy and enstrophy. The efficiency of our method is greater than one and increases with the number of processors, with the optimal speed up of 29 and efficiency 3.7 in 8 processors. Efficiency greater than one was obtained due to the reduction the degrees of freedom in each sub-domain that reduces the spectral operational count and also due to a larger time step allowed in the sub-domain method. The communication overhead begins to dominate when the number of processors further increases, but the method still results in an efficiency of 0.9 in 16 processors. As a result, the parallel domain-decomposition Chebyshev method may serve as an efficient alternative for atmospheric and oceanic modeling.
AMDLIBAE, IBM 360 Subroutine Library, Special Function, Polynomials, Differential Equation
International Nuclear Information System (INIS)
158S P ANC4P: Adap. quad. using 4-th order Newton-Cotes; D161S F GAUSS: Arbitrary Gaussian weights and nodes; D162S F SQUANK: Simpson's quad. used adaptively; D252S F DDFSUB: DP Neville or Stoer sol. lin. dif. eqns.; D253S F DDFSYS: Driver for D252S; D255S F DFBND: Stoer sol. dif. eqs. with error bounds; D256S F DFBDRV: Driver for D255S; D257S F GEARDV: Gear's sol. of init. value problem; D452S F ENDACE: Numerical derivatives real analytic fn.; E206S F LSQPOL: Least squares weighted polynomial fit; E208S F1: Arbitrary function fit, least squares; E209S F CALLSQ: Driver for E206S; E212S F: General least squares fit + function eval.; E250S F VA02A: Least squares funct. min. w/o derivatives; E252S F MINMAX: Chebyshev line fit; E253S F: Arbitrary functional fit II; E256S F BGPOL: Least squares fit with polynomials; E257S F BGLSSQ: Least squares fit with arbitrary function; E350S F SMOOTH: Smoothing by cubic splines
Efficient generation of correlated random numbers using Chebyshev-optimal magnitude-only IIR filters
Rodríguez, A; Johnson, Steven G.; Rodriguez, Alejandro
2007-01-01
We compare several methods for the efficient generation of correlated random sequences (colored noise) by filtering white noise to achieve a desired correlation spectrum. We argue that a class of IIR filter-design techniques developed in the 1970s, which obtain the global Chebyshev-optimum minimum-phase filter with a desired magnitude and arbitrary phase, are uniquely suited for this problem but have seldom been used. The short filters that result from such techniques are crucial for applications of colored noise in physical simulations involving random processes, for which many long random sequences must be generated and computational time and memory are at a premium.
Directory of Open Access Journals (Sweden)
Zahra Masouri
2014-04-01
Full Text Available The focus of this paper is on the numerical solution of linear systems of Fredhlom integral equations of the second kind. For this purpose, the Chebyshev cardinal functions with Gauss-Lobatto points are used. By combination of properties of these functions and the effective Clenshaw-Curtis quadrature rule, an applicable numerical method for solving the mentioned systems is formulated. Some error bounds for the method are computed and its convergence rate is estimated. The method is numerically evaluated by solving some test problems caught from the literature by which the accuracy and computational efficiency of the method will be demonstrated.
CONSTRUCTION OF THE ENCRYPTION MATRIX BASED ON CHEBYSHEV CHAOTIC NEURAL NETWORKS
Institute of Scientific and Technical Information of China (English)
Zou Ajin; Wu Wei; Li Renfa; Li Yongjiang
2012-01-01
The paper proposes a novel algorithm to get the encryption matrix.Firstly,a chaotic sequence generated by Chebyshev chaotic neural networks is converted into a series of low-order integer matrices from which available encryption matrices are selected.Then,a higher order encryption matrix relating real world application is constructed by means of tensor production method based on selected encryption matrices.The results show that the proposed algorithm can produce a "one-time pad cipher" encryption matrix with high security; and the encryption results have good chaos and autocorrelation with the natural frequency of the plaintext being hidden and homogenized.
Institute of Scientific and Technical Information of China (English)
魏麒
2014-01-01
The floating point computing power of early graphic processing unit(GPU) is not strong. So it can’t do data processing. In other words, in early graphics processing, central processing unit(CPU) do data processing chain firstly, then GPU do image processing chain. But until now, much more GPU having stronger performance have been designed. In accordance with this trend, in near future, these powerful GPU can accomplish most work previously accomplished by CPU such as data processing. Prospectively studying how reasonable dispatch CPU and GPU to faster processing graphics issues in such a new case is interesting. Actually, this new problem is equal to a two-stage two-machine hybrid flow shop problem: There are two machines and n jobs. Each job has two tasks, the first task can be processed on either machine, called flexible task, while the second task can’t be processed unless the first task is completed and must be processed on the second machine. The objective of the problem is minimizing the makespan. A polynomial time approximation scheme (PTAS) for this problem is given in this paper.%由于早期的图形处理器浮点运算能力不强，所以在处理图形问题时一般由中央处理器处理数据运算环节，然后再由图形处理器进行图像处理。但是最近几年图形处理器的浮点运算能力得到很大提高，相信很快就能胜任原先只有中央处理器才能完成的图形问题中的数据运算任务，为此前瞻性的研究在这样一种新情况下如何合理调度中央处理器和图形处理器来更快的处理图形问题是很有必要的。事实上该问题其实相当于一个两阶段两台处理器的混合流水作业问题：有两台处理器和一批需要加工的工件，每个工件都包含两个任务，前一个任务是为第二个任务做准备的。第一个任务可以选择在任何一台处理器上处理，而第二个任务则必须当第一个任务完成后，在第二台处理
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.
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
In this paper, we mainly study the relation of two cyclically reduced words w and w' on the condition they have the same trace polynomial (i.e., tr w= tr w' ). By defining an equivalence relation through such operators on words as inverse, cyclically left shift, and mirror, it is straightforward to get that w ～ w' implies tr w = tr w'. We show by a counter example that tr w = tr w' does not imply w ～ w'. And in two special cases, we prove that tr w = tr w' if and only if w ～ w'.
Zeroes of random Reinhardt polynomials
Karami, Arash
2012-01-01
For a Reinhardt domain $\\Omega$ with the smooth boundary in $\\mathbb{C}^{m+1}$ and a positive smooth measure $\\mu$ on the boundary of $\\Omega$, we consider the ensemble $P_{N}$ of polynomials of degree $N$ with the Gaussian probability measure $\\gamma_{N}$ which is induced by $L^{2}(\\partial\\Omega,d\\mu)$. Our aim is to compute scaling limit distribution function and scaling limit pair correlation function between zeros when $z\\in\\partial\\Omega$. First of all we apply stationary phase method t...
Chromatic Polynomials of Mixed Hypercycles
Directory of Open Access Journals (Sweden)
Allagan Julian A.
2014-08-01
Full Text Available We color the vertices of each of the edges of a C-hypergraph (or cohypergraph in such a way that at least two vertices receive the same color and in every proper coloring of a B-hypergraph (or bihypergraph, we forbid the cases when the vertices of any of its edges are colored with the same color (monochromatic or when they are all colored with distinct colors (rainbow. In this paper, we determined explicit formulae for the chromatic polynomials of C-hypercycles and B-hypercycles
An Improved Volumetric Estimation Using Polynomial Regression
Directory of Open Access Journals (Sweden)
Noraini Abdullah
2011-12-01
Full Text Available The polynomial regression (PR technique is used to estimate the parameters of the dependent variable having a polynomial relationship with the independent variable. Normality and nonlinearity exhibit polynomial characterization of power terms greater than 2. Polynomial Regression models (PRM with the auxiliary variables are considered up to their third order interactions. Preliminary, multicollinearity between the independent variables is minimized and statistical tests involving the Global, Correlation Coefficient, Wald, and Goodness-of-Fit tests, are carried out to select significant variables with their possible interactions. Comparisons between the polynomial regression models (PRM are made using the eight selection criteria (8SC. The best regression model is identified based on the minimum value of the eight selection criteria (8SC. The use of an appropriate transformation will increase in the degree of a statistically valid polynomial, hence, providing a better estimation for the model.
Control to Facet for Polynomial Systems
DEFF Research Database (Denmark)
Sloth, Christoffer; Wisniewski, Rafael
2014-01-01
This paper presents a solution to the control to facet problem for arbitrary polynomial vector fields defined on simplices. The novelty of the work is to use Bernstein coefficients of polynomials for determining certificates of positivity. Specifically, the constraints that are set up for the con......This paper presents a solution to the control to facet problem for arbitrary polynomial vector fields defined on simplices. The novelty of the work is to use Bernstein coefficients of polynomials for determining certificates of positivity. Specifically, the constraints that are set up...... for the controller design are solved by searching for polynomials in Bernstein form. This allows the controller design problem to be formulated as a linear programming problem. Examples are provided that demonstrate the efficiency of the method for designing controls for polynomial systems....
Directory of Open Access Journals (Sweden)
Sohrab Bazm
2016-02-01
Full Text Available In this study, the Bernoulli polynomials are used to obtain an approximate solution of a class of nonlinear two-dimensional integral equations. To this aim, the operational matrices of integration and the product for Bernoulli polynomials are derived and utilized to reduce the considered problem to a system of nonlinear algebraic equations. Some examples are presented to illustrate the efficiency and accuracy of the method.
Methods of Approximation Theory in Complex Analysis and Mathematical Physics
Saff, Edward
1993-01-01
The book incorporates research papers and surveys written by participants ofan International Scientific Programme on Approximation Theory jointly supervised by Institute for Constructive Mathematics of University of South Florida at Tampa, USA and the Euler International Mathematical Instituteat St. Petersburg, Russia. The aim of the Programme was to present new developments in Constructive Approximation Theory. The topics of the papers are: asymptotic behaviour of orthogonal polynomials, rational approximation of classical functions, quadrature formulas, theory of n-widths, nonlinear approximation in Hardy algebras,numerical results on best polynomial approximations, wavelet analysis. FROM THE CONTENTS: E.A. Rakhmanov: Strong asymptotics for orthogonal polynomials associated with exponential weights on R.- A.L. Levin, E.B. Saff: Exact Convergence Rates for Best Lp Rational Approximation to the Signum Function and for Optimal Quadrature in Hp.- H. Stahl: Uniform Rational Approximation of x .- M. Rahman, S.K. ...
Matrix-valued polynomials in Lanczos type methods
Energy Technology Data Exchange (ETDEWEB)
Simoncini, V. [Universita di Padova (Italy); Gallopoulos, E. [Univ. of Illinois, Urbana, IL (United States)
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.
Exceptional polynomials and SUSY quantum mechanics
Indian Academy of Sciences (India)
K V S Shiv Chaitanya; S Sree Ranjani; Prasanta K Panigrahi; R Radhakrishnan; V Srinivasan
2015-07-01
We show that for the quantum mechanical problem which admit classical Laguerre/Jacobi polynomials as solutions for the Schrödinger equations (SE), will also admit exceptional Laguerre/Jacobi polynomials as solutions having the same eigenvalues but with the ground state missing after a modification of the potential. Then, we claim that the existence of these exceptional polynomials leads to the presence of non-trivial supersymmetry.
A new Arnoldi approach for polynomial eigenproblems
Energy Technology Data Exchange (ETDEWEB)
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.
Landau and Kolmogoroff type polynomial inequalities
Alves Claudia RR; Dimitrov Dimitar K
1999-01-01
Let be integers. Denote by the norm . For various positive values of and we establish Kolmogoroff type inequalities with certain constants , which hold for every ( denotes the space of real algebraic polynomials of degree not exceeding ). For the particular case and , we provide a complete characterisation of the positive constants and , for which the corresponding Landau type polynomial inequalities hold. In each case we determine the corresponding extremal polynomials for which e...
Quantum Schubert polynomials and quantum Schur functions
Kirillov, Anatol N.
1997-01-01
We introduce the quantum multi-Schur functions, quantum factorial Schur functions and quantum Macdonald polynomials. We prove that for restricted vexillary permutations the quantum double Schubert polynomial coincides with some quantum multi-Schur function and prove a quantum analog of the Nagelsbach-Kostka and Jacobi-Trudi formulae for the quantum double Schubert polynomials in the case of Grassmannian permutations. We prove, also, an analog of the Billey-Jockusch-Stanley formula for quantum...
Accelerated graph-based spectral polynomial filters
Knyazev, Andrew; Malyshev, Alexander,
2015-01-01
Graph-based spectral denoising is a low-pass filtering using the eigendecomposition of the graph Laplacian matrix of a noisy signal. Polynomial filtering avoids costly computation of the eigendecomposition by projections onto suitable Krylov subspaces. Polynomial filters can be based, e.g., on the bilateral and guided filters. We propose constructing accelerated polynomial filters by running flexible Krylov subspace based linear and eigenvalue solvers such as the Block Locally Optimal Precond...
Generalizations of Bernoulli numbers and polynomials
Qiu-Ming Luo; Bai-Ni Guo; Feng Qi; Lokenath Debnath
2003-01-01
The concepts of Bernoulli numbers Bn, Bernoulli polynomials Bn(x), and the generalized Bernoulli numbers Bn(a,b) are generalized to the one Bn(x;a,b,c) which is called the generalized Bernoulli polynomials depending on three positive real parameters. Numerous properties of these polynomials and some relationships between Bn, Bn(x), Bn(a,b), and Bn(x;a,b,c) are established.
Constructing Polynomial Spectral Models for Stars
Rix, Hans-Walter; Conroy, Charlie; Hogg, David W
2016-01-01
Stellar spectra depend on the stellar parameters and on dozens of photospheric elemental abundances. Simultaneous fitting of these $\\mathcal{N}\\sim \\,$10-40 model labels to observed spectra has been deemed unfeasible, because the number of ab initio spectral model grid calculations scales exponentially with $\\mathcal{N}$. We suggest instead the construction of a polynomial spectral model (PSM) of order $\\mathcal{O}$ for the model flux at each wavelength. Building this approximation requires a minimum of only ${\\mathcal{N}+\\mathcal{O}\\choose\\mathcal{O}}$ calculations: e.g. a quadratic spectral model ($\\mathcal{O}=\\,$2), which can then fit $\\mathcal{N}=\\,$20 labels simultaneously, can be constructed from as few as 231 ab initio spectral model calculations; in practice, a somewhat larger number ($\\sim\\,$300-1000) of randomly chosen models lead to a better performing PSM. Such a PSM can be a good approximation to ab initio spectral models only over a limited portion of label space, which will vary case by case. Y...
Constructing Polynomial Spectral Models for Stars
Rix, Hans-Walter; Ting, Yuan-Sen; Conroy, Charlie; Hogg, David W.
2016-08-01
Stellar spectra depend on the stellar parameters and on dozens of photospheric elemental abundances. Simultaneous fitting of these { N } ˜ 10-40 model labels to observed spectra has been deemed unfeasible because the number of ab initio spectral model grid calculations scales exponentially with { N }. We suggest instead the construction of a polynomial spectral model (PSM) of order { O } for the model flux at each wavelength. Building this approximation requires a minimum of only ≤ft(≥nfrac{}{}{0em}{}{{ N }+{ O }}{{ O }}\\right) calculations: e.g., a quadratic spectral model ({ O }=2) to fit { N }=20 labels simultaneously can be constructed from as few as 231 ab initio spectral model calculations; in practice, a somewhat larger number (˜300-1000) of randomly chosen models lead to a better performing PSM. Such a PSM can be a good approximation only over a portion of label space, which will vary case-by-case. Yet, taking the APOGEE survey as an example, a single quadratic PSM provides a remarkably good approximation to the exact ab initio spectral models across much of this survey: for random labels within that survey the PSM approximates the flux to within 10-3 and recovers the abundances to within ˜0.02 dex rms of the exact models. This enormous speed-up enables the simultaneous many-label fitting of spectra with computationally expensive ab initio models for stellar spectra, such as non-LTE models. A PSM also enables the simultaneous fitting of observational parameters, such as the spectrum’s continuum or line-spread function.
Asymptotics for a generalization of Hermite polynomials
Alfaro, M; Peña, A; Rezola, M L
2009-01-01
We consider a generalization of the classical Hermite polynomials by the addition of terms involving derivatives in the inner product. This type of generalization has been studied in the literature from the point of view of the algebraic properties. Thus, our aim is to study the asymptotics of this sequence of nonstandard orthogonal polynomials. In fact, we obtain Mehler--Heine type formulas for these polynomials and, as a consequence, we prove that there exists an acceleration of the convergence of the smallest positive zeros of these generalized Hermite polynomials towards the origin.
Relative risk regression models with inverse polynomials.
Ning, Yang; Woodward, Mark
2013-08-30
The proportional hazards model assumes that the log hazard ratio is a linear function of parameters. In the current paper, we model the log relative risk as an inverse polynomial, which is particularly suitable for modeling bounded and asymmetric functions. The parameters estimated by maximizing the partial likelihood are consistent and asymptotically normal. The advantages of the inverse polynomial model over the ordinary polynomial model and the fractional polynomial model for fitting various asymmetric log relative risk functions are shown by simulation. The utility of the method is further supported by analyzing two real data sets, addressing the specific question of the location of the minimum risk threshold.
Multi-indexed (q)-Racah Polynomials
Odake, Satoru
2012-01-01
As the second stage of the project $multi-indexed orthogonal polynomials$, we present, in the framework of `discrete quantum mechanics' with real shifts in one dimension, the multi-indexed (q)-Racah polynomials. They are obtained from the (q)-Racah polynomials by multiple application of the discrete analogue of the Darboux transformations or the Crum-Krein-Adler deletion of `virtual state' vectors of type I and II, in a similar way to the multi-indexed Laguerre and Jacobi polynomials reported earlier. The virtual state vectors are the `solutions' of the matrix Schr\\"odinger equation with negative `eigenvalues', except for one of the two boundary points.
On the verification of polynomial system solvers
Institute of Scientific and Technical Information of China (English)
Changbo CHEN; Marc MORENO MAZA; Wei PAN; Yuzhen XI
2008-01-01
We discuss the verification of mathematical software solving polynomial systems symbolically by way of triangular decomposition. Standard verification techniques are highly resource consuming and apply only to polynomial systems which are easy to solve. We exhibit a new approach which manipulates constructible sets represented by regular systems. We provide comparative benchmarks of different verification procedures applied to four solvers on a large set of well-known polynomial systems. Our experimental results illustrate the high effi-ciency of our new approach. In particular, we are able to verify triangular decompositions of polynomial systems which are not easy to solve.
Directory of Open Access Journals (Sweden)
Ryoo CS
2010-01-01
Full Text Available The purpose of this paper is to give some properties of several Bernstein type polynomials to represent the fermionic -adic integral on . From these properties, we derive some interesting identities on the Euler numbers and polynomials.
Genus expansion of HOMFLY polynomials
Mironov, A; Sleptsov, A
2013-01-01
In the planar limit of the 't Hooft expansion, the Wilson-loop average in 3d Chern-Simons theory (i.e. the HOMFLY polynomial) depends in a very simple way on representation (the Young diagram), so that the (knot-dependent) Ooguri-Vafa partition function becomes a trivial KP tau-function. We study higher genus corrections to this formula in the form of expansion in powers of z = q-q^{-1}. Expansion coefficients are expressed through the eigenvalues of the cut-and-join operators, i.e. symmetric group characters. Moreover, the z-expansion is naturally exponentiated. Representation through cut-and-join operators makes contact with Hurwitz theory and its sophisticated integrability properties. Our formulas describe the shape of genus expansion for the HOMFLY polynomials, which for their matrix model counterparts is usually controlled by Virasoro like constraints and AMM/EO topological recursion. The genus expansion differs from the better studied weak coupling expansion at finite number of colors N, which is descr...
Algorithms in Solving Polynomial Inequalities
Directory of Open Access Journals (Sweden)
Christopher M. Cordero
2015-11-01
Full Text Available A new method to solve the solution set of polynomial inequalities was conducted. When −1 −2 >0 ℎ 1,2∈ ℝ 10 if n is even. Then, the solution set is ∈ ℝ ∈ −∞,1 ∪ ,+∞ ∪ ,+1 : }. Thus, when −1−2…−≥0, the solution is ∈ ℝ ∈−∞, 1∪ ,+∞∪, +1: }. If is odd, then the solution set is ∈ ℝ ∈ ,+∞ ∪ ,+1 : }. Thus, when −1 −2…−≥0, the solution set is ∈ ℝ ∈ ,+∞∪, +1: }. Let −1−2…−<0 if n is even. Then, the solution set is ∈ ℝ ∈ ,+1 ∶ }. Thus, when −1 −2…−≤0, then the solution set is ∈ ℝ ∈, +1: }. If is an odd, then the solution set is ∈ ℝ ∈ −∞,1 ∪ ,+1 : }. Thus, when −1 −2 … − ≤0, the solution set is ∈ ℝ ∈ −∞,1 ∪ ,+1 : }. This research provides a novel method in solving the solution set of polynomial inequalities, in addition to other existing methods.
Assaf, Sami; Searles, Dominic
2016-01-01
We introduce two new bases for polynomials that lift monomial and fundamental quasisymmetric functions to the full polynomial ring. By defining a new condition on pipe dreams, called quasi-Yamanouchi, we give a positive combinatorial rule for expanding Schubert polynomials into these new bases that parallels the expansion of Schur functions into fundamental quasisymmetric functions. As a result, we obtain a refinement of the stable limits of Schubert polynomials to Stanley symmetric functions...
Kravchuk functions for the finite oscillator approximation
Atakishiyev, Natig M.; Wolf, Kurt Bernardo
1995-01-01
Kravchuk orthogonal functions - Kravchuk polynomials multiplied by the square root of the weight function - simplify the inversion algorithm for the analysis of discrete, finite signals in harmonic oscillator components. They can be regarded as the best approximation set. As the number of sampling points increases, the Kravchuk expansion becomes the standard oscillator expansion.
De Raedt, H; Michielsen, K; Kole, JS; Figge, MT
2003-01-01
We present a one-step algorithm that solves the Maxwell equations for systems with spatially varying permittivity and permeability by the Chebyshev method. We demonstrate that this algorithm may be orders of magnitude more efficient than current finite-difference time-domain (FDTD) algorithms.
Decomposition and Approximation of Multivariate Functions on the Cube
Institute of Scientific and Technical Information of China (English)
Zhi Hua ZHANG
2013-01-01
In this paper,we present a decomposition method of multivariate functions.This method shows that any multivariate function f on [0,1]d is a finite sum of the form ∑j φjΨj,where each φj can be extended to a smooth periodic function,each Ψj is an algebraic polynomial,and each φjΨj is a product of separated variable type and its smoothness is same as f.Since any smooth periodic function can be approximated well by trigonometric polynomials,using our decomposition method,we find that any smooth multivariate function on [0,1]d can be approximated well by a combination of algebraic polynomials and trigonometric polynomials.Meanwhile,we give a precise estimate of the approximation error.
Niven, Ivan
2008-01-01
This self-contained treatment originated as a series of lectures delivered to the Mathematical Association of America. It covers basic results on homogeneous approximation of real numbers; the analogue for complex numbers; basic results for nonhomogeneous approximation in the real case; the analogue for complex numbers; and fundamental properties of the multiples of an irrational number, for both the fractional and integral parts.The author refrains from the use of continuous fractions and includes basic results in the complex case, a feature often neglected in favor of the real number discuss
Klerk, E. de; Laurent, M.
2010-01-01
We consider the problem of minimizing a polynomial on the hypercube [0, 1]n and derive new error bounds for the hierarchy of semidefinite programming approximations to this problem corresponding to the Positivstellensatz of Schmu ̈dgen [26]. The main tool we employ is Bernstein approximations of pol
Iterative methods for simultaneous inclusion of polynomial zeros
Petković, Miodrag
1989-01-01
The simultaneous inclusion of polynomial complex zeros is a crucial problem in numerical analysis. Rapidly converging algorithms are presented in these notes, including convergence analysis in terms of circular regions, and in complex arithmetic. Parallel circular iterations, where the approximations to the zeros have the form of circular regions containing these zeros, are efficient because they also provide error estimates. There are at present no book publications on this topic and one of the aims of this book is to collect most of the algorithms produced in the last 15 years. To decrease the high computational cost of interval methods, several effective iterative processes for the simultaneous inclusion of polynomial zeros which combine the efficiency of ordinary floating-point arithmetic with the accuracy control that may be obtained by the interval methods, are set down, and their computational efficiency is described. The rate of these methods is of interest in designing a package for the simultaneous ...
2006-01-01
This interactive tutorial presents the following concepts of Approximation Techniques: Methods of Weighted Residual (MWR), Weak Formulatioin, Piecewise Continuous Function, Galerkin Finite Element FormulationExplanations especially for mathematical statements are provided using mouseover the highlight equations. ME4613 Finite Element Methods
BOUNDS FOR THE ZEROS OF POLYNOMIALS
Institute of Scientific and Technical Information of China (English)
W. M. Shah; A.Liman
2004-01-01
Let P(z) =n∑j=0 ajzj be a polynomial of degree n. In this paper we prove a more general result which interalia improves upon the bounds of a class of polynomials. We also prove a result which includes some extensions and generalizations of Enestrom-Kakeya theorem.
Distortion control of conjugacies between quadratic polynomials
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
We use a new type of distortion control of univalent functions to give an alternative proof of Douady-Hubbard’s ray-landing theorem for quadratic Misiurewicz polynomials. The univalent maps arise from Thurston’s iterated algorithm on perturbation of such polynomials.
Uniqueness of meromorphic functions concerning differential polynomials
Institute of Scientific and Technical Information of China (English)
QIAO Lei
2007-01-01
Based on a unicity theorem for entire funcitions concerning differential polynomials proposed by M. L. Fang and W. Hong, we studied the uniqueness problem of two meromorphic functions whose differential polynomials share the same 1-point by proving two theorems and their related lemmas. The results extend and improve given by Fang and Hong's theorem.
Fostering Connections between Classes of Polynomial Functions.
Buck, Judy Curran
The typical path of instruction in high school algebra courses for the study of polynomial functions has been from linear functions, to quadratic functions, to polynomial functions of degree greater than two. This paper reports results of clinical interviews with an Algebra II student. The interviews were used to probe into the student's…
On Polynomial Functions over Finite Commutative Rings
Institute of Scientific and Technical Information of China (English)
Jian Jun JIANG; Guo Hua PENG; Qi SUN; Qi Fan ZHANG
2006-01-01
Let R be an arbitrary finite commutative local ring. In this paper, we obtain a necessary and sufficient condition for a function over R to be a polynomial function. Before this paper, necessary and sufficient conditions for a function to be a polynomial function over some special finite commutative local rings were obtained.
Point vortex equilibria related to Bessel polynomials
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.
Large degree asymptotics of generalized Bessel polynomials
López, J.L.; Temme, N.M.
2011-01-01
Asymptotic expansions are given for large values of $n$ of the generalized Bessel polynomials $Y_n^\\mu(z)$. The analysis is based on integrals that follow from the generating functions of the polynomials. A new simple expansion is given that is valid outside a compact neighborhood of the origin in t
The Approach of Moments for Polynomial Equations
Laurent, M.; Rostalski, P.
2010-01-01
In this article we present the moment based approach for computing all real solutions of a given system of polynomial equations. This approach builds upon a lifting method for constructing semidefinite relaxations of several nonconvex optimization problems, using sums of squares of polynomials and t
A generalization of the Bernoulli polynomials
Pierpaolo Natalini; Angela Bernardini
2003-01-01
A generalization of the Bernoulli polynomials and, consequently, of the Bernoulli numbers, is defined starting from suitable generating functions. Furthermore, the differential equations of these new classes of polynomials are derived by means of the factorization method introduced by Infeld and Hull (1951).
A generalization of the Bernoulli polynomials
Directory of Open Access Journals (Sweden)
Pierpaolo Natalini
2003-01-01
Full Text Available A generalization of the Bernoulli polynomials and, consequently, of the Bernoulli numbers, is defined starting from suitable generating functions. Furthermore, the differential equations of these new classes of polynomials are derived by means of the factorization method introduced by Infeld and Hull (1951.
Reliability polynomials crossing more than twice
Brown, J.I.; Koç, Y.; Kooij, R.E.
2011-01-01
In this paper we study all-terminal reliability polynomials of networks having the same number of nodes and the same number of links. First we show that the smallest possible size for a pair of networks that allows for two crossings of their reliability polynomials have seven nodes and fifteen edges
Indian Academy of Sciences (India)
V K Jain
2009-02-01
For a polynomial of degree , we have obtained an upper bound involving coefficients of the polynomial, for moduli of its zeros of smallest moduli, and then a refinement of the well-known Eneström–Kakeya theorem (under certain conditions).
Sums of Powers of Fibonacci Polynomials
Indian Academy of Sciences (India)
Helmut Prodinger
2009-11-01
Using the explicit (Binet) formula for the Fibonacci polynomials, a summation formula for powers of Fibonacci polynomials is derived straightforwardly, which generalizes a recent result for squares that appeared in Proc. Ind. Acad. Sci. (Math. Sci.) 118 (2008) 27--41.
A Note on Solvable Polynomial Algebras
Directory of Open Access Journals (Sweden)
Huishi Li
2014-03-01
Full Text Available In terms of their defining relations, solvable polynomial algebras introduced by Kandri-Rody and Weispfenning [J. Symbolic Comput., 9(1990] are characterized by employing Gr\\"obner bases of ideals in free algebras, thereby solvable polynomial algebras are completely determinable and constructible in a computational way.
Polynomial invariants of quantum codes
Rains, E M
1997-01-01
The weight enumerators (quant-ph/9610040) of a quantum code are quite powerful tools for exploring its structure. As the weight enumerators are quadratic invariants of the code, this suggests the consideration of higher-degree polynomial invariants. We show that the space of degree k invariants of a code of length n is spanned by a set of basic invariants in one-to-one correspondence with S_k^n. We then present a number of equations and inequalities in these invariants; in particular, we give a higher-order generalization of the shadow enumerator of a code, and prove that its coefficients are nonnegative. We also prove that the quartic invariants of a ((4,4,2)) are uniquely determined, an important step in a proof that any ((4,4,2)) is additive ([2]).
Algebras, dialgebras, and polynomial identities
Bremner, Murray R
2012-01-01
This is a survey of some recent developments in the theory of associative and nonassociative dialgebras, with an emphasis on polynomial identities and multilinear operations. We discuss associative, Lie, Jordan, and alternative algebras, and the corresponding dialgebras; the KP algorithm for converting identities for algebras into identities for dialgebras; the BSO algorithm for converting operations in algebras into operations in dialgebras; Lie and Jordan triple systems, and the corresponding disystems; and a noncommutative version of Lie triple systems based on the trilinear operation abc-bca. The paper concludes with a conjecture relating the KP and BSO algorithms, and some suggestions for further research. Most of the original results are joint work with Raul Felipe, Luiz A. Peresi, and Juana Sanchez-Ortega.
Fuzzy Morphological Polynomial Image Representation
Directory of Open Access Journals (Sweden)
Chin-Pan Huang
2010-01-01
Full Text Available A novel signal representation using fuzzy mathematical morphology is developed. We take advantage of the optimum fuzzy fitting and the efficient implementation of morphological operators to extract geometric information from signals. The new representation provides results analogous to those given by the polynomial transform. Geometrical decomposition of a signal is achieved by windowing and applying sequentially fuzzy morphological opening with structuring functions. The resulting representation is made to resemble an orthogonal expansion by constraining the results of opening to equate adapted structuring functions. Properties of the geometric decomposition are considered and used to calculate the adaptation parameters. Our procedure provides an efficient and flexible representation which can be efficiently implemented in parallel. The application of the representation is illustrated in data compression and fractal dimension estimation temporal signals and images.
赋β-范空间中的最佳逼近问题%The Problems of Best Approximation in β-Normed Spaces(0＜β＜1)
Institute of Scientific and Technical Information of China (English)
王见勇
2008-01-01
This paper deals with the problems of best approximation in β-normed spaces.With the tool of conjugate cone introduced in [1] and via the Hahn-Banach extension theorem of β-subseminorm in [2],the characteristics that an element in a closed subspace is the best approximation are given in Section 2.It is obtained in Section 3 that all convex sets or subspaces of a β-normed space are semi-Chebyshev if and only if the space is itself strictly convex.The fact that every finite dimensional subspace of a strictly convex β-normed space must be Chebyshev is proved at last.
Sobolev orthogonal polynomials on a simplex
Aktas, Rabia
2011-01-01
The Jacobi polynomials on the simplex are orthogonal polynomials with respect to the weight function $W_\\bg(x) = x_1^{\\g_1} ... x_d^{\\g_d} (1- |x|)^{\\g_{d+1}}$ when all $\\g_i > -1$ and they are eigenfunctions of a second order partial differential operator $L_\\bg$. The singular cases that some, or all, $\\g_1,...,\\g_{d+1}$ are -1 are studied in this paper. Firstly a complete basis of polynomials that are eigenfunctions of $L_\\bg$ in each singular case is found. Secondly, these polynomials are shown to be orthogonal with respect to an inner product which is explicitly determined. This inner product involves derivatives of the functions, hence the name Sobolev orthogonal polynomials.
Polynomials with Palindromic and Unimodal Coeﬃ cients
Institute of Scientific and Technical Information of China (English)
Hua SUN; Yi WANG; Hai Xia ZHANG
2015-01-01
Let f(q) = arqr +· · ·+asqs, with ar = 0 and as = 0, be a real polynomial. It is a palindromic polynomial of darga n if r+s = n and ar+i = as−i for all i. Polynomials of darga n form a linear subspace Pn(q) of R(q)n+1 of dimension ? n2 ?+1. We give transition matrices between two bases ?qj(1+q+· · ·+qn−2j)? , ?qj(1+q)n−2j? and the standard basis ?qj(1+qn−2j)? of Pn(q). We present some characterizations and sufficcient conditions for palindromic polynomials that can be expressed in terms of these two bases with nonnegative coefficients. We also point out the link between such polynomials and rank-generating functions of posets.
Tutte polynomial in functional magnetic resonance imaging
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.
Orthogonal Polynomials from Hermitian Matrices II
Odake, Satoru
2016-01-01
This is the second part of the project `unified theory of classical orthogonal polynomials of a discrete variable derived from the eigenvalue problems of hermitian matrices.' In a previous paper, orthogonal polynomials having Jackson integral measures were not included, since such measures cannot be obtained from single infinite dimensional hermitian matrices. Here we show that Jackson integral measures for the polynomials of the big $q$-Jacobi family are the consequence of the recovery of self-adjointness of the unbounded Jacobi matrices governing the difference equations of these polynomials. The recovery of self-adjointness is achieved in an extended $\\ell^2$ Hilbert space on which a direct sum of two unbounded Jacobi matrices acts as a Hamiltonian or a difference Schr\\"odinger operator for an infinite dimensional eigenvalue problem. The polynomial appearing in the upper/lower end of Jackson integral constitutes the eigenvector of each of the two unbounded Jacobi matrix of the direct sum. We also point out...
Matrix product formula for Macdonald polynomials
International Nuclear Information System (INIS)
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. (paper)
Wang, Deming; Yang, Zhengyi
2008-03-01
The use of polynomial functions for modeling geometric distortion in magnetic resonance imaging (MRI) that arises from scanner's hardware imperfection is studied in detail. In this work, the geometric distortion data from four representative MRI systems were used. Modeling of these data using polynomial functions of the fourth, fifth, sixth, and seventh orders was carried out. In order to investigate how this modeling performed for different size and shape of the volume of interest, the modeling was carried out for three different volumes of interest (VOI): a cube, a cylinder, and a sphere. The modeling's goodness was assessed using both the maximum and mean absolute errors. The modeling results showed that (i) for the cube VOI there appears to be an optimal polynomial function that gives the least modeling errors and the sixth order polynomial was found to be the optimal polynomial function for the size of the cubic VOI considered in the present work; (ii) for the cylinder VOI, all four polynomials performed approximately equally well but a trend of a slight decrease in the mean absolute error with the increasing order of the polynomial was noted; and (iii) for the sphere VOI, the maximum absolute error showed some variations with the order of the polynomial, with the fourth order polynomial producing the smallest maximum absolute errors. It is further noted that extrapolation could lead to very large errors so any extrapolation needs to be avoided. A detailed analysis on the modeling errors is presented.
Välimäki, Vesa; Pekonen, Jussi; Nam, Juhan
2012-01-01
Digital subtractive synthesis is a popular music synthesis method, which requires oscillators that are aliasing-free in a perceptual sense. It is a research challenge to find computationally efficient waveform generation algorithms that produce similar-sounding signals to analog music synthesizers but which are free from audible aliasing. A technique for approximately bandlimited waveform generation is considered that is based on a polynomial correction function, which is defined as the difference of a non-bandlimited step function and a polynomial approximation of the ideal bandlimited step function. It is shown that the ideal bandlimited step function is equivalent to the sine integral, and that integrated polynomial interpolation methods can successfully approximate it. Integrated Lagrange interpolation and B-spline basis functions are considered for polynomial approximation. The polynomial correction function can be added onto samples around each discontinuity in a non-bandlimited waveform to suppress aliasing. Comparison against previously known methods shows that the proposed technique yields the best tradeoff between computational cost and sound quality. The superior method amongst those considered in this study is the integrated third-order B-spline correction function, which offers perceptually aliasing-free sawtooth emulation up to the fundamental frequency of 7.8 kHz at the sample rate of 44.1 kHz.
Bernoulli-like polynomials associated with Stirling Numbers
Bender, Carl M; Brody, Dorje C.; BERNHARD K. MEISTER
2005-01-01
The Stirling numbers of the first kind can be represented in terms of a new class of polynomials that are closely related to the Bernoulli polynomials. Recursion relations for these polynomials are given.
HIGHER ORDER MULTIVARIABLE NORLUND EULER-BERNOULLI POLYNOMIALS
Institute of Scientific and Technical Information of China (English)
刘国栋
2002-01-01
The definitions of higher order multivariable Norlund Euler polynomials and Norlund Bernoulli polynomials are presented and some of their important properties are expounded. Some identities involving recurrence sequences and higher order multivariable Norlund Euler-Bernoulli polynomials are established.
Zhang, Yiqiang; Alexander, J. I. D.; Ouazzani, J.
1994-01-01
Free and moving boundary problems require the simultaneous solution of unknown field variables and the boundaries of the domains on which these variables are defined. There are many technologically important processes that lead to moving boundary problems associated with fluid surfaces and solid-fluid boundaries. These include crystal growth, metal alloy and glass solidification, melting and name propagation. The directional solidification of semi-conductor crystals by the Bridgman-Stockbarger method is a typical example of such a complex process. A numerical model of this growth method must solve the appropriate heat, mass and momentum transfer equations and determine the location of the melt-solid interface. In this work, a Chebyshev pseudospectra collocation method is adapted to the problem of directional solidification. Implementation involves a solution algorithm that combines domain decomposition, finite-difference preconditioned conjugate minimum residual method and a Picard type iterative scheme.
Directory of Open Access Journals (Sweden)
S. S. Motsa
2014-01-01
Full Text Available This paper presents a new method for solving higher order nonlinear evolution partial differential equations (NPDEs. The method combines quasilinearisation, the Chebyshev spectral collocation method, and bivariate Lagrange interpolation. In this paper, we use the method to solve several nonlinear evolution equations, such as the modified KdV-Burgers equation, highly nonlinear modified KdV equation, Fisher's equation, Burgers-Fisher equation, Burgers-Huxley equation, and the Fitzhugh-Nagumo equation. The results are compared with known exact analytical solutions from literature to confirm accuracy, convergence, and effectiveness of the method. There is congruence between the numerical results and the exact solutions to a high order of accuracy. Tables were generated to present the order of accuracy of the method; convergence graphs to verify convergence of the method and error graphs are presented to show the excellent agreement between the results from this study and the known results from literature.
Institute of Scientific and Technical Information of China (English)
无
2003-01-01
The design and analysis of special type beamformer, the Butler matrix, to achieve orthogonal beamforming networks is presented in this paper. A 4×4 microstrip planar array antenna is assumed to simulate a 4×4 Butler matrix to demonstrate orthogonal beamforming and beam steering. The dimensions of rectangular patches in the planar array are chosen according to the Dolph-Chebyshev current distribution in order to minimize the side-lobe level ratio for a given value of beamwidth. The simulations are carried out using an antenna design and analysis software PCAAD. It is shown that orthogonal beams can be formed to cover about 163° angle with a constant beam crossover level and high directivity.
A third-order multistep time discretization for a Chebyshev tau spectral method
Vreman, A. W.; Kuerten, J. G. M.
2016-01-01
A time discretization scheme based on the third-order backward difference formula has been embedded into a Chebyshev tau spectral method for the Navier-Stokes equations. The time discretization is a variant of the second-order backward scheme proposed by Krasnov et al. (2008) [3]. High-resolution direct numerical simulations of turbulent incompressible channel flow have been performed to compare the backward scheme to the Runge-Kutta scheme proposed by Spalart et al. (1991) [2]. It is shown that the Runge-Kutta scheme leads to a poor convergence of some third-order spatial derivatives in the direct vicinity of the wall, derivatives that represent the diffusion of wall-tangential vorticity. The convergence at the wall is shown to be significantly improved if the backward scheme is applied.
On the Approximation and Smoothed Complexity of Leontief Market Equilibria
Huang, Li-Sha; Teng, Shang-Hua
2006-01-01
We show that the problem of finding an \\epsilon-approximate Nash equilibrium of an n by n two-person games can be reduced to the computation of an (\\epsilon/n)^2-approximate market equilibrium of a Leontief economy. Together with a recent result of Chen, Deng and Teng, this polynomial reduction implies that the Leontief market exchange problem does not have a fully polynomial-time approximation scheme, that is, there is no algorithm that can compute an \\epsilon-approximate market equilibrium ...
Polynomial Interpolation in the Elliptic Curve Cryptosystem
Directory of Open Access Journals (Sweden)
Liew K. Jie
2011-01-01
Full Text Available Problem statement: In this research, we incorporate the polynomial interpolation method in the discrete logarithm problem based cryptosystem which is the elliptic curve cryptosystem. Approach: In this study, the polynomial interpolation method to be focused is the Lagrange polynomial interpolation which is the simplest polynomial interpolation method. This method will be incorporated in the encryption algorithm of the elliptic curve ElGamal cryptosystem. Results: The scheme modifies the elliptic curve ElGamal cryptosystem by adding few steps in the encryption algorithm. Two polynomials are constructed based on the encrypted points using Lagrange polynomial interpolation and encrypted for the second time using the proposed encryption method. We believe it is safe from the theoretical side as it still relies on the discrete logarithm problem of the elliptic curve. Conclusion/Recommendations: The modified scheme is expected to be more secure than the existing scheme as it offers double encryption techniques. On top of the existing encryption algorithm, we managed to encrypt one more time using the polynomial interpolation method. We also have provided detail examples based on the described algorithm.
Approximate Representations and Approximate Homomorphisms
Moore, Cristopher
2010-01-01
Approximate algebraic structures play a defining role in arithmetic combinatorics and have found remarkable applications to basic questions in number theory and pseudorandomness. Here we study approximate representations of finite groups: functions f:G -> U_d such that Pr[f(xy) = f(x) f(y)] is large, or more generally Exp_{x,y} ||f(xy) - f(x)f(y)||^2$ is small, where x and y are uniformly random elements of the group G and U_d denotes the unitary group of degree d. We bound these quantities in terms of the ratio d / d_min where d_min is the dimension of the smallest nontrivial representation of G. As an application, we bound the extent to which a function f : G -> H can be an approximate homomorphism where H is another finite group. We show that if H's representations are significantly smaller than G's, no such f can be much more homomorphic than a random function. We interpret these results as showing that if G is quasirandom, that is, if d_min is large, then G cannot be embedded in a small number of dimensi...
Directory of Open Access Journals (Sweden)
Muhammed Çetin
2015-01-01
Full Text Available An approximation method based on Lucas polynomials is presented for the solution of the system of high-order linear differential equations with variable coefficients under the mixed conditions. This method transforms the system of ordinary differential equations (ODEs to the linear algebraic equations system by expanding the approximate solutions in terms of the Lucas polynomials with unknown coefficients and by using the matrix operations and collocation points. In addition, the error analysis based on residual function is developed for present method. To demonstrate the efficiency and accuracy of the method, numerical examples are given with the help of computer programmes written in Maple and Matlab.
Bounds for variable degree rational L∞ approximations to the matrix cosine
Tsitouras, Ch.; Katsikis, V. N.
2014-11-01
In this work we derive new alternatives for efficient computation of the matrix cosine which is useful when solving second order Initial Value Problems such as free vibration. We focus especially on the two classes of normal and nonnegative matrices and we present intervals of applications for rational L∞ approximations of various degrees for these types of matrices in the lines of Hargreaves and Higham (2005). Our method relies on Remez algorithm for rational approximation while the innovation here is the choice of the starting set of non-symmetrical Chebyshev points. Only one Remez iteration is then usually enough to quickly approach the actual L∞ approximant.
Polynomial threshold functions and Boolean threshold circuits
DEFF Research Database (Denmark)
Hansen, Kristoffer Arnsfelt; Podolskii, Vladimir V.
2013-01-01
We study the complexity of computing Boolean functions on general Boolean domains by polynomial threshold functions (PTFs). A typical example of a general Boolean domain is 12n . We are mainly interested in the length (the number of monomials) of PTFs, with their degree and weight being...... of secondary interest. We show that PTFs on general Boolean domains are tightly connected to depth two threshold circuits. Our main results in regard to this connection are: PTFs of polynomial length and polynomial degree compute exactly the functions computed by THRMAJ circuits. An exponential length lower...
On function compositions that are polynomials
Aichinger, Erhard
2015-01-01
For a polynomial map $\\tupBold{f} : k^n \\to k^m$ ($k$ a field), we investigate those polynomials $g \\in k[t_1,\\ldots, t_n]$ that can be written as a composition $g = h \\circ \\tupBold{f}$, where $h: k^m \\to k$ is an arbitrary function. In the case that $k$ is algebraically closed of characteristic~$0$ and $\\tupBold{f}$ is surjective, we will show that $g = h \\circ \\tupBold{f}$ implies that $h$ is a polynomial.
More on rotations as spin matrix polynomials
Energy Technology Data Exchange (ETDEWEB)
Curtright, Thomas L. [Department of Physics, University of Miami, Coral Gables, Florida 33124-8046 (United States)
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 virial theorem for nonlinear problems
Energy Technology Data Exchange (ETDEWEB)
Amore, Paolo [Facultad de Ciencias, Universidad de Colima, Bernal DIaz del Castillo 340, Colima (Mexico); Fernandez, Francisco M [INIFTA (UNLP, CCT La Plata-CONICET), Division Quimica Teorica, Blvd 113 S/N, Sucursal 4, Casilla de Correo 16, 1900 La Plata (Argentina)], E-mail: paolo.amore@gmail.com, E-mail: fernande@quimica.unlp.edu.ar
2009-09-15
We show that the virial theorem provides a useful simple tool for approximating nonlinear problems. In particular, we consider conservative nonlinear oscillators and obtain the same main result derived earlier from the expansion in Chebyshev polynomials. (letters and comments)
CERN. Geneva
2015-01-01
Most physics results at the LHC end in a likelihood ratio test. This includes discovery and exclusion for searches as well as mass, cross-section, and coupling measurements. The use of Machine Learning (multivariate) algorithms in HEP is mainly restricted to searches, which can be reduced to classification between two fixed distributions: signal vs. background. I will show how we can extend the use of ML classifiers to distributions parameterized by physical quantities like masses and couplings as well as nuisance parameters associated to systematic uncertainties. This allows for one to approximate the likelihood ratio while still using a high dimensional feature vector for the data. Both the MEM and ABC approaches mentioned above aim to provide inference on model parameters (like cross-sections, masses, couplings, etc.). ABC is fundamentally tied Bayesian inference and focuses on the “likelihood free” setting where only a simulator is available and one cannot directly compute the likelihood for the dat...
Schmidt, Wolfgang M
1980-01-01
"In 1970, at the U. of Colorado, the author delivered a course of lectures on his famous generalization, then just established, relating to Roth's theorem on rational approxi- mations to algebraic numbers. The present volume is an ex- panded and up-dated version of the original mimeographed notes on the course. As an introduction to the author's own remarkable achievements relating to the Thue-Siegel-Roth theory, the text can hardly be bettered and the tract can already be regarded as a classic in its field."(Bull.LMS) "Schmidt's work on approximations by algebraic numbers belongs to the deepest and most satisfactory parts of number theory. These notes give the best accessible way to learn the subject. ... this book is highly recommended." (Mededelingen van het Wiskundig Genootschap)
Modular polynomials via isogeny volcanoes
Broker, Reinier; Sutherland, Andrew V
2010-01-01
We present a new algorithm to compute the classical modular polynomial Phi_n in the rings Z[X,Y] and (Z/mZ)[X,Y], for a prime n and any positive integer m. Our approach uses the graph of n-isogenies to efficiently compute Phi_n mod p for many primes p of a suitable form, and then applies the Chinese Remainder Theorem (CRT). Under the Generalized Riemann Hypothesis (GRH), we achieve an expected running time of O(n^3 (log n)^3 log log n), and compute Phi_n mod m using O(n^2 (log n)^2 + n^2 log m) space. We have used the new algorithm to compute Phi_n with n over 5000, and Phi_n mod m with n over 20000. We also consider several modular functions g for which Phi_n^g is smaller than Phi_n, allowing us to handle n over 60000.
The Jones polynomial: quantum algorithms and applications in quantum complexity theory
Yard, J; Yard, Jon; Wocjan, Pawel
2006-01-01
We analyze relationships between the Jones polynomial and quantum computation. Our first result is a polynomial-time quantum algorithm which gives an additive approximation of the Jones polynomial, in the sense of Bordewich, Freedman, Lovasz and Welsh, of any link obtained from a certain general family of closures of braids, evaluated at any primitive root of unity. This family encompasses the well-known plat and trace closures, generalizing results recently obtained by Aharonov, Jones and Landau. We base our algorithm on a local qubit implementation of the unitary Jones-Wenzl representations of the braid group which makes the underlying representation theory apparent, allowing us to provide an algorithm for approximating the HOMFLYPT two-variable polynomial of the trace closure of a braid at certain pairs of values as well. Next, we provide a self-contained proof that any quantum computation can be replaced by an additive approximation of the Jones polynomial, evaluated at almost any primitive root of unity....
Institute of Scientific and Technical Information of China (English)
刘天宝; 吕显瑞
2011-01-01
提出一族求解非线性方程的修正Chebyshev-Halley迭代方法.该方法避免了计算函数的二阶导数,且具有至少三阶收敛的性质,当参数选取特殊值时,可以得到四阶收敛方法.收敛性分析和数值实验结果表明,该方法与具有同阶收敛性质的算法相比效率更高.%We presented a family of modified Chebyshev-Halley iterative methods for nonlinear equations, themethods avoid calculating the second derivative, and have third-order convergent at least, if we choose somevalues of the parameters, these methods will be of order four. Several examples were given to illustrate theefficiency and stability of the new methods.
Curves and surfaces represented by polynomial support functions
DEFF Research Database (Denmark)
Sir, Z.; Gravesen, Jens; Juttler, B.
2008-01-01
This paper studies shapes (curves and surfaces) which can be described by (piecewise) polynomial support functions. The class of these shapes is closed under convolutions, offsetting, rotations and translations. We give a geometric discussion of these shapes and present methods for the approximat...... for the approximation of general curves and surfaces by them. Based on the rich theory of spherical spline functions, this leads to computational techniques for rational curves and surfaces with rational offsets, which can deal with shapes without inflections/parabolic points....
Superconformal minimal models and admissible Jack polynomials
Blondeau-Fournier, Olivier; Ridout, David; Wood, Simon
2016-01-01
We give new proofs of the rationality of the N=1 superconformal minimal model vertex operator superalgebras and of the classification of their modules in both the Neveu-Schwarz and Ramond sectors. For this, we combine the standard free field realisation with the theory of Jack symmetric functions. A key role is played by Jack symmetric polynomials with a certain negative parameter that are labelled by admissible partitions. These polynomials are shown to describe free fermion correlators, suitably dressed by a symmetrising factor. The classification proofs concentrate on explicitly identifying Zhu's algebra and its twisted analogue. Interestingly, these identifications do not use an explicit expression for the non-trivial vacuum singular vector. While the latter is known to be expressible in terms of an Uglov symmetric polynomial or a linear combination of Jack superpolynomials, it turns out that standard Jack polynomials (and functions) suffice to prove the classification.
Inequalities for a polynomial and its derivative
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 z0, then for 0Yadav and Pukhta [K.K. Dewan, R.S. Yadav, M.S. Pukhta, Inequalities for a polynomial and its derivative, Math. Inequal. Appl. 2 (2) (1999) 203-205] proved Equality holds for the polynomial where n is a multiple of [mu]E In this paper, we obtain an improvement of the above inequality by involving some of the coefficients. As an application of our result, we further improve upon a result recently proved by Aziz and Shah [A. Aziz, W.M. Shah, Inequalities for a polynomial and its derivative, Math. Inequal. Appl. 7 (3) (2004) 379-391].
Tutte Polynomial of Scale-Free Networks
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.
Twisted Polynomials and Forgery Attacks on GCM
DEFF Research Database (Denmark)
Abdelraheem, Mohamed Ahmed A. M. A.; Beelen, Peter; Bogdanov, Andrey;
2015-01-01
nonce misuse resistance, such as POET. The algebraic structure of polynomial hashing has given rise to security concerns: At CRYPTO 2008, Handschuh and Preneel describe key recovery attacks, and at FSE 2013, Procter and Cid provide a comprehensive framework for forgery attacks. Both approaches rely...... heavily on the ability to construct forgery polynomials having disjoint sets of roots, with many roots (“weak keys”) each. Constructing such polynomials beyond naïve approaches is crucial for these attacks, but still an open problem. In this paper, we comprehensively address this issue. We propose to use...... in an improved key recovery algorithm. As cryptanalytic applications of our twisted polynomials, we develop the first universal forgery attacks on GCM in the weak-key model that do not require nonce reuse. Moreover, we present universal weak-key forgeries for the nonce-misuse resistant AE scheme POET, which...
Transversals of Complex Polynomial Vector Fields
DEFF Research Database (Denmark)
Dias, Kealey
Vector fields in the complex plane are defined by assigning the vector determined by the value P(z) to each point z in the complex plane, where P is a polynomial of one complex variable. We consider special families of so-called rotated vector fields that are determined by a polynomial multiplied...... by rotational constants. Transversals are a certain class of curves for such a family of vector fields that represent the bifurcation states for this family of vector fields. More specifically, transversals are curves that coincide with a homoclinic separatrix for some rotation of the vector field. Given...... a concrete polynomial, it seems to take quite a bit of work to prove that it is generic, i.e. structurally stable. This has been done for a special class of degree d polynomial vector fields having simple equilibrium points at the d roots of unity, d odd. In proving that such vector fields are generic...
Characteristic Polynomials of Sample Covariance Matrices
Kösters, Holger
2009-01-01
We investigate the second-order correlation function of the characteristic polynomial of a sample covariance matrix. Starting from an explicit formula for the generating function, we re-obtain several well-known kernels from random matrix theory.
Local Polynomial Estimation of Distribution Functions
Institute of Scientific and Technical Information of China (English)
LI Yong-hong; ZENG Xia
2007-01-01
Under the condition that the total distribution function is continuous and bounded on (-∞,∞), we constructed estimations for distribution and hazard functions with local polynomial method, and obtained the rate of strong convergence of the estimations.
Concentration for noncommutative polynomials in random matrices
Meckes, Mark W.; Szarek, Stanislaw J.
2011-01-01
We present a concentration inequality for linear functionals of noncommutative polynomials in random matrices. Our hypotheses cover most standard ensembles, including Gaussian matrices, matrices with independent uniformly bounded entries and unitary or orthogonal matrices.
Bergman orthogonal polynomials and the Grunsky matrix
Beckermann, Bernhard; Stylianopoulos, Nikos
2016-01-01
By exploiting a link between Bergman orthogonal polynomials and the Grunsky matrix, probably first observed by Kühnau in 1985, we improve some recent results on strong asymptotics of Bergman polynomials outside the domain G of orthogonality, and on entries of the Bergman shift operator. In our proofs we suggest a new matrix approach involving the Grunsky matrix, and use well-established results in the literature relating properties of the Grunsky matrix to the regularity of the boundary of G,...
Positive maps, positive polynomials and entanglement witnesses
Skowronek, Lukasz
2009-01-01
We link the study of positive quantum maps, block positive operators, and entanglement witnesses with problems related to multivariate polynomials. For instance, we show how indecomposable block positive operators relate to biquadratic forms that are not sums of squares. Although the general problem of describing the set of positive maps remains open, in some particular cases we solve the corresponding polynomial inequalities and obtain explicit conditions for positivity.
Positive maps, positive polynomials and entanglement witnesses
Energy Technology Data Exchange (ETDEWEB)
Skowronek, Lukasz; Zyczkowski, Karol [Institute of Physics, Jagiellonian University, Krakow (Poland)], E-mail: lukasz.skowronek@uj.edu.pl, E-mail: karol@tatry.if.uj.edu.pl
2009-08-14
We link the study of positive quantum maps, block positive operators and entanglement witnesses with problems related to multivariate polynomials. For instance, we show how indecomposable block positive operators relate to biquadratic forms that are not sums of squares. Although the general problem of describing the set of positive maps remains open, in some particular cases we solve the corresponding polynomial inequalities and obtain explicit conditions for positivity.
Equivalence of polynomial conjectures in additive combinatorics
Lovett, Shachar
2010-01-01
We study two conjectures in additive combinatorics. The first is the polynomial Freiman-Ruzsa conjecture, which relates to the structure of sets with small doubling. The second is the inverse Gowers conjecture for $U^3$, which relates to functions which locally look like quadratics. In both cases a weak form, with exponential decay of parameters is known, and a strong form with only a polynomial loss of parameters is conjectured. Our main result is that the two conjectures are in fact equivalent.
On Sharing, Memoization, and Polynomial Time
Avanzini, Martin; Dal Lago, Ugo
2015-01-01
We study how the adoption of an evaluation mechanism with sharing and memoization impacts the class of functions which can be computed in polynomial time. We first show how a natural cost model in which lookup for an already computed result has no cost is indeed invariant. As a corollary, we then prove that the most general notion of ramified recurrence is sound for polynomial time, this way settling an open problem in implicit computational complexity.
Stochastic processes with orthogonal polynomial eigenfunctions
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.
Quantum group invariants and link polynomials
International Nuclear Information System (INIS)
A general method is developed for constructing quantum group invariants and determining their eigenvalues. Applied to the universal R-matrix this method leads to the construction of a closed formula for link polynomials. To illustrate the application of this formula, the quantum groups Uq(E8), Uq(so(2m+1)) and Uq(gl(m)) are considered as examples, and corresponding link polynomials are obtained. (orig.)
ON ABEL-GONTSCHAROFF-GOULD'S POLYNOMIALS
Institute of Scientific and Technical Information of China (English)
He Tianxiao; Leetsch C. Hsu; Peter J. S. Shiue
2003-01-01
In this paper a connective study of Gould's annihilation coefficients and Abel-Gontscharoff polynomials is presented. It is shown that Gould's annihilation coefficients and Abel-Gontscharoff polynomials are actually equivalent to each other under certain linear substitutions for the variables. Moreover, a pair of related expansion formulas involving Gontscharoff's remainder and a new form of it are demonstrated, and also illustrated with several examples.
Local fibred right adjoints are polynomial
DEFF Research Database (Denmark)
Kock, Anders; Kock, Joachim
2013-01-01
For any locally cartesian closed category E, we prove that a local fibred right adjoint between slices of E is given by a polynomial. The slices in question are taken in a well known fibred sense......For any locally cartesian closed category E, we prove that a local fibred right adjoint between slices of E is given by a polynomial. The slices in question are taken in a well known fibred sense...
Ferrers Matrices Characterized by the Rook Polynomials
Institute of Scientific and Technical Information of China (English)
MAHai-cheng; HUSheng-biao
2003-01-01
In this paper,we show that there exist precisely W(A) Ferrers matrices F(C1,C2,…,cm)such that the rook polynomials is equal to the rook polynomial of Ferrers matrix F(b1,b2,…,bm), where A={b1,b2-1,…,bm-m+1} is a repeated set,W(A) is weight of A.
Laguerre polynomials method in the valon model
Boroun, G R
2014-01-01
We used the Laguerre polynomials method for determination of the proton structure function in the valon model. We have examined the applicability of the valon model with respect to a very elegant method, where the structure of the proton is determined by expanding valon distributions and valon structure functions on Laguerre polynomials. We compared our results with the experimental data, GJR parameterization and DL model. Having checked, this method gives a good description for the proton structure function in valon model.
Vector-Valued Jack Polynomials from Scratch
Directory of Open Access Journals (Sweden)
Jean-Gabriel Luque
2011-03-01
Full Text Available Vector-valued Jack polynomials associated to the symmetric group S_N are polynomials with multiplicities in an irreducible module of S_N and which are simultaneous eigenfunctions of the Cherednik-Dunkl operators with some additional properties concerning the leading monomial. These polynomials were introduced by Griffeth in the general setting of the complex reflections groups G(r,p,N and studied by one of the authors (C. Dunkl in the specialization r=p=1 (i.e. for the symmetric group. By adapting a construction due to Lascoux, we describe an algorithm allowing us to compute explicitly the Jack polynomials following a Yang-Baxter graph. We recover some properties already studied by C. Dunkl and restate them in terms of graphs together with additional new results. In particular, we investigate normalization, symmetrization and antisymmetrization, polynomials with minimal degree, restriction etc. We give also a shifted version of the construction and we discuss vanishing properties of the associated polynomials.
Diffusion tensor image registration using polynomial expansion
International Nuclear Information System (INIS)
In this paper, we present a deformable registration framework for the diffusion tensor image (DTI) using polynomial expansion. The use of polynomial expansion in image registration has previously been shown to be beneficial due to fast convergence and high accuracy. However, earlier work was developed only for 3D scalar medical image registration. In this work, it is shown how polynomial expansion can be applied to DTI registration. A new measurement is proposed for DTI registration evaluation, which seems to be robust and sensitive in evaluating the result of DTI registration. We present the algorithms for DTI registration using polynomial expansion by the fractional anisotropy image, and an explicit tensor reorientation strategy is inherent to the registration process. Analytic transforms with high accuracy are derived from polynomial expansion and used for transforming the tensor's orientation. Three measurements for DTI registration evaluation are presented and compared in experimental results. The experiments for algorithm validation are designed from simple affine deformation to nonlinear deformation cases, and the algorithms using polynomial expansion give a good performance in both cases. Inter-subject DTI registration results are presented showing the utility of the proposed method. (paper)
Reliable Computational Predictions by Modeling Uncertainties Using Arbitrary Polynomial Chaos
Witteveen, J.A.S.; Bijl, H
2006-01-01
Inherent physical uncertainties can have a significant influence on computational predictions. It is therefore important to take physical uncertainties into account to obtain more reliable computational predictions. The Galerkin polynomial chaos method is a commonly applied uncertainty quantification method. However, the polynomial chaos expansion has some limitations. Firstly, the polynomial chaos expansion based on classical polynomials can achieve exponential convergence for a limited set ...
Directory of Open Access Journals (Sweden)
Chih-Hong Lin
2016-06-01
Full Text Available A permanent magnet (PM synchronous generator system driven by wind turbine (WT, connected with smart grid via AC-DC converter and DC-AC converter, are controlled by the novel recurrent Chebyshev neural network (NN and amended particle swarm optimization (PSO to regulate output power and output voltage in two power converters in this study. Because a PM synchronous generator system driven by WT is an unknown non-linear and time-varying dynamic system, the on-line training novel recurrent Chebyshev NN control system is developed to regulate DC voltage of the AC-DC converter and AC voltage of the DC-AC converter connected with smart grid. Furthermore, the variable learning rate of the novel recurrent Chebyshev NN is regulated according to discrete-type Lyapunov function for improving the control performance and enhancing convergent speed. Finally, some experimental results are shown to verify the effectiveness of the proposed control method for a WT driving a PM synchronous generator system in smart grid.
Chakraborty, Debananda
2011-01-01
We consider the Klein-Gordon and sine-Gordon type equations with a point-like potential, which describes the wave phenomenon in disordered media with a defect. The singular potential term yields a critical phenomenon--that is, the solution behavior around the critical parameter value bifurcates into two extreme cases. Pinpointing the critical value with arbitrary accuracy is even more challenging. In this work, we adopt the generalized polynomial chaos (gPC) method to determine the critical values and the mean solutions around such values. First, we consider the critical value associated with the strength of the singular potential for the Klein-Gordon equation. We expand the solution in the random variable associated with the parameter. The obtained partial differential equations are solved using the Chebyshev collocation method. Due to the existence of the singularity, the Gibbs phenomenon appears in the solution, yielding a slow convergence of the numerically computed critical value. To deal with the singul...
Approximation of Hardy space on the unit sphere
Institute of Scientific and Technical Information of China (English)
余纯武; 陈莘萌; 王昆扬; 戴峰
2003-01-01
The authors discuss the boundedness and approximation properties of translation and mean operator on H1(∑) by the estimates of high degree difference on ultraspherical polynomials, atom de-composition and construct properties on sphere. Also the boundedness and approximation of linear means at all kinds of indexes on Hp(0 < p < 1) and the almost everywhere convergence of Cesaro means are established.
Regression with Sparse Approximations of Data
DEFF Research Database (Denmark)
Noorzad, Pardis; Sturm, Bob L.
2012-01-01
We propose sparse approximation weighted regression (SPARROW), a method for local estimation of the regression function that uses sparse approximation with a dictionary of measurements. SPARROW estimates the regression function at a point with a linear combination of a few regressands selected by a...... sparse approximation of the point in terms of the regressors. We show SPARROW can be considered a variant of \\(k\\)-nearest neighbors regression (\\(k\\)-NNR), and more generally, local polynomial kernel regression. Unlike \\(k\\)-NNR, however, SPARROW can adapt the number of regressors to use based on the...
Approximation of the semi-infinite interval
Directory of Open Access Journals (Sweden)
A. McD. Mercer
1980-01-01
Full Text Available The approximation of a function f∈C[a,b] by Bernstein polynomials is well-known. It is based on the binomial distribution. O. Szasz has shown that there are analogous approximations on the interval [0,∞ based on the Poisson distribution. Recently R. Mohapatra has generalized Szasz' result to the case in which the approximating function is αe−ux∑k=N∞(uxkα+β−1Γ(kα+βf(kαuThe present note shows that these results are special cases of a Tauberian theorem for certain infinite series having positive coefficients.
Explicit classes of permutation polynomials of F33m
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
Permutation polynomials have been an interesting subject of study for a long time and have applications in many areas of mathematics and engineering. However, only a small number of specific classes of permutation polynomials are known so far. In this paper, six classes of linearized permutation polynomials and six classes of nonlinearized permutation polynomials over F33m are presented. These polynomials have simple shapes, and they are related to planar functions.
Explicit classes of permutation polynomials of F33m
Institute of Scientific and Technical Information of China (English)
DING CunSheng; XIANG Qing; YUAN Jin; YUAN PingZhi
2009-01-01
Permutation polynomials have been an interesting subject of study for a long time and have applications in many areas of mathematics and engineering. However, only a small number of specific classes of permutation polynomials are known so far. In this paper, six classes of linearized permutation polynomials and six classes of nonlinearized permutation polynomials over F33 are pre-sented. These polynomials have simple shapes, and they are related to planar functions.
Long-time uncertainty propagation using generalized polynomial chaos and flow map composition
International Nuclear Information System (INIS)
We present an efficient and accurate method for long-time uncertainty propagation in dynamical systems. Uncertain initial conditions and parameters are both addressed. The method approximates the intermediate short-time flow maps by spectral polynomial bases, as in the generalized polynomial chaos (gPC) method, and uses flow map composition to construct the long-time flow map. In contrast to the gPC method, this approach has spectral error convergence for both short and long integration times. The short-time flow map is characterized by small stretching and folding of the associated trajectories and hence can be well represented by a relatively low-degree basis. The composition of these low-degree polynomial bases then accurately describes the uncertainty behavior for long integration times. The key to the method is that the degree of the resulting polynomial approximation increases exponentially in the number of time intervals, while the number of polynomial coefficients either remains constant (for an autonomous system) or increases linearly in the number of time intervals (for a non-autonomous system). The findings are illustrated on several numerical examples including a nonlinear ordinary differential equation (ODE) with an uncertain initial condition, a linear ODE with an uncertain model parameter, and a two-dimensional, non-autonomous double gyre flow
Approximation by Chebyshevian Bernstein Operators versus Convergence of Dimension Elevation
Ait-Haddou, Rachid
2016-03-18
On a closed bounded interval, consider a nested sequence of Extended Chebyshev spaces possessing Bernstein bases. This situation automatically generates an infinite dimension elevation algorithm transforming control polygons of any given level into control polygons of the next level. The convergence of these infinite sequences of polygons towards the corresponding curves is a classical issue in computer-aided geometric design. Moreover, according to recent work proving the existence of Bernstein-type operators in such Extended Chebyshev spaces, this nested sequence is automatically associated with an infinite sequence of Bernstein operators which all reproduce the same two-dimensional space. Whether or not this sequence of operators converges towards the identity on the space of all continuous functions is a natural issue in approximation theory. In the present article, we prove that the two issues are actually equivalent. Not only is this result interesting on the theoretical side, but it also has practical implications. For instance, it provides us with a Korovkin-type theorem of convergence of any infinite dimension elevation algorithm. It also enables us to tackle the question of convergence of the dimension elevation algorithm for any nested sequence obtained by repeated integration of the kernel of a given linear differential operator with constant coefficients. © 2016 Springer Science+Business Media New York
Vyas, Bhargav Y; Das, Biswarup; Maheshwari, Rudra Prakash
2016-08-01
This paper presents the Chebyshev neural network (ChNN) as an improved artificial intelligence technique for power system protection studies and examines the performances of two ChNN learning algorithms for fault classification of series compensated transmission line. The training algorithms are least-square Levenberg-Marquardt (LSLM) and recursive least-square algorithm with forgetting factor (RLSFF). The performances of these algorithms are assessed based on their generalization capability in relating the fault current parameters with an event of fault in the transmission line. The proposed algorithm is fast in response as it utilizes postfault samples of three phase currents measured at the relaying end corresponding to half-cycle duration only. After being trained with only a small part of the generated fault data, the algorithms have been tested over a large number of fault cases with wide variation of system and fault parameters. Based on the studies carried out in this paper, it has been found that although the RLSFF algorithm is faster for training the ChNN in the fault classification application for series compensated transmission lines, the LSLM algorithm has the best accuracy in testing. The results prove that the proposed ChNN-based method is accurate, fast, easy to design, and immune to the level of compensations. Thus, it is suitable for digital relaying applications. PMID:25314714
Che, Cheng-Xuan; Wang, Xiu-Ming; Lin, Wei-Jun
2010-06-01
Based on strong and weak forms of elastic wave equations, a Chebyshev spectral element method (SEM) using the Galerkin variational principle is developed by discretizing the wave equation in the spatial and time domains and introducing the preconditioned conjugate gradient (PCG)-element by element (EBE) method in the spatial domain and the staggered predictor/corrector method in the time domain. The accuracy of our proposed method is verified by comparing it with a finite-difference method (FDM) for a homogeneous solid medium and a double layered solid medium with an inclined interface. The modeling results using the two methods are in good agreement with each other. Meanwhile, to show the algorithm capability, the suggested method is used to simulate the wave propagation in a layered medium with a topographic traction free surface. By introducing the EBE algorithm with an optimized tensor product technique, the proposed SEM is especially suitable for numerical simulation of wave propagations in complex models with irregularly free surfaces at a fast convergence rate, while keeping the advantage of the finite element method.
Certain non-linear differential polynomials sharing a non zero polynomial
Directory of Open Access Journals (Sweden)
Majumder Sujoy
2015-10-01
functions sharing a nonzero polynomial and obtain two results which improves and generalizes the results due to L. Liu [Uniqueness of meromorphic functions and differential polynomials, Comput. Math. Appl., 56 (2008, 3236-3245.] and P. Sahoo [Uniqueness and weighted value sharing of meromorphic functions, Applied. Math. E-Notes., 11 (2011, 23-32.].
A new class of generalized polynomials associated with Hermite and Bernoulli polynomials
Directory of Open Access Journals (Sweden)
M. A. Pathan
2015-05-01
Full Text Available In this paper, we introduce a new class of generalized polynomials associated with the modified Milne-Thomson's polynomials Φ_{n}^{(α}(x,ν of degree n and order α introduced by Derre and Simsek.The concepts of Bernoulli numbers B_n, Bernoulli polynomials B_n(x, generalized Bernoulli numbers B_n(a,b, generalized Bernoulli polynomials B_n(x;a,b,c of Luo et al, Hermite-Bernoulli polynomials {_HB}_n(x,y of Dattoli et al and {_HB}_n^{(α} (x,y of Pathan are generalized to the one {_HB}_n^{(α}(x,y,a,b,c which is called the generalized polynomial depending on three positive real parameters. Numerous properties of these polynomials and some relationships between B_n, B_n(x, B_n(a,b, B_n(x;a,b,c and {}_HB_n^{(α}(x,y;a,b,c are established. Some implicit summation formulae and general symmetry identities are derived by using different analytical means and applying generating functions. These results extend some known summations and identities of generalized Bernoulli numbers and polynomials
Directory of Open Access Journals (Sweden)
Kunio Ichinobe
2015-01-01
Full Text Available We study the \\(k\\-summability of divergent formal solutions for the Cauchy problem of certain linear partial differential operators with coefficients which are polynomial in \\(t\\. We employ the method of successive approximation in order to construct the formal solutions and to obtain the properties of analytic continuation of the solutions of convolution equations and their exponential growth estimates.
Dynamic Approximate Vertex Cover and Maximum Matching
Onak, Krzysztof; Rubinfeld, Ronitt
2010-01-01
We consider the problem of maintaining a large matching or a small vertex cover in a dynamically changing graph. Each update to the graph is either an edge deletion or an edge insertion. We give the first randomized data structure that simultaneously achieves a constant approximation factor and handles a sequence of k updates in k. polylog(n) time. Previous data structures require a polynomial amount of computation per update. The starting point of our construction is a distributed algorit...
Twisted Alexander polynomials of hyperbolic knots
Dunfield, Nathan M; Jackson, Nicholas
2011-01-01
We study a twisted Alexander polynomial naturally associated to a hyperbolic knot in an integer homology 3-sphere via a lift of the holonomy representation to SL(2, C). It is an unambiguous symmetric Laurent polynomial whose coefficients lie in the trace field of the knot. It contains information about genus, fibering, and chirality, and moreover is powerful enough to sometimes detect mutation. We calculated this invariant numerically for all 313,209 hyperbolic knots in S^3 with at most 15 crossings, and found that in all cases it gave a sharp bound on the genus of the knot and determined both fibering and chirality. We also study how such twisted Alexander polynomials vary as one moves around in an irreducible component X_0 of the SL(2, C)-character variety of the knot group. We show how to understand all of these polynomials at once in terms of a polynomial whose coefficients lie in the function field of X_0. We use this to help explain some of the patterns observed for knots in S^3, and explore a potential...
Algorithms for Testing Monomials in Multivariate Polynomials
Chen, Zhixiang; Liu, Yang; Schweller, Robert
2010-01-01
This paper is our second step towards developing a theory of testing monomials in multivariate polynomials. The central question is to ask whether a polynomial represented by an arithmetic circuit has some types of monomials in its sum-product expansion. The complexity aspects of this problem and its variants have been investigated in our first paper by Chen and Fu (2010), laying a foundation for further study. In this paper, we present two pairs of algorithms. First, we prove that there is a randomized $O^*(p^k)$ time algorithm for testing $p$-monomials in an $n$-variate polynomial of degree $k$ represented by an arithmetic circuit, while a deterministic $O^*(6.4^k + p^k)$ time algorithm is devised when the circuit is a formula, here $p$ is a given prime number. Second, we present a deterministic $O^*(2^k)$ time algorithm for testing multilinear monomials in $\\Pi_m\\Sigma_2\\Pi_t\\times \\Pi_k\\Pi_3$ polynomials, while a randomized $O^*(1.5^k)$ algorithm is given for these polynomials. The first algorithm extends...
SIMULATED ANNEALING BASED POLYNOMIAL TIME QOS ROUTING ALGORITHM FOR MANETS
Institute of Scientific and Technical Information of China (English)
Liu Lianggui; Feng Guangzeng
2006-01-01
Multi-constrained Quality-of-Service (QoS) routing is a big challenge for Mobile Ad hoc Networks (MANETs) where the topology may change constantly. In this paper a novel QoS Routing Algorithm based on Simulated Annealing (SA_RA) is proposed. This algorithm first uses an energy function to translate multiple QoS weights into a single mixed metric and then seeks to find a feasible path by simulated annealing. The paper outlines simulated annealing algorithm and analyzes the problems met when we apply it to Qos Routing (QoSR) in MANETs. Theoretical analysis and experiment results demonstrate that the proposed method is an effective approximation algorithms showing better performance than the other pertinent algorithm in seeking the (approximate) optimal configuration within a period of polynomial time.
Uncertainty Analysis via Failure Domain Characterization: Polynomial Requirement Functions
Crespo, Luis G.; Munoz, Cesar A.; Narkawicz, Anthony J.; Kenny, Sean P.; Giesy, Daniel P.
2011-01-01
This paper proposes an uncertainty analysis framework based on the characterization of the uncertain parameter space. This characterization enables the identification of worst-case uncertainty combinations and the approximation of the failure and safe domains with a high level of accuracy. Because these approximations are comprised of subsets of readily computable probability, they enable the calculation of arbitrarily tight upper and lower bounds to the failure probability. A Bernstein expansion approach is used to size hyper-rectangular subsets while a sum of squares programming approach is used to size quasi-ellipsoidal subsets. These methods are applicable to requirement functions whose functional dependency on the uncertainty is a known polynomial. Some of the most prominent features of the methodology are the substantial desensitization of the calculations from the uncertainty model assumed (i.e., the probability distribution describing the uncertainty) as well as the accommodation for changes in such a model with a practically insignificant amount of computational effort.
Optimal stability polynomials for numerical integration of initial value problems
Ketcheson, David I.
2013-01-08
We consider the problem of finding optimally stable polynomial approximations to the exponential for application to one-step integration of initial value ordinary and partial differential equations. The objective is to find the largest stable step size and corresponding method for a given problem when the spectrum of the initial value problem is known. The problem is expressed in terms of a general least deviation feasibility problem. Its solution is obtained by a new fast, accurate, and robust algorithm based on convex optimization techniques. Global convergence of the algorithm is proven in the case that the order of approximation is one and in the case that the spectrum encloses a starlike region. Examples demonstrate the effectiveness of the proposed algorithm even when these conditions are not satisfied.
A polynomial analytical method for one-group slab-geometry discrete ordinates heterogeneous problems
International Nuclear Information System (INIS)
In this work we evaluate polynomial approximations to obtain the transfer functions that appear in SGF auxiliary equations (Green's Functions) for monoenergetic linearly anisotropic scattering SN equations in one-dimensional Cartesian geometry. For this task we use Lagrange Polynomials in order to compare the numerical results with the ones generated by the standard SGF method applied to SN problems in heterogeneous domains. This work is a preliminary investigation of a new proposal for handling the transverse leakage terms that appear in the transverse-integrated one-dimensional SN equations when we use the SGF - exponential nodal method (SGF-ExpN) in multidimensional rectangular geometry. (author)
New development in theory of Laguerre polynomials
Guseinov, I I
2012-01-01
The new complete orthonormal sets of -Laguerre type polynomials (-LTP,) are suggested. Using Schr\\"odinger equation for complete orthonormal sets of -exponential type orbitals (-ETO) introduced by the author, it is shown that the origin of these polynomials is the centrally symmetric potential which contains the core attraction potential and the quantum frictional potential of the field produced by the particle itself. The quantum frictional forces are the analog of radiation damping or frictional forces suggested by Lorentz in classical electrodynamics. The new -LTP are complete without the inclusion of the continuum states of hydrogen like atoms. It is shown that the nonstandard and standard conventions of -LTP and their weight functions are the same. As an application, the sets of infinite expansion formulas in terms of -LTP and L-Generalized Laguerre polynomials (L-GLP) for atomic nuclear attraction integrals of Slater type orbitals (STO) and Coulomb-Yukawa like correlated interaction potentials (CIP) wit...
Quantum chaotic dynamics and random polynomials
Energy Technology Data Exchange (ETDEWEB)
Bogomolny, E.; Bohigas, O.; Leboeuf, P.
1995-11-01
The distribution of roots of polynomials of high degree with random coefficients is investigated which, among others, appear naturally in the context of `quantum chaotic dynamics`. It is shown that under quite general conditions their roots tend to concentrate near the unit circle in the complex plane. In order to further increase this tendency, the particular case of self-inverse random polynomials is studied, and it is shown that for them a finite portion of all roots lies exactly on the unit circle. Correlation functions of these roots are also computed analytically, and compared to the correlations of eigenvalues of random matrices. The problem of ergodicity of chaotic wavefunctions is also considered. Special attention is devoted to the role of symmetries in the distribution of roots of random polynomials. (author). 32 refs.
Uniform trigonometric polynomial B-spline curves
Institute of Scientific and Technical Information of China (English)
吕勇刚; 汪国昭; 杨勋年
2002-01-01
This paper presents a new kind of uniform spline curve, named trigonometric polynomialB-splines, over space Ω = span{sint, cost, tk-3,tk-4,…,t,1} of which k is an arbitrary integerlarger than or equal to 3. We show that trigonometric polynomial B-spline curves have many similarV properties to traditional B-splines. Based on the explicit representation of the curve we have also presented the subdivision formulae for this new kind of curve. Since the new spline can include both polynomial curves and trigonometric curves as special cases without rational form, it can be used as an efficient new model for geometric design in the fields of CAD/CAM.
A complete discrimination system for polynomials
Institute of Scientific and Technical Information of China (English)
杨路; 侯晓荣; 曾振柄
1996-01-01
Given a polynomial with symbolic/literal coefficients,a complete discrimination system is a set of explicit expressions in terms of the coefficients,which is sufficient for determining the numbers and multiplicities of the real and imaginary roots.Though it is of great significance,such a criterion for root-classification has never been given for polynomials with degrees greater than 4.The lack of efficient tools in this aspect extremely prevents computer implementations for Tarski’s and other methods in automated theorem proving.To remedy this defect,a generic algorithm is proposed to produce a complete discrimination system for a polynomial with any degrees.This result has extensive applications in various fields,and its efficiency was demonstrated by computer implementations.
Fast beampattern evaluation by polynomial rooting
Häcker, P.; Uhlich, S.; Yang, B.
2011-07-01
Current automotive radar systems measure the distance, the relative velocity and the direction of objects in their environment. This information enables the car to support the driver. The direction estimation capabilities of a sensor array depend on its beampattern. To find the array configuration leading to the best angle estimation by a global optimization algorithm, a huge amount of beampatterns have to be calculated to detect their maxima. In this paper, a novel algorithm is proposed to find all maxima of an array's beampattern fast and reliably, leading to accelerated array optimizations. The algorithm works for arrays having the sensors on a uniformly spaced grid. We use a general version of the gcd (greatest common divisor) function in order to write the problem as a polynomial. We differentiate and root the polynomial to get the extrema of the beampattern. In addition, we show a method to reduce the computational burden even more by decreasing the order of the polynomial.
Dominating Sets and Domination Polynomials of Paths
Directory of Open Access Journals (Sweden)
Saeid Alikhani
2009-01-01
Full Text Available Let G=(V,E be a simple graph. A set S⊆V is a dominating set of G, if every vertex in V\\S is adjacent to at least one vertex in S. Let 𝒫ni be the family of all dominating sets of a path Pn with cardinality i, and let d(Pn,j=|𝒫nj|. In this paper, we construct 𝒫ni, and obtain a recursive formula for d(Pn,i. Using this recursive formula, we consider the polynomial D(Pn,x=∑i=⌈n/3⌉nd(Pn,ixi, which we call domination polynomial of paths and obtain some properties of this polynomial.
Minimal residual method stronger than polynomial preconditioning
Energy Technology Data Exchange (ETDEWEB)
Faber, V.; Joubert, W.; Knill, E. [Los Alamos National Lab., NM (United States)] [and others
1994-12-31
Two popular methods for solving symmetric and nonsymmetric systems of equations are the minimal residual method, implemented by algorithms such as GMRES, and polynomial preconditioning methods. In this study results are given on the convergence rates of these methods for various classes of matrices. It is shown that for some matrices, such as normal matrices, the convergence rates for GMRES and for the optimal polynomial preconditioning are the same, and for other matrices such as the upper triangular Toeplitz matrices, it is at least assured that if one method converges then the other must converge. On the other hand, it is shown that matrices exist for which restarted GMRES always converges but any polynomial preconditioning of corresponding degree makes no progress toward the solution for some initial error. The implications of these results for these and other iterative methods are discussed.
Some Results on the Simultaneous Approximation
Institute of Scientific and Technical Information of China (English)
M. R. Haddadi∗
2014-01-01
In this paper, we give some result on the simultaneous proximinal subset and simultaneous Chebyshev in the uniformly convex Banach space. Also we give relation between fixed point theory and simultaneous proximity.
A Quasi-Interpolation Satisfying Quadratic Polynomial Reproduction with Radial Basis Functions
Institute of Scientific and Technical Information of China (English)
Li Zha; Renzhong Feng
2007-01-01
In this paper, a new quasi-interpolation with radial basis functions which satisfies quadratic polynomial reproduction is constructed on the infinite set of equally spaced data. A new basis function is constructed by making convolution integral with a constructed spline and a given radial basis function. In particular, for twicely differentiable function the proposed method provides better approximation and also takes care of derivatives approximation.
Large Degree Asymptotics of Generalized Bessel Polynomials
López, J. L.; Temme, Nico
2011-01-01
Asymptotic expansions are given for large values of $n$ of the generalized Bessel polynomials $Y_n^\\mu(z)$. The analysis is based on integrals that follow from the generating functions of the polynomials. A new simple expansion is given that is valid outside a compact neighborhood of the origin in the $z-$plane. New forms of expansions in terms of elementary functions valid in sectors not containing the turning points $z=\\pm i/n$ are derived, and a new expansion in terms of modified Bessel fu...
Polynomial Vector Fields in One Complex Variable
DEFF Research Database (Denmark)
Branner, Bodil
In recent years Adrien Douady was interested in polynomial vector fields, both in relation to iteration theory and as a topic on their own. This talk is based on his work with Pierrette Sentenac, work of Xavier Buff and Tan Lei, and my own joint work with Kealey Dias.......In recent years Adrien Douady was interested in polynomial vector fields, both in relation to iteration theory and as a topic on their own. This talk is based on his work with Pierrette Sentenac, work of Xavier Buff and Tan Lei, and my own joint work with Kealey Dias....
The chromatic polynomial and list colorings
DEFF Research Database (Denmark)
Thomassen, Carsten
2009-01-01
We prove that, if a graph has a list of k available colors at every vertex, then the number of list-colorings is at least the chromatic polynomial evaluated at k when k is sufficiently large compared to the number of vertices of the graph.......We prove that, if a graph has a list of k available colors at every vertex, then the number of list-colorings is at least the chromatic polynomial evaluated at k when k is sufficiently large compared to the number of vertices of the graph....
Incomplete Bivariate Fibonacci and Lucas -Polynomials
Directory of Open Access Journals (Sweden)
Dursun Tasci
2012-01-01
Full Text Available We define the incomplete bivariate Fibonacci and Lucas -polynomials. In the case =1, =1, we obtain the incomplete Fibonacci and Lucas -numbers. If =2, =1, we have the incomplete Pell and Pell-Lucas -numbers. On choosing =1, =2, we get the incomplete generalized Jacobsthal number and besides for =1 the incomplete generalized Jacobsthal-Lucas numbers. In the case =1, =1, =1, we have the incomplete Fibonacci and Lucas numbers. If =1, =1, =1, =⌊(−1/(+1⌋, we obtain the Fibonacci and Lucas numbers. Also generating function and properties of the incomplete bivariate Fibonacci and Lucas -polynomials are given.
Genus Polynomials of Cycles with Double Edges
Institute of Scientific and Technical Information of China (English)
Eunyoung BAEK; Jongyook PARK
2011-01-01
Two cellular embeddings i : G → S and j : G → S of a connected graph G into a closed orientable surface S are equivalent if there is an orientation-preserving surface homeomorphism h: S → S such that hi = j. The genus polynomial of a graph G is defined by g[G](x)=∑∞g=0agxg,where ag is the number of equivalence classes of embeddings of G into the orientable surface Sg with g genera.In this paper, we compute the genus polynomial of a graph obtained from a cycle by replacing each edge by two multiple edges.
The classification of polynomial basins of infinity
DeMarco, Laura
2011-01-01
We consider the problem of classifying the dynamics of complex polynomials $f: \\mathbb{C} \\to \\mathbb{C}$ restricted to their basins of infinity. We synthesize existing combinatorial tools --- tableaux, trees, and laminations --- into a new invariant of basin dynamics we call the pictograph. For polynomials with all critical points escaping to infinity, we obtain a complete description of the set of topological conjugacy classes. We give an algorithm for constructing abstract pictographs, and we provide an inductive algorithm for counting topological conjugacy classes with a given pictograph.
Perturbations around the zeros of classical orthogonal polynomials
Sasaki, Ryu
2014-01-01
Starting from degree N solutions of a time dependent Schroedinger-like equation for classical orthogonal polynomials, a linear matrix equation describing perturbations around the N zeros of the polynomial is derived. The matrix has remarkable Diophantine properties. Its eigenvalues are independent of the zeros. The corresponding eigenvectors provide the representations of the lower degree (0,1,...,N-1) polynomials in terms of the zeros of the degree N polynomial. The results are valid universally for all the classical orthogonal polynomials, including the Askey scheme of hypergeometric orthogonal polynomials and its q-analogues.
Perturbations around the zeros of classical orthogonal polynomials
Sasaki, Ryu
2015-04-01
Starting from degree N solutions of a time dependent Schrödinger-like equation for classical orthogonal polynomials, a linear matrix equation describing perturbations around the N zeros of the polynomial is derived. The matrix has remarkable Diophantine properties. Its eigenvalues are independent of the zeros. The corresponding eigenvectors provide the representations of the lower degree ( 0 , 1 , … , N - 1 ) polynomials in terms of the zeros of the degree N polynomial. The results are valid universally for all the classical orthogonal polynomials, including the Askey scheme of hypergeometric orthogonal polynomials and its q-analogues.
Institute of Scientific and Technical Information of China (English)
Tan Xiaogang; Wei Ping; Li Liping
2009-01-01
To detect higher order polynomial phase signals (HOPPSs), the smoothed-pseudo polynomial Wigner-Ville distribution (SP-PWVD), an improved version of the polynomial Wigner-Ville distribution (PWVD), is pre-sented using a separable kernel. By adjusting the lengths of the functions in the kernel, the balance between resolution retaining and interference suppressing can be adjusted conveniently. The proposed method with merits of interference terms reduction and noise suppression can provide time frequency representation of better readability and more accurate instantaneous frequency (IF) estimation with higher order SP-PWVD. The performance of the SP-PWVD is verified by computer simulations.
Directory of Open Access Journals (Sweden)
S.Padma
2013-06-01
Full Text Available A new wavelet based approximation method for solving the second order differential equations arising science and engineering is presented in this paper. Such differential equation is often applied to model phenomena in various fields of science and engineering. In this study, shifted second kind Chebyshev wavelet (CW operational matrices of derivatives is introduced and applied for solvingthe second order differential equations with various initial conditions. The key idea for getting the numerical solutions for these equations is to convert the differential equations (linear or nonlinear to a system of linear or nonlinear algebraic equations in the unknown expansion coefficients. Some illustrative examples are given to demonstrate the validity and applicability of the proposed method. The power of the manageable method is confirmed. Moreover the use of the shifted second kind Chebyshev wavelet method (CWM is found to be simple, flexible, efficient, small computation costs and computationally attractive.
Computation of the Likelihood in Biallelic Diffusion Models Using Orthogonal Polynomials
Directory of Open Access Journals (Sweden)
Claus Vogl
2014-11-01
Full Text Available In population genetics, parameters describing forces such as mutation, migration and drift are generally inferred from molecular data. Lately, approximate methods based on simulations and summary statistics have been widely applied for such inference, even though these methods waste information. In contrast, probabilistic methods of inference can be shown to be optimal, if their assumptions are met. In genomic regions where recombination rates are high relative to mutation rates, polymorphic nucleotide sites can be assumed to evolve independently from each other. The distribution of allele frequencies at a large number of such sites has been called “allele-frequency spectrum” or “site-frequency spectrum” (SFS. Conditional on the allelic proportions, the likelihoods of such data can be modeled as binomial. A simple model representing the evolution of allelic proportions is the biallelic mutation-drift or mutation-directional selection-drift diffusion model. With series of orthogonal polynomials, specifically Jacobi and Gegenbauer polynomials, or the related spheroidal wave function, the diffusion equations can be solved efficiently. In the neutral case, the product of the binomial likelihoods with the sum of such polynomials leads to finite series of polynomials, i.e., relatively simple equations, from which the exact likelihoods can be calculated. In this article, the use of orthogonal polynomials for inferring population genetic parameters is investigated.
Approximations of continuous Newton's method: An extension of Cayley's problem
Directory of Open Access Journals (Sweden)
Jon Jacobsen
2007-02-01
Full Text Available Continuous Newton's Method refers to a certain dynamical system whose associated flow generically tends to the roots of a given polynomial. An Euler approximation of this system, with step size $h=1$, yields the discrete Newton's method algorithm for finding roots. In this note we contrast Euler approximations with several different approximations of the continuous ODE system and, using computer experiments, consider their impact on the associated fractal basin boundaries of the roots.
ON APPROXIMATION BY REPRODUCING KERNEL SPACES IN WEIGHTED Lp SPACES
Institute of Scientific and Technical Information of China (English)
Baohuai SHENG
2007-01-01
In this paper, we investigate the order of approximation by reproducing kernel spaces on (-1, 1) in weighted Lp spaces. We first restate the translation network from the view of reproducing kernel spaces and then construct a sequence of approximating operators with the help of Jacobi orthogonal polynomials, with which we establish a kind of Jackson inequality to describe the error estimate.Finally, The results are used to discuss an approximation problem arising from learning theory.
Quantum Hilbert matrices and orthogonal polynomials
DEFF Research Database (Denmark)
Andersen, Jørgen Ellegaard; Berg, Christian
2009-01-01
Using the notion of quantum integers associated with a complex number q≠0 , we define the quantum Hilbert matrix and various extensions. They are Hankel matrices corresponding to certain little q -Jacobi polynomials when |q|... of reciprocal Fibonacci numbers called Filbert matrices. We find a formula for the entries of the inverse quantum Hilbert matrix....
On "the matching polynomial of a polygraph"
Broersma, H.J.; Li, Xueliang
1993-01-01
In this note we give an explanation for two phenomena mentioned in the concluding remarks of “The matching polynomial of a polygraph” by Babić et al. The following results are obtained: 1. Although three matrices for given polygraphs defined in the above article in general have different orders, th
Optimization of Cubic Polynomial Functions without Calculus
Taylor, Ronald D., Jr.; Hansen, Ryan
2008-01-01
In algebra and precalculus courses, students are often asked to find extreme values of polynomial functions in the context of solving an applied problem; but without the notion of derivative, something is lost. Either the functions are reduced to quadratics, since students know the formula for the vertex of a parabola, or solutions are…
Bernoulli Polynomials, Fourier Series and Zeta Numbers
DEFF Research Database (Denmark)
Scheufens, Ernst E
2013-01-01
Fourier series for Bernoulli polynomials are used to obtain information about values of the Riemann zeta function for integer arguments greater than one. If the argument is even we recover the well-known exact values, if the argument is odd we find integral representations and rapidly convergent...
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/.
Ideals in Polynomial Near-rings
Institute of Scientific and Technical Information of China (English)
Mark Farag
2002-01-01
In this paper, we further explore the relationship between the ideals of N and those of N[x], where N is a zero-symmetric right near-ring with identity and N[x] is the polynomial near-ring introduced by Bagley in 1993.
Miscellaneous identities of generalized Hermite polynomials
Directory of Open Access Journals (Sweden)
G. Dattoli
1997-11-01
Full Text Available We extend a number of identities valid for the ordinary case to generalized Hermite polynomials with two indices and two variables. These identities, new to the authors knowledge, are obtained by using an operatorial procedure based on the properties of the Weyl group.
Polynomial computation of Hankel singular values
Kwakernaak, Huibert
1992-01-01
A revised and improved version of a polynomial algorithm is presented. It was published by N.J. Young (1990) for the computation of the singular values and vectors of the Hankel operator defined by a linear time-invariant system with a rotational transfer matrix. Tentative numerical experiments indi
Euler Polynomials, Fourier Series and Zeta Numbers
DEFF Research Database (Denmark)
Scheufens, Ernst E
2012-01-01
Fourier series for Euler polynomials is used to obtain information about values of the Riemann zeta function for integer arguments greater than one. If the argument is even we recover the well-known exact values, if the argument is odd we find integral representations and rapidly convergent series....
On the Schinzel Identity of Cyclotomic Polynomial
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
@@For integer n>0, let n(x) denote the nth cyclotomic polynomial n(x)=tackrel{01 be an odd square-free number.Aurifeuille and Le Lasseur［1］ proved thatequationn(x)=An2(x)-(-1)n-12)nxBn2(x).equation
Algebraic polynomial system solving and applications
Bleylevens, I.W.M.
2010-01-01
The problem of computing the solutions of a system of multivariate polynomial equations can be approached by the Stetter-Möller matrix method which casts the problem into a large eigenvalue problem. This Stetter-Möller matrix method forms the starting point for the development of computational proce
The GCD property and irreduciable quadratic polynomials
Directory of Open Access Journals (Sweden)
Saroj Malik
1986-01-01
Full Text Available The proof of the following theorem is presented: If D is, respectively, a Krull domain, a Dedekind domain, or a Prüfer domain, then D is correspondingly a UFD, a PID, or a Bezout domain if and only if every irreducible quadratic polynomial in D[X] is a prime element.
UNIQUENESS OF DIFFERENCE POLYNOMIALS OF MEROMORPHIC FUNCTIONS
Institute of Scientific and Technical Information of China (English)
刘永; 祁晓光
2014-01-01
In this article, we investigate the uniqueness problems of difference polynomials of meromorphic functions and obtain some results which can be viewed as discrete analogues of the results given by Shibazaki. Some examples are given to show the results in this article are best possible.
Polynomial Asymptotes of the Second Kind
Dobbs, David E.
2011-01-01
This note uses the analytic notion of asymptotic functions to study when a function is asymptotic to a polynomial function. Along with associated existence and uniqueness results, this kind of asymptotic behaviour is related to the type of asymptote that was recently defined in a more geometric way. Applications are given to rational functions and…
Scalar Field Theories with Polynomial Shift Symmetries
Griffin, Tom; Horava, Petr; Yan, Ziqi
2014-01-01
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 (essen...
Dynamic system uncertainty propagation using polynomial chaos
Institute of Scientific and Technical Information of China (English)
Xiong Fenfen; Chen Shishi; Xiong Ying
2014-01-01
The classic polynomial chaos method (PCM), characterized as an intrusive methodology, has been applied to uncertainty propagation (UP) in many dynamic systems. However, the intrusive polynomial chaos method (IPCM) requires tedious modification of the governing equations, which might introduce errors and can be impractical. Alternative to IPCM, the non-intrusive polynomial chaos method (NIPCM) that avoids such modifications has been developed. In spite of the frequent application to dynamic problems, almost all the existing works about NIPCM for dynamic UP fail to elaborate the implementation process in a straightforward way, which is important to readers who are unfamiliar with the mathematics of the polynomial chaos theory. Meanwhile, very few works have compared NIPCM to IPCM in terms of their merits and applicability. Therefore, the mathematic procedure of dynamic UP via both methods considering parametric and initial condition uncertainties are comparatively discussed and studied in the present paper. Comparison of accuracy and efficiency in statistic moment estimation is made by applying the two methods to several dynamic UP problems. The relative merits of both approaches are discussed and summarized. The detailed description and insights gained with the two methods through this work are expected to be helpful to engineering designers in solving dynamic UP problems.
Takahasi, Hiroki
2016-09-01
It is well-known that the geometric pressure function tin {{R}}mapsto sup _{μ }{ h_μ (T_2)-tint log |dT_2(x)|dμ (x)} of the Chebyshev quadratic map T_2(x)=1-2x^2 (xin {{R}}) is not differentiable at t=-1. We show that this phase transition can be "removed", by an arbitrarily small singular perturbation of the map T_2 into Hénon-like diffeomorphisms. A proof of this result relies on an elaboration of the well-known inducing techniques adapted to Hénon-like dynamics near the first bifurcation.
Maximum of the Characteristic Polynomial of Random Unitary Matrices
Arguin, Louis-Pierre; Belius, David; Bourgade, Paul
2016-09-01
It was recently conjectured by Fyodorov, Hiary and Keating that the maximum of the characteristic polynomial on the unit circle of a {N× N} random unitary matrix sampled from the Haar measure grows like {CN/(log N)^{3/4}} for some random variable C. In this paper, we verify the leading order of this conjecture, that is, we prove that with high probability the maximum lies in the range {[N^{1 - ɛ},N^{1 + ɛ}]} , for arbitrarily small ɛ. The method is based on identifying an approximate branching random walk in the Fourier decomposition of the characteristic polynomial, and uses techniques developed to describe the extremes of branching random walks and of other log-correlated random fields. A key technical input is the asymptotic analysis of Toeplitz determinants with dimension-dependent symbols. The original argument for these asymptotics followed the general idea that the statistical mechanics of 1/f-noise random energy models is governed by a freezing transition. We also prove the conjectured freezing of the free energy for random unitary matrices.
Polynomial Chaos Expansion Approach to Interest Rate Models
Directory of Open Access Journals (Sweden)
Luca Di Persio
2015-01-01
Full Text Available The Polynomial Chaos Expansion (PCE technique allows us to recover a finite second-order random variable exploiting suitable linear combinations of orthogonal polynomials which are functions of a given stochastic quantity ξ, hence acting as a kind of random basis. The PCE methodology has been developed as a mathematically rigorous Uncertainty Quantification (UQ method which aims at providing reliable numerical estimates for some uncertain physical quantities defining the dynamic of certain engineering models and their related simulations. In the present paper, we use the PCE approach in order to analyze some equity and interest rate models. In particular, we take into consideration those models which are based on, for example, the Geometric Brownian Motion, the Vasicek model, and the CIR model. We present theoretical as well as related concrete numerical approximation results considering, without loss of generality, the one-dimensional case. We also provide both an efficiency study and an accuracy study of our approach by comparing its outputs with the ones obtained adopting the Monte Carlo approach, both in its standard and its enhanced version.
Irreducibility Results for Compositions of Polynomials in Several Variables
Indian Academy of Sciences (India)
Anca Iuliana Bonciocat; Alexandru Zaharescu
2005-05-01
We obtain explicit upper bounds for the number of irreducible factors for a class of compositions of polynomials in several variables over a given field. In particular, some irreducibility criteria are given for this class of compositions of polynomials.
Combinatorial polynomials as moments, Hankel transforms and exponential Riordan arrays
Barry, Paul
2011-01-01
In the case of two combinatorial polynomials, we show that they can exhibited as moments of paramaterized families of orthogonal polynomials, and hence derive their Hankel transforms. Exponential Riordan arrays are the main vehicles used for this.
Representations of Knot Groups and Twisted Alexander Polynomials
Institute of Scientific and Technical Information of China (English)
Xiao Song LIN
2001-01-01
We present a twisted version of the Alexander polynomial associated with a matrix representation of the knot group. Examples of two knots with the same Alexander module but differenttwisted Alexander polynomials are given.
Identities involving Bessel polynomials arising from linear differential equations
Kim, Taekyun; Kim, Dae San
2016-01-01
In this paper, we study linear di?erential equations arising from Bessel polynomials and their applications. From these linear differential equations, we give some new and explicit identities for Bessel polynomials.
On an Inequality Concerning the Polar Derivative of a Polynomial
Indian Academy of Sciences (India)
A Aziz; N A Rather
2007-08-01
In this paper, we present a correct proof of an -inequality concerning the polar derivative of a polynomial with restricted zeros. We also extend Zygmund’s inequality to the polar derivative of a polynomial.
Characterization and Enumeration of Good Punctured Polynomials over Finite Fields
Directory of Open Access Journals (Sweden)
Somphong Jitman
2016-01-01
Full Text Available A family of good punctured polynomials is introduced. The complete characterization and enumeration of such polynomials are given over the binary field F2. Over a nonbinary finite field Fq, the set of good punctured polynomials of degree less than or equal to 2 are completely determined. For n≥3, constructive lower bounds of the number of good punctured polynomials of degree n over Fq are given.
Institute of Scientific and Technical Information of China (English)
程桂贤; 何国龙
2012-01-01
给出了求解非线性方程的一族新的带单参数β的免求二阶导数的Chebyshev-Halley型迭代法.新的迭代法在每次迭代过程中只需计算2次函数值和1次一阶导数值,其收敛阶至少为3.若参数β=3/2,则新的迭代法收敛阶为4.数值实验结果验证了此方法的有效性.%It was presented a new family of modified Chebyshev-Halley-type iterative methods free from second derivative for solving nonlinear equations. The convergence analysis showed that the new methods were at least third-order convergent, and it only required two functions and one first derivative evaluations per iteration. Especially when β = 3/2, the new method was fourth-order convergent. Several numerical examples were given to demonstrate the effectiveness of the presented methods by comparing with some other methods.
On the Lorentz degree of a product of polynomials
Ait-Haddou, Rachid
2015-01-01
In this note, we negatively answer two questions of T. Erdélyi (1991, 2010) on possible lower bounds on the Lorentz degree of product of two polynomials. We show that the correctness of one question for degree two polynomials is a direct consequence of a result of Barnard et al. (1991) on polynomials with nonnegative coefficients.
Does the polynomial hierarchy collapse if onto functions are invertible?
H. Buhrman; L. Fortnow; M. Koucký; J.D. Rogers; N. Vereshchagin
2010-01-01
The class TFNP, defined by Megiddo and Papadimitriou, consists of multivalued functions with values that are polynomially verifiable and guaranteed to exist. Do we have evidence that such functions are hard, for example, if TFNP is computable in polynomial-time does this imply the polynomial-time hi
A Determinant Expression for the Generalized Bessel Polynomials
Sheng-liang Yang; Sai-nan Zheng
2013-01-01
Using the exponential Riordan arrays, we show that a variation of the generalized Bessel polynomial sequence is of Sheffer type, and we obtain a determinant formula for the generalized Bessel polynomials. As a result, the Bessel polynomial is represented as determinant the entries of which involve Catalan numbers.
On conformal measures for infinitely renormalizable quadratic polynomials
Institute of Scientific and Technical Information of China (English)
HUANG Zhiyong; JIANG Yunping; WANG Yuefei
2005-01-01
We study a conformal measure for an infinitely renormalizable quadratic polynomial. We prove that the conformal measure is ergodic if the polynomial is unbranched and has complex bounds. The main technique we use in the proof is the three-dimensional puzzle for an infinitely renormalizable quadratic polynomial.
Some advances in tensor analysis and polynomial optimization
Li, Zhening; Ling, Chen; Wang, Yiju; Yang, Qingzhi
2014-01-01
Tensor analysis (also called as numerical multilinear algebra) mainly includes tensor decomposition, tensor eigenvalue theory and relevant algorithms. Polynomial optimization mainly includes theory and algorithms for solving optimization problems with polynomial objects functions under polynomial constrains. This survey covers the most of recent advances in these two fields. For tensor analysis, we introduce some properties and algorithms concerning the spectral radius of nonnegative tensors'...
Universality for polynomial invariants on ribbon graphs with flags
Avohou, Remi C.; Geloun, Joseph Ben; Hounkonnou , Mahouton N.
2013-01-01
In this paper, we analyze the Bollobas and Riordan polynomial for ribbon graphs with flags introduced in arXiv:1301.1987 and prove its universality. We also show that this polynomial can be defined on some equivalence classes of ribbon graphs involving flag moves and that the new polynomial is still universal on these classes.
Transmitting electric power system dynamics in SCADA using polynomial fitting
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
The paper proposes an approach to transmit electric power system dynamics in the SCADA. With the prevalent application of digital substation automation system, it is feasible for the remote terminal units (RTUs) to collect phasors within a substation. However, limited communication capacity remains the bottleneck that prevents SCADA from transmitting system dynamics. This paper proposes to compress dynamics data with curve fitting in the RTUs and reconstruct the dynamics in the SCADA server for reducing communication demand. Dispatchers in the control center can thus get system dynamics with a delay of several seconds. Simulation result shows that for a power system under disturbance with short-circuit that once occurred and was cleared, the SCADA can approximate the original dynamics with satisfying precision using limited degree polynomial fitting. The approach is highly scalable and adaptable, and can be implemented on existing communication infrastructure with a few software modifications. The approach has extensive application potential.
Dolgov, Sergey
2015-11-03
We apply the tensor train (TT) decomposition to construct the tensor product polynomial chaos expansion (PCE) of a random field, to solve the stochastic elliptic diffusion PDE with the stochastic Galerkin discretization, and to compute some quantities of interest (mean, variance, and exceedance probabilities). We assume that the random diffusion coefficient is given as a smooth transformation of a Gaussian random field. In this case, the PCE is delivered by a complicated formula, which lacks an analytic TT representation. To construct its TT approximation numerically, we develop the new block TT cross algorithm, a method that computes the whole TT decomposition from a few evaluations of the PCE formula. The new method is conceptually similar to the adaptive cross approximation in the TT format but is more efficient when several tensors must be stored in the same TT representation, which is the case for the PCE. In addition, we demonstrate how to assemble the stochastic Galerkin matrix and to compute the solution of the elliptic equation and its postprocessing, staying in the TT format. We compare our technique with the traditional sparse polynomial chaos and the Monte Carlo approaches. In the tensor product polynomial chaos, the polynomial degree is bounded for each random variable independently. This provides higher accuracy than the sparse polynomial set or the Monte Carlo method, but the cardinality of the tensor product set grows exponentially with the number of random variables. However, when the PCE coefficients are implicitly approximated in the TT format, the computations with the full tensor product polynomial set become possible. In the numerical experiments, we confirm that the new methodology is competitive in a wide range of parameters, especially where high accuracy and high polynomial degrees are required.
Dolgov, S.
2015-03-11
We apply the Tensor Train (TT) decomposition to construct the tensor product Polynomial Chaos Expansion (PCE) of a random field, to solve the stochastic elliptic diffusion PDE with the stochastic Galerkin discretization, and to compute some quantities of interest (mean, variance, exceedance probabilities). We assume that the random diffusion coefficient is given as a smooth transformation of a Gaussian random field. In this case, the PCE is delivered by a complicated formula, which lacks an analytic TT representation. To construct its TT approximation numerically, we develop the new block TT cross algorithm, a method that computes the whole TT decomposition from a few evaluations of the PCE formula. The new method is conceptually similar to the adaptive cross approximation in the TT format, but is more efficient when several tensors must be stored in the same TT representation, which is the case for the PCE. Besides, we demonstrate how to assemble the stochastic Galerkin matrix and to compute the solution of the elliptic equation and its post-processing, staying in the TT format. We compare our technique with the traditional sparse polynomial chaos and the Monte Carlo approaches. In the tensor product polynomial chaos, the polynomial degree is bounded for each random variable independently. This provides higher accuracy than the sparse polynomial set or the Monte Carlo method, but the cardinality of the tensor product set grows exponentially with the number of random variables. However, when the PCE coefficients are implicitly approximated in the TT format, the computations with the full tensor product polynomial set become possible. In the numerical experiments, we confirm that the new methodology is competitive in a wide range of parameters, especially where high accuracy and high polynomial degrees are required.
Forecasting with Universal Approximators and a Learning Algorithm
DEFF Research Database (Denmark)
Kock, Anders Bredahl
2011-01-01
This paper applies three universal approximators for forecasting. They are the Artificial Neural Networks, the Kolmogorov-Gabor polynomials, as well as the Elliptic Basis Function Networks. We are particularly interested in the relative performance and stability of these. Even though forecast...
Forecasting with Universal Approximators and a Learning Algorithm
DEFF Research Database (Denmark)
Kock, Anders Bredahl
This paper applies three universal approximators for forecasting. They are the Artificial Neural Networks, the Kolmogorov- Gabor polynomials, as well as the Elliptic Basis Function Networks. Even though forecast combination has a long history in econometrics focus has not been on proving loss...
New approximating results for data with errors in both variables
Bogdanova, N.; Todorov, S.
2015-05-01
We introduce new data from mineral water probe Lenovo Bulgaria, measured with errors in both variables. We apply our Orthonormal Polynomial Expansion Method (OPEM), based on Forsythe recurrence formula to describe the data in the new error corridor. The development of OPEM gives the approximating curves and their derivatives in optimal orthonormal and usual expansions including the errors in both variables with special criteria.
Approximation algorithms for multi-criteria traveling salesman problems
Manthey, B.; Shankar Ram, L.
2009-01-01
We analyze approximation algorithms for several variants of the traveling salesman problem with multiple objective functions. First, we consider the symmetric TSP (STSP) with $\\gamma$-triangle inequality. For this problem, we present a deterministic polynomial-time algorithm that achieves an approxi
Polynomial Operators on Classes of Regular Languages
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.
Eigenvalue conjecture and colored Alexander polynomials
Mironov, A
2016-01-01
We connect two important conjectures in the theory of knot polynomials. The first one is the property Al_R(q) = Al_{[1]}(q^{|R|}) for all single hook Young diagrams R, which is known to hold for all knots. The second conjecture claims that all the mixing matrices U_{i} in the relation {\\cal R}_i = U_i{\\cal R}_1U_i^{-1} between the i-th and the first generators {\\cal R}_i of the braid group are universally expressible through the eigenvalues of {\\cal R}_1. Since the above property of Alexander polynomials is very well tested, this relation provides a new support to the eigenvalue conjecture, especially for i>2, when its direct check by evaluation of the Racah matrices and their convolutions is technically difficult.
On Polynomial Sized MDP Succinct Policies
Liberatore, P
2011-01-01
Policies of Markov Decision Processes (MDPs) determine the next action to execute from the current state and, possibly, the history (the past states). When the number of states is large, succinct representations are often used to compactly represent both the MDPs and the policies in a reduced amount of space. In this paper, some problems related to the size of succinctly represented policies are analyzed. Namely, it is shown that some MDPs have policies that can only be represented in space super-polynomial in the size of the MDP, unless the polynomial hierarchy collapses. This fact motivates the study of the problem of deciding whether a given MDP has a policy of a given size and reward. Since some algorithms for MDPs work by finding a succinct representation of the value function, the problem of deciding the existence of a succinct representation of a value function of a given size and reward is also considered.
Venereau polynomials and related fiber bundles
Kaliman, Shulim; ZAIDENBERG, MIKHAIL
2003-01-01
The Venereau polynomials v-n:=y+x^n(xz+y(yu+z^2)), n>= 1, on A4 have all fibers isomorphic to the affine space A3. Moreover, for all n>= 1 the map (v-n, x) : A4 -> A2 yields a flat family of affine planes over A2. In the present note we show that over the punctured plane A2\\0, this family is a fiber bundle. This bundle is trivial if and only if v-n is a variable of the ring C[x][y,z,u] over C[x]. It is an open question whether v1 and v2 are variables of the polynomial ring C[x,y,z,u]. S. Vene...
On computing factors of cyclotomic polynomials
Brent, Richard P.
1993-07-01
For odd square-free n > 1 the cyclotomic polynomial {Φ_n}(x) satisfies the identity of Gauss, 4{Φ_n}(x) = A_n^2 - {( - 1)^{(n - 1)/2}}nB_n^2. A similar identity of Aurifeuille, Le Lasseur, and Lucas is {Φ_n}({( - 1)^{(n - 1)/2}}x) = C_n^2 - nxD_n^2 or, in the case that n is even and square-free, ± {Φ_{n/2}}( - {x^2}) = C_n^2 - nxD_n^2. Here, {A_n}(x), ldots ,{D_n}(x) are polynomials with integer coefficients. We show how these coefficients can be computed by simple algorithms which require O({n^2}) arithmetic operations and work over the integers. We also give explicit formulae and generating functions for {A_n}(x), ldots ,{D_n}(x) , and illustrate the application to integer factorization with some numerical examples.