New pole placement algorithm - Polynomial matrix approach
Shafai, B.; Keel, L. H.
1990-01-01
A simple and direct pole-placement algorithm is introduced for dynamical systems having a block companion matrix A. The algorithm utilizes well-established properties of matrix polynomials. Pole placement is achieved by appropriately assigning coefficient matrices of the corresponding matrix polynomial. This involves only matrix additions and multiplications without requiring matrix inversion. A numerical example is given for the purpose of illustration.
The q-Laguerre matrix polynomials.
Salem, Ahmed
2016-01-01
The Laguerre polynomials have been extended to Laguerre matrix polynomials by means of studying certain second-order matrix differential equation. In this paper, certain second-order matrix q-difference equation is investigated and solved. Its solution gives a generalized of the q-Laguerre polynomials in matrix variable. Four generating functions of this matrix polynomials are investigated. Two slightly different explicit forms are introduced. Three-term recurrence relation, Rodrigues-type formula and the q-orthogonality property are given.
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.
Characteristic Polynomials of Complex Random Matrix Models
Akemann, G
2003-01-01
We calculate the expectation value of an arbitrary product of characteristic polynomials of complex random matrices and their hermitian conjugates. Using the technique of orthogonal polynomials in the complex plane our result can be written in terms of a determinant containing these polynomials and their kernel. It generalizes the known expression for hermitian matrices and it also provides a generalization of the Christoffel formula to the complex plane. The derivation we present holds for complex matrix models with a general weight function at finite-N, where N is the size of the matrix. We give some explicit examples at finite-N for specific weight functions. The characteristic polynomials in the large-N limit at weak and strong non-hermiticity follow easily and they are universal in the weak limit. We also comment on the issue of the BMN large-N limit.
Representations of the Schroedinger group and matrix orthogonal polynomials
Energy Technology Data Exchange (ETDEWEB)
Vinet, Luc [Centre de recherches mathematiques, Universite de Montreal, CP 6128, succ. Centre-ville, Montreal, QC H3C 3J7 (Canada); Zhedanov, Alexei, E-mail: luc.vinet@umontreal.ca, E-mail: zhedanov@fti.dn.ua [Donetsk Institute for Physics and Technology, Donetsk 83114 (Ukraine)
2011-09-02
The representations of the Schroedinger group in one space dimension are explicitly constructed in the basis of the harmonic oscillator states. These representations are seen to involve matrix orthogonal polynomials in a discrete variable that have Charlier and Meixner polynomials as building blocks. The underlying Lie-theoretic framework allows for a systematic derivation of the structural formulas (recurrence relations, difference equations, Rodrigues' formula, etc) that these matrix orthogonal polynomials satisfy. (paper)
Strict Positivstellens\\" atze for matrix polynomials with scalar constraints
Cimpric, Jaka
2010-01-01
We extend Krivine's strict positivstellensatz for usual (real multivariate) polynomials to symmetric matrix polynomials with scalar constraints. The proof is an elementary computation with Schur complements. Analogous extensions of Schm\\" udgen's and Putinar's strict positivstellensatz were recently proved by Hol and Scherer using methods from optimization theory.
Polynomial Solutions to the Matrix Equation X−AXTB=C
Directory of Open Access Journals (Sweden)
Caiqin Song
2014-01-01
Full Text Available Solutions are constructed for the Kalman-Yakubovich-transpose equation X−AXTB=C. The solutions are stated as a polynomial of parameters of the matrix equation. One of the polynomial solutions is expressed by the symmetric operator matrix, controllability matrix, and observability matrix. Moreover, the explicit solution is proposed when the Kalman-Yakubovich-transpose matrix equation has a unique solution. The provided approach does not require the coefficient matrices to be in canonical form. In addition, the numerical example is given to illustrate the effectiveness of the derived method. Some applications in control theory are discussed at the end of this paper.
An Elementary Proof of the Polynomial Matrix Spectral Factorization Theorem
Ephremidze, Lasha
2010-01-01
A very simple and short proof of the polynomial matrix spectral factorization theorem (on the unit circle as well as on the real line) is presented, which relies on elementary complex analysis and linear algebra.
Pure states, positive matrix polynomials and sums of hermitian squares
Klep, Igor
2009-01-01
Let M be an archimedean quadratic module of real t-by-t matrix polynomials in n variables, and let S be the set of all real n-tuples where each element of M is positive semidefinite. Our key finding is a natural bijection between the set of pure states of M and the cartesian product of S with the real projective (t-1)-space. This leads us to conceptual proofs of positivity certificates for matrix polynomials, including the recent seminal result of Hol and Scherer: If a symmetric matrix polynomial is positive definite on S, then it belongs to M. We also discuss what happens for non-symmetric matrix polynomials or in the absence of the archimedean assumption, and review some of the related classical results. The methods employed are both algebraic and functional analytic.
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. .
Alvarez-Fernandez, Carlos; Manas, Manuel
2009-01-01
We consider the relation of the multi-component 2D Toda hierarchy with matrix orthogonal and biorthogonal polynomials. The multi-graded Hankel reduction of this hierarchy is considered and the corresponding generalized matrix orthogonal polynomials are studied. In particular for these polynomials we consider the recursion relations, and for rank one weights its relation with multiple orthogonal polynomials of mixed type with a type II normalization and the corresponding link with a Riemann--Hilbert problem.
Skew-orthogonal polynomials and random matrix theory
Ghosh, Saugata
2009-01-01
Orthogonal polynomials satisfy a three-term recursion relation irrespective of the weight function with respect to which they are defined. This gives a simple formula for the kernel function, known in the literature as the Christoffel-Darboux sum. The availability of asymptotic results of orthogonal polynomials and the simple structure of the Christoffel-Darboux sum make the study of unitary ensembles of random matrices relatively straightforward. In this book, the author develops the theory of skew-orthogonal polynomials and obtains recursion relations which, unlike orthogonal polynomials, depend on weight functions. After deriving reduced expressions, called the generalized Christoffel-Darboux formulas (GCD), he obtains universal correlation functions and non-universal level densities for a wide class of random matrix ensembles using the GCD. The author also shows that once questions about higher order effects are considered (questions that are relevant in different branches of physics and mathematics) the ...
A NOTE ON SOBOLEV ORTHOGONALITY FOR LAGUERRE MATRIX POLYNOMIALS
Institute of Scientific and Technical Information of China (English)
Zhihui Zhu; Zhongkai Li
2007-01-01
Let {Ln(A,λ)(x)}n≥0 be the sequence of monic Laguerre matrix polynomials defined on [0, ∞) by Ln(A,λ)(x)=n!/(-λ)n∑nk=0(-λ)κ/k!(n-1)! (A+I)n[(A+I)k]-1 xk,where A ∈ Cr×r. It is known that {Ln(A,λ)(x)}n≥0 is orthogonal with respect to a matrix moment functional when A satisfies the spectral condition that Re(z) ＞ - 1 for every z ∈σ(A).In this note we show that forA such that σ(A) does not contain negative integers, the Laguerre matrix polynomials Ln(A,λ) (x) are orthogonal with respect to a non-diagonal SobolevLaguerre matrix moment functional, which extends two cases: the above matrix case and the known scalar case.
Institute of Scientific and Technical Information of China (English)
Bin ZHOU; Guangren DUAN
2006-01-01
In this paper, an explicit solution to polynomial matrix right coprime factorization of input-state transfer function is obtained in terms of the Krylov matrix and the Pseudo-controllability indices of the pair of coefficient matrices.The proposed approach only needs to solve a series of line ar equations. Applications of this solution to a type of generalized Sylvester matrix equations and the problem of parametric eigenstructure assignment by state feedback are investigated.These new solutions are simple, they possess better structural properties and are very convenient to use. An example shows the effect of the proposed results.
Cassatella-Contra, Giovanni A
2011-01-01
In this paper matrix orthogonal polynomials in the real line are described in terms of a Riemann--Hilbert problem. This approach provides an easy derivation of discrete equations for the corresponding matrix recursion coefficients. The discrete equation is explicitly derived in the matrix Freud case, associated with matrix quartic potentials. It is shown that, when the initial condition and the measure are simultaneously triangularizable, this matrix discrete equation possesses the singularity confinement property, independently if the solution under consideration is given by recursion coefficients to quartic Freud matrix orthogonal polynomials or not.
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.
Transfer matrix computation of generalized critical polynomials in percolation
Scullard, Christian R.; Lykke Jacobsen, Jesper
2012-12-01
Percolation thresholds have recently been studied by means of a graph polynomial PB(p), henceforth referred to as the critical polynomial, that may be defined on any periodic lattice. The polynomial depends on a finite subgraph B, called the basis, and the way in which the basis is tiled to form the lattice. The unique root of PB(p) in [0, 1] either gives the exact percolation threshold for the lattice, or provides an approximation that becomes more accurate with appropriately increasing size of B. Initially PB(p) was defined by a contraction-deletion identity, similar to that satisfied by the Tutte polynomial. Here, we give an alternative probabilistic definition of PB(p), which allows for much more efficient computations, by using the transfer matrix, than was previously possible with contraction-deletion. We present bond percolation polynomials for the (4, 82), kagome, and (3, 122) lattices for bases of up to respectively 96, 162 and 243 edges, much larger than the previous limit of 36 edges using contraction-deletion. We discuss in detail the role of the symmetries and the embedding of B. For the largest bases, we obtain the thresholds pc(4, 82) = 0.676 803 329…, pc(kagome) = 0.524 404 998…, pc(3, 122) = 0.740 420 798…, comparable to the best simulation results. We also show that the alternative definition of PB(p) can be applied to study site percolation problems. This article is part of ‘Lattice models and integrability’, a special issue of Journal of Physics A: Mathematical and Theoretical in honour of F Y Wu's 80th birthday.
Apostol, Tom M. (Editor)
1991-01-01
In this 'Project Mathematics! series, sponsored by California Institute for Technology (CalTech), the mathematical concept of polynomials in rectangular coordinate (x, y) systems are explored. sing film footage of real life applications and computer animation sequences, the history of, the application of, and the different linear coordinate systems for quadratic, cubic, intersecting, and higher degree of polynomials are discussed.
The Schur algorithm for generalized Schur functions III : J-unitary matrix polynomials on the circle
Alpay, Daniel; Azizov, Tomas; Dijksma, Aad; Langer, Heinz
2003-01-01
The main result is that for J = ((1)(0) (0)(-1)) every J-unitary 2 x 2-matrix polynomial on the unit circle is an essentially unique product of elementary J-unitary 2 x 2-matrix polynomials which are either of degree 1 or 2k. This is shown by means of the generalized Schur transformation introduced
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.
Ridge Polynomial Neural Network with Error Feedback for Time Series Forecasting.
Waheeb, Waddah; Ghazali, Rozaida; Herawan, Tutut
2016-01-01
Time series forecasting has gained much attention due to its many practical applications. Higher-order neural network with recurrent feedback is a powerful technique that has been used successfully for time series forecasting. It maintains fast learning and the ability to learn the dynamics of the time series over time. Network output feedback is the most common recurrent feedback for many recurrent neural network models. However, not much attention has been paid to the use of network error feedback instead of network output feedback. In this study, we propose a novel model, called Ridge Polynomial Neural Network with Error Feedback (RPNN-EF) that incorporates higher order terms, recurrence and error feedback. To evaluate the performance of RPNN-EF, we used four univariate time series with different forecasting horizons, namely star brightness, monthly smoothed sunspot numbers, daily Euro/Dollar exchange rate, and Mackey-Glass time-delay differential equation. We compared the forecasting performance of RPNN-EF with the ordinary Ridge Polynomial Neural Network (RPNN) and the Dynamic Ridge Polynomial Neural Network (DRPNN). Simulation results showed an average 23.34% improvement in Root Mean Square Error (RMSE) with respect to RPNN and an average 10.74% improvement with respect to DRPNN. That means that using network errors during training helps enhance the overall forecasting performance for the network.
Institute of Scientific and Technical Information of China (English)
XU Xiu-Wei; REN Ting-Qi; LIU Shu-Yan; MA Qiu-Ming; LIU Sheng-Dian
2007-01-01
Making use of the transformation relation among usual, normal, and antinormal ordering for the multimode boson exponential quadratic polynomial operators (BEQPO's), we present the analytic expression of arbitrary matrix elements for BEQPO's. As a preliminary application, we obtain the exact expressions of partition function about the boson quadratic polynomial system, matrix elements in particle-number, coordinate, and momentum representation, and P representation for the BEQPO's.
Matrix exponentials, SU(N) group elements, and real polynomial roots
Van Kortryk, T S
2015-01-01
The exponential of an NxN matrix can always be expressed as a matrix polynomial of order N-1. In particular, a general group element for the fundamental representation of SU(N) can be expressed as a matrix polynomial of order N-1 in a traceless NxN hermitian generating matrix, with polynomial coefficients consisting of elementary trigonometric functions dependent on N-2 invariants in addition to the group parameter. These invariants are just angles determined by the direction of a real N-vector whose components are the eigenvalues of the hermitian matrix. Equivalently, the eigenvalues are given by projecting the vertices of an (N-1)-simplex onto a particular axis passing through the center of the simplex. The orientation of the simplex relative to this axis determines the angular invariants and hence the real eigenvalues of the matrix.
Skew-orthogonal polynomials, differential systems and random matrix theory
Energy Technology Data Exchange (ETDEWEB)
Ghosh, Saugata [Abdus Salam ICTP, Strada Costiera 11, 34100, Trieste (Italy)
2007-01-26
We study skew-orthogonal polynomials with respect to the weight function exp [ - 2V(x)], with V(x) = {sigma}{sup 2d}{sub K=1}(u{sub K}/K)x{sup K}, u{sub 2d} > 0, d > 0. A finite subsequence of such skew-orthogonal polynomials arising in the study of orthogonal and symplectic ensembles of random matrices satisfies a system of differential-difference-deformation equation. The vectors formed by such subsequence have the rank equal to the degree of the potential in the quaternion sense. These solutions satisfy certain compatibility condition and hence admit a simultaneous fundamental system of solutions.
Properties of Matrix Orthogonal Polynomials via their Riemann-Hilbert Characterization
Directory of Open Access Journals (Sweden)
F. Alberto Grünbaum
2011-10-01
Full Text Available We give a Riemann-Hilbert approach to the theory of matrix orthogonal polynomials. We will focus on the algebraic aspects of the problem, obtaining difference and differential relations satisfied by the corresponding orthogonal polynomials. We will show that in the matrix case there is some extra freedom that allows us to obtain a family of ladder operators, some of them of 0-th order, something that is not possible in the scalar case. The combination of the ladder operators will lead to a family of second-order differential equations satisfied by the orthogonal polynomials, some of them of 0-th and first order, something also impossible in the scalar setting. This shows that the differential properties in the matrix case are much more complicated than in the scalar situation. We will study several examples given in the last years as well as others not considered so far.
Institute of Scientific and Technical Information of China (English)
Feng Yi-Fu; Zhang Qing-Ling; Feng De-Zhi
2012-01-01
The global stability problem of Takagi-Sugeno (T S) fuzzy Hopfield neural networks (FHNNs) with time delays is investigated.Novel LMI-based stability criteria are obtained by using Lyapunov functional theory to guarantee the asymptotic stability of the FHNNs with less conservatism.Firstly,using both Finsler's lemma and an improved homogeneous matrix polynomial technique,and applying an affine parameter-dependent Lyapunov-Krasovskii functional,we obtain the convergent LMI-based stability criteria.Algebraic properties of the fuzzy membership functions in the unit simplex are considered in the process of stability analysis via the homogeneous matrix polynomials technique.Secondly,to further reduce the conservatism,a new right-hand-side slack variables introducing technique is also proposed in terms of LMIs,which is suitable to the homogeneous matrix polynomials setting.Finally,two illustrative examples are given to show the efficiency of the proposed approaches.
On tau functions for orthogonal polynomials and matrix models
Blower, Gordon
2010-01-01
Let v be a real polynomial of even degree, and let \\rho be the equilibrium probability measure for v with support S; so that v(x)\\geq 2\\int \\log |x-y| \\rho (dy)+C_v for some constant C_v with support S. Then S is the union of finitely many bounded intervals with endpoints delta_j, and \\rho is given by an algebrais weight w(x) on S. The system of orthogonal polynomials for w gives rise to the Magnus--Schlesinger differential equations. This paper identifies the tau function of this system with the Hankel determinant det[\\in x^{j+k}\\rho (dx)] of \\rho. The solutions of the Magnus--Schlesinger equations are realised by a linear system, which is used to compute the tau function in terms of a Gelfand--Levitan equaiton. The tau function is associated with a potential q and a scattering problem for the Schrodinger operator with potential q. For some algebro-geometric potentials, the paper solves the scattering problem in terms of linear systems. The theory extends naturally to elliptic curves and resolves the case wh...
Energy Technology Data Exchange (ETDEWEB)
Tumelero, Fernanda, E-mail: fernanda.tumelero@yahoo.com.br [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Programa de Pos-Graduacao em Engenharia Mecanica; Petersen, Claudio Z.; Goncalves, Glenio A.; Lazzari, Luana, E-mail: claudiopeteren@yahoo.com.br, E-mail: gleniogoncalves@yahoo.com.br, E-mail: luana-lazzari@hotmail.com [Universidade Federal de Pelotas (DME/UFPEL), Capao do Leao, RS (Brazil). Instituto de Fisica e Matematica
2015-07-01
In this work, we present a solution of the Neutron Point Kinetics Equations with temperature feedback effects applying the Polynomial Approach Method. For the solution, we consider one and six groups of delayed neutrons precursors with temperature feedback effects and constant reactivity. The main idea is to expand the neutron density, delayed neutron precursors and temperature as a power series considering the reactivity as an arbitrary function of the time in a relatively short time interval around an ordinary point. In the first interval one applies the initial conditions of the problem and the analytical continuation is used to determine the solutions of the next intervals. With the application of the Polynomial Approximation Method it is possible to overcome the stiffness problem of the equations. In such a way, one varies the time step size of the Polynomial Approach Method and performs an analysis about the precision and computational time. Moreover, we compare the method with different types of approaches (linear, quadratic and cubic) of the power series. The answer of neutron density and temperature obtained by numerical simulations with linear approximation are compared with results in the literature. (author)
Institute of Scientific and Technical Information of China (English)
Hossain O. Yakhlef; Francisco Marcellán
2002-01-01
Given a positive definite matrix measure Ω supported on the unit circle T, then main purpose of this paper is to study the asymptotic behavior of Ln(Ω)Ln(Ω)-1 and Φn(z;Ω)Φn(z;Ω)-1 where Ω(z) = Ω(z) + zδ(z - w); |w| ＞ 1,M is a positive definite matrix and δ is the Dirac matrix measure. Here, Ln ( @ ) means the leading coefficient of the orthonormal matrix polynomials Φn(z; @ ).Finally, we deduce the asymptotic behavior of Φn (w;Ω)Φn (w;Ω) * in the case when M=I.
A survey on orthogonal matrix polynomials satisfying second order differential equations
Duran, Antonio J.; Grunbaum, F. Alberto
2005-06-01
The subject of orthogonal polynomials cuts across a large piece of mathematics and its applications. Two notable examples are mathematical physics in the 19th and 20th centuries, as well as the theory of spherical functions for symmetric spaces. It is also clear that many areas of mathematics grew out of the consideration of problems like the moment problem that are intimately associated to the study of (scalar valued) orthogonal polynomials.Matrix orthogonality on the real line has been sporadically studied during the last half century since Krein devoted some papers to the subject in 1949, see (AMS Translations, Series 2, vol. 97, Providence, Rhode Island, 1971, pp. 75-143, Dokl. Akad. Nauk SSSR 69(2) (1949) 125). In the last decade this study has been made more systematic with the consequence that many basic results of scalar orthogonality have been extended to the matrix case. The most recent of these results is the discovery of important examples of orthogonal matrix polynomials: many families of orthogonal matrix polynomials have been found that (as the classical families of Hermite, Laguerre and Jacobi in the scalar case) satisfy second order differential equations with coefficients independent of n. The aim of this paper is to give an overview of the techniques that have led to these examples, a small sample of the examples themselves and a small step in the challenging direction of finding applications of these new examples.
Affine arithmetic in matrix form for polynomial evaluation and algebraic curve drawing
Institute of Scientific and Technical Information of China (English)
无
2002-01-01
This paper shows how tight bounds for the range of a bivariate polynomial can be found using a matrix method based on affine arithmetic. Then, this method is applied to drawing an algebraic curve with a hierarchical algorithm, which demonstrates that more accurate answers can be obtained more rapidly than using conventional interval arithmetic.
The Ehrlich-Aberth method for palindromic matrix polynomials represented in the Dickson basis
Gemignani, Luca
2011-01-01
An algorithm based on the Ehrlich-Aberth root-finding method is presented for the computation of the eigenvalues of a T-palindromic matrix polynomial. A structured linearization of the polynomial represented in the Dickson basis is introduced in order to exploit the symmetry of the roots by halving the total number of the required approximations. The rank structure properties of the linearization allow the design of a fast and numerically robust implementation of the root-finding iteration. Numerical experiments that confirm the effectiveness and the robustness of the approach are provided.
Finding all real roots of a polynomial by matrix algebra and the Adomian decomposition method
Directory of Open Access Journals (Sweden)
Hooman Fatoorehchi
2014-10-01
Full Text Available In this paper, we put forth a combined method for calculation of all real zeroes of a polynomial equation through the Adomian decomposition method equipped with a number of developed theorems from matrix algebra. These auxiliary theorems are associated with eigenvalues of matrices and enable convergence of the Adomian decomposition method toward different real roots of the target polynomial equation. To further improve the computational speed of our technique, a nonlinear convergence accelerator known as the Shanks transform has optionally been employed. For the sake of illustration, a number of numerical examples are given.
Nonmonotonic Recursive Polynomial Expansions for Linear Scaling Calculation of the Density Matrix.
Rubensson, Emanuel H
2011-05-10
As it stands, density matrix purification is a powerful tool for linear scaling electronic structure calculations. The convergence is rapid and depends only weakly on the band gap. However, as will be shown in this letter, there is room for improvements. The key is to allow for nonmonotonicity in the recursive polynomial expansion. On the basis of this idea, new purification schemes are proposed that require only half the number of matrix-matrix multiplications compared to previous schemes. The speedup is essentially independent of the location of the chemical potential and increases with decreasing band gap.
Orthogonal polynomials in the normal matrix model with a cubic potential
Bleher, Pavel M
2011-01-01
We consider the normal matrix model with a cubic potential. The model is ill-defined, and in order to reguralize it, Elbau and Felder introduced a model with a cut-off and corresponding system of orthogonal polynomials with respect to a varying exponential weight on the cut-off region on the complex plane. In the present paper we show how to define orthogonal polynomials on a specially chosen system of infinite contours on the complex plane, without any cut-off, which satisfy the same recurrence algebraic identity that is asymptotically valid for the orthogonal polynomials of Elbau and Felder. The main goal of this paper is to develop the Riemann-Hilbert (RH) approach to the orthogonal polynomials under consideration and to obtain their asymptotic behavior on the complex plane as the degree $n$ of the polynomial goes to infinity. As the first step in the RH approach, we introduce an auxiliary vector equilibrium problem for a pair of measures $(\\mu_1,\\mu_2)$ on the complex plane. We then formulate a $3\\times 3...
Institute of Scientific and Technical Information of China (English)
LI Chunxiang; ZHOU Dai
2004-01-01
The polynomial matrix using the block coefficient matrix representation auto-regressive moving average (referred to as the PM-ARMA) model is constructed in this paper for actively controlled multi-degree-of-freedom (MDOF) structures with time-delay through equivalently transforming the preliminary state space realization into the new state space realization. The PM-ARMA model is a more general formulation with respect to the polynomial using the coefficient representation auto-regressive moving average (ARMA) model due to its capability to cope with actively controlled structures with any given structural degrees of freedom and any chosen number of sensors and actuators. (The sensors and actuators are required to maintain the identical number.) under any dimensional stationary stochastic excitation.
Vector-Valued Polynomials and a Matrix Weight Function with B2-Action
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Charles F. Dunkl
2013-01-01
Full Text Available The structure of orthogonal polynomials on $mathbb{R}^{2}$ with the weight function $vert x_{1}^{2}-x_{2}^{2}vert ^{2k_{0}}vertx_{1}x_{2}vert ^{2k_{1}}e^{-( x_{1}^{2}+x_{2}^{2}/2}$ is based on the Dunkl operators of type $B_{2}$. This refers to the full symmetry group of the square, generated by reflections in the lines $x_{1}=0$ and $x_{1}-x_{2}=0$. The weight function is integrable if $k_{0},k_{1},k_{0} +k_{1}>-frac{1}{2}$. Dunkl operators can be defined for polynomials taking values in a module of the associated reflection group, that is, a vector space on which the group has an irreducible representation. The unique $2$-dimensional representation of the group $B_{2}$ is used here. The specific operators for this group and an analysis of the inner products on the harmonic vector-valued polynomials are presented in this paper. An orthogonal basis for the harmonic polynomials is constructed, and is used to define an exponential-type kernel. In contrast to the ordinary scalar case the inner product structure is positive only when $( k_{0},k_{1}$ satisfy $-frac{1}{2} < k_{0}pm k_{1} < frac{1}{2}$. For vector polynomials $(f_{i}_{i=1}^{2}$, $(g_{i}_{i=1}^{2}$ the inner product has the form $iint_{mathbb{R}^{2}}f(x K(x g(x^{T}e^{-( x_{1}^{2}+x_{2}^{2}/2}dx_{1}dx_{2}$ where the matrix function $K(x$ has to satisfy various transformation and boundary conditions. The matrix $K$ is expressed in terms of hypergeometric functions.
A local construction of the Smith normal form of a matrix polynomial
Energy Technology Data Exchange (ETDEWEB)
Wilkening, Jon; Yu, Jia
2008-09-01
We present an algorithm for computing a Smith form with multipliers of a regular matrix polynomial over a field. This algorithm differs from previous ones in that it computes a local Smith form for each irreducible factor in the determinant separately and then combines them into a global Smith form, whereas other algorithms apply a sequence of unimodular operations to the original matrix row by row (or column by column). The performance of the algorithm in exact arithmetic is reported for several test cases.
Directory of Open Access Journals (Sweden)
Mario Faliva
2007-10-01
Full Text Available In this paper the issue of the inversion of a matrix polynomial about a unit root is tackled by resorting to Laurent expansion: The principal-part matrix coefficients associated with a simple and a second order pole are properly characterized and closed-form expressions are derived by virtue of a recent result on partitioned inversion (Faliva and Zoia, 2002. This eventually sheds more light on the analytical foundations of unit-root econometrics which in turn paves the way to an elegant unified representation theorem for (cointegrated processes up to the second order.
Transfer matrix computation of critical polynomials for two-dimensional Potts models
Lykke Jacobsen, Jesper; Scullard, Christian R.
2013-02-01
In our previous work [1] we have shown that critical manifolds of the q-state Potts model can be studied by means of a graph polynomial PB(q, v), henceforth referred to as the critical polynomial. This polynomial may be defined on any periodic two-dimensional lattice. It depends on a finite subgraph B, called the basis, and the manner in which B is tiled to construct the lattice. The real roots v = eK - 1 of PB(q, v) either give the exact critical points for the lattice, or provide approximations that, in principle, can be made arbitrarily accurate by increasing the size of B in an appropriate way. In earlier work, PB(q, v) was defined by a contraction-deletion identity, similar to that satisfied by the Tutte polynomial. Here, we give a probabilistic definition of PB(q, v), which facilitates its computation, using the transfer matrix, on much larger B than was previously possible. We present results for the critical polynomial on the (4, 82), kagome, and (3, 122) lattices for bases of up to respectively 96, 162, and 243 edges, compared to the limit of 36 edges with contraction-deletion. We discuss in detail the role of the symmetries and the embedding of B. The critical temperatures vc obtained for ferromagnetic (v > 0) Potts models are at least as precise as the best available results from Monte Carlo simulations or series expansions. For instance, with q = 3 we obtain vc(4, 82) = 3.742 489 (4), vc(kagome) = 1.876 459 7 (2), and vc(3, 122) = 5.033 078 49 (4), the precision being comparable or superior to the best simulation results. More generally, we trace the critical manifolds in the real (q, v) plane and discuss the intricate structure of the phase diagram in the antiferromagnetic (v < 0) region.
P A M Dirac meets M G Krein: matrix orthogonal polynomials and Dirac's equation
Energy Technology Data Exchange (ETDEWEB)
Duran, Antonio J [Departamento de Analisis Matematico, Universidad de Sevilla, Apdo (PO BOX) 1160, 41080 Sevilla (Spain); Gruenbaum, F Alberto [Department of Mathematics, University of California, Berkeley, CA 94720 (United States)
2006-04-07
The solution of several instances of the Schroedinger equation (1926) is made possible by using the well-known orthogonal polynomials associated with the names of Hermite, Legendre and Laguerre. A relativistic alternative to this equation was proposed by Dirac (1928) involving differential operators with matrix coefficients. In 1949 Krein developed a theory of matrix-valued orthogonal polynomials without any reference to differential equations. In Duran A J (1997 Matrix inner product having a matrix symmetric second order differential operator Rocky Mt. J. Math. 27 585-600), one of us raised the question of determining instances of these matrix-valued polynomials going along with second order differential operators with matrix coefficients. In Duran A J and Gruenbaum F A (2004 Orthogonal matrix polynomials satisfying second order differential equations Int. Math. Res. Not. 10 461-84), we developed a method to produce such examples and observed that in certain cases there is a connection with the instance of Dirac's equation with a central potential. We observe that the case of the central Coulomb potential discussed in the physics literature in Darwin C G (1928 Proc. R. Soc. A 118 654), Nikiforov A F and Uvarov V B (1988 Special Functions of Mathematical Physics (Basle: Birkhauser) and Rose M E 1961 Relativistic Electron Theory (New York: Wiley)), and its solution, gives rise to a matrix weight function whose orthogonal polynomials solve a second order differential equation. To the best of our knowledge this is the first instance of a connection between the solution of the first order matrix equation of Dirac and the theory of matrix-valued orthogonal polynomials initiated by M G Krein.
Matrix biorthogonal polynomials on the unit circle and non-Abelian Ablowitz-Ladik hierarchy
Energy Technology Data Exchange (ETDEWEB)
Cafasso, Mattia [Departement de Mathematiques, Universite Catholique de Louvain, Chemin du Cyclotron, 2, 1348 Louvain-la Neuve (Belgium)
2009-09-11
Adler and van Moerbeke (2001 Commun. Pure Appl. Math. 54 153-205) described a reduction of the 2D-Toda hierarchy called the Toeplitz lattice. This hierarchy turns out to be equivalent to the one originally described by Ablowitz and Ladik (1975 J. Math. Phys. 16 598-603) using semidiscrete zero- curvature equations. In this paper, we obtain the original semidiscrete zero-curvature equations starting directly from the Toeplitz lattice and we generalize these computations to the matrix case. This generalization leads us to the semidiscrete zero-curvature equations for the non-Abelian (or multicomponent) version of the Ablowitz-Ladik equations (Gerdzhikov and Ivanov 1982 Theor. Math. Phys. 52 676-85). In this way, we extend the link between biorthogonal polynomials on the unit circle and the Ablowitz-Ladik hierarchy to the matrix case.
Siminovitch, David; Untidt, Thomas; Nielsen, Niels Chr
2004-01-01
Our recent exact effective Hamiltonian theory (EEHT) for exact analysis of nuclear magnetic resonance (NMR) experiments relied on a novel entanglement of unitary exponential operators via finite expansion of the logarithmic mapping function. In the present study, we introduce simple alternant quotient expressions for the coefficients of the polynomial matrix expansion of these entangled operators. These expressions facilitate an extension of our previous closed solution to the Baker-Campbell-Hausdorff problem for SU(N) systems from Nfunction. The general applicability of these expressions is demonstrated by several examples with relevance for NMR spectroscopy. The specific form of the alternant quotients is also used to demonstrate the fundamentally important equivalence of Sylvester's theorem (also known as the spectral theorem) and the EEHT expansion.
Alpay, D.; Dijksma, A.; Langer, H.
2004-01-01
We prove that a 2 × 2 matrix polynomial which is J-unitary on the real line can be written as a product of normalized elementary J-unitary factors and a J-unitary constant. In the second part we give an algorithm for this factorization using an analog of the Schur transformation.
Towards R-matrix construction of Khovanov-Rozansky polynomials. I. Primary $T$-deformation of HOMFLY
Anokhina, A
2014-01-01
We elaborate on the simple alternative from arXiv:1308.5759 to the matrix-factorization construction of Khovanov-Rozansky (KR) polynomials for arbitrary knots and links in the fundamental representation of arbitrary SL(N). Construction consists of two steps: with every link diagram with m vertices one associates an m-dimensional hypercube with certain q-graded vector spaces, associated to its 2^m vertices. A generating function for q-dimensions of these spaces is what we suggest to call the primary T-deformation of HOMFLY polynomial -- because, as we demonstrate, it can be explicitly reduced to calculations of ordinary HOMFLY polynomials, i.e. to manipulations with quantum R-matrices. The second step is a certain minimization of residues of this new polynomial with respect to T+1. Minimization is ambiguous and is actually specified by the choice of commuting cut-and-join morphisms, acting along the edges of the hypercube -- this promotes it to Abelian quiver, and KR polynomial is a Poincare polynomial of asso...
Directory of Open Access Journals (Sweden)
Waleed M. Abd-Elhameed
2016-09-01
Full Text Available Herein, two numerical algorithms for solving some linear and nonlinear fractional-order differential equations are presented and analyzed. For this purpose, a novel operational matrix of fractional-order derivatives of Fibonacci polynomials was constructed and employed along with the application of the tau and collocation spectral methods. The convergence and error analysis of the suggested Fibonacci expansion were carefully investigated. Some numerical examples with comparisons are presented to ensure the efficiency, applicability and high accuracy of the proposed algorithms. Two accurate semi-analytic polynomial solutions for linear and nonlinear fractional differential equations are the result.
Polynomial Fibonacci-Hessenberg matrices
Energy Technology Data Exchange (ETDEWEB)
Esmaeili, Morteza [Dept. of Mathematical Sciences, Isfahan University of Technology, 84156-83111 Isfahan (Iran, Islamic Republic of)], E-mail: emorteza@cc.iut.ac.ir; Esmaeili, Mostafa [Dept. of Electrical and Computer Engineering, Isfahan University of Technology, 84156-83111 Isfahan (Iran, Islamic Republic of)
2009-09-15
A Fibonacci-Hessenberg matrix with Fibonacci polynomial determinant is referred to as a polynomial Fibonacci-Hessenberg matrix. Several classes of polynomial Fibonacci-Hessenberg matrices are introduced. The notion of two-dimensional Fibonacci polynomial array is introduced and three classes of polynomial Fibonacci-Hessenberg matrices satisfying this property are given.
Bounds for the variation of the roots of a polynomial and the eigenvalues of a matrix
Bhatia, Rajendra; Elsner, Ludwig; Krause, G.
1990-01-01
We derive some new bounds for the distance between the roots of two polynomials in terms of their coefficients and for the distance between the eigenvalues of two matrices in terms of the norm of their difference.
Generalized Christoffel-Darboux formula for skew-orthogonal polynomials and random matrix theory
Energy Technology Data Exchange (ETDEWEB)
Ghosh, Saugata [Abdus Salam ICTP, Strada Costiera 11, 34100, Trieste (Italy)
2006-07-14
We obtain a generalized Christoffel-Darboux (GCD) formula for skew-orthogonal polynomials. Using this, we present an alternative derivation of the level density and two-point function for Gaussian orthogonal ensembles and Gaussian symplectic ensembles of random matrices.
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....
Some New Formulae for Genocchi Numbers and Polynomials Involving Bernoulli and Euler Polynomials
Directory of Open Access Journals (Sweden)
Serkan Araci
2014-01-01
Full Text Available We give some new formulae for product of two Genocchi polynomials including Euler polynomials and Bernoulli polynomials. Moreover, we derive some applications for Genocchi polynomials to study a matrix formulation.
New linear codes from matrix-product codes with polynomial units
DEFF Research Database (Denmark)
Hernando, Fernando; Ruano Benito, Diego
2010-01-01
A new construction of codes from old ones is considered, it is an extension of the matrix-product construction. Several linear codes that improve the parameters of the known ones are presented.......A new construction of codes from old ones is considered, it is an extension of the matrix-product construction. Several linear codes that improve the parameters of the known ones are presented....
A Polynomial Greatest Common Factor Of Numerical Matrix%多项式最大公因式的数值矩阵求法
Institute of Scientific and Technical Information of China (English)
韩建玲
2012-01-01
求多项式的最大公因式常用分解因式和辗转相除法,分解因式对次数较高的多项式有一定难度,而辗转相除法又比较繁琐,根据矩阵的性质提出了一种求两个及两个以上多项式的最大公因式的方法——数值矩阵法。%Find the greatest common divisor of polynomials commonly used decomposition type and Euclidean algorithm, factored some difficulty on the number of higher polynomial, Euclidean algorithm is more complicated. According to the nature of the matrix to a request of two or more for the greatest common divisor of polynomials - numerical matrix method.
Directory of Open Access Journals (Sweden)
Josep Rubió-Massegú
2013-01-01
Full Text Available In this paper, a new strategy to design static output-feedback controllers for a class of vehicle suspension systems is presented. A theoretical background on recent advances in output-feedback control is first provided, which makes possible an effective synthesis of static output-feedback controllers by solving a single linear matrix inequality optimization problem. Next, a simplified model of a quarter-car suspension system is proposed, taking the ride comfort, suspension stroke, road holding ability, and control effort as the main performance criteria in the vehicle suspension design. The new approach is then used to design a static output-feedback H∞ controller that only uses the suspension deflection and the sprung mass velocity as feedback information. Numerical simulations indicate that, despite the restricted feedback information, this static output-feedback H∞ controller exhibits an excellent behavior in terms of both frequency and time responses, when compared with the corresponding state-feedback H∞ controller.
Directory of Open Access Journals (Sweden)
Ayşe Betül Koç
2014-01-01
Full Text Available A pseudospectral method based on the Fibonacci operational matrix is proposed to solve generalized pantograph equations with linear functional arguments. By using this method, approximate solutions of the problems are easily obtained in form of the truncated Fibonacci series. Some illustrative examples are given to verify the efficiency and effectiveness of the proposed method. Then, the numerical results are compared with other methods.
The Quadratic Matrix Inequality in Singular H∞ Control with State Feedback
Stoorvogel, A.A.; Trentelman, H.L.
1990-01-01
In this paper the standard H∞ control problem using state feedback is considered. Given a linear, time-invariant, finite-dimensional system, this problem consists of finding a static state feedback such that the resulting closed-loop transfer matrix has H∞ norm smaller than some a priori given upper
Energy Technology Data Exchange (ETDEWEB)
Albuquerque, Marilia Cordeiro C. de; Lima, Thaysa Araujo de; Aquino, Katia Aparecida da Silva; Araujo, Elmo S., E-mail: aquino@ufpe.b [Universidade Federal de Pernambuco (DEN/UFPE), Recife, PE (Brazil). Dept. de Energia Nuclear
2011-07-01
Stibnite (Sb{sub 2}S{sub 3}) was synthesized by sonochemical method. Amorphous powder of Sb{sub 2}S{sub 3} was obtained and exhibit nanospheres structure with an average size in the range of 300-500 nm. Commercial Polyvinyl Chloride (PVC) containing Sb{sub 2}S{sub 3} nanoparticles (PVC/Sb) at concentrations of 0.10; 0.30 and 0.50 wt% were investigated. The samples were irradiated with gamma radiation ({sup 60}Co) at room temperature and air atmosphere. The viscosity-average molar mass (M{sub v}) was measured for PVC systems without nanoparticles and with nanoparticles. Decreases in molar mass observed when the systems were gamma irradiated reflect the random scission effects that take place in the main chain. Degradation index (DI) value was also obtained by viscosity analysis. DI results showed that the addition of Sb{sub 2}S{sub 3} nanoparticles at 0.3 wt% into PVC matrix irradiated at dose of 25 kGy decreased the number of main chain scissions and was calculated a protection index of 66,5% in PVC matrix. Results about the free radical scavenger action of the Sb{sub 2}S{sub 3} were obtained by use of 2,2-diphenyl-1-(2,4,6-trinitrophenyl)-hydrazyl radical (DPPH) and are discussed in this study. Changes in the infrared spectra of PVC systems indicated that polymer molecules interact with Sb{sub 2}S{sub 3} nanoparticles. (author)
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.
Extracellular matrix proteins: A positive feedback loop in lung fibrosis?
Blaauboer, M.E.; Boeijen, F.R.; Emson, C.L.; Turner, S.M.; Zandieh-Doulabi, B.; Hanemaaijer, R.; Smit, T.H.; Stoop, R.; Everts, V.
2014-01-01
Lung fibrosis is characterized by excessive deposition of extracellular matrix. This not only affects tissue architecture and function, but it also influences fibroblast behavior and thus disease progression. Here we describe the expression of elastin, type V collagen and tenascin C during the
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.
Determinants and Polynomial Root Structure
De Pillis, L. G.
2005-01-01
A little known property of determinants is developed in a manner accessible to beginning undergraduates in linear algebra. Using the language of matrix theory, a classical result by Sylvester that describes when two polynomials have a common root is recaptured. Among results concerning the structure of polynomial roots, polynomials with pairs of…
Prime power polynomial maps over finite fields
Berson, Joost
2012-01-01
We consider polynomial maps described by so-called prime power polynomials. These polynomials are defined using a fixed power of a prime number, say q. Considering invertible polynomial maps of this type over a characteristic zero field, we will only obtain (up to permutation of the variables) triangular maps, which are the most basic examples of polynomial automorphisms. However, over the finite field F_q automorphisms of this type have (in general) an entirely different structure. Namely, we will show that the prime power polynomial maps over F_q are in one-to-one correspondence with matrices having coefficients in a univariate polynomial ring over F_q. Furthermore, composition of polynomial maps translates to matrix multiplication, implying that invertible prime power polynomial maps correspond to invertible matrices. This alternate description of the prime power polynomial automorphism subgroup leads to the solution of many famous conjectures for this kind of polynomials and polynomial maps.
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
Directory of Open Access Journals (Sweden)
D. Jabari Sabeg
2016-10-01
Full Text Available In this paper, we present a new computational method for solving nonlinear singular boundary value problems of fractional order arising in biology. To this end, we apply the operational matrices of derivatives of shifted Legendre polynomials to reduce such problems to a system of nonlinear algebraic equations. To demonstrate the validity and applicability of the presented method, we present some numerical examples.
Ghosh, Saugata
2008-01-01
We derive bulk asymptotics of skew-orthogonal polynomials (sop) $\\pi^{\\bt}_{m}$, $\\beta=1$, 4, defined w.r.t. the weight $\\exp(-2NV(x))$, $V (x)=gx^4/4+tx^2/2$, $g>0$ and $t 0$, such that $\\epsilon\\leq (m/N)\\leq \\lambda_{\\rm cr}-\\epsilon$, where $\\lambda_{\\rm cr}$ is the critical value which separates sop with two cuts from those with one cut. Simultaneously we derive asymptotics for the recursive coefficients of skew-orthogonal polynomials. The proof is based on obtaining a finite term recursion relation between sop and orthogonal polynomials (op) and using asymptotic results of op derived in \\cite{bleher}. Finally, we apply these asymptotic results of sop and their recursion coefficients in the generalized Christoffel-Darboux formula (GCD) \\cite{ghosh3} to obtain level densities and sine-kernels in the bulk of the spectrum for orthogonal and symplectic ensembles of random matrices.
A matrix transformation approach to H∞ control via static output feedback for input delay systems
Du, B; Shu, Z; Lam, J.
2009-01-01
This paper addresses the static output feedback (SOF) H∞ control for continuous-time linear systems with an unknown input delay from a novel perspective. New equivalent characterizations on the stability and H∞ performance of the closed-loop system are established in terms of nonlinear matrix inequalities with free parametrization matrices. These delay-dependent characterizations possess a special monotonic structure, which leads to linearized iterative computation. The effectiveness and meri...
Tensor Bezoutian for matrix polynomials with respect to a general basis%一般基下矩阵多项式的张量Bezoutian
Institute of Scientific and Technical Information of China (English)
马超; 吴化璋
2011-01-01
给出矩阵多项式在一般基下的张量Bezoutian的定义,推广了标准幂基下的古典张量Bezoutian.讨论了该矩阵的Barnett型分解,缠绕关系和关于可控制/可观测矩阵的表示等重要性质.%In this paper, the definition of tensor Bezoutian for matrix polynomials with respect to a general basis is given, which generalizes the classical tensor Bezoutian under a standard power basis. Some important properties such as the Barnett-type factorization, an intertwining relation, and the controllability/observability matrix expressions are discussed.
Generalized Gegenbauer Koornwinder's type polynomials change of bases
AlQudah, Mohammad; AlMheidat, Maalee
2017-07-01
In this paper we characterize the generalized Gegenbauer polynomials using Bernstein basis, and derive the matrix of transformation of the generalized Gegenbauer polynomial basis form into the Bernstein polynomial basis and vice versa.
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...
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…
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.
Eikenhout, Nelson; Austin, John
2005-01-01
This study employed an ABAC and multiple baseline design to evaluate the effects of (B) feedback and (C) a package of feedback, goalsetting, and reinforcement (supervisor praise and an area-wide celebration as managed through a performance matrix, on a total of 14 various customer service behaviors for a total of 115 employees at a large…
Institute of Scientific and Technical Information of China (English)
童长飞; 章辉; 孙优贤
2007-01-01
By means of polynomial decomposition, a control scheme for polynomial nonlinear systems with affine timevarying uncertain parameters is presented. The idea of polynomial decomposition is to convert the coefficients of polynomial into a matrix with free variables, so that the nonnegativity of polynomials with even orders can be checked by linear matrix inequality (LMI) solvers or bilinear matrix inequality (BMI)solvers. Control synthesis for polynomial nonlinear system is based on Lyapunov stability theorem in this paper. Constructing Lyapunov function and finding feedback controller are automatically finished by computer programming with algorithms given in this paper. For multidimension systems with relatively high-order controller, the controller constructed with full monomial base will be in numerous terms. To overcome this problem,the reduced-form controller with minimum monomial terms is derived by the proposed algorithm. Then a suboptimal control aiming at minimum cost performance with gain constraints is advanced. The control scheme achieves effective performance as illustrated by numerical examples.
TCSC controller design based on output feedback control with linear matrix inequality
Energy Technology Data Exchange (ETDEWEB)
Ishimaru, Masachika; Shirai, Goro [Hosei University, Tokyo (Japan). Dept. of Electrical Engineering; Niioka, Satoru; Yokoyama, Ryuichi [Tokyo Metropolitan University (Japan). Dept. of Electrical Engineering
2000-07-01
The authors aim at designing the fast responsible and robust stabilizing controller. Recently, many researches propose robust stabilizing compensators based on H{sub {infinity}} control theory. Especiady, the LMI (Linear Matrix Inequality) solving efficient convex problems is very effective. LMI is based on a linear function composed by matrices, and it is expansion of conventional H{sub {infinity}} control. In addition to the LMI approach, authors pay attention to the output-feedback control for stabilizing a system using observable output values. This paper presents a stabilizing control using measurable values by using the output-feedback method. In order to discuss the advantage of the proposed method, 3-machine 9-bus system is used. Moreover, this system is applied TCSC (Thyristor Controlled Series Capacitor) controllers, and H{sub {infinity}} control based on the LMI is proposed for the design method of TCSC controllers to attain the robust stability. (author)
Narkiewicz, Wŀadysŀaw
1995-01-01
The book deals with certain algebraic and arithmetical questions concerning polynomial mappings in one or several variables. Algebraic properties of the ring Int(R) of polynomials mapping a given ring R into itself are presented in the first part, starting with classical results of Polya, Ostrowski and Skolem. The second part deals with fully invariant sets of polynomial mappings F in one or several variables, i.e. sets X satisfying F(X)=X . This includes in particular a study of cyclic points of such mappings in the case of rings of algebrai integers. The text contains several exercises and a list of open problems.
Near real-time ORM measurements and SVD matrix generation for 10 Hz global orbit feedback in RHIC
Energy Technology Data Exchange (ETDEWEB)
Liu, C.; Hulsart, R.; MacKay, W.; Marusic, A.; Mernick, K.; Michnoff, R.; Minty, M.
2011-03-28
To reduce the effect of trajectory perturbations ({approx}10 Hz) due to vibrations of the final focusing quadrupoles at RHIC, global orbit feedback was successfully prototyped during run-10. After upgraded to a system with 36 BPMs and 12 correctors, 10 Hz feedback was tested successfully in Run-11 and is in operational status for physics program. The test and operation of the system has been performed using transfer functions between the beam position monitors and correctors obtained from the online optics model and a correction algorithm based on singular value decomposition (SVD). One of our goals is to self-calibrate the system using SVD matrices derived from orbit response matrix (ORM) measurements acquired real-time using the new FPGA-based signal processing. Comparisons between measurement matrix and model matrix and the generation of SVD matrix for the feedback operation are presented.
A New Generalisation of Macdonald Polynomials
Garbali, Alexandr; de Gier, Jan; Wheeler, Michael
2017-01-01
We introduce a new family of symmetric multivariate polynomials, whose coefficients are meromorphic functions of two parameters (q, t) and polynomial in a further two parameters (u, v). We evaluate these polynomials explicitly as a matrix product. At u = v = 0 they reduce to Macdonald polynomials, while at q = 0, u = v = s they recover a family of inhomogeneous symmetric functions originally introduced by Borodin.
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-
Stochastic Estimation via Polynomial Chaos
2015-10-01
TΨ is a vector with P+1 elements. With these dimensions, (29) is solvable by standard numerical linear algebra techniques. The specific matrix...initial conditions for partial differential equations. Here, the elementary theory of the polynomial chaos is presented followed by the details of a...the elementary theory of the polynomial chaos is presented followed by the details of a number of example calculations where the statistical mean and
Richardson, Barbara K
2004-12-01
The emergency department provides a rich environment for diverse patient encounters, rapid clinical decision making, and opportunities to hone procedural skills. Well-prepared faculty can utilize this environment to teach residents and medical students and gain institutional recognition for their incomparable role and teamwork. Giving effective feedback is an essential skill for all teaching faculty. Feedback is ongoing appraisal of performance based on direct observation aimed at changing or sustaining a behavior. Tips from the literature and the author's experience are reviewed to provide formats for feedback, review of objectives, and elements of professionalism and how to deal with poorly performing students. Although the following examples pertain to medical student education, these techniques are applicable to the education of all adult learners, including residents and colleagues. Specific examples of redirection and reflection are offered, and pitfalls are reviewed. Suggestions for streamlining verbal and written feedback and obtaining feedback from others in a fast-paced environment are given. Ideas for further individual and group faculty development are presented.
d-Orthogonal Charlier Polynomials and the Weyl Algebra
Energy Technology Data Exchange (ETDEWEB)
Vinet, Luc [Centre de recherches mathematiques, Universite de Montreal, C.P. 6128, succ. Centre-ville, Montreal, Qc, H3C 3J7 (Canada); Zhedanov, Alexei, E-mail: luc.vinet@umontreal.ca, E-mail: zhedanov@yahoo.com [Department of Electronic and Kinetic Properties, Donetsk Institute for Physics and Technology R.Luxemburg str. 72, Donetsk, 83114 (Ukraine)
2011-03-01
It is shown that d-orthogonal Charlier polynomials arise as matrix elements of non unitary automorphisms of the Weyl algebra. The structural formulas that these polynomials obey are derived from this algebraic setting.
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.
Abelian avalanches and Tutte polynomials
Gabrielov, Andrei
1993-04-01
We introduce a class of deterministic lattice models of failure, Abelian avalanche (AA) models, with continuous phase variables, similar to discrete Abelian sandpile (ASP) models. We investigate analytically the structure of the phase space and statistical properties of avalanches in these models. We show that the distributions of avalanches in AA and ASP models with the same redistribution matrix and loading rate are identical. For an AA model on a graph, statistics of avalanches is linked to Tutte polynomials associated with this graph and its subgraphs. In the general case, statistics of avalanches is linked to an analog of a Tutte polynomial defined for any symmetric matrix.
Polynomial Subtraction Method for Disconnected Quark Loops
Liu, Quan; Morgan, Ron
2014-01-01
The polynomial subtraction method, a new numerical approach for reducing the noise variance of Lattice QCD disconnected matrix elements calculation, is introduced in this paper. We use the MinRes polynomial expansion of the QCD matrix as the approximation to the matrix inverse and get a significant reduction in the variance calculation. We compare our results with that of the perturbative subtraction and find that the new strategy yields a faster decrease in variance which increases with quark mass.
Weak lensing tomography with orthogonal polynomials
Schaefer, Bjoern Malte
2011-01-01
The topic of this article is weak cosmic shear tomography where the line of sight-weighting is carried out with a set of specifically constructed orthogonal polynomials, dubbed TaRDiS (Tomography with orthogonAl Radial Distance polynomIal Systems). We investigate the properties of these polynomials and employ weak convergence spectra, which have been obtained by weighting with these polynomials, for the estimation of cosmological parameters. We quantify their power in constraining parameters in a Fisher-matrix technique and demonstrate how each polynomial projects out statistically independent information, and how the combination of multiple polynomials lifts degeneracies. The assumption of a reference cosmology is needed for the construction of the polynomials, and as a last point we investigate how errors in the construction with a wrong cosmological model propagate to misestimates in cosmological parameters. TaRDiS performs on a similar level as traditional tomographic methods and some key features of tomo...
Weak lensing tomography with orthogonal polynomials
Schäfer, Björn Malte; Heisenberg, Lavinia
2012-07-01
The topic of this paper is weak cosmic shear tomography where the line-of-sight weighting is carried out with a set of specifically constructed orthogonal polynomials, dubbed Tomography with Orthogonal Radial Distance Polynomial Systems (TaRDiS). We investigate the properties of these polynomials and employ weak convergence spectra, which have been obtained by weighting with these polynomials, for the estimation of cosmological parameters. We quantify their power in constraining parameters in a Fisher matrix technique and demonstrate how each polynomial projects out statistically independent information, and how the combination of multiple polynomials lifts degeneracies. The assumption of a reference cosmology is needed for the construction of the polynomials, and as a last point we investigate how errors in the construction with a wrong cosmological model propagate to misestimates in cosmological parameters. TaRDiS performs on a similar level as traditional tomographic methods and some key features of tomography are made easier to understand.
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.
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.
Application of polynomial preconditioners to conservation laws
Geurts, Bernardus J.; van Buuren, R.; Lu, H.
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
Koornwinder, T.H.
2012-01-01
Askey-Wilson polynomial refers to a four-parameter family of q-hypergeometric orthogonal polynomials which contains all families of classical orthogonal polynomials (in the wide sense) as special or limit cases.
STABILITY SYSTEMS VIA HURWITZ POLYNOMIALS
Directory of Open Access Journals (Sweden)
BALTAZAR AGUIRRE HERNÁNDEZ
2017-01-01
Full Text Available To analyze the stability of a linear system of differential equations ẋ = Ax we can study the location of the roots of the characteristic polynomial pA(t associated with the matrix A. We present various criteria - algebraic and geometric - that help us to determine where the roots are located without calculating them directly.
Polynomial J-spectral factorization
Kwakernaak, Huibert; Sebek, Michael
1994-01-01
Several algorithms are presented for the J-spectral factorization of a para-Hermitian polynomial matrix. The four algorithms that are discussed are based on diagonalization, successive factor extraction, interpolation, and the solution of an algebraic Riccati equation, respectively. The paper includ
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.
Robust Control Synthesis of Polynomial Nonlinear Systems Using Sum of Squares Technique
Institute of Scientific and Technical Information of China (English)
HUANG Wen-Chao; SUN Hong-Fei; ZENG Jian-Ping
2013-01-01
In this paper,sum of squares (SOS) technique is used to analyze the robust state feedback synthesis problem for a class of uncertain affine nonlinear systems with polynomial vector fields.Sufficient conditions are given to obtain the solutions to the above control problem either without or with guaranteed cost or H∞ performance objectives.Moreover,such solvable conditions can be formulated as SOS programming problems in terms of state dependent linear matrix inequalities (LMIs) which can be dealt with by the SOS technique directly.Besides,an idea is provided to describe the inverse of polynomial or even rational matrices by introducing some extra polynomials.A numerical example is presented to illustrate the effectiveness of the approach.
The stable computation of formal orthogonal polynomials
Beckermann, Bernhard
1996-12-01
For many applications - such as the look-ahead variants of the Lanczos algorithm - a sequence of formal (block-)orthogonal polynomials is required. Usually, one generates such a sequence by taking suitable polynomial combinations of a pair of basis polynomials. These basis polynomials are determined by a look-ahead generalization of the classical three term recurrence, where the polynomial coefficients are obtained by solving a small system of linear equations. In finite precision arithmetic, the numerical orthogonality of the polynomials depends on a good choice of the size of the small systems; this size is usually controlled by a heuristic argument such as the condition number of the small matrix of coefficients. However, quite often it happens that orthogonality gets lost.
DEFF Research Database (Denmark)
Ribard, Nicolas; Wisniewski, Rafael; Sloth, Christoffer
2016-01-01
In the paper, we strive to develop an algorithm that simultaneously computes a polynomial control and a polynomial Lyapunov function. This ensures asymptotic stability of the designed feedback system. The above problem is translated to a certificate of positivity. To this end, we use the represen......In the paper, we strive to develop an algorithm that simultaneously computes a polynomial control and a polynomial Lyapunov function. This ensures asymptotic stability of the designed feedback system. The above problem is translated to a certificate of positivity. To this end, we use...... the representation of the given control system in Bernstein basis. Subsequently, the control synthesis problem is reduced to finite number of evaluations of a polynomial on vertices of cubes in the space of parameters representing admissible controls and Lyapunov functions....
Reddy, A Satyanarayana
2011-01-01
A graph $X$ is said to be a pattern polynomial graph if its adjacency algebra is a coherent algebra. In this study we will find a necessary and sufficient condition for a graph to be a pattern polynomial graph. Some of the properties of the graphs which are polynomials in the pattern polynomial graph have been studied. We also identify known graph classes which are pattern polynomial graphs.
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 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...
Improved polynomial remainder sequences for Ore polynomials.
Jaroschek, Maximilian
2013-11-01
Polynomial remainder sequences contain the intermediate results of the Euclidean algorithm when applied to (non-)commutative polynomials. The running time of the algorithm is dependent on the size of the coefficients of the remainders. Different ways have been studied to make these as small as possible. The subresultant sequence of two polynomials is a polynomial remainder sequence in which the size of the coefficients is optimal in the generic case, but when taking the input from applications, the coefficients are often larger than necessary. We generalize two improvements of the subresultant sequence to Ore polynomials and derive a new bound for the minimal coefficient size. Our approach also yields a new proof for the results in the commutative case, providing a new point of view on the origin of the extraneous factors of the coefficients.
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)
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...
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|matrices...... of reciprocal Fibonacci numbers called Filbert matrices. We find a formula for the entries of the inverse quantum Hilbert matrix....
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
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
Polynomial Datapaths Optimization
Parta, Hojat
2014-01-01
The research presented focuses on optimization of polynomials using algebraic manipulations at the high level and digital arithmetic techniques at the implementation level. Previous methods lacked any algebraic understanding of the polynomials or only exposed limited potential. We have treated the polynomial optimization problem in abstract algebra allowing us algebraic freedom to transform polynomials. Unlike previous attempts where only a set of limited benchmarks have been used, we have fo...
Palindromic random trigonometric polynomials
Conrey, J. Brian; Farmer, David W.; Imamoglu, Özlem
2008-01-01
We show that if a real trigonometric polynomial has few real roots, then the trigonometric polynomial obtained by writing the coefficients in reverse order must have many real roots. This is used to show that a class of random trigonometric polynomials has, on average, many real roots. In the case that the coefficients of a real trigonometric polynomial are independently and identically distributed, but with no other assumptions on the distribution, the expected fraction of real zeros is at l...
Branched polynomial covering maps
DEFF Research Database (Denmark)
Hansen, Vagn Lundsgaard
1999-01-01
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...
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...
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.
Rational Convolution Roots of Isobaric Polynomials
Conci, Aura; Li, Huilan; MacHenry, Trueman
2014-01-01
In this paper, we exhibit two matrix representations of the rational roots of generalized Fibonacci polynomials (GFPs) under convolution product, in terms of determinants and permanents, respectively. The underlying root formulas for GFPs and for weighted isobaric polynomials (WIPs), which appeared in an earlier paper by MacHenry and Tudose, make use of two types of operators. These operators are derived from the generating functions for Stirling numbers of the first kind and second kind. Hen...
Multiplication of a Schubert polynomial by a Stanley symmetric polynomial
Assaf, Sami
2017-01-01
We prove, combinatorially, that the product of a Schubert polynomial by a Stanley symmetric polynomial is a truncated Schubert polynomial. Using Monk's rule, we derive a nonnegative combinatorial formula for the Schubert polynomial expansion of a truncated Schubert polynomial. Combining these results, we give a nonnegative combinatorial rule for the product of a Schubert and a Schur polynomial in the Schubert basis.
Orthogonal polynomials and random matrices
Deift, Percy
2000-01-01
This volume expands on a set of lectures held at the Courant Institute on Riemann-Hilbert problems, orthogonal polynomials, and random matrix theory. The goal of the course was to prove universality for a variety of statistical quantities arising in the theory of random matrix models. The central question was the following: Why do very general ensembles of random n {\\times} n matrices exhibit universal behavior as n {\\rightarrow} {\\infty}? The main ingredient in the proof is the steepest descent method for oscillatory Riemann-Hilbert problems.
Mechanical cell-matrix feedback explains pairwise and collective endothelial cell behavior in vitro.
Directory of Open Access Journals (Sweden)
René F M van Oers
2014-08-01
Full Text Available In vitro cultures of endothelial cells are a widely used model system of the collective behavior of endothelial cells during vasculogenesis and angiogenesis. When seeded in an extracellular matrix, endothelial cells can form blood vessel-like structures, including vascular networks and sprouts. Endothelial morphogenesis depends on a large number of chemical and mechanical factors, including the compliancy of the extracellular matrix, the available growth factors, the adhesion of cells to the extracellular matrix, cell-cell signaling, etc. Although various computational models have been proposed to explain the role of each of these biochemical and biomechanical effects, the understanding of the mechanisms underlying in vitro angiogenesis is still incomplete. Most explanations focus on predicting the whole vascular network or sprout from the underlying cell behavior, and do not check if the same model also correctly captures the intermediate scale: the pairwise cell-cell interactions or single cell responses to ECM mechanics. Here we show, using a hybrid cellular Potts and finite element computational model, that a single set of biologically plausible rules describing (a the contractile forces that endothelial cells exert on the ECM, (b the resulting strains in the extracellular matrix, and (c the cellular response to the strains, suffices for reproducing the behavior of individual endothelial cells and the interactions of endothelial cell pairs in compliant matrices. With the same set of rules, the model also reproduces network formation from scattered cells, and sprouting from endothelial spheroids. Combining the present mechanical model with aspects of previously proposed mechanical and chemical models may lead to a more complete understanding of in vitro angiogenesis.
Directory of Open Access Journals (Sweden)
Thomas Bertero
2015-11-01
Full Text Available Pulmonary hypertension (PH is a deadly vascular disease with enigmatic molecular origins. We found that vascular extracellular matrix (ECM remodeling and stiffening are early and pervasive processes that promote PH. In multiple pulmonary vascular cell types, such ECM stiffening induced the microRNA-130/301 family via activation of the co-transcription factors YAP and TAZ. MicroRNA-130/301 controlled a PPARγ-APOE-LRP8 axis, promoting collagen deposition and LOX-dependent remodeling and further upregulating YAP/TAZ via a mechanoactive feedback loop. In turn, ECM remodeling controlled pulmonary vascular cell crosstalk via such mechanotransduction, modulation of secreted vasoactive effectors, and regulation of associated microRNA pathways. In vivo, pharmacologic inhibition of microRNA-130/301, APOE, or LOX activity ameliorated ECM remodeling and PH. Thus, ECM remodeling, as controlled by the YAP/TAZ-miR-130/301 feedback circuit, is an early PH trigger and offers combinatorial therapeutic targets for this devastating disease.
Structure relations for monic orthogonal polynomials in two discrete variables
Rodal, J.; Area, I.; Godoy, E.
2008-04-01
In this paper, extensions of several relations linking differences of bivariate discrete orthogonal polynomials and polynomials themselves are given, by using an appropriate vector-matrix notation. Three-term recurrence relations are presented for the partial differences of the monic polynomial solutions of admissible second order partial difference equation of hypergeometric type. Structure relations, difference representations as well as lowering and raising operators are obtained. Finally, expressions for all matrix coefficients appearing in these finite-type relations are explicitly presented for a finite set of Hahn and Kravchuk orthogonal polynomials.
Higher-Order Singular Systems and Polynomial Matrices
2005-01-01
There is a one-to-one correspondence between the set of quadruples of matrices defining singular linear time-invariant dynamical systems and a subset of the set of polynomial matrices. This correspondence preserves the equivalence relations introduced in both sets (feedback-similarity and strict equivalence): two quadruples of matrices are feedback-equivalent if, and only if, the polynomial matrices associated to them are also strictly equivalent. Los sistemas lineales singulares...
On the reproducing kernel of a Pontryagin space of vector valued polynomials
Ćurgus, B.; Dijksma, A.
2012-01-01
We give necessary and sufficient conditions under which the reproducing kernel of a Pontryagin space of d x 1 vector polynomials is determined by a generalized Nevanlinna pair of d x d matrix polynomials. Published by Elsevier Inc.
Weierstrass polynomials for links
DEFF Research Database (Denmark)
Hansen, Vagn Lundsgaard
1997-01-01
There is a natural way of identifying links in3-space with polynomial covering spaces over thecircle. Thereby any link in 3-space can be definedby a Weierstrass polynomial over the circle. Theequivalence relation for covering spaces over thecircle is, however, completely different from...... that for links in 3-space. This paper initiates a study of the connections between polynomial covering spaces over the circle and links in 3-space....
Blankertz, Raoul
2011-01-01
This diploma thesis is concerned with functional decomposition $f = g \\circ h$ of polynomials. First an algorithm is described which computes decompositions in polynomial time. This algorithm was originally proposed by Zippel (1991). A bound for the number of minimal collisions is derived. Finally a proof of a conjecture in von zur Gathen, Giesbrecht & Ziegler (2010) is given, which states a classification for a special class of decomposable polynomials.
Towards synthesis of polynomial controllers - a multi-affine approach
DEFF Research Database (Denmark)
Ribard, Nicolas; Wisniewski, Rafael; Sloth, Christoffer
2016-01-01
In the paper, we strive to develop an algorithm that simultaneously computes a polynomial control and a polynomial Lyapunov function. This ensures asymptotic stability of the designed feedback system. The above problem is translated to a certificate of positivity. To this end, we use the represen......In the paper, we strive to develop an algorithm that simultaneously computes a polynomial control and a polynomial Lyapunov function. This ensures asymptotic stability of the designed feedback system. The above problem is translated to a certificate of positivity. To this end, we use...... the representation of the given control system in Bernstein basis. Subsequently, the control synthesis problem is reduced to finite number of evaluations of a polynomial on vertices of cubes in the space of parameters representing admissible controls and Lyapunov functions....
Liu, Chuang; Lam, H. K.
2015-01-01
In this paper, we propose a polynomial fuzzy observer controller for nonlinear systems, where the design is achieved through the stability analysis of polynomial-fuzzy-model-based (PFMB) observer-control system. The polynomial fuzzy observer estimates the system states using estimated premise variables. The estimated states are then employed by the polynomial fuzzy controller for the feedback control of nonlinear systems represented by the polynomial fuzzy model. The system stability of the P...
Polynomial solution of quantum Grassmann matrices
Tierz, Miguel
2017-05-01
We study a model of quantum mechanical fermions with matrix-like index structure (with indices N and L) and quartic interactions, recently introduced by Anninos and Silva. We compute the partition function exactly with q-deformed orthogonal polynomials (Stieltjes-Wigert polynomials), for different values of L and arbitrary N. From the explicit evaluation of the thermal partition function, the energy levels and degeneracies are determined. For a given L, the number of states of different energy is quadratic in N, which implies an exponential degeneracy of the energy levels. We also show that at high-temperature we have a Gaussian matrix model, which implies a symmetry that swaps N and L, together with a Wick rotation of the spectral parameter. In this limit, we also write the partition function, for generic L and N, in terms of a single generalized Hermite polynomial.
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…
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…
On Period of the Sequence of Fibonacci Polynomials Modulo
Directory of Open Access Journals (Sweden)
İnci Gültekin
2013-01-01
Full Text Available It is shown that the sequence obtained by reducing modulo coefficient and exponent of each Fibonacci polynomials term is periodic. Also if is prime, then sequences of Fibonacci polynomial are compared with Wall numbers of Fibonacci sequences according to modulo . It is found that order of cyclic group generated with matrix is equal to the period of these sequences.
The Further Study of the Matrix Polynomials Space%矩阵多项式空间的进一步研究
Institute of Scientific and Technical Information of China (English)
杨闻起
2011-01-01
If A is a square matrix of number field F,denoted by F[A] = {f{A)\\ f(x) € ffc]}, The F[A] is subspace of Fnxn obviouly.The base and dimension of F[A] are discussed, the coordinate of f(A) and factoring subspaces of F[A] are introduced,and the structure of F[A] are given by factoring subspaces.%设A为数域F上的n级矩阵,记F[A]={f(A)｜f(x)∈F[x]｝,它显然是Fn×n的子空间.讨论了F[A]的基和维数,引入了f(A)的坐标和F[A]的因式子空间的概念,给出了用因式子空间表示F[A]的几个定理,刻画了F[A]的结构.
On the Tutte-Krushkal-Renardy polynomial for cell complexes
Bajo, Carlos; Chmutov, Sergei
2012-01-01
Recently V.Krushkal and D.Renardy generalized the Tutte polynomial from graphs to cell complexes. We show that evaluating this polynomial at the origin gives the number of cellular spanning trees in the sense of A.Duval, C.Klivans, and J.Martin. Moreover, after a slight modification, the Tutte-Krushkal-Renardy polynomial evaluated at the origin gives a weighted count of cellular spanning trees, and therefore its free term can be calculated by the cellular matrix-tree theorem of Duval et al. In the case of cell decomposition of a sphere, this modified polynomial satisfies the same duality identity as before. We find that evaluating the Tutte-Krushkal-Renardy along a certain line is the Bott polynomial. Finally we prove skein relations for the Tutte-Krushkal-Renardy polynomial.
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.
Control synthesis for polynomial nonlinear systems and application in attitude control
Institute of Scientific and Technical Information of China (English)
Chang-fei TONG; Hui ZHANG; You-xian SUN
2008-01-01
A method for positive polynomial validation based on polynomial decomposition is proposed to deal with control synthesis problems. Detailed algorithms for decomposition are given which mainly consider how to convert coefficients of a polynomial to a matrix with free variables. Then, the positivity of a polynomial is checked by the decomposed matrix with semidefinite programming solvers. A nonlinear control law is presented for single input polynomial systems based on the Lyapunov stability theorem. The control synthesis method is advanced to multi-input systems further. An application in attitude control is finally presented. The proposed control law achieves effective performance as illustrated by the numerical example.
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
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.
Tricubic polynomial interpolation.
Birkhoff, G
1971-06-01
A new triangular "finite element" is described; it involves the 12-parameter family of all quartic polynomial functions that are "tricubic" in that their variation is cubic along any parallel to any side of the triangle. An interpolation scheme is described that approximates quite accurately any smooth function on any triangulated domain by a continuously differentiable function, tricubic on each triangular element.
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.…
On Generalized Bell Polynomials
Directory of Open Access Journals (Sweden)
Roberto B. Corcino
2011-01-01
Full Text Available It is shown that the sequence of the generalized Bell polynomials Sn(x is convex under some restrictions of the parameters involved. A kind of recurrence relation for Sn(x is established, and some numbers related to the generalized Bell numbers and their properties are investigated.
Complexity of Ising Polynomials
Kotek, Tomer
2011-01-01
This paper deals with the partition function of the Ising model from statistical mechanics, which is used to study phase transitions in physical systems. A special case of interest is that of the Ising model with constant energies and external field. One may consider such an Ising system as a simple graph together with vertex and edge weight values. When these weights are considered indeterminates, the partition function for the constant case is a trivariate polynomial Z(G;x,y,z). This polynomial was studied with respect to its approximability by L. A. Goldberg, M. Jerrum and M. Patersonin 2003. Z(G;x,y,z) generalizes a bivariate polynomial Z(G;t,y), which was studied in by D. Andr\\'{e}n and K. Markstr\\"{o}m in 2009. We consider the complexity of Z(G;t,y) and Z(G;x,y,z) in comparison to that of the Tutte polynomial, which is well-known to be closely related to the Potts model in the absence of an external field. We show that Z(G;\\x,\\y,\\z) is #P-hard to evaluate at all points in $mathbb{Q}^3$, except those in ...
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 ...
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).
Explicit Formulas for Meixner Polynomials
Directory of Open Access Journals (Sweden)
Dmitry V. Kruchinin
2015-01-01
Full Text Available Using notions of composita and composition of generating functions, we show an easy way to obtain explicit formulas for some current polynomials. Particularly, we consider the Meixner polynomials of the first and second kinds.
Numerical solution of stochastic SIR model by Bernstein polynomials
Directory of Open Access Journals (Sweden)
N. Rahmani
2016-01-01
Full Text Available In this paper, we present numerical method based on Bernstein polynomials for solving the stochastic SIR model. By use of Bernstein operational matrix and its stochastic operational matrix we convert stochastic SIR model to a nonlinear system that can be solved by Newton method. Finally, a test problem of SIR model is presented to illustrate our mathematical findings.
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...
Interpolation and Polynomial Curve Fitting
Yang, Yajun; Gordon, Sheldon P.
2014-01-01
Two points determine a line. Three noncollinear points determine a quadratic function. Four points that do not lie on a lower-degree polynomial curve determine a cubic function. In general, n + 1 points uniquely determine a polynomial of degree n, presuming that they do not fall onto a polynomial of lower degree. The process of finding such a…
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
R.J. Stroeker (Roel)
2006-01-01
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
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.
Callier, F. M.; Nahum, C. D.
1975-01-01
The series connection of two linear time-invariant systems that have minimal state space system descriptions is considered. From these descriptions, strict-system-equivalent polynomial matrix system descriptions in the manner of Rosenbrock are derived. They are based on the factorization of the transfer matrix of the subsystems as a ratio of two right or left coprime polynomial matrices. They give rise to a simple polynomial matrix system description of the tandem connection. Theorem 1 states that for the complete controllability and observability of the state space system description of the series connection, it is necessary and sufficient that certain 'denominator' and 'numerator' groups are coprime. Consequences for feedback systems are drawn in Corollary 1. The role of pole-zero cancellations is explained by Lemma 3 and Corollaires 2 and 3.
Full Static Output Feedback Equivalence
Directory of Open Access Journals (Sweden)
Aristotle G. Yannakoudakis
2013-01-01
Full Text Available We present a constructive solution to the problem of full output feedback equivalence, of linear, minimal, time-invariant systems. The equivalence relation on the set of systems is transformed to another on the set of invertible block Bezout/Hankel matrices using the isotropy subgroups of the full state feedback group and the full output injection group. The transformation achieving equivalence is calculated solving linear systems of equations. We give a polynomial version of the results proving that two systems are full output feedback equivalent, if and only if they have the same family of generalized Bezoutians. We present a new set of output feedback invariant polynomials that generalize the breakaway polynomial of scalar systems.
Feynman graph polynomials and iterative algorithms
Energy Technology Data Exchange (ETDEWEB)
Bogner, Christian [Johannes Gutenberg-Universitaet, Mainz (Germany)
2009-07-01
I briefly report on recent work with Stefan Weinzierl, where we have proven a theorem, stating that the Laurent coefficients of scalar Feynman integrals are periods in the sense of Kontsevich and Zagier, if they are evaluated at kinematical invariants taking rational values in Euclidean momentum space. Our proof uses the (extended) sector decomposition algorithm by Binoth and Heinrich. Our result is related to the appearance of multiple zeta values in coefficients of Feynman integrals which has recently been investigated by Francis Brown, using another iterative algorithm. Both of these algorithms apply to the Feynman parametric representation of the integral and perform iterative manipulations of the polynomials in the integrand, which originate from the Symanzik polynomials. Motivated by the success of these methods I give a brief review on some more and some less well-known combinatorial properties of Symanzik polynomials. I focus on their accessibility to generalized theorems of the matrix-tree type and their relation to the multivariate Tutte polynomial.
Schemes for Deterministic Polynomial Factoring
Ivanyos, Gábor; Saxena, Nitin
2008-01-01
In this work we relate the deterministic complexity of factoring polynomials (over finite fields) to certain combinatorial objects we call m-schemes. We extend the known conditional deterministic subexponential time polynomial factoring algorithm for finite fields to get an underlying m-scheme. We demonstrate how the properties of m-schemes relate to improvements in the deterministic complexity of factoring polynomials over finite fields assuming the generalized Riemann Hypothesis (GRH). In particular, we give the first deterministic polynomial time algorithm (assuming GRH) to find a nontrivial factor of a polynomial of prime degree n where (n-1) is a smooth number.
Complex Polynomial Vector Fields
DEFF Research Database (Denmark)
Dias, Kealey
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...... 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....
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...... 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.
Spreading lengths of Hermite polynomials
Sánchez-Moreno, P; Manzano, D; Yáñez, R; 10.1016/j.cam.2009.09.043
2009-01-01
The Renyi, Shannon and Fisher spreading lengths of the classical or hypergeometric orthogonal polynomials, which are quantifiers of their distribution all over the orthogonality interval, are defined and investigated. These information-theoretic measures of the associated Rakhmanov probability density, which are direct measures of the polynomial spreading in the sense of having the same units as the variable, share interesting properties: invariance under translations and reflections, linear scaling and vanishing in the limit that the variable tends towards a given definite value. The expressions of the Renyi and Fisher lengths for the Hermite polynomials are computed in terms of the polynomial degree. The combinatorial multivariable Bell polynomials, which are shown to characterize the finite power of an arbitrary polynomial, play a relevant role for the computation of these information-theoretic lengths. Indeed these polynomials allow us to design an error-free computing approach for the entropic moments (w...
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 interpolation methods for viscous flow calculations
Rubin, S. G.; Khosla, P. K.
1977-01-01
Higher-order collocation procedures which result in block-tridiagonal matrix systems are derived from (1) Taylor series expansions and from (2) polynomial interpolation, and the relationships between the two formulations, called respectively Hermite and spline collocation, are investigated. A Hermite block-tridiagonal system for a nonuniform mesh is derived, and the Hermite approach is extended in order to develop a variable-mesh sixth-order block-tridiagonal procedure. It is shown that all results obtained by Hermite development can be recovered by appropriate spline polynomial interpolation. The additional boundary conditions required for these higher-order procedures are also given. Comparative solutions using second-order accurate finite difference and spline and Hermite formulations are presented for the boundary layer on a flat plate, boundary layers with uniform and variable mass transfer, and the viscous incompressible Navier-Stokes equations describing flow in a driven cavity.
Polynomial interpolation methods for viscous flow calculations
Rubin, S. G.; Khosla, P. K.
1977-01-01
Higher-order collocation procedures which result in block-tridiagonal matrix systems are derived from (1) Taylor series expansions and from (2) polynomial interpolation, and the relationships between the two formulations, called respectively Hermite and spline collocation, are investigated. A Hermite block-tridiagonal system for a nonuniform mesh is derived, and the Hermite approach is extended in order to develop a variable-mesh sixth-order block-tridiagonal procedure. It is shown that all results obtained by Hermite development can be recovered by appropriate spline polynomial interpolation. The additional boundary conditions required for these higher-order procedures are also given. Comparative solutions using second-order accurate finite difference and spline and Hermite formulations are presented for the boundary layer on a flat plate, boundary layers with uniform and variable mass transfer, and the viscous incompressible Navier-Stokes equations describing flow in a driven cavity.
Circular β ensembles, CMV representation, characteristic polynomials
Institute of Scientific and Technical Information of China (English)
SU ZhongGen
2009-01-01
In this note we first briefly review some recent progress in the study of the circular β ensemble on the unit circle, where 0 > 0 is a model parameter. In the special cases β = 1,2 and 4, this ensemble describes the joint probability density of eigenvalues of random orthogonal, unitary and sympletic matrices, respectively. For general β, Killip and Nenciu discovered a five-diagonal sparse matrix model, the CMV representation. This representation is new even in the case β = 2; and it has become a powerful tool for studying the circular β ensemble. We then give an elegant derivation for the moment identities of characteristic polynomials via the link with orthogonal polynomials on the unit circle.
Tabulating knot polynomials for arborescent knots
Mironov, A; Morozov, An; Sleptsov, A; Ramadevi, P; Singh, Vivek Kumar
2016-01-01
Arborescent knots are the ones which can be represented in terms of double fat graphs or equivalently as tree Feynman diagrams. This is the class of knots for which the present knowledge is enough for lifting topological description to the level of effective analytical formulas. The paper describes the origin and structure of the new tables of colored knot polynomials, which will be posted at the dedicated site. Even if formal expressions are known in terms of modular transformation matrices, the computation in finite time requires additional ideas. We use the "family" approach, and apply it to arborescent knots in Rolfsen table by developing a Feynman diagram technique, associated with an auxiliary matrix model field theory. Gauge invariance in this theory helps to provide meaning to Racah matrices in the case of non-trivial multiplicities and explains the need for peculiar sign prescriptions in the calculation of [21]-colored HOMFLY polynomials.
Complex Polynomial Vector Fields
DEFF Research Database (Denmark)
Dias, Kealey
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...... 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...
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...
Identification of nonlinear vibrating structures by polynomial expansion in the z-domain
Fasana, Alessandro; Garibaldi, Luigi; Marchesiello, Stefano
2017-02-01
A new method in the frequency domain for the identification of nonlinear vibrating structures is described, by adopting the perspective of nonlinearities as internal feedback forces. The technique is based on a polynomial expansion representation of the frequency response function of the underlying linear system, relying on a z-domain formulation. A least squares approach is adopted to take into account the information of all the frequency response functions but, when large data sets are used, the solution of the resulting system of algebraic linear equations can be a difficult task. A procedure to drastically reduce the matrix dimensions and consequently the computational cost - which largely depends on the number of spectral lines - is adopted, leading to a compact and well conditioned problem. The robustness and numerical performances of the method are demonstrated by its implementation on simulated data from single and two degree of freedom systems with typical nonlinear characteristics.
Multivariable q-Racah polynomials
Van Diejen, J F
1996-01-01
The Koornwinder-Macdonald multivariable generalization of the Askey-Wilson polynomials is studied for parameters satisfying a truncation condition such that the orthogonality measure becomes discrete with support on a finite grid. For this parameter regime the polynomials may be seen as a multivariable counterpart of the (one-variable) q-Racah polynomials. We present the discrete orthogonality measure, expressions for the normalization constants converting the polynomials into an orthonormal system (in terms of the normalization constant for the unit polynomial), and we discuss the limit q\\rightarrow 1 leading to multivariable Racah type polynomials. Of special interest is the situation that q lies on the unit circle; in that case it is found that there exists a natural parameter domain for which the discrete orthogonality measure (which is complex in general) becomes real-valued and positive. We investigate the properties of a finite-dimensional discrete integral transform for functions over the grid, whose ...
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...
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.
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.
Log-concavity of the genus polynomials of Ringel Ladders
Directory of Open Access Journals (Sweden)
Jonathan L Gross
2015-10-01
Full Text Available A Ringel ladder can be formed by a self-bar-amalgamation operation on a symmetric ladder, that is, by joining the root vertices on its end-rungs. The present authors have previously derived criteria under which linear chains of copies of one or more graphs have log-concave genus polyno- mials. Herein we establish Ringel ladders as the first significant non-linear infinite family of graphs known to have log-concave genus polynomials. We construct an algebraic representation of self-bar-amalgamation as a matrix operation, to be applied to a vector representation of the partitioned genus distribution of a symmetric ladder. Analysis of the resulting genus polynomial involves the use of Chebyshev polynomials. This paper continues our quest to affirm the quarter-century-old conjecture that all graphs have log-concave genus polynomials.
In, Hai‑Jung; Choi, Byong‑Deok; Chung, Ho‑Kyoon; Kwon, Oh‑Kyong
2006-05-01
There is the problem of picture quality nonuniformity due to thin film transistor (TFT) characteristic variations throughout a panel of large-area high-resolution active matrix organic light emitting diodes. The current programming method could solve this issue, but it also requires very long charging time of a data line at low gray shades. Therefore, we propose a new driving method and a pixel circuit with emission-current sensing and feedback operation in order to resolve these problems. The proposed driving method and pixel circuit successfully compensate threshold voltage and mobility variations of TFTs and overcome the data line charging problem. Simulation results show that emission current deviations of the proposed driving method are less than 1.7% with ± 10.0% mobility and ± 0.3 V threshold voltage variations of pixel-driving TFTs, which means the proposed driving method is applicable to large-area high-resolution applications.
Superoscillations with arbitrary polynomial shape
Chremmos, Ioannis; Fikioris, George
2015-07-01
We present a method for constructing superoscillatory functions the superoscillatory part of which approximates a given polynomial with arbitrarily small error in a fixed interval. These functions are obtained as the product of the polynomial with a sufficiently flat, bandlimited envelope function whose Fourier transform has at least N-1 continuous derivatives and an Nth derivative of bounded variation, N being the order of the polynomial. Polynomials of arbitrarily high order can be approximated if the Fourier transform of the envelope is smooth, i.e. a bump function.
Hall Polynomials for Affine Quivers of Type Am(m≥1)
Institute of Scientific and Technical Information of China (English)
Tao XIE
2008-01-01
It is well known that Hall polynomials as structural coe .cients play an important role in the structure of Lie algebras and quantum groups.By using the properties of representation categories of affine quivers,the task of computing Hall polynomials for affine quivers can be reduced to counting the numbers of solutions of some matrix equations.This method has been applied to obtain Hall polynomials for indecomposable representations of quivers of type Am(m≥1).
Mixing of orthogonal and skew-orthogonal polynomials and its relation to Wilson RMT
Kieburg, Mario
2012-01-01
The unitary Wilson random matrix theory is an interpolation between the chiral Gaussian unitary ensemble and the Gaussian unitary ensemble. This new way of interpolation is also reflected in the orthogonal polynomials corresponding to such a random matrix ensemble. Although the chiral Gaussian unitary ensemble as well as the Gaussian unitary ensemble are associated to the Dyson index $\\beta=2$ the intermediate ensembles exhibit a mixing of orthogonal polynomials and skew-orthogonal polynomials. We consider the Hermitian as well as the non-Hermitian Wilson random matrix and derive the corresponding polynomials, their recursion relations, Christoffel-Darboux-like formulas, Rodrigues formulas and representations as random matrix averages in a unifying way. With help of these results we derive the unquenched $k$-point correlation function of the Hermitian and then non-Hermitian Wilson random matrix in terms of two flavour partition functions only. This representation is due to a Pfaffian factorization drastically...
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.
Polynomial identities for ternary intermolecular recombination
Bremner, Murray R
2010-01-01
The operation of binary intermolecular recombination, originating in the theory of DNA computing, permits a natural generalization to n-ary operations which perform simultaneous recombination of n molecules. In the case n = 3, we use computer algebra to determine the polynomial identities of degree <= 9 satisfied by this trilinear nonassociative operation. Our approach requires computing a basis for the nullspace of a large integer matrix, and for this we compare two methods: (i) the row canonical form, and (ii) the Hermite normal form with lattice basis reduction. In the conclusion, we formulate some conjectures for the general case of n-ary intermolecular recombination.
Characteristic polynomials of pseudo-Anosov maps
Birman, Joan; Kawamuro, Keiko
2010-01-01
We study the relationship between three different approaches to the action of a pseudo-Anosov mapping class $[F]$ on a surface: the original theorem of Thurston, its algorithmic proof by Bestvina-Handel, and related investigations of Penner-Harer. Bestvina and Handel represent $[F]$ as a suitably chosen homotopy equivalence $f: G\\to G$ of a finite graph, with an associated transition matrix $T$ whose largest eigenvalue is the dilatation of $[F]$. Extending a skew-symmetric form introduced by Penner and Harer to the setting of Bestvina and Handel, we show that the characteristic polynomial of $T$ is a monic and palindromic or anti-palindromic polynomial, possibly multiplied by a power of $x$. Moreover, it factors as a product of three polynomials. One of them reflects the action of $[F]$ on a certain symplectic space, the second one relates to the degeneracies of the skew-symmetric form, and the third one reflects the restriction of $f$ to the vertices of $G$. We give an application to the problem of deciding ...
Optimization over polynomials: Selected topics
M. Laurent (Monique); S.Y. Jang; Y.R. Kim; D.-W. Lee; I. Yie
2014-01-01
htmlabstractMinimizing a polynomial function over a region defined by polynomial inequalities models broad classes of hard problems from combinatorics, geometry and optimization. New algorithmic approaches have emerged recently for computing the global minimum, by combining tools from real algebra
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 methods in combinatorics
Guth, Larry
2016-01-01
This book explains some recent applications of the theory of polynomials and algebraic geometry to combinatorics and other areas of mathematics. One of the first results in this story is a short elegant solution of the Kakeya problem for finite fields, which was considered a deep and difficult problem in combinatorial geometry. The author also discusses in detail various problems in incidence geometry associated to Paul Erdős's famous distinct distances problem in the plane from the 1940s. The proof techniques are also connected to error-correcting codes, Fourier analysis, number theory, and differential geometry. Although the mathematics discussed in the book is deep and far-reaching, it should be accessible to first- and second-year graduate students and advanced undergraduates. The book contains approximately 100 exercises that further the reader's understanding of the main themes of the book. Some of the greatest advances in geometric combinatorics and harmonic analysis in recent years have been accompl...
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...... 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...
The number of polynomial solutions of polynomial Riccati equations
Gasull, Armengol; Torregrosa, Joan; Zhang, Xiang
2016-11-01
Consider real or complex polynomial Riccati differential equations a (x) y ˙ =b0 (x) +b1 (x) y +b2 (x)y2 with all the involved functions being polynomials of degree at most η. We prove that the maximum number of polynomial solutions is η + 1 (resp. 2) when η ≥ 1 (resp. η = 0) and that these bounds are sharp. For real trigonometric polynomial Riccati differential equations with all the functions being trigonometric polynomials of degree at most η ≥ 1 we prove a similar result. In this case, the maximum number of trigonometric polynomial solutions is 2η (resp. 3) when η ≥ 2 (resp. η = 1) and, again, these bounds are sharp. Although the proof of both results has the same starting point, the classical result that asserts that the cross ratio of four different solutions of a Riccati differential equation is constant, the trigonometric case is much more involved. The main reason is that the ring of trigonometric polynomials is not a unique factorization domain.
The Algebra Theory for PolynomialInterpolation Method
Institute of Scientific and Technical Information of China (English)
2015-01-01
In this paper, several usually used polynomial interpolation methods are explained in view of vector basis and dimension in linearalgebra theory. Using transition matrixes, general conversion formula between the basis function sets of these polynomialinterpolation methods are given. An example also shows the effectiveness of the results.
On Zeros of Self-Reciprocal Random Algebraic Polynomials
Directory of Open Access Journals (Sweden)
K. Farahmand
2008-01-01
Full Text Available This paper provides an asymptotic estimate for the expected number of level crossings of a trigonometric polynomial TN(ÃŽÂ¸=Ã¢ÂˆÂ‘j=0NÃ¢ÂˆÂ’1{ÃŽÂ±NÃ¢ÂˆÂ’jcos(j+1/2ÃŽÂ¸+ÃŽÂ²NÃ¢ÂˆÂ’jsin(j+1/2ÃŽÂ¸}, where ÃŽÂ±j and ÃŽÂ²j, j=0,1,2,Ã¢Â€Â¦, NÃ¢ÂˆÂ’1, are sequences of independent identically distributed normal standard random variables. This type of random polynomial is produced in the study of random algebraic polynomials with complex variables and complex random coefficients, with a self-reciprocal property. We establish the relation between this type of random algebraic polynomials and the above random trigonometric polynomials, and we show that the required level crossings have the functionality form of cos(N+ÃŽÂ¸/2. We also discuss the relationship which exists and can be explored further between our random polynomials and random matrix theory.
Universal Racah matrices and adjoint knot polynomials: Arborescent knots
Mironov, A.; Morozov, A.
2016-04-01
By now it is well established that the quantum dimensions of descendants of the adjoint representation can be described in a universal form, independent of a particular family of simple Lie algebras. The Rosso-Jones formula then implies a universal description of the adjoint knot polynomials for torus knots, which in particular unifies the HOMFLY (SUN) and Kauffman (SON) polynomials. For E8 the adjoint representation is also fundamental. We suggest to extend the universality from the dimensions to the Racah matrices and this immediately produces a unified description of the adjoint knot polynomials for all arborescent (double-fat) knots, including twist, 2-bridge and pretzel. Technically we develop together the universality and the "eigenvalue conjecture", which expresses the Racah and mixing matrices through the eigenvalues of the quantum R-matrix, and for dealing with the adjoint polynomials one has to extend it to the previously unknown 6 × 6 case. The adjoint polynomials do not distinguish between mutants and therefore are not very efficient in knot theory, however, universal polynomials in higher representations can probably be better in this respect.
A new derivation of the highest-weight polynomial of a unitary lie algebra
Energy Technology Data Exchange (ETDEWEB)
P Chau, Huu-Tai; P Van, Isacker [Grand Accelerateur National d' Ions Lourds (GANIL), 14 - Caen (France)
2000-07-01
A new method is presented to derive the expression of the highest-weight polynomial used to build the basis of an irreducible representation (IR) of the unitary algebra U(2J+1). After a brief reminder of Moshinsky's method to arrive at the set of equations defining the highest-weight polynomial of U(2J+1), an alternative derivation of the polynomial from these equations is presented. The method is less general than the one proposed by Moshinsky but has the advantage that the determinantal expression of the highest-weight polynomial is arrived at in a direct way using matrix inversions. (authors)
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...
Hadamard Factorization of Stable Polynomials
Loredo-Villalobos, Carlos Arturo; Aguirre-Hernández, Baltazar
2011-11-01
The stable (Hurwitz) polynomials are important in the study of differential equations systems and control theory (see [7] and [19]). A property of these polynomials is related to Hadamard product. Consider two polynomials p,q ∈ R[x]:p(x) = anxn+an-1xn-1+...+a1x+a0q(x) = bmx m+bm-1xm-1+...+b1x+b0the Hadamard product (p × q) is defined as (p×q)(x) = akbkxk+ak-1bk-1xk-1+...+a1b1x+a0b0where k = min(m,n). Some results (see [16]) shows that if p,q ∈R[x] are stable polynomials then (p×q) is stable, also, i.e. the Hadamard product is closed; however, the reciprocal is not always true, that is, not all stable polynomial has a factorization into two stable polynomials the same degree n, if n> 4 (see [15]).In this work we will give some conditions to Hadamard factorization existence for stable polynomials.
Characteristic polynomial assignment in F-M model Ⅱ of 2-D systems
Institute of Scientific and Technical Information of China (English)
唐万生; 亢京力
2004-01-01
The problems of characteristic polynomial assignment in Fornasini-Marchesini (F-M) model Ⅱ of 2-D systems are investigated. The corresponding closed-loop systems described by F-M model Ⅱ are obtained via the state feedback.Using the algebraic geometry method, the characteristic polynomial assignment in the closed-loop systems is discussed. In terms of the theory of algebraic geometry, the problem of characteristic polynomial assignment is transferred to the one whether a rational mapping is onto. Sufficient conditions for almost arbitrary assignment coefficients of characteristic polynomial in F-M model Ⅱ of 2-D systems via state feedback are derived, and they are available for multi-input cases. It also has been shown that this method can be applied to assign the characteristic polynomial with output feedback. The sufficient conditions for almost arbitrary assignment coefficients of characteristic polynomial of multi-input 2-D systems described by F-M model Ⅱ with output feedback are established.
Directory of Open Access Journals (Sweden)
Veysel Hatipoglu
2015-09-01
Full Text Available In this study, we present a practical matrix method to find an approximate solution of higher order linear difference equation with constant coefficients under the initial-boundary conditions in terms of Taylor polynomials. To obtain this goal, we first present time scale extension of previous polynomial approach, then restrict the formula to the Integers with h step. This method converts the difference equation to a matrix equation, which may be considered as a system of linear algebraic equations.
Numerical solutions of multi-order fractional differential equations by Boubaker polynomials
Directory of Open Access Journals (Sweden)
Bolandtalat A.
2016-01-01
Full Text Available In this paper, we have applied a numerical method based on Boubaker polynomials to obtain approximate numerical solutions of multi-order fractional differential equations. We obtain an operational matrix of fractional integration based on Boubaker polynomials. Using this operational matrix, the given problem is converted into a set of algebraic equations. Illustrative examples are are given to demonstrate the efficiency and simplicity of this technique.
Polynomial Regressions and Nonsense Inference
Directory of Open Access Journals (Sweden)
Daniel Ventosa-Santaulària
2013-11-01
Full Text Available Polynomial specifications are widely used, not only in applied economics, but also in epidemiology, physics, political analysis and psychology, just to mention a few examples. In many cases, the data employed to estimate such specifications are time series that may exhibit stochastic nonstationary behavior. We extend Phillips’ results (Phillips, P. Understanding spurious regressions in econometrics. J. Econom. 1986, 33, 311–340. by proving that 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 results, therefore, lead to a call to be cautious whenever practitioners estimate polynomial regressions.
Locally tame plane polynomial automorphisms
Berson, Joost; Furter, Jean-Philippe; Maubach, Stefan
2010-01-01
For automorphisms of a polynomial ring in two variables over a domain R, we show that local tameness implies global tameness provided that every 2-generated invertible R-module is free. We give many examples illustrating this property.
On the Hermite-Apostol-Genocchi Polynomials
Kurt, Veli; Kurt, Burak
2011-09-01
In this study, we introduce and investigate the Hermite-Apostol-Genocchi polynomials by means of a suitable generating function. We establish several interesting properties of these general polynomials. Also, we prove two theorems between 2-dimensional Hermite polynomials and Hermite-Apostol-Genocchi polynomials.
On chromatic and flow polynomial unique graphs
National Research Council Canada - National Science Library
Duan, Yinghua; Wu, Haidong; Yu, Qinglin
2008-01-01
... research on graphs uniquely determined by their chromatic polynomials and more recently on their Tutte polynomials, but rather spotty research on graphs uniquely determined by their flow polynomials or the combination of both chromatic and flow polynomials. This article is an initiation of investigation on graphs uniquely determin...
Properties of Leach-Flessas-Gorringe polynomials
Pursey, D. L.
1990-09-01
A generating function is obtained for the polynomials recently introduced by Leach, Flessas, and Gorringe [J. Math. Phys. 30, 406 (1989)], and is then used to relate the Leach-Flessas-Gorringe (or LFG) polynomials to Hermite polynomials. The generating function is also used to express a number of integrals involving the LFG polynomials as finite sums of parabolic cylinder functions.
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
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.
Tracking control of piezoelectric actuators using a polynomial-based hysteresis model
Gan, Jinqiang; Zhang, Xianmin; Wu, Heng
2016-06-01
A polynomial-based hysteresis model that describes hysteresis behavior in piezoelectric actuators is presented. The polynomial-based model is validated by comparing with the classic Prandtl-Ishlinskii model. Taking the advantages of the proposed model into consideration, inverse control using the polynomial-based model is proposed. To achieve better tracking performance, a hybrid control combining the developed inverse control and a proportional-integral-differential feedback loop is then proposed. To demonstrate the effectiveness of the proposed tracking controls, several comparative experiments of the polynomial-based model and Prandtl-Ishlinskii model are conducted. The experimental results show that inverse control and hybrid control using the polynomial-based model in trajectory-tracking applications are effective and meaningful.
On the Complexity of the Interlace Polynomial
Bläser, Markus
2007-01-01
We consider the two-variable interlace polynomial introduced by Arratia, Bollob\\'as and Sorkin. For this graph polynomial we derive two graph transformations yielding point-to-point reductions similar to the thickening transformation in the context of the Tutte polynomial. This enables us to prove that the two-variable interlace polynomial is #P-hard to evaluate at every algebraic point of R^2, except at one line, where it is trivially polynomial time computable, and four lines and two points, where the complexity is still open. As a consequence, three specializations of the two-variable interlace polynomial, the vertex-nullity interlace polynomial, the vertex-rank interlace polynomial and the independent set polynomial, are #P-hard to evaluate almost everywhere, too. For the independent set polynomial, our graph transformations allow us to prove that it is even hard to approximate at every algebraic point except at -1 and 0.
Vector-valued Jack polynomials and wavefunctions on the torus
Dunkl, Charles F.
2017-06-01
The Hamiltonian of the quantum Calogero-Sutherland model of N identical particles on the circle with 1/r 2 interactions has eigenfunctions consisting of Jack polynomials times the base state. By use of the generalized Jack polynomials taking values in modules of the symmetric group and the matrix solution of a system of linear differential equations one constructs novel eigenfunctions of the Hamiltonian. Like the usual wavefunctions each eigenfunction determines a symmetric probability density on the N-torus. The construction applies to any irreducible representation of the symmetric group. The methods depend on the theory of generalized Jack polynomials due to Griffeth, and the Yang-Baxter graph approach of Luque and the author.
Performance of a recursive algorithm for polynomial predistorter design
Institute of Scientific and Technical Information of China (English)
XU Ling-jun; WU Xiao-guang; WANG Yong; ZHANG Ping
2008-01-01
In this article, based on least square estimation, a recursive algorithm for indirect learning structure predistorter is introduced. Simulation results show that of all polynomial predistorter nonlinear terms, higher-order (higher than 7th-order) nonlinear terms are so minor that they can be omitted in practical predistorter design. So, it is unnecessary to construct predistorter with higher-order polynomials, and the algorithm will always be stable. Further results show that even when 15th-order polynomial model is used, the algorithm is convergent after 10 iterations, and it can improve out-band spectrum of 20 MHz bandwidth signal by 64 dB, with a 1.2×1011 matrix condition number.
Multi-particle dynamical systems and polynomials
Demina, Maria V.; Kudryashov, Nikolai A.
2016-05-01
Polynomial dynamical systems describing interacting particles in the plane are studied. A method replacing integration of a polynomial multi-particle dynamical system by finding polynomial solutions of partial differential equations is introduced. The method enables one to integrate a wide class of polynomial multi-particle dynamical systems. The general solutions of certain dynamical systems related to linear second-order partial differential equations are found. As a by-product of our results, new families of orthogonal polynomials are derived.
Alvarez-Fernandez, Carlos
2012-01-01
Orthogonal Laurent polynomials in the unit circle and the theory of Toda-like integrable systems are connected using the Gauss--Borel factorization of a Cantero-Moral-Velazquez moment matrix, which is constructed in terms of a complex quasi-definite measure supported in the unit circle. The factorization of the moment matrix leads to orthogonal Laurent polynomials in the unit circle and the corresponding second kind functions. Jacobi operators, 5-term recursion relations and Christoffel-Darboux kernels, projecting to particular spaces of truncated Laurent polynomials, and corresponding Christoffel-Darboux formulae are obtained within this point of view in a completely algebraic way. Cantero-Moral-Velazquez sequence of Laurent monomials is generalized and recursion relations, Christoffel-Darboux kernels, projecting to general spaces of truncated Laurent polynomials and corresponding Christoffel-Darboux formulae are found in this extended context. Continuous deformations of the moment matrix are introduced and ...
Universal Racah matrices and adjoint knot polynomials. I. Arborescent knots
Mironov, A
2015-01-01
By now it is well established that the quantum dimensions of descendants of the adjoint representation can be described in a universal form, independent of a particular family of simple Lie algebras. The Rosso-Jones formula then implies a universal description of the adjoint knot polynomials for torus knots, which in particular unifies the HOMFLY (SU_N) and Kauffman (SO_N) polynomials. For E_8 the adjoint representation is also fundamental. We suggest to extend the universality from the dimensions to the Racah matrices and this immediately produces a unified description of the adjoint knot polynomials for all arborescent (double-fat) knots, including twist, 2-bridge and pretzel. Technically we develop together the universality and the "eigenvalue conjecture", which expresses the Racah and mixing matrices through the eigenvalues of the quantum R-matrix, and for dealing with the adjoint polynomials one has to extend it to the previously unknown 6x6 case. The adjoint polynomials do not distinguish between mutant...
Chromatic polynomials of random graphs
Van Bussel, Frank; Ehrlich, Christoph; Fliegner, Denny; Stolzenberg, Sebastian; Timme, Marc
2010-04-01
Chromatic polynomials and related graph invariants are central objects in both graph theory and statistical physics. Computational difficulties, however, have so far restricted studies of such polynomials to graphs that were either very small, very sparse or highly structured. Recent algorithmic advances (Timme et al 2009 New J. Phys. 11 023001) now make it possible to compute chromatic polynomials for moderately sized graphs of arbitrary structure and number of edges. Here we present chromatic polynomials of ensembles of random graphs with up to 30 vertices, over the entire range of edge density. We specifically focus on the locations of the zeros of the polynomial in the complex plane. The results indicate that the chromatic zeros of random graphs have a very consistent layout. In particular, the crossing point, the point at which the chromatic zeros with non-zero imaginary part approach the real axis, scales linearly with the average degree over most of the density range. While the scaling laws obtained are purely empirical, if they continue to hold in general there are significant implications: the crossing points of chromatic zeros in the thermodynamic limit separate systems with zero ground state entropy from systems with positive ground state entropy, the latter an exception to the third law of thermodynamics.
Orthogonal Polynomials and $S$-curves
Rakhmanov, E A
2011-01-01
This paper is devoted to a study of $S$-curves, that is systems of curves in the complex plane whose equilibrium potential in a harmonic external field satisfies a special symmetry property ($S$-property). Such curves have many applications. In particular, they play a fundamental role in the theory of complex (non-hermitian) orthogonal polynomials. One of the main theorems on zero distribution of such polynomials asserts that the limit zero distribution is presented by an equilibrium measure of an $S$-curve associated with the problem if such a curve exists. These curves are also the starting point of the matrix Riemann-Hilbert approach to srtong asymptotics. Other approaches to the problem of strong asymptotics (differential equations, Riemann surfaces) are also related to $S$-curves or may be interpreted this way. Existence problem $S$-curve in a given class of curves in presence of a nontrivial external field presents certain challenge. We formulate and prove a version of existence theorem for the case whe...
Positive trigonometric polynomials and signal processing applications
Dumitrescu, Bogdan
2017-01-01
This revised edition is made up of two parts: theory and applications. Though many of the fundamental results are still valid and used, new and revised material is woven throughout the text. As with the original book, the theory of sum-of-squares trigonometric polynomials is presented unitarily based on the concept of Gram matrix (extended to Gram pair or Gram set). The programming environment has also evolved, and the books examples are changed accordingly. The applications section is organized as a collection of related problems that use systematically the theoretical results. All the problems are brought to a semi-definite programming form, ready to be solved with algorithms freely available, like those from the libraries SeDuMi, CVX and Pos3Poly. A new chapter discusses applications in super-resolution theory, where Bounded Real Lemma for trigonometric polynomials is an important tool. This revision is written to be more appealing and easier to use for new readers. < Features updated information on LMI...
Tabulating knot polynomials for arborescent knots
Mironov, A.; Morozov, A.; Morozov, A.; Ramadevi, P.; Singh, Vivek Kumar; Sleptsov, A.
2017-02-01
Arborescent knots are those which can be represented in terms of double fat graphs or equivalently as tree Feynman diagrams. This is the class of knots for which the present knowledge is sufficient for lifting topological description to the level of effective analytical formulas. The paper describes the origin and structure of the new tables of colored knot polynomials, which will be posted at the dedicated site (http://knotebook.org). Even if formal expressions are known in terms of modular transformation matrices, the computation in finite time requires additional ideas. We use the ‘family’ approach, suggested in Mironov and Morozov (2015 Nucl. Phys. B 899 395–413), and apply it to arborescent knots in the Rolfsen table by developing a Feynman diagram technique, associated with an auxiliary matrix model field theory. Gauge invariance in this theory helps to provide meaning to Racah matrices in the case of non-trivial multiplicities and explains the need for peculiar sign prescriptions in the calculation of [21]-colored HOMFLY-PT polynomials.
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
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.
Derivations and identities for Fibonacci and Lucas polynomials
Bedratyuk, Leonid
2012-01-01
We introduce the notion of Fibonacci and Lucas derivations of the polynomial algebras and prove that any element of kernel of the derivations defines a polynomial identity for the Fibonacci and Lucas polynomials. Also, we prove that any polynomial identity for Appel polynomial yields a polynomial identity for the Fibonacci and Lucas polynomials and describe the corresponding intertwining maps.
Audio Feedback -- Better Feedback?
Voelkel, Susanne; Mello, Luciane V.
2014-01-01
National Student Survey (NSS) results show that many students are dissatisfied with the amount and quality of feedback they get for their work. This study reports on two case studies in which we tried to address these issues by introducing audio feedback to one undergraduate (UG) and one postgraduate (PG) class, respectively. In case study one…
Tree modules and counting polynomials
Kinser, Ryan
2011-01-01
We give a formula for counting tree modules for the quiver S_g with g loops and one vertex in terms of tree modules on its universal cover. This formula, along with work of Helleloid and Rodriguez-Villegas, is used to show that the number of d-dimensional tree modules for S_g is polynomial in g with the same degree and leading coefficient as the counting polynomial A_{S_g}(d, q) for absolutely indecomposables over F_q, evaluated at q=1.
Orthogonal polynomials and operator orderings
Hamdi, Adel; 10.1063/1.3372526
2010-01-01
An alternative and combinatorial proof is given for a connection between a system of Hahn polynomials and identities for symmetric elements in the Heisenberg algebra, which was first observed by Bender, Mead, and Pinsky [Phys. Rev. Lett. 56 (1986), J. Math. Phys. 28, 509 (1987)] and proved by Koornwinder [J. Phys. Phys. 30(4), 1989]. In the same vein two results announced by Bender and Dunne [J. Math. Phys. 29 (8), 1988] connecting a special one-parameter class of Hermitian operator orderings and the continuous Hahn polynomials are also proved.
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).
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 ...
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.
Polynomial Linear Programming with Gaussian Belief Propagation
Bickson, Danny; Shental, Ori; Dolev, Danny
2008-01-01
Interior-point methods are state-of-the-art algorithms for solving linear programming (LP) problems with polynomial complexity. Specifically, the Karmarkar algorithm typically solves LP problems in time O(n^{3.5}), where $n$ is the number of unknown variables. Karmarkar's celebrated algorithm is known to be an instance of the log-barrier method using the Newton iteration. The main computational overhead of this method is in inverting the Hessian matrix of the Newton iteration. In this contribution, we propose the application of the Gaussian belief propagation (GaBP) algorithm as part of an efficient and distributed LP solver that exploits the sparse and symmetric structure of the Hessian matrix and avoids the need for direct matrix inversion. This approach shifts the computation from realm of linear algebra to that of probabilistic inference on graphical models, thus applying GaBP as an efficient inference engine. Our construction is general and can be used for any interior-point algorithm which uses the Newt...
Recursive Polynomial Remainder Sequence and the Nested Subresultants
Terui, Akira
2008-01-01
We give two new expressions of subresultants, nested subresultant and reduced nested subresultant, for the recursive polynomial remainder sequence (PRS) which has been introduced by the author. The reduced nested subresultant reduces the size of the subresultant matrix drastically compared with the recursive subresultant proposed by the authors before, hence it is much more useful for investigation of the recursive PRS. Finally, we discuss usage of the reduced nested subresultant in approxima...
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...
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...
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.
Polynomial Regressions and Nonsense Inference
DEFF Research Database (Denmark)
Ventosa-Santaulària, Daniel; Rodríguez-Caballero, Carlos Vladimir
Polynomial specifications are widely used, not only in applied economics, but also in epidemiology, physics, political analysis, and psychology, just to mention a few examples. In many cases, the data employed to estimate such estimations are time series that may exhibit stochastic nonstationary ...
Uniform approximation by (quantum) polynomials
Drucker, A.; de Wolf, R.
2011-01-01
We show that quantum algorithms can be used to re-prove a classical theorem in approximation theory, Jackson's Theorem, which gives a nearly-optimal quantitative version of Weierstrass's Theorem on uniform approximation of continuous functions by polynomials. We provide two proofs, based respectivel
Method of applying single higher order polynomial basis function over multiple domains
CSIR Research Space (South Africa)
Lysko, AA
2010-03-01
Full Text Available simple shift in the coordinate. The work shows that the matrix M can be written in the form M · X ¢G ¢X¡1: Here, the matrix X relates the set of original basis functions, BFold, and their decomposition into terms of a polynomial, Pold as BFold = X... ¢Pold; while the matrix G deflnes a conversion from the hierarchical polynomials deflned in the original local coordinate system, Pold, into the new local co-ordinate system, Pnew (i.e., the relationship between the coe–cients ak and Ak): Pold = G...
Herman's condition and Siegel disks of polynomials
Chéritat, Arnaud
2011-01-01
Herman proved the presence of critical points on the boundary of Siegel disks of unicritical polynomials under some diophantine condition now called the Herman condition. We extend this result to polynomials with two critical points.
An analysis on the inversion of polynomials
M. F. González-Cardel; R. Díaz-Uribe
2006-01-01
In this work the application and the intervals of validity of an inverse polynomial, according to the method proposed by Arfken [1] for the inversion of series, is analyzed. It is shown that, for the inverse polynomial there exists a restricted domain whose longitude depends on the magnitude of the acceptable error when the inverse polynomial is used to approximate the inverse function of the original polynomial. A method for calculating the error of the approximation and its use in determini...
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.
A Summation Formula for Macdonald Polynomials
de Gier, Jan; Wheeler, Michael
2016-03-01
We derive an explicit sum formula for symmetric Macdonald polynomials. Our expression contains multiple sums over the symmetric group and uses the action of Hecke generators on the ring of polynomials. In the special cases {t = 1} and {q = 0}, we recover known expressions for the monomial symmetric and Hall-Littlewood polynomials, respectively. Other specializations of our formula give new expressions for the Jack and q-Whittaker polynomials.
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.
Fixed-point bifurcation analysis in biological models using interval polynomials theory.
Rigatos, Gerasimos G
2014-06-01
The paper proposes a systematic method for fixed-point bifurcation analysis in circadian cells and similar biological models using interval polynomials theory. The stages for performing fixed-point bifurcation analysis in such biological systems comprise (i) the computation of fixed points as functions of the bifurcation parameter and (ii) the evaluation of the type of stability for each fixed point through the computation of the eigenvalues of the Jacobian matrix that is associated with the system's nonlinear dynamics model. Stage (ii) requires the computation of the roots of the characteristic polynomial of the Jacobian matrix. This problem is nontrivial since the coefficients of the characteristic polynomial are functions of the bifurcation parameter and the latter varies within intervals. To obtain a clear view about the values of the roots of the characteristic polynomial and about the stability features they provide to the system, the use of interval polynomials theory and particularly of Kharitonov's stability theorem is proposed. In this approach, the study of the stability of a characteristic polynomial with coefficients that vary in intervals is equivalent to the study of the stability of four polynomials with crisp coefficients computed from the boundaries of the aforementioned intervals. The efficiency of the proposed approach for the analysis of fixed-point bifurcations in nonlinear models of biological neurons is tested through numerical and simulation experiments.
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.
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
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.
Frobenious-Euler Type Polynomials Related to Hermite-Bernoulli Polynomials
Kurt, Burak; Simsek, Yilmaz
2011-09-01
The aim of this paper is to define and investigate a new generating functions of the Frobenious-Euler polynomials and numbers. We establish some fundamental properties of these numbers and polynomials. We also derive relationship between these polynomials and Hermite-Apostol-Bernoulli polynomials and numbers. We also give some remarks and applications.
Energy Technology Data Exchange (ETDEWEB)
Vinet, Luc [Universite de Montreal, PO Box 6128, Station Centre-ville, Montreal QC H3C 3J7 (Canada); Zhedanov, Alexei [Donetsk Institute for Physics and Technology, Donetsk 83114 (Ukraine)
2009-10-30
We construct new families of elliptic solutions of the restricted Toda chain. The main tool is a special (so-called Stieltjes) ansatz for the moments of corresponding orthogonal polynomials. We show that the moments thus obtained are related to three types of Lame polynomials. The corresponding orthogonal polynomials can be considered as a generalization of the Stieltjes-Carlitz elliptic polynomials.
2015-01-01
In this paper we present a deterministic polynomial time algorithm for testing if a symbolic matrix in non-commuting variables over $\\mathbb{Q}$ is invertible or not. The analogous question for commuting variables is the celebrated polynomial identity testing (PIT) for symbolic determinants. In contrast to the commutative case, which has an efficient probabilistic algorithm, the best previous algorithm for the non-commutative setting required exponential time (whether or not randomization is ...
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 ...
Multiple orthogonal polynomials, string equations and the large-n limit
Energy Technology Data Exchange (ETDEWEB)
Alonso, L MartInez [Departamento de Fisica Teorica II, Universidad Complutense, E28040 Madrid (Spain); Medina, E [Departamento de Matematicas, Universidad de Cadiz, E11510 Puerto Real, Cadiz (Spain)
2009-05-22
The Riemann-Hilbert problems for multiple orthogonal polynomials of types I and II are used to derive string equations associated with pairs of Lax-Orlov operators. A method for determining the quasiclassical limit of string equations in the phase space of the Whitham hierarchy of dispersionless integrable systems is provided. Applications to the analysis of the large-n limit of multiple orthogonal polynomials and their associated random matrix ensembles and models of non-intersecting Brownian motions are given.
Bipartition Polynomials, the Ising Model, and Domination in Graphs
Directory of Open Access Journals (Sweden)
Dod Markus
2015-05-01
Full Text Available This paper introduces a trivariate graph polynomial that is a common generalization of the domination polynomial, the Ising polynomial, the matching polynomial, and the cut polynomial of a graph. This new graph polynomial, called the bipartition polynomial, permits a variety of interesting representations, for instance as a sum ranging over all spanning forests. As a consequence, the bipartition polynomial is a powerful tool for proving properties of other graph polynomials and graph invariants. We apply this approach to show that, analogously to the Tutte polynomial, the Ising polynomial introduced by Andrén and Markström in [3], can be represented as a sum over spanning forests.
Polynomial fitting of tight-binding method in carbon
Haa, Wai Kang; Yeak, Su Hoe
2017-04-01
Carbon is very unique in among the elements and its ability to form strong chemical bonds with a variety number such as two carbons (graphene) and four carbons (diamond). This combination of strong bonds with tight mass and high melting point makes them technologically and scientifically important in nanoscience development. Tight-binding model (TB) is one of the semi-empirical approximations used in quantum mechanical world which is restricted to the Linear Combinations of Localized Atomic Orbitals (LCAO). Currently, there are many approaches in tight-binding calculation. In this paper, we have reproduced a polynomial scaling function by fitting to the TB model. The model is then applied into carbon molecules and obtained the energy bands of the system. The elements of the overlap Hamiltonian matrix in the model will be depending on the parameter of the polynomials. Our purpose is to find out a set of parameters in the polynomial which were commonly fit to an independently calculated band structure. We used minimization approach to calculate the polynomial coefficients which involves differentiation of eigenvalues in the eigensystem. The algorithm of fitting the parameters is carried out in FORTRAN.
A polynomial criterion for adaptive stabilizability of discrete-time nonlinear systems
Li, Chanying; Xie, Liang-Liang; Guo, Lei
2006-01-01
In this paper, we will investigate the maximum capability of adaptive feedback in stabilizing a basic class of discrete-time nonlinear systems with both multiple unknown parameters and bounded noises. We will present a complete proof of the polynomial criterion for feedback capability as stated in "Robust stability of discrete-time adaptive nonlinear control" (C. Li, L.-L. Xie. and L. Guo, IFAC World Congress, Prague, July 3-8, 2005), by providing both the necessity and sufficiency analyze...
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.
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.
Polynomial-Chaos-based Kriging
Schöbi, R; Sudret, B.; Wiart, J.
2015-01-01
International audience; Computer simulation has become the standard tool in many engineering fields for designing and optimizing systems, as well as for assessing their reliability. Optimization and uncertainty quantification problems typically require a large number of runs of the computational model at hand, which may not be feasible with high-fidelity models directly. Thus surrogate models (a.k.a metamodels) have been increasingly investigated in the last decade. Polynomial Chaos Expansion...
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
Tutte polynomial of the Apollonian network
Liao, Yunhua; Hou, Yaoping; Shen, Xiaoling
2014-10-01
The Tutte polynomial of a graph, or equivalently the q-state Potts model partition function, is a two-variable polynomial graph invariant of considerable importance in both combinatorics and statistical physics. The computation of this invariant for a graph is, in general, NP-hard. The aim of this paper is to compute the Tutte polynomial of the Apollonian network. Based on the well-known duality property of the Tutte polynomial, we extend the subgraph-decomposition method. In particular, we do not calculate the Tutte polynomial of the Apollonian network directly, instead we calculate the Tutte polynomial of the Apollonian dual graph. By using the close relation between the Apollonian dual graph and the Hanoi graph, we express the Tutte polynomial of the Apollonian dual graph in terms of that of the Hanoi graph. As an application, we also give the number of spanning trees of the Apollonian network.
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.
Institute of Scientific and Technical Information of China (English)
胡敏; 罗珣; 马韵洁
2012-01-01
为解决相关反馈三维模型检索方法存在用户不能确定模型是否相似的问题,提出了一种基于语义矩阵反馈的多特征融合三维模型检索方法.首先,采用形状分布和球面调和两种特征提取算法进行多特征提取.然后,对每种特征进行检索计算,将得到的相似度进行基于语义的反馈,根据反馈结果对不同特征分配不同的权值.最后,对迭代反馈结果的权求和得到检索模型的相似度.实验结果表明,本方法的检索结果比用单一的特征提取方法得到的结果准确.%When the relevance feedback method is used to retrieve the 3D model, it has the problem that the user unclear whether the models are similar or not similar. In order to solve this problem, an integrated method of 3D model retrieval is proposed which based on the combination of semantic matrix feedback and feature. Firstly, 3D model features are extracted by using shape distributions and spherical harmonics methods. Then these features similarity of the 3D mod-els are involved to calculate the assessment weights. These assessment weights are combined with semantic matrix feed-back in the 3D model retrieval. Finally, the similarity of the 3D models is calculated based on the iterative feedback result. The experiment results show that this method is more accurate than the single feature extraction method.
Circular β ensembles,CMV representation,characteristic polynomials
Institute of Scientific and Technical Information of China (English)
2009-01-01
In this note we first briefly review some recent progress in the study of the circular β ensemble on the unit circle,where β > 0 is a model parameter.In the special cases β = 1,2 and 4,this ensemble describes the joint probability density of eigenvalues of random orthogonal,unitary and sympletic matrices,respectively.For general β,Killip and Nenciu discovered a five-diagonal sparse matrix model,the CMV representation.This representation is new even in the case β = 2;and it has become a powerful tool for studying the circular β ensemble.We then give an elegant derivation for the moment identities of characteristic polynomials via the link with orthogonal polynomials on the unit circle.
HOMFLY polynomials in representation [3, 1] for 3-strand braids
Mironov, A.; Morozov, A.; Morozov, An.; Sleptsov, A.
2016-09-01
This paper is a new step in the project of systematic description of colored knot polynomials started in [1]. In this paper, we managed to explicitly find the inclusive Racah matrix, i.e. the whole set of mixing matrices in channels R ⊗3 -→ Q with all possible Q, for R = [3 , 1]. The calculation is made possible by the use of a newly-developed efficient highest-weight method, still it remains tedious. The result allows one to evaluate and investigate [3 , 1]-colored polynomials for arbitrary 3-strand knots, and this confirms many previous conjectures on various factorizations, universality, and differential expansions. We consider in some detail the next-to-twist-knots three-strand family ( n, -1 | 1 , -1) and deduce its colored HOMFLY. Also confirmed and clarified is the eigenvalue hypothesis for the Racah matrices, which promises to provide a shortcut to generic formulas for arbitrary representations.
Correlations of RMT Characteristic Polynomials and Integrability: Hermitean Matrices
Osipov, Vladimir Al
2010-01-01
Integrable theory is formulated for correlation functions of characteristic polynomials associated with invariant non-Gaussian ensembles of Hermitean random matrices. By embedding the correlation functions of interest into a more general theory of tau-functions, we (i) identify a zoo of hierarchical relations satisfied by tau-functions in an abstract infinite-dimensional space, and (ii) present a technology to translate these relations into hierarchically structured nonlinear differential equations describing the correlation functions of characteristic polynomials in the physical, spectral space. Implications of this formalism for fermionic, bosonic, and supersymmetric variations of zero-dimensional replica field theories are discussed at length. A particular emphasis is placed on the phenomenon of fermionic-bosonic factorisation of random-matrix-theory correlation functions.
Directory of Open Access Journals (Sweden)
Hiroshi Miki
2012-02-01
Full Text Available Discrete spectral transformations of skew orthogonal polynomials are presented. From these spectral transformations, it is shown that the corresponding discrete integrable systems are derived both in 1+1 dimension and in 2+1 dimension. Especially in the (2+1-dimensional case, the corresponding system can be extended to 2×2 matrix form. The factorization theorem of the Christoffel kernel for skew orthogonal polynomials in random matrix theory is presented as a by-product of these transformations.
Hayman, Marilyn J.
1981-01-01
Investigated the effectiveness of supervisor feedback in contributing to learning counseling skills. Counselor trainees (N=64) were assigned to supervisor feedback, no supervisor feedback, or control groups for three training sessions. Results indicated counseling skills were learned best by students with no supervisor feedback but self and peer…
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'.
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
Optimizing polynomials for floating-point implementation
De Dinechin, Florent
2008-01-01
The floating-point implementation of a function on an interval often reduces to polynomial approximation, the polynomial being typically provided by Remez algorithm. However, the floating-point evaluation of a Remez polynomial sometimes leads to catastrophic cancellations. This happens when some of the polynomial coefficients are very small in magnitude with respects to others. In this case, it is better to force these coefficients to zero, which also reduces the operation count. This technique, classically used for odd or even functions, may be generalized to a much larger class of functions. An algorithm is presented that forces to zero the smaller coefficients of the initial polynomial thanks to a modified Remez algorithm targeting an incomplete monomial basis. One advantage of this technique is that it is purely numerical, the function being used as a numerical black box. This algorithm is implemented within a larger polynomial implementation tool that is demonstrated on a range of examples, resulting in ...
Transversals of Complex Polynomial Vector Fields
DEFF Research Database (Denmark)
Dias, Kealey
, an important step was proving that the transversals possessed a certain characteristic. Understanding transversals might be the key to proving other polynomial vector fields are generic, and they are important in understanding bifurcations of polynomial vector fields in general. We consider two important......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...... 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...
A Cubic Kernel for Feedback Vertex Set
Bodlaender, H.L.
2006-01-01
The FEEDBACK VERTEX SET problem on unweighted, undirected graphs is considered. Improving upon a result by Burrage et al. [7], we show that this problem has a kernel with O(κ3) vertices, i.e., there is a polynomial time algorithm, that given a graph G and an integer κ, finds a graph G' and integer
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.
Haglund's conjecture on 3-column Macdonald polynomials
Blasiak, Jonah
2014-01-01
We prove a positive combinatorial formula for the Schur expansion of LLT polynomials indexed by a 3-tuple of skew shapes. This verifies a conjecture of Haglund. The proof requires expressing a noncommutative Schur function as a positive sum of monomials in Lam's algebra of ribbon Schur operators. Combining this result with the expression of Haglund, Haiman, and Loehr for transformed Macdonald polynomials in terms of LLT polynomials then yields a positive combinatorial rule for transformed Mac...
Discrete-time filtering for nonlinear polynomial systems over linear observations
Hernandez-Gonzalez, M.; Basin, M. V.
2014-07-01
This paper designs a discrete-time filter for nonlinear polynomial systems driven by additive white Gaussian noises over linear observations. The solution is obtained by computing the time-update and measurement-update equations for the state estimate and the error covariance matrix. A closed form of this filter is obtained by expressing the conditional expectations of polynomial terms as functions of the estimate and the error covariance. As a particular case, a third-degree polynomial is considered to obtain the finite-dimensional filtering equations. Numerical simulations are performed for a third-degree polynomial system and an induction motor model. Performance of the designed filter is compared with the extended Kalman one to verify its effectiveness.
Matrix theory selected topics and useful results
Mehta, Madan Lal
1989-01-01
Matrices and operations on matrices ; determinants ; elementary operations on matrices (continued) ; eigenvalues and eigenvectors, diagonalization of normal matrices ; functions of a matrix ; positive definiteness, various polar forms of a matrix ; special matrices ; matrices with quaternion elements ; inequalities ; generalised inverse of a matrix ; domain of values of a matrix, location and dispersion of eigenvalues ; symmetric functions ; integration over matrix variables ; permanents of doubly stochastic matrices ; infinite matrices ; Alexander matrices, knot polynomials, torsion numbers.
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.
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.
Polynomial chaotic inflation in supergravity revisited
Directory of Open Access Journals (Sweden)
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.
Chromatic polynomials, Potts models and all that
Sokal, Alan D.
2000-04-01
The q-state Potts model can be defined on an arbitrary finite graph, and its partition function encodes much important information about that graph, including its chromatic polynomial, flow polynomial and reliability polynomial. The complex zeros of the Potts partition function are of interest both to statistical mechanicians and to combinatorists. I give a pedagogical introduction to all these problems, and then sketch two recent results: (a) Construction of a countable family of planar graphs whose chromatic zeros are dense in the whole complex q-plane except possibly for the disc | q-1|chromatic polynomial (or antiferromagnetic Potts-model partition function) in terms of the graph's maximum degree.
Control to Facet for Polynomial Systems
DEFF Research Database (Denmark)
Sloth, Christoffer; Wisniewski, Rafael
2014-01-01
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.......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...
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.
Aspects of the Tutte polynomial
DEFF Research Database (Denmark)
Ok, Seongmin
This thesis studies various aspects of the Tutte polynomial, especially focusing on the Merino-Welsh conjecture. We write T(G;x,y) for the Tutte polynomial of a graph G with variables x and y. In 1999, Merino and Welsh conjectured that if G is a loopless 2-connected graph, then T(G;1,1) ≤ max{T(G;2......-Welsh conjecture. Assume the graph G is loopless, bridgeless and has n vertices and m edges. If m ≤ 1.066 n then T(G;1,1) ≤ T(G;2,0). If m ≥ 4(n-1) then T(G;1,1) ≤ T(G;0,2). I improve in this thesis Thomassen's result as follows: If m ≤ 1.29(n-1) then T(G;1,1) ≤ T(G;2,0). If m ≥ 3.58(n-1) and G is 3-edge...
Classification based polynomial image interpolation
Lenke, Sebastian; Schröder, Hartmut
2008-02-01
Due to the fast migration of high resolution displays for home and office environments there is a strong demand for high quality picture scaling. This is caused on the one hand by large picture sizes and on the other hand due to an enhanced visibility of picture artifacts on these displays [1]. There are many proposals for an enhanced spatial interpolation adaptively matched to picture contents like e.g. edges. The drawback of these approaches is the normally integer and often limited interpolation factor. In order to achieve rational factors there exist combinations of adaptive and non adaptive linear filters, but due to the non adaptive step the overall quality is notably limited. We present in this paper a content adaptive polyphase interpolation method which uses "offline" trained filter coefficients and an "online" linear filtering depending on a simple classification of the input situation. Furthermore we present a new approach to a content adaptive interpolation polynomial, which allows arbitrary polyphase interpolation factors at runtime and further improves the overall interpolation quality. The main goal of our new approach is to optimize interpolation quality by adapting higher order polynomials directly to the image content. In addition we derive filter constraints for enhanced picture quality. Furthermore we extend the classification based filtering to the temporal dimension in order to use it for an intermediate image interpolation.
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.
Gabor-based kernel PCA with fractional power polynomial models for face recognition.
Liu, Chengjun
2004-05-01
This paper presents a novel Gabor-based kernel Principal Component Analysis (PCA) method by integrating the Gabor wavelet representation of face images and the kernel PCA method for face recognition. Gabor wavelets first derive desirable facial features characterized by spatial frequency, spatial locality, and orientation selectivity to cope with the variations due to illumination and facial expression changes. The kernel PCA method is then extended to include fractional power polynomial models for enhanced face recognition performance. A fractional power polynomial, however, does not necessarily define a kernel function, as it might not define a positive semidefinite Gram matrix. Note that the sigmoid kernels, one of the three classes of widely used kernel functions (polynomial kernels, Gaussian kernels, and sigmoid kernels), do not actually define a positive semidefinite Gram matrix either. Nevertheless, the sigmoid kernels have been successfully used in practice, such as in building support vector machines. In order to derive real kernel PCA features, we apply only those kernel PCA eigenvectors that are associated with positive eigenvalues. The feasibility of the Gabor-based kernel PCA method with fractional power polynomial models has been successfully tested on both frontal and pose-angled face recognition, using two data sets from the FERET database and the CMU PIE database, respectively. The FERET data set contains 600 frontal face images of 200 subjects, while the PIE data set consists of 680 images across five poses (left and right profiles, left and right half profiles, and frontal view) with two different facial expressions (neutral and smiling) of 68 subjects. The effectiveness of the Gabor-based kernel PCA method with fractional power polynomial models is shown in terms of both absolute performance indices and comparative performance against the PCA method, the kernel PCA method with polynomial kernels, the kernel PCA method with fractional power
Polynomial-time solutions to image segmentation
Energy Technology Data Exchange (ETDEWEB)
Asano, Tetsuo [Osaka Electro-Communication Univ., Neyagawa (Japan); Chen, D.Z. [Notre Dame, South Bend, IN (United States); Katoh, Naoki [Kobe Univ. of Commerce (Japan)
1996-12-31
Separating an object in an image from its background is a central problem (called segmentation) in pattern recognition and computer vision. In this paper, we study the complexity of the segmentation problem, assuming that the object forms a connected region in an intensity image. We show that the optimization problem of separating a connected region in an n-pixel grid is NP-hard under the interclass variance, a criterion that is used in discriminant analysis. More importantly, we consider the basic case in which the object is separated by two x-monotone curves (i.e., the object itself is x-monotone), and present polynomial-time algorithms for computing exact and approximate optimal segmentation. Our main algorithm for exact optimal segmentation by two x-monotone curves runs in O(n{sup 2}) time; this algorithm is based on several techniques such as a parametric optimization formulation, a hand-probing algorithm for the convex hull of an unknown point set, and dynamic programming using fast matrix searching. Our efficient approximation scheme obtains an {epsilon}-approximate solution in O({epsilon}{sup -1} n log L) time, where {epsilon} is any fixed constant with 1 > {epsilon} > 0, and L is the total sum of the absolute values of brightness levels of the image.
Institute of Scientific and Technical Information of China (English)
王静; 史志华; 王炎; 尤肖虎
2009-01-01
针对多天线广播信道(MIMO BC)块对角化(BD)预编码空分多址接入(SDMA)系统对角化预编码时,直接量化信道矩阵的有限反馈方法复杂度过高的问题,利用系统中用户配置多个天线的特点,提出了基于天线合并逐列量化(PCQ)信道矩阵的有限反馈方法.该方法对接收信号做多次天线合并,将传统算法中直接量化信道矩阵转变为多次量化等效信道矢量,仿真结果表明,在总的反馈比特数相同的情况下,该方法以较少的和容量损失换来了复杂度的大幅降低.%In consideration of the very high complexity of the limited feedback method to directly quantize the channel matrix for block diagonalization(BD)precoding in multiple input multiple output broadcast channel (MIMO BC) space division multiple access (SDMA) systems, the paper proposes a new limited feedback method based on the antenna combining and the per column quantization (PCO) of channel matrixes according to the characteristic that each user has more than one antennas. The new method converts the direct matrix quantization into a series of vector quantizations, thus reducing the complexity considerably. The simulations show that the PCQ algorithm is efficient in complexity reduction, only with a slight performance degradation.
Feedback Loop Gains and Feedback Behavior (1996)
DEFF Research Database (Denmark)
Kampmann, Christian Erik
2012-01-01
Linking feedback loops and system behavior is part of the foundation of system dynamics, yet the lack of formal tools has so far prevented a systematic application of the concept, except for very simple systems. Having such tools at their disposal would be a great help to analysts in understanding...... large, complicated simulation models. The paper applies tools from graph theory formally linking individual feedback loop strengths to the system eigenvalues. The significance of a link or a loop gain and an eigenvalue can be expressed in the eigenvalue elasticity, i.e., the relative change...... of an eigenvalue resulting from a relative change in the gain. The elasticities of individual links and loops may be found through simple matrix operations on the linearized system. Even though the number of feedback loops can grow rapidly with system size, reaching astronomical proportions even for modest systems...
Global Monte Carlo Simulation with High Order Polynomial Expansions
Energy Technology Data Exchange (ETDEWEB)
William R. Martin; James Paul Holloway; Kaushik Banerjee; Jesse Cheatham; Jeremy Conlin
2007-12-13
The functional expansion technique (FET) was recently developed for Monte Carlo simulation. The basic idea of the FET is to expand a Monte Carlo tally in terms of a high order expansion, the coefficients of which can be estimated via the usual random walk process in a conventional Monte Carlo code. If the expansion basis is chosen carefully, the lowest order coefficient is simply the conventional histogram tally, corresponding to a flat mode. This research project studied the applicability of using the FET to estimate the fission source, from which fission sites can be sampled for the next generation. The idea is that individual fission sites contribute to expansion modes that may span the geometry being considered, possibly increasing the communication across a loosely coupled system and thereby improving convergence over the conventional fission bank approach used in most production Monte Carlo codes. The project examined a number of basis functions, including global Legendre polynomials as well as “local” piecewise polynomials such as finite element hat functions and higher order versions. The global FET showed an improvement in convergence over the conventional fission bank approach. The local FET methods showed some advantages versus global polynomials in handling geometries with discontinuous material properties. The conventional finite element hat functions had the disadvantage that the expansion coefficients could not be estimated directly but had to be obtained by solving a linear system whose matrix elements were estimated. An alternative fission matrix-based response matrix algorithm was formulated. Studies were made of two alternative applications of the FET, one based on the kernel density estimator and one based on Arnoldi’s method of minimized iterations. Preliminary results for both methods indicate improvements in fission source convergence. These developments indicate that the FET has promise for speeding up Monte Carlo fission source
Energy Technology Data Exchange (ETDEWEB)
Myers, N.J. [Univ. of Durham (United Kingdom)
1994-12-31
The author gives a hybrid method for the iterative solution of linear systems of equations Ax = b, where the matrix (A) is nonsingular, sparse and nonsymmetric. As in a method developed by Starke and Varga the method begins with a number of steps of the Arnoldi method to produce some information on the location of the spectrum of A. This method then switches to an iterative method based on the Faber polynomials for an annular sector placed around these eigenvalue estimates. The Faber polynomials for an annular sector are used because, firstly an annular sector can easily be placed around any eigenvalue estimates bounded away from zero, and secondly the Faber polynomials are known analytically for an annular sector. Finally the author gives three numerical examples, two of which allow comparison with Starke and Varga`s results. The third is an example of a matrix for which many iterative methods would fall, but this method converges.
Ochoa, Agustin
2016-01-01
This book describes a consistent and direct methodology to the analysis and design of analog circuits with particular application to circuits containing feedback. The analysis and design of circuits containing feedback is generally presented by either following a series of examples where each circuit is simplified through the use of insight or experience (someone else’s), or a complete nodal-matrix analysis generating lots of algebra. Neither of these approaches leads to gaining insight into the design process easily. The author develops a systematic approach to circuit analysis, the Driving Point Impedance and Signal Flow Graphs (DPI/SFG) method that does not require a-priori insight to the circuit being considered and results in factored analysis supporting the design function. This approach enables designers to account fully for loading and the bi-directional nature of elements both in the feedback path and in the amplifier itself, properties many times assumed negligible and ignored. Feedback circuits a...
Exact relaxation for polynomial optimization on semi-algebraic sets
Abril Bucero, Marta; Mourrain, Bernard
2014-01-01
In this paper, we study the problem of computing by relaxation hierarchies the infimum of a real polynomial function f on a closed basic semialgebraic set and the points where this infimum is reached, if they exist. We show that when the infimum is reached, a relaxation hierarchy constructed from the Karush-Kuhn-Tucker ideal is always exact and that the vanishing ideal of the KKT minimizer points is generated by the kernel of the associated moment matrix in that degree, even if this ideal is ...
A robust polynomial principal component analysis for seismic noise attenuation
Wang, Yuchen; Lu, Wenkai; Wang, Benfeng; Liu, Lei
2016-12-01
Random and coherent noise attenuation is a significant aspect of seismic data processing, especially for pre-stack seismic data flattened by normal moveout correction or migration. Signal extraction is widely used for pre-stack seismic noise attenuation. Principle component analysis (PCA), one of the multi-channel filters, is a common tool to extract seismic signals, which can be realized by singular value decomposition (SVD). However, when applying the traditional PCA filter to seismic signal extraction, the result is unsatisfactory with some artifacts when the seismic data is contaminated by random and coherent noise. In order to directly extract the desired signal and fix those artifacts at the same time, we take into consideration the amplitude variation with offset (AVO) property and thus propose a robust polynomial PCA algorithm. In this algorithm, a polynomial constraint is used to optimize the coefficient matrix. In order to simplify this complicated problem, a series of sub-optimal problems are designed and solved iteratively. After that, the random and coherent noise can be effectively attenuated simultaneously. Applications on synthetic and real data sets note that our proposed algorithm can better suppress random and coherent noise and have a better performance on protecting the desired signals, compared with the local polynomial fitting, conventional PCA and a L1-norm based PCA method.
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…
Fractal Trigonometric Polynomials for Restricted Range Approximation
Chand, A. K. B.; Navascués, M. A.; Viswanathan, P.; Katiyar, S. K.
2016-05-01
One-sided approximation tackles the problem of approximation of a prescribed function by simple traditional functions such as polynomials or trigonometric functions that lie completely above or below it. In this paper, we use the concept of fractal interpolation function (FIF), precisely of fractal trigonometric polynomials, to construct one-sided uniform approximants for some classes of continuous functions.
Elementary combinatorics of the HOMFLYPT polynomial
Chmutov, Sergei
2009-01-01
We explore Jaeger's state model for the HOMFLYPT polynomial. We reformulate this model in the language of Gauss diagrams and use it to obtain Gauss diagram formulas for a two-parameter family of Vassiliev invariants coming from the HOMFLYPT polynomial. These formulas are new already for invariants of degree 3.
ON FIRST INTEGRALS OF POLYNOMIAL AUTONOMOUS SYSTEMS
Institute of Scientific and Technical Information of China (English)
WANG Yuzhen; CHENG Daizhan; LI Chunwen
2002-01-01
Using Carleman linearization procedure, this paper investigates the problemof first integrals of polynomial autonomous systems and proposes a procedure to find thefirst integrals of polynomial family for the systems. A generalized eigenequation is obtainedand then the problem is reduced to the solvability of the eigenequation. The result is ageneralization of some known results.
Large degree asymptotics of generalized Bessel polynomials
J.L. López; N.M. Temme (Nico)
2011-01-01
textabstractAsymptotic 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
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.
A polynomial approach to nonlinear system controllability
Zheng, YF; Willems, JC; Zhang, CH
2001-01-01
This note uses a polynomial approach to present a necessary and sufficient condition for local controllability of single-input-single-output (SISO) nonlinear systems. The condition is presented in terms of common factors of a noncommutative polynomial expression. This result exposes controllability
Connections between the matching and chromatic polynomials
Directory of Open Access Journals (Sweden)
E. J. Farrell
1992-01-01
Full Text Available The main results established are (i a connection between the matching and chromatic polynomials and (ii a formula for the matching polynomial of a general complement of a subgraph of a graph. Some deductions on matching and chromatic equivalence and uniqueness are made.
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.
The topology of Julia sets for polynomials
Institute of Scientific and Technical Information of China (English)
尹永成
2002-01-01
We prove that wandering components of the Julia set of a polynomial are singletons provided each critical point in a wandering Julia component is non-recurrent. This means a conjecture of Branner-Hubbard is true for this kind of polynomials.
Percolation critical polynomial as a graph invariant
Scullard, Christian R.
2012-10-01
Every lattice for which the bond percolation critical probability can be found exactly possesses a critical polynomial, with the root in [0,1] providing the threshold. Recent work has demonstrated that this polynomial may be generalized through a definition that can be applied on any periodic lattice. The polynomial depends on the lattice and on its decomposition into identical finite subgraphs, but once these are specified, the polynomial is essentially unique. On lattices for which the exact percolation threshold is unknown, the polynomials provide approximations for the critical probability with the estimates appearing to converge to the exact answer with increasing subgraph size. In this paper, I show how this generalized critical polynomial can be viewed as a graph invariant, similar to the Tutte polynomial. In particular, the critical polynomial is computed on a finite graph and may be found using the recursive deletion-contraction algorithm. This allows calculation on a computer, and I present such results for the kagome lattice using subgraphs of up to 36 bonds. For one of these, I find the prediction pc=0.52440572⋯, which differs from the numerical value, pc=0.52440503(5), by only 6.9×10-7.
Tutte Polynomial of Multi-Bridge Graphs
Directory of Open Access Journals (Sweden)
Julian A. Allagan
2013-10-01
Full Text Available In this paper, using a well-known recursion for computing the Tutte polynomial of any graph, we found explicit formulae for the Tutte polynomials of any multi-bridge graph and some $2-$tree graphs. Further, several recursive formulae for other graphs such as the fan and the wheel graphs are also discussed.
Several explicit formulae for Bernoulli polynomials
Komatsu, Takao; Pita Ruiz V., Claudio de J.
2016-01-01
We prove several explicit formulae for the $n$-th Bernoulli polynomial $B_{n}(x)$, in which $B_{n}(x)$ is equal to an affine combination of the polynomials $(x-1)^{n}$, $(x-2)^{n}$, $ldots$, $(x-k-1)^{n}$, where $k$ is any fixed positive integer greater or equal than $n$.
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
Notes on Schubert, Grothendieck and Key Polynomials
Kirillov, Anatol N.
2016-03-01
We introduce common generalization of (double) Schubert, Grothendieck, Demazure, dual and stable Grothendieck polynomials, and Di Francesco-Zinn-Justin polynomials. Our approach is based on the study of algebraic and combinatorial properties of the reduced rectangular plactic algebra and associated Cauchy kernels.
Differential Krull dimension in differential polynomial extensions
Smirnov, Ilya
2011-01-01
We investigate the differential Krull dimension of differential polynomials over a differential ring. We prove a differential analogue of Jaffard's Special Chain Theorem and show that differential polynomial extensions of certain classes of differential rings have no anomaly of differential Krull dimension.
Colored HOMFLY polynomials can distinguish mutant knots
Nawata, Satoshi; Singh, Vivek Kumar
2015-01-01
We illustrate from the viewpoint of braiding operations on WZNW conformal blocks how colored HOMFLY polynomials with multiplicity structure can detect mutations. As an example, we explicitly evaluate the (2,1)-colored HOMFLY polynomials that distinguish a famous mutant pair, Kinoshita-Terasaka and Conway knot.
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).
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.
Polynomial weights and code constructions
DEFF Research Database (Denmark)
Massey, J; Costello, D; Justesen, Jørn
1973-01-01
polynomial included. This fundamental property is then used as the key to a variety of code constructions including 1) a simplified derivation of the binary Reed-Muller codes and, for any primepgreater than 2, a new extensive class ofp-ary "Reed-Muller codes," 2) a new class of "repeated-root" cyclic codes...... that are subcodes of the binary Reed-Muller codes and can be very simply instrumented, 3) a new class of constacyclic codes that are subcodes of thep-ary "Reed-Muller codes," 4) two new classes of binary convolutional codes with large "free distance" derived from known binary cyclic codes, 5) two new classes...... of long constraint length binary convolutional codes derived from2^r-ary Reed-Solomon codes, and 6) a new class ofq-ary "repeated-root" constacyclic codes with an algebraic decoding algorithm....
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.
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.
Baxter operator formalism for Macdonald polynomials
Gerasimov, Anton; Oblezin, Sergey
2012-01-01
We develop basic constructions of the Baxter operator formalism for the Macdonald polynomials. Precisely we construct a dual pair of mutually commuting Baxter operators such that the Macdonald polynomials are their common eigenfunctions. The dual pair of Baxter operators is closely related to the dual pair of recursive operators for Macdonald polynomials leading to various families of their integral representations. We also construct the Baxter operator formalism for the q-deformed Whittaker functions and the Jack polynomials obtained by degenerations of the Macdonald polynomials. This note provides a generalization of our previous results on the Baxter operator formalism for the Whittaker functions. It was demonstrated previously that Baxter operator formalism for the Whittaker functions has deep connections with representation theory. In particular the Baxter operators should be considered as elements of appropriate spherical Hecke algebras and their eigenvalues are identified with local Archimedean L-facto...
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.
On Chebyshev polynomials and torus knots
Gavrilik, A M
2009-01-01
In this work we demonstrate that the q-numbers and their two-parameter generalization, the q,p-numbers, can be used to obtain some polynomial invariants for torus knots and links. First, we show that the q-numbers, which are closely connected with the Chebyshev polynomials, can also be related with the Alexander polynomials for the class T(s,2) of torus knots, s being an odd integer, and used for finding the corresponding skein relation. Then, we develop this procedure in order to obtain, with the help of q,p-numbers, the generalized two-variable Alexander polynomials, and prove their direct connection with the HOMFLY polynomials and the skein relation of the latter.
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.
Jacob's ladders and new orthogonal systems generated by Jacobi polynomials
Moser, Jan
2010-01-01
Is is shown in this paper that there is a connection between the Riemann zeta-function $\\zf$ and the classical Jacobi's polynomials, i.e. the Legendre polynomials, Chebyshev polynomials of the first and the second kind,...
Liu, Yang; Chen, Zhenyu; Yang, Zhile; Li, Kang; Tan, Jiubin
2016-12-01
The accuracy of surface measurement determines the manufacturing quality of membrane mirrors. Thus, an efficient and accurate measuring method is critical in membrane mirror fabrication. This paper formulates this measurement issue as a surface reconstruction problem and employs two-stage trained Zernike polynomials as an inline measuring tool to solve the optical surface measurement problem in the membrane mirror manufacturing process. First, all terms of the Zernike polynomial are generated and projected to a non-circular region as the candidate model pool. The training data are calculated according to the measured values of distance sensors and the geometrical relationship between the ideal surface and the installed sensors. Then the terms are selected by minimizing the cost function each time successively. To avoid the problem of ill-conditioned matrix inversion by the least squares method, the coefficient of each model term is achieved by modified elitist teaching-learning-based optimization. Subsequently, the measurement precision is further improved by a second stage of model refinement. Finally, every point on the membrane surface can be measured according to this model, providing more the subtle feedback information needed for the precise control of membrane mirror fabrication. Experimental results confirm that the proposed method is effective in a membrane mirror manufacturing system driven by negative pressure, and the measurement accuracy can achieve 15 µm.
Graph Polynomials: From Recursive Definitions To Subset Expansion Formulas
Godlin, Benny; Makowsky, Johann A
2008-01-01
Many graph polynomials, such as the Tutte polynomial, the interlace polynomial and the matching polynomial, have both a recursive definition and a defining subset expansion formula. In this paper we present a general, logic-based framework which gives a precise meaning to recursive definitions of graph polynomials. We then prove that in this framework every recursive definition of a graph polynomial can be converted into a subset expansion formula.
Symmetric polynomials, quantum Jacobi-Trudi identities and \\tau-functions
Harnad, J
2013-01-01
An element [\\Phi] of the Grassmannian of n-dimensional subspaces of the Hardy space H^2, extended over the field C(x_1,..., x_n), may be associated to any polynomial basis {\\phi} for C(x). The Pl\\"ucker coordinates S^\\phi_{\\lambda,n}(x_1,..., x_n) of \\Phi, labelled by partitions \\lambda, provide an analog of Jacobi's bi-alternant formula, defining a generalization of Schur polynomials. Applying the recursion relations satisfied by the polynomial system to the analog of the complete symmetric functions generates a doubly infinite matrix of symmetric polynomials that determine an element [H] of the Grasmannian. This is shown to coincide with [\\Phi], implying a set of {\\it quantum Jacobi-Trudi identities} that generalize a result obtained by Sergeev and Veselov for the case of orthogonal polynomials. The symmetric polynomials S^\\phi_{\\lambda,n}(x_1,..., x_n) are shown to be KP (Kadomtsev-Petviashvili) tau-functions in terms of the monomial sums in the parameters x_a, viewed as KP flow variables. A fermionic oper...
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.
2008-06-01
stream output from both registers is identical at certain offsets, the Fibonacci register is computationally more efficient at producing the m- sequence ...5 3. Fibonacci LFSR for f(x) = x7 + x3 + x2 + x+ 1 . . . . . . . . . . . . . . . 15 4. Galois LFSR for f(x) = x7 + x3 + x2 + x+ 1...generation is via a maximum period Linear-Feedback Shift Register (LFSR). The authoritative source on the topic is Shift Register Sequences by Solomon W
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...
The Translated Dowling Polynomials and Numbers.
Mangontarum, Mahid M; Macodi-Ringia, Amila P; Abdulcarim, Normalah S
2014-01-01
More properties for the translated Whitney numbers of the second kind such as horizontal generating function, explicit formula, and exponential generating function are proposed. Using the translated Whitney numbers of the second kind, we will define the translated Dowling polynomials and numbers. Basic properties such as exponential generating functions and explicit formula for the translated Dowling polynomials and numbers are obtained. Convexity, integral representation, and other interesting identities are also investigated and presented. We show that the properties obtained are generalizations of some of the known results involving the classical Bell polynomials and numbers. Lastly, we established the Hankel transform of the translated Dowling numbers.
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.
Laurent polynomial moment problem: a case study
Pakovich, F; Zvonkin, A
2009-01-01
In recent years, the so-called polynomial moment problem, motivated by the classical Poincare center-focus problem, was thoroughly studied, and the answers to the main questions have been found. The study of a similar problem for rational functions is still at its very beginning. In this paper, we make certain progress in this direction; namely, we construct an example of a Laurent polynomial for which the solutions of the corresponding moment problem behave in a significantly more complicated way than it would be possible for a polynomial.
On Calculation of Adomian Polynomials by MATLAB
Directory of Open Access Journals (Sweden)
Hossein ABOLGHASEMI
2011-01-01
Full Text Available Adomian Decomposition Method (ADM is an elegant technique to handle an extensive class of linear or nonlinear differential and integral equations. However, in case of nonlinear equations, ADM demands a special representation of each nonlinear term, namely, Adomian polynomials. The present paper introduces a novel MATLAB code which computes Adomian polynomials associated with several types of nonlinearities. The code exploits symbolic programming incorporated with a recently proposed alternative scheme to be straightforward and fast. For the sake of exemplification, Adomian polynomials of famous nonlinear operators, computed by the code, are given.
ECG data compression using Jacobi polynomials.
Tchiotsop, Daniel; Wolf, Didier; Louis-Dorr, Valérie; Husson, René
2007-01-01
Data compression is a frequent signal processing operation applied to ECG. We present here a method of ECG data compression utilizing Jacobi polynomials. ECG signals are first divided into blocks that match with cardiac cycles before being decomposed in Jacobi polynomials bases. Gauss quadratures mechanism for numerical integration is used to compute Jacobi transforms coefficients. Coefficients of small values are discarded in the reconstruction stage. For experimental purposes, we chose height families of Jacobi polynomials. Various segmentation approaches were considered. We elaborated an efficient strategy to cancel boundary effects. We obtained interesting results compared with ECG compression by wavelet decomposition methods. Some propositions are suggested to improve the results.
Limits of zeros of polynomial sequences
Zhu, Xinyun; Grossman, George
2007-01-01
In the present paper we consider $F_k(x)=x^{k}-\\sum_{t=0}^{k-1}x^t,$ the characteristic polynomial of the $k$-th order Fibonacci sequence, the latter denoted $G(k,l).$ We determine the limits of the real roots of certain odd and even degree polynomials related to the derivatives and integrals of $F_k(x),$ that form infinite sequences of polynomials, of increasing degree. In particular, as $k \\to \\infty,$ the limiting values of the zeros are determined, for both odd and even cases. It is also ...
Cycles are determined by their domination polynomials
Akbari, Saieed
2009-01-01
Let $G$ be a simple graph of order $n$. A dominating set of $G$ is a set $S$ of vertices of $G$ so that every vertex of $G$ is either in $S$ or adjacent to a vertex in $S$. The domination polynomial of $G$ is the polynomial $D(G,x)=\\sum_{i=1}^{n} d(G,i) x^{i}$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. In this paper we show that cycles are determined by their domination polynomials.
Empowering Polynomial Theory Conjectures with Spreadsheets
Directory of Open Access Journals (Sweden)
Chris Petersdinh
2017-06-01
Full Text Available Polynomial functions and their properties are fundamental to algebra, calculus, and mathematical modeling. Students who do not have a strong understanding of the relationship between factoring and solving equations can have difficulty with optimization problems in calculus and solving application problems in any field. Understanding function transformations is important in trigonometry, the idea of the general antiderivative, and describing the geometry of a problem mathematically. This paper presents spreadsheet activities designed to bolster students' conceptualization of the factorization theorem for polynomials, complex zeros of polynomials, and function transformations. These activities were designed to use a constructivist approach involving student experimentation and conjectures.
A Polynomial Preconditioner for the CMRH Algorithm
Directory of Open Access Journals (Sweden)
Jiangzhou Lai
2011-01-01
Full Text Available Many large and sparse linear systems can be solved efficiently by restarted GMRES and CMRH methods Sadok 1999. The CMRH(m method is less expensive and requires slightly less storage than GMRES(m. But like GMRES, the restarted CMRH method may not converge. In order to remedy this defect, this paper presents a polynomial preconditioner for CMRH-based algorithm. Numerical experiments are given to show that the polynomial preconditioner is quite simple and easily constructed and the preconditioned CMRH(m with the polynomial preconditioner has better performance than CMRH(m.
On Chebyshev polynomials and torus knots
Gavrilik, A. M.; Pavlyuk, A. M.
2009-01-01
In this work we demonstrate that the q-numbers and their two-parameter generalization, the q,p-numbers, can be used to obtain some polynomial invariants for torus knots and links. First, we show that the q-numbers, which are closely connected with the Chebyshev polynomials, can also be related with the Alexander polynomials for the class T(s,2) of torus knots, s being an odd integer, and used for finding the corresponding skein relation. Then, we develop this procedure in order to obtain, wit...
A bivariate chromatic polynomial for signed graphs
Beck, Matthias
2012-01-01
We study Dohmen--P\\"onitz--Tittmann's bivariate chromatic polynomial $c_\\Gamma(k,l)$ which counts all $(k+l)$-colorings of a graph $\\Gamma$ such that adjacent vertices get different colors if they are $\\le k$. Our first contribution is an extension of $c_\\Gamma(k,l)$ to signed graphs, for which we obtain an inclusion--exclusion formula and several special evaluations giving rise, e.g., to polynomials that encode balanced subgraphs. Our second goal is to derive combinatorial reciprocity theorems for $c_\\Gamma(k,l)$ and its signed-graph analogues, reminiscent of Stanley's reciprocity theorem linking chromatic polynomials to acyclic orientations.
Polynomials for Crystal Frameworks and the Rigid Unit Mode Spectrum
Power, S C
2011-01-01
Two derivations are given for a matrix-valued function $\\Phi_\\C(z)$, defined on the $d$-torus, that can be associated with a discrete, translationally periodic bar-joint framework $\\C$ in $\\bR^d$. The rigid unit mode (RUM) spectrum of $\\C$ is defined in terms of the phases of phase-periodic infinitesimal flexes and is identified in terms of the singular points of the function $z \\to \\rank \\Phi_\\C(z)$ and also in terms of the wave vectors of excitations with vanishing energy in the long wavelength limit. To a crystal framework $\\C$ in Maxwell counting equilibrium we associate a unique multi-variable monic polynomial $p_\\C(z_1,..,z_d)$ and for ideal zeolites the algebraic variety of zeros of $p_\\C(z)$ on the $d$-torus determines the RUM spectrum. The matrix function is related to periodic "floppy modes" and their asymptotic order and an explicit formula is obtained for the number of periodic floppy modes for a given supercell. The crystal polynomial, RUM spectrum and the mode multiplicity are computed for a num...
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.
Energy Technology Data Exchange (ETDEWEB)
Dorey, Nick [Department of Applied Mathematics and Theoretical Physics, University of Cambridge,Wilberforce Road, Cambridge, CB3 OWA (United Kingdom); Tong, David [Department of Applied Mathematics and Theoretical Physics, University of Cambridge,Wilberforce Road, Cambridge, CB3 OWA (United Kingdom); Department of Theoretical Physics, TIFR,Homi Bhabha Road, Mumbai 400 005 (India); Stanford Institute for Theoretical Physics,Via Pueblo, Stanford, CA 94305 (United States); Turner, Carl [Department of Applied Mathematics and Theoretical Physics, University of Cambridge,Wilberforce Road, Cambridge, CB3 OWA (United Kingdom)
2016-08-01
We study a U(N) gauged matrix quantum mechanics which, in the large N limit, is closely related to the chiral WZW conformal field theory. This manifests itself in two ways. First, we construct the left-moving Kac-Moody algebra from matrix degrees of freedom. Secondly, we compute the partition function of the matrix model in terms of Schur and Kostka polynomials and show that, in the large N limit, it coincides with the partition function of the WZW model. This same matrix model was recently shown to describe non-Abelian quantum Hall states and the relationship to the WZW model can be understood in this framework.
Dorey, Nick; Turner, Carl
2016-01-01
We study a U(N) gauged matrix quantum mechanics which, in the large N limit, is closely related to the chiral WZW conformal field theory. This manifests itself in two ways. First, we construct the left-moving Kac-Moody algebra from matrix degrees of freedom. Secondly, we compute the partition function of the matrix model in terms of Schur and Kostka polynomials and show that, in the large $N$ limit, it coincides with the partition function of the WZW model. This same matrix model was recently shown to describe non-Abelian quantum Hall states and the relationship to the WZW model can be understood in this framework.
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.
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...
Handbook on semidefinite, conic and polynomial optimization
Anjos, Miguel F
2012-01-01
This book offers the reader a snapshot of the state-of-the-art in the growing and mutually enriching areas of semidefinite optimization, conic optimization and polynomial optimization. It covers theory, algorithms, software and applications.
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.
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.
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.
Hermite polynomials and quasi-classical asymptotics
Energy Technology Data Exchange (ETDEWEB)
Ali, S. Twareque, E-mail: twareque.ali@concordia.ca [Department of Mathematics and Statistics, Concordia University, Montréal, Québec H3G 1M8 (Canada); Engliš, Miroslav, E-mail: englis@math.cas.cz [Mathematics Institute, Silesian University in Opava, Na Rybníčku 1, 74601 Opava, Czech Republic and Mathematics Institute, Žitná 25, 11567 Prague 1 (Czech Republic)
2014-04-15
We study an unorthodox variant of the Berezin-Toeplitz type of quantization scheme, on a reproducing kernel Hilbert space generated by the real Hermite polynomials and work out the associated quasi-classical asymptotics.
Generation of multivariate Hermite interpolating polynomials
Tavares, Santiago Alves
2005-01-01
Generation of Multivariate Hermite Interpolating Polynomials advances the study of approximate solutions to partial differential equations by presenting a novel approach that employs Hermite interpolating polynomials and bysupplying algorithms useful in applying this approach.Organized into three sections, the book begins with a thorough examination of constrained numbers, which form the basis for constructing interpolating polynomials. The author develops their geometric representation in coordinate systems in several dimensions and presents generating algorithms for each level number. He then discusses their applications in computing the derivative of the product of functions of several variables and in the construction of expression for n-dimensional natural numbers. Section II focuses on the construction of Hermite interpolating polynomials, from their characterizing properties and generating algorithms to a graphical analysis of their behavior. The final section of the book is dedicated to the applicatio...
Concentration for noncommutative polynomials in random matrices
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.
Limits of zeros of polynomial sequences
Zhu, Xinyun
2007-01-01
In the present paper we consider $F_k(x)=x^{k}-\\sum_{t=0}^{k-1}x^t,$ the characteristic polynomial of the $k$-th order Fibonacci sequence, the latter denoted $G(k,l).$ We determine the limits of the real roots of certain odd and even degree polynomials related to the derivatives and integrals of $F_k(x),$ that form infinite sequences of polynomials, of increasing degree. In particular, as $k \\to \\infty,$ the limiting values of the zeros are determined, for both odd and even cases. It is also shown, in both cases, that the convergence is monotone for sufficiently large degree. We give an upper bound for the modulus of the complex zeros of the polynomials for each sequence. This gives a general solution related to problems considered by Dubeau 1989, 1993, Miles 1960, Flores 1967, Miller 1971 and later by the second author in the present paper, and Narayan 1997.
Recursive Polynomial Remainder Sequence and its Subresultants
Terui, Akira
2008-01-01
We introduce concepts of "recursive polynomial remainder sequence (PRS)" and "recursive subresultant," along with investigation of their properties. A recursive PRS is defined as, if there exists the GCD (greatest common divisor) of initial polynomials, a sequence of PRSs calculated "recursively" for the GCD and its derivative until a constant is derived, and recursive subresultants are defined by determinants representing the coefficients in recursive PRS as functions of coefficients of init...
Subresultants in Recursive Polynomial Remainder Sequence
Terui, Akira
2008-01-01
We introduce concepts of "recursive polynomial remainder sequence (PRS)" and "recursive subresultant," and investigate their properties. In calculating PRS, if there exists the GCD (greatest common divisor) of initial polynomials, we calculate "recursively" with new PRS for the GCD and its derivative, until a constant is derived. We call such a PRS a recursive PRS. We define recursive subresultants to be determinants representing the coefficients in recursive PRS by coefficients of initial po...
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.
Blind Signature Scheme Based on Chebyshev Polynomials
Maheswara Rao Valluri
2011-01-01
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.
On Certain Divisibility Property of Polynomials
Caceres, Luis F
2010-01-01
We review the definition of D-rings introduced by H. Gunji & D. L. MacQuillan. We provide an alternative characterization for such rings that allows us to give an elementary proof of that a ring of algebraic integers is a D-ring. Moreover, we give a characterization for D-rings that are also unique factorization domains to determine divisibility of polynomials using polynomial evaluations.
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.
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...
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.
A generalization of the dichromatic polynomial of a graph
1981-01-01
The Subgraph polynomial fo a graph pair (G,H), where H⫅G, is defined. By assigning particular weights to the variables, it is shown that this polynomial reduces to the dichromatic polynomial of G. This idea of a graph pair leads to a dual generalization of the dichromatic polynomial.
Interpolation on Real Algebraic Curves to Polynomial Data
Directory of Open Access Journals (Sweden)
Len Bos
2013-04-01
Full Text Available We discuss a polynomial interpolation problem where the data are of the form of a set of algebraic curves in R^2 on each of which is prescribed a polynomial. The object is then to construct a global bivariate polynomial that agrees with the given polynomials when restricted to the corresponding curves.
On λ-Bell polynomials associated with umbral calculus
Kim, T.; Kim, D. S.
2017-01-01
In this paper, we introduce some new λ-Bell polynomials and Bell polynomials of the second kind and investigate properties of these polynomials. Using our investigation, we derive some new identities for the two kinds of λ-Bell polynomials arising from umbral calculus.
Interpolation Functions of -Extensions of Apostol's Type Euler Polynomials
Directory of Open Access Journals (Sweden)
Kim Young-Hee
2009-01-01
Full Text Available The main purpose of this paper is to present new -extensions of Apostol's type Euler polynomials using the fermionic -adic integral on . We define the - -Euler polynomials and obtain the interpolation functions and the Hurwitz type zeta functions of these polynomials. We define -extensions of Apostol type's Euler polynomials of higher order using the multivariate fermionic -adic integral on . We have the interpolation functions of these - -Euler polynomials. We also give -extensions of Apostol's type Euler polynomials of higher order and have the multiple Hurwitz type zeta functions of these - -Euler polynomials.
DEFF Research Database (Denmark)
Hyldahl, Kirsten Kofod
Denne bog undersøger, hvordan lærere kan anvende feedback til at forbedre undervisningen i klasselokalet. I denne sammenhæng har John Hattie, professor ved Melbourne Universitet, udviklet en model for feedback, hvilken er baseret på synteser af meta-analyser. I 2009 udgav han bogen "Visible...
Dawson, Nathan J.; Kuzyk, Mark G.
2016-12-01
We attempt to get a polynomial solution to the inverse problem, that is, to determine the form of the mechanical Hamiltonian when given the energy spectrum and transition dipole moment matrix. Our approach is to determine the potential in the form of a polynomial by finding an approximate solution to the inverse problem, then to determine the hyperpolarizability for that system's Hamiltonian. We find that the largest hyperpolarizabilities approach the apparent limit of previous potential optimization studies, but we do not find real potentials for the parameter values necessary to exceed this apparent limit. We also explore half potentials with positive exponent, which cannot be expressed as a polynomial except for integer powers. This yields a simple closed potential with only one parameter that scans nearly the full range of the intrinsic hyperpolarizability. The limiting case of vanishing exponent yields the largest intrinsic hyperpolarizability.
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.
Zeros of Jones polynomials for families of knots and links
Chang, S.-C.; Shrock, R.
2001-12-01
We calculate Jones polynomials VL( t) for several families of alternating knots and links by computing the Tutte polynomials T( G, x, y) for the associated graphs G and then obtaining VL( t) as a special case of the Tutte polynomial. For each of these families we determine the zeros of the Jones polynomial, including the accumulation set in the limit of infinitely many crossings. A discussion is also given of the calculation of Jones polynomials for non-alternating links.
Blumthaler, Ingrid; Oberst, Ulrich
2012-03-01
Control design belongs to the most important and difficult tasks of control engineering and has therefore been treated by many prominent researchers and in many textbooks, the systems being generally described by their transfer matrices or by Rosenbrock equations and more recently also as behaviors. Our approach to controller design uses, in addition to the ideas of our predecessors on coprime factorizations of transfer matrices and on the parametrization of stabilizing compensators, a new mathematical technique which enables simpler design and also new theorems in spite of the many outstanding results of the literature: (1) We use an injective cogenerator signal module ℱ over the polynomial algebra [Formula: see text] (F an infinite field), a saturated multiplicatively closed set T of stable polynomials and its quotient ring [Formula: see text] of stable rational functions. This enables the simultaneous treatment of continuous and discrete systems and of all notions of stability, called T-stability. We investigate stabilizing control design by output feedback of input/output (IO) behaviors and study the full feedback IO behavior, especially its autonomous part and not only its transfer matrix. (2) The new technique is characterized by the permanent application of the injective cogenerator quotient signal module [Formula: see text] and of quotient behaviors [Formula: see text] of [Formula: see text]-behaviors B. (3) For the control tasks of tracking, disturbance rejection, model matching, and decoupling and not necessarily proper plants we derive necessary and sufficient conditions for the existence of proper stabilizing compensators with proper and stable closed loop behaviors, parametrize all such compensators as IO behaviors and not only their transfer matrices and give new algorithms for their construction. Moreover we solve the problem of pole placement or spectral assignability for the complete feedback behavior. The properness of the full feedback behavior
Roots of the derivative of the Riemann zeta function and of characteristic polynomials
Dueñez, Eduardo; Froehlich, Sara; Hughes, Chris; Mezzadri, Francesco; Phan, Toan
2010-01-01
We investigate the horizontal distribution of zeros of the derivative of the Riemann zeta function and compare this to the radial distribution of zeros of the derivative of the characteristic polynomial of a random unitary matrix. Both cases show a surprising bimodal distribution which has yet to be explained. We show by example that the bimodality is a general phenomenon. For the unitary matrix case we prove a conjecture of Mezzadri concerning the leading order behavior, and we show that the same follows from the random matrix conjectures for the zeros of the zeta function.
Extending a Property of Cubic Polynomials to Higher-Degree Polynomials
Miller, David A.; Moseley, James
2012-01-01
In this paper, the authors examine a property that holds for all cubic polynomials given two zeros. This property is discovered after reviewing a variety of ways to determine the equation of a cubic polynomial given specific conditions through algebra and calculus. At the end of the article, they will connect the property to a very famous method…
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
Extending a Property of Cubic Polynomials to Higher-Degree Polynomials
Miller, David A.; Moseley, James
2012-01-01
In this paper, the authors examine a property that holds for all cubic polynomials given two zeros. This property is discovered after reviewing a variety of ways to determine the equation of a cubic polynomial given specific conditions through algebra and calculus. At the end of the article, they will connect the property to a very famous method…
Sandryhaila, Aliaksei; Pueschel, Markus
2010-01-01
A polynomial transform is the multiplication of an input vector $x\\in\\C^n$ by a matrix $\\PT_{b,\\alpha}\\in\\C^{n\\times n},$ whose $(k,\\ell)$-th element is defined as $p_\\ell(\\alpha_k)$ for polynomials $p_\\ell(x)\\in\\C[x]$ from a list $b=\\{p_0(x),\\dots,p_{n-1}(x)\\}$ and sample points $\\alpha_k\\in\\C$ from a list $\\alpha=\\{\\alpha_0,\\dots,\\alpha_{n-1}\\}$. Such transforms find applications in the areas of signal processing, data compression, and function interpolation. Important examples include the discrete Fourier and cosine transforms. In this paper we introduce a novel technique to derive fast algorithms for polynomial transforms. The technique uses the relationship between polynomial transforms and the representation theory of polynomial algebras. Specifically, we derive algorithms by decomposing the regular modules of these algebras as a stepwise induction. As an application, we derive novel $O(n\\log{n})$ general-radix algorithms for the discrete Fourier transform and the discrete cosine transform of type 4.
High order overlay modeling and APC simulation with Zernike-Legendre polynomials
Ju, JawWuk; Kim, MinGyu; Lee, JuHan; Sherwin, Stuart; Hoo, George; Choi, DongSub; Lee, Dohwa; Jeon, Sanghuck; Lee, Kangsan; Tien, David; Pierson, Bill; Robinson, John C.; Levy, Ady; Smith, Mark D.
2015-03-01
Feedback control of overlay errors to the scanner is a well-established technique in semiconductor manufacturing [1]. Typically, overlay errors are measured, and then modeled by least-squares fitting to an overlay model. Overlay models are typically Cartesian polynomial functions of position within the wafer (Xw, Yw), and of position within the field (Xf, Yf). The coefficients from the data fit can then be fed back to the scanner to reduce overlay errors in future wafer exposures, usually via a historically weighted moving average. In this study, rather than using the standard Cartesian formulation, we examine overlay models using Zernike polynomials to represent the wafer-level terms, and Legendre polynomials to represent the field-level terms. Zernike and Legendre polynomials can be selected to have the same fitting capability as standard polynomials (e.g., second order in X and Y, or third order in X and Y). However, Zernike polynomials have the additional property of being orthogonal over the unit disk, which makes them appropriate for the wafer-level model, and Legendre polynomials are orthogonal over the unit square, which makes them appropriate for the field-level model. We show several benefits of Zernike/Legendre-based models in this investigation in an Advanced Process Control (APC) simulation using highly-sampled fab data. First, the orthogonality property leads to less interaction between the terms, which makes the lot-to-lot variation in the fitted coefficients smaller than when standard polynomials are used. Second, the fitting process itself is less coupled - fitting to a lower-order model, and then fitting the residuals to a higher order model gives very similar results as fitting all of the terms at once. This property makes fitting techniques such as dual pass or cascading [2] unnecessary, and greatly simplifies the options available for the model recipe. The Zernike/Legendre basis gives overlay performance (mean plus 3 sigma of the residuals
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...
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...
Chemical Reaction Networks for Computing Polynomials.
Salehi, Sayed Ahmad; Parhi, Keshab K; Riedel, Marc D
2017-01-20
Chemical reaction networks (CRNs) provide a fundamental model in the study of molecular systems. Widely used as formalism for the analysis of chemical and biochemical systems, CRNs have received renewed attention as a model for molecular computation. This paper demonstrates that, with a new encoding, CRNs can compute any set of polynomial functions subject only to the limitation that these functions must map the unit interval to itself. These polynomials can be expressed as linear combinations of Bernstein basis polynomials with positive coefficients less than or equal to 1. In the proposed encoding approach, each variable is represented using two molecular types: a type-0 and a type-1. The value is the ratio of the concentration of type-1 molecules to the sum of the concentrations of type-0 and type-1 molecules. The proposed encoding naturally exploits the expansion of a power-form polynomial into a Bernstein polynomial. Molecular encoders for converting any input in a standard representation to the fractional representation as well as decoders for converting the computed output from the fractional to a standard representation are presented. The method is illustrated first for generic CRNs; then chemical reactions designed for an example are mapped to DNA strand-displacement reactions.
Direct discriminant locality preserving projection with Hammerstein polynomial expansion.
Chen, Xi; Zhang, Jiashu; Li, Defang
2012-12-01
Discriminant locality preserving projection (DLPP) is a linear approach that encodes discriminant information into the objective of locality preserving projection and improves its classification ability. To enhance the nonlinear description ability of DLPP, we can optimize the objective function of DLPP in reproducing kernel Hilbert space to form a kernel-based discriminant locality preserving projection (KDLPP). However, KDLPP suffers the following problems: 1) larger computational burden; 2) no explicit mapping functions in KDLPP, which results in more computational burden when projecting a new sample into the low-dimensional subspace; and 3) KDLPP cannot obtain optimal discriminant vectors, which exceedingly optimize the objective of DLPP. To overcome the weaknesses of KDLPP, in this paper, a direct discriminant locality preserving projection with Hammerstein polynomial expansion (HPDDLPP) is proposed. The proposed HPDDLPP directly implements the objective of DLPP in high-dimensional second-order Hammerstein polynomial space without matrix inverse, which extracts the optimal discriminant vectors for DLPP without larger computational burden. Compared with some other related classical methods, experimental results for face and palmprint recognition problems indicate the effectiveness of the proposed HPDDLPP.
Multi-indexed Meixner and little q-Jacobi (Laguerre) polynomials
Odake, Satoru; Sasaki, Ryu
2017-04-01
As the fourth stage of the project multi-indexed orthogonal polynomials, we present the multi-indexed Meixner and little q-Jacobi (Laguerre) polynomials in the framework of ‘discrete quantum mechanics’ with real shifts defined on the semi-infinite lattice in one dimension. They are obtained, in a similar way to the multi-indexed Laguerre and Jacobi polynomials reported earlier, from the quantum mechanical systems corresponding to the original orthogonal polynomials by multiple application of the discrete analogue of the Darboux transformations or the Crum–Krein–Adler deletion of virtual state vectors. The virtual state vectors are the solutions of the matrix Schrödinger equation on all the lattice points having negative energies and infinite norm. This is in good contrast to the (q-)Racah systems defined on a finite lattice, in which the ‘virtual state’ vectors satisfy the matrix Schrödinger equation except for one of the two boundary points.
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.
Transversals of Complex Polynomial Vector Fields
DEFF Research Database (Denmark)
Dias, Kealey
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......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...... examples of rotated families to argue this. There will be discussed several open questions concerning the number of transversals that can appear for a certain degree d of a polynomial vector field, and furthermore how transversals are analyzed with respect to bifurcations around multiple equilibrium points....
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.
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.
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...
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.
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.
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.
High degree interpolation polynomial in Newton form
Tal-Ezer, Hillel
1988-01-01
Polynomial interpolation is an essential subject in numerical analysis. Dealing with a real interval, it is well known that even if f(x) is an analytic function, interpolating at equally spaced points can diverge. On the other hand, interpolating at the zeroes of the corresponding Chebyshev polynomial will converge. Using the Newton formula, this result of convergence is true only on the theoretical level. It is shown that the algorithm which computes the divided differences is numerically stable only if: (1) the interpolating points are arranged in a different order, and (2) the size of the interval is 4.
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.
Some Inequalities for the Derivative of Polynomials
Directory of Open Access Journals (Sweden)
Sunil Hans
2014-01-01
Full Text Available If pz=∑υ=0ncυzυ is a polynomial of degree n, having no zeros in z<1, then Aziz (1989 proved maxz=1p′z≤n/2Mα2+Mα+π21/2, where Mα=max1≤k≤npeiα+2kπ/n. In this paper, we consider a class of polynomial Pnμ of degree n, defined as pz=a0+∑υ=μnaυzυ and present certain generalizations of above inequality and some other well-known results.
Knot polynomial identities and quantum group coincidences
Morrison, Scott; Snyder, Noah
2010-01-01
We construct link invariants using the D_2n subfactor planar algebras, and use these to prove new identities relating certain specializations of colored Jones polynomials to specializations of other quantum knot polynomials. These identities can also be explained by coincidences between small modular categories involving the even parts of the D_2n planar algebras. We discuss the origins of these coincidences, explaining the role of SO level-rank duality, Kirby-Melvin symmetry, and properties of small Dynkin diagrams. One of these coincidences involves G_2 and does not appear to be related to level-rank duality.
Polynomial Kernelizations for $\\MINF_1$ and $\\MNP$
Kratsch, Stefan
2009-01-01
The relation of constant-factor approximability to fixed-parameter tractability and kernelization is a long-standing open question. We prove that two large classes of constant-factor approximable problems, namely $\\MINF_1$ and $\\MNP$, including the well-known subclass $\\MSNP$, admit polynomial kernelizations for their natural decision versions. This extends results of Cai and Chen (JCSS 1997), stating that the standard parameterizations of problems in $\\MSNP$ and $\\MINF_1$ are fixed-parameter tractable, and complements recent research on problems that do not admit polynomial kernelizations (Bodlaender et al. ICALP 2008).
Spread polynomials, rotations and the butterfly effect
Goh, Shuxiang
2009-01-01
The spread between two lines in rational trigonometry replaces the concept of angle, allowing the complete specification of many geometrical and dynamical situations which have traditionally been viewed approximately. This paper investigates the case of powers of a rational spread rotation, and in particular, a curious periodicity in the prime power decomposition of the associated values of the spread polynomials, which are the analogs in rational trigonometry of the Chebyshev polynomials of the first kind. Rational trigonometry over finite fields plays a role, together with non-Euclidean geometries.
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.
The Potts model and the Tutte polynomial
Welsh, D. J. A.; Merino, C.
2000-03-01
This is an invited survey on the relation between the partition function of the Potts model and the Tutte polynomial. On the assumption that the Potts model is more familiar we have concentrated on the latter and its interpretations. In particular we highlight the connections with Abelian sandpiles, counting problems on random graphs, error correcting codes, and the Ehrhart polynomial of a zonotope. Where possible we use the mean field and square lattice as illustrations. We also discuss in some detail the complexity issues involved.
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....
Reaching Nonlinear Consensus via Non-Autonomous Polynomial Stochastic Operators
Saburov, Mansoor; Saburov, Khikmat
2017-03-01
This paper is a continuation of our previous studies on nonlinear consensus which unifies and generalizes all previous results. We consider a nonlinear protocol for a structured time-varying synchronous multi-agent system. We present an opinion sharing dynamics of the multi-agent system as a trajectory of non-autonomous polynomial stochastic operators associated with multidimensional stochastic hyper-matrices. We show that the multi-agent system eventually reaches to a nonlinear consensus if either one of the following two conditions is satisfied: (i) every member of the group people has a positive subjective distribution on the given task after some revision steps or (ii) all entries of some multidimensional stochastic hyper-matrix are positive.
On ambiguity in knot polynomials for virtual knots
Morozov, A; Popolitov, A
2015-01-01
We claim that HOMFLY polynomials for virtual knots, defined with the help of the matrix-model recursion relations, contain more parameters, than just the usual $q$ and $A = q^N$. These parameters preserve topological invariance and do not show up in the case of ordinary (non-virtual) knots and links. They are most conveniently observed in the hypercube formalism: then they substitute $q$-dimensions of certain fat graphs, which are not constrained by recursion and can be chosen arbitrarily. The number of these new topological invariants seems to grow fast with the number of non-virtual crossings: 0, 1, 1, 5, 15, 91, 784, 9160, ... This number can be decreased by imposing the factorization requirement for composites, in addition to topological invariance -- still freedom remains. None of these new parameters, however, appear in HOMFLY for Kishino unknot, which thus remains unseparated from the ordinary unknots even by this enriched set of knot invariants.
On ambiguity in knot polynomials for virtual knots
Morozov, A.; Morozov, And.; Popolitov, A.
2016-06-01
We claim that HOMFLY polynomials for virtual knots, defined with the help of the matrix-model recursion relations, contain more parameters, than just the usual q and A =qN. These parameters preserve topological invariance and do not show up in the case of ordinary (non-virtual) knots and links. They are most conveniently observed in the hypercube formalism: then they substitute q-dimensions of certain fat graphs, which are not constrained by recursion and can be chosen arbitrarily. The number of these new topological invariants seems to grow fast with the number of non-virtual crossings: 0, 1, 1, 5, 15, 91, 784, 9160, ... This number can be decreased by imposing the factorization requirement for composites, in addition to topological invariance - still freedom remains. None of these new parameters, however, appears in HOMFLY for Kishino unknot, which thus remains unseparated from the ordinary unknots even by this enriched set of knot invariants.
Polynomial Representations for a Wavelet Model of Interest Rates
Directory of Open Access Journals (Sweden)
Dennis G. Llemit
2015-12-01
Full Text Available In this paper, we approximate a non – polynomial function which promises to be an essential tool in interest rates forecasting in the Philippines. We provide two numerical schemes in order to generate polynomial functions that approximate a new wavelet which is a modification of Morlet and Mexican Hat wavelets. The first is the Polynomial Least Squares method which approximates the underlying wavelet according to desired numerical errors. The second is the Chebyshev Polynomial approximation which generates the required function through a sequence of recursive and orthogonal polynomial functions. We seek to determine the lowest order polynomial representations of this wavelet corresponding to a set of error thresholds.
Painleve V and time-dependent Jacobi polynomials
Energy Technology Data Exchange (ETDEWEB)
Basor, Estelle [American Institute of Mathematics, Palo Alto, CA 94306 (United States); Chen Yang [Department of Mathematics, Imperial College London, 180 Queen' s Gates, London SW7 2BZ (United Kingdom); Ehrhardt, Torsten [Department of Mathematics, University of California, Santa Cruz, CA 95064 (United States)], E-mail: ebasor@aimath.org, E-mail: ychen@imperial.ac.uk, E-mail: ehrhardt@math.ucsc.edu
2010-01-08
In this paper we study the simplest deformation on a sequence of orthogonal polynomials. This in turn induces a deformation on the moment matrix of the polynomials and associated Hankel determinant. We replace the original (or reference) weight w{sub 0}(x) (supported on R or subsets of R) by w{sub 0}(x) e{sup -tx}. It is a well-known fact that under such a deformation the recurrence coefficients denoted as {alpha}{sub n} and {beta}{sub n} evolve in t according to the Toda equations, giving rise to the time-dependent orthogonal polynomials and time-dependent determinants, using Sogo's terminology. If w{sub 0} is the normal density e{sup -x{sup 2}}, x element of R, or the gamma density x{sup {alpha}} e{sup -x}, x element of R{sub +}, {alpha} > -1, then the initial value problem of the Toda equations can be trivially solved. This is because under elementary scaling and translation the orthogonality relations reduce to the original ones. However, if w{sub 0} is the beta density (1 - x){sup {alpha}}(1 + x){sup {beta}}, x in [ - 1, 1], {alpha}, {beta} > -1, the resulting 'time-dependent' Jacobi polynomials will again satisfy a linear second-order ode, but no longer in the Sturm-Liouville form, which is to be expected. This deformation induces an irregular singular point at infinity in addition to three regular singular points of the hypergeometric equation satisfied by the Jacobi polynomials. We will show that the coefficients of this ode, as well as the Hankel determinant, are intimately related to a particular Painleve V. In particular we show that p{sub 1}(n,t), where p{sub 1}(n,t) is the coefficient of z{sup n-1} of the monic orthogonal polynomials associated with the 'time-dependent' Jacobi weight, satisfies, up to a translation in t, the Jimbo-Miwa {sigma}-form of the same P{sub V}; while a recurrence coefficient {alpha}{sub n}(t) is up to a translation in t and a linear fractional transformation P{sub V}({alpha}{sup 2}/2, - {beta}{sup 2
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.
DEFF Research Database (Denmark)
Eriksson, Tor Viking; Poulsen, Anders; Villeval, Marie Claire
2009-01-01
This paper experimentally investigates the impact of different pay schemes and relative performance feedback policies on employee effort. We explore three feedback rules: no feedback on relative performance, feedback given halfway through the production period, and continuously updated feedback. ...
Global stability and quadratic Hamiltonian structure in Lotka-Volterra and quasi-polynomial systems
Energy Technology Data Exchange (ETDEWEB)
Szederkenyi, Gabor; Hangos, Katalin M
2004-04-26
We show that the global stability of quasi-polynomial (QP) and Lotka-Volterra (LV) systems with the well-known logarithmic Lyapunov function is equivalent to the existence of a local generalized dissipative Hamiltonian description of the LV system with a diagonal quadratic form as a Hamiltonian function. The Hamiltonian function can be calculated and the quadratic dissipativity neighborhood of the origin can be estimated by solving linear matrix inequalities.
The 6 Vertex Model and Schubert Polynomials
Directory of Open Access Journals (Sweden)
Alain Lascoux
2007-02-01
Full Text Available We enumerate staircases with fixed left and right columns. These objects correspond to ice-configurations, or alternating sign matrices, with fixed top and bottom parts. The resulting partition functions are equal, up to a normalization factor, to some Schubert polynomials.
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...
A recursive algorithm for Zernike polynomials
Davenport, J. W.
1982-01-01
The analysis of a function defined on a rotationally symmetric system, with either a circular or annular pupil is discussed. In order to numerically analyze such systems it is typical to expand the given function in terms of a class of orthogonal polynomials. Because of their particular properties, the Zernike polynomials are especially suited for numerical calculations. Developed is a recursive algorithm that can be used to generate the Zernike polynomials up to a given order. The algorithm is recursively defined over J where R(J,N) is the Zernike polynomial of degree N obtained by orthogonalizing the sequence R(J), R(J+2), ..., R(J+2N) over (epsilon, 1). The terms in the preceding row - the (J-1) row - up to the N+1 term is needed for generating the (J,N)th term. Thus, the algorith generates an upper left-triangular table. This algorithm was placed in the computer with the necessary support program also included.
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…
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.
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/.
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....
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....
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…
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
On Arithmetic-Geometric-Mean Polynomials
Griffiths, Martin; MacHale, Des
2017-01-01
We study here an aspect of an infinite set "P" of multivariate polynomials, the elements of which are associated with the arithmetic-geometric-mean inequality. In particular, we show in this article that there exist infinite subsets of probability "P" for which every element may be expressed as a finite sum of squares of real…
A note on Fibonacci-type polynomials
Amdeberhan, Tewodros
2008-01-01
We opt to study the convergence of maximal real roots of certain Fibonacci-type polynomials given by $G_n=x^kG_{n-1}+G_{n-2}$. The special cases $k=1$ and $k=2$ are found in [4] and [7], respectively.
The approach of moments for polynomial equations
M. Laurent (Monique); P. Rostalski
2012-01-01
htmlabstractIn this chapter 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
Dynamic system uncertainty propagation using polynomial chaos
Directory of Open Access Journals (Sweden)
Xiong Fenfen
2014-10-01
Full Text Available 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.
Modeling Power Amplifiers using Memory Polynomials
Kokkeler, Andre B.J.
2005-01-01
In this paper we present measured in- and output data of a power amplifier (PA). We compare this data with an AM-AM and AM-PM model. We conclude that a more sophisticated PA model is needed to cope with severe memory effects. We suggest to use memory polynomials and introduce two approaches to
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.
Orthogonality Relations for Multivariate Krawtchouk Polynomials
Directory of Open Access Journals (Sweden)
Hiroshi Mizukawa
2011-02-01
Full Text Available The orthogonality relations of multivariate Krawtchouk polynomials are discussed. In case of two variables, the necessary and sufficient conditions of orthogonality is given by Grünbaum and Rahman in [SIGMA 6 (2010, 090, 12 pages]. In this study, a simple proof of the necessary and sufficient condition of orthogonality is given for a general case.
The Tutte Polynomial of Some Matroids
Directory of Open Access Journals (Sweden)
Criel Merino
2012-01-01
graphs or matroids. In this work, we compile known formulas for the Tutte polynomial of some families of graphs and matroids. Also, we give brief explanations of the techniques that were used to find the formulas. Hopefully, this will be useful for researchers in Combinatorics and elsewhere.
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.
Inverse polynomial reconstruction method in DCT domain
Dadkhahi, Hamid; Gotchev, Atanas; Egiazarian, Karen
2012-12-01
The discrete cosine transform (DCT) offers superior energy compaction properties for a large class of functions and has been employed as a standard tool in many signal and image processing applications. However, it suffers from spurious behavior in the vicinity of edge discontinuities in piecewise smooth signals. To leverage the sparse representation provided by the DCT, in this article, we derive a framework for the inverse polynomial reconstruction in the DCT expansion. It yields the expansion of a piecewise smooth signal in terms of polynomial coefficients, obtained from the DCT representation of the same signal. Taking advantage of this framework, we show that it is feasible to recover piecewise smooth signals from a relatively small number of DCT coefficients with high accuracy. Furthermore, automatic methods based on minimum description length principle and cross-validation are devised to select the polynomial orders, as a requirement of the inverse polynomial reconstruction method in practical applications. The developed framework can considerably enhance the performance of the DCT in sparse representation of piecewise smooth signals. Numerical results show that denoising and image approximation algorithms based on the proposed framework indicate significant improvements over wavelet counterparts for this class of signals.
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.
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…
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....
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 ...
Polynomial Structure of Topological String Partition Functions
Zhou, Jie
2015-01-01
We review the polynomial structure of the topological string partition functions as solutions to the holomorphic anomaly equations. We also explain the connection between the ring of propagators defined from special K\\"ahler geometry and the ring of almost-holomorphic modular forms defined on modular curves.
Intrinsic Diophantine approximation on general polynomial surfaces
DEFF Research Database (Denmark)
Tiljeset, Morten Hein
2017-01-01
We study the Hausdorff measure and dimension of the set of intrinsically simultaneously -approximable points on a curve, surface, etc, given as a graph of integer polynomials. We obtain complete answers to these questions for algebraically “nice” manifolds. This generalizes earlier work done...
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.
Density of Real Zeros of the Tutte Polynomial
DEFF Research Database (Denmark)
Ok, Seongmin; Perrett, Thomas
2017-01-01
The Tutte polynomial of a graph is a two-variable polynomial whose zeros and evaluations encode many interesting properties of the graph. In this article we investigate the real zeros of the Tutte polynomials of graphs, and show that they form a dense subset of certain regions of the plane. This ....... This is the first density result for the real zeros of the Tutte polynomial in a region of positive volume. Our result almost confirms a conjecture of Jackson and Sokal except for one region which is related to an open problem on flow polynomials.......The Tutte polynomial of a graph is a two-variable polynomial whose zeros and evaluations encode many interesting properties of the graph. In this article we investigate the real zeros of the Tutte polynomials of graphs, and show that they form a dense subset of certain regions of the plane...
Polynomial Eigenvalue Solutions to Minimal Problems in Computer Vision.
Kukelova, Zuzana; Bujnak, Martin; Pajdla, Tomas
2012-07-01
We present a method for solving systems of polynomial equations appearing in computer vision. This method is based on polynomial eigenvalue solvers and is more straightforward and easier to implement than the state-of-the-art Gröbner basis method since eigenvalue problems are well studied, easy to understand, and efficient and robust algorithms for solving these problems are available. We provide a characterization of problems that can be efficiently solved as polynomial eigenvalue problems (PEPs) and present a resultant-based method for transforming a system of polynomial equations to a polynomial eigenvalue problem. We propose techniques that can be used to reduce the size of the computed polynomial eigenvalue problems. To show the applicability of the proposed polynomial eigenvalue method, we present the polynomial eigenvalue solutions to several important minimal relative pose problems.
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.
Khan, Waseem A; Haroon, Hiba
2016-01-01
In 2008, Liu and Wang established various symmetric identities for Bernoulli, Euler and Genocchi polynomials. In this paper, we extend these identities in a unified and generalized form to families of Hermite-Bernoulli, Euler and Genocchi polynomials. The procedure followed is that of generating functions. Some relevant connections of the general theory developed here with the results obtained earlier by Pathan and Khan are also pointed out.
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
Optical feedback structures and methods of making
Snee, Preston T; Chan, Yin Thai; Nocera, Daniel G; Bawendi, Moungi G
2014-11-18
An optical resonator can include an optical feedback structure disposed on a substrate, and a composite including a matrix including a chromophore. The composite disposed on the substrate and in optical communication with the optical feedback structure. The chromophore can be a semiconductor nanocrystal. The resonator can provide laser emission when excited.
A 'missing' family of classical orthogonal polynomials
Energy Technology Data Exchange (ETDEWEB)
Vinet, Luc [Centre de Recherches Mathematiques, Universite de Montreal, PO Box 6128, Centre-ville Station, Montreal, Quebec, H3C 3J7 (Canada); Zhedanov, Alexei, E-mail: zhedanov@fti.dn.ua [Donetsk Institute for Physics and Technology, Donetsk 83114 (Ukraine)
2011-02-25
We study a family of 'classical' orthogonal polynomials which satisfy (apart from a three-term recurrence relation) an eigenvalue problem with a differential operator of Dunkl type. These polynomials can be obtained from the little q-Jacobi polynomials in the limit q = -1. We also show that these polynomials provide a nontrivial realization of the Askey-Wilson algebra for q = -1.
q-Extensions for the Apostol Type Polynomials
Directory of Open Access Journals (Sweden)
Nazim I. Mahmudov
2014-01-01
Full Text Available The aim of this work is to introduce an extension for q-standard notations. The q-Apostol type polynomials and study some of their properties. Besides, some relations between the mentioned polynomials and some other known polynomials are obtained.
Symmetry identities for 2-variable Apostol type and related polynomials
2015-01-01
In this article, certain symmetry identities for the 2-variable Apostol type polynomials are derived. By taking suitable values of parameters and indices, the symmetry identities for the special cases of the 2-variable Apostol type polynomials are established. Further, the symmetry identities for certain members belonging to the 2-variable Apostol type polynomials are also considered.
The Roots of Adjoint Polynomial of the Graphs Contain Triangles
Institute of Scientific and Technical Information of China (English)
YECheng-fu
2004-01-01
We denote h(G,x) as the adjoint polynomial of graph G. In [5], Ma obtained the interpolation properties of the roots of adjoint polynomial of graphs containing triangles. By the properties, we prove the non-zero root of adjoint polynomial of Dn and Fn are single multiple.
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
Approximating Exponential and Logarithmic Functions Using Polynomial Interpolation
Gordon, Sheldon P.; Yang, Yajun
2017-01-01
This article takes a closer look at the problem of approximating the exponential and logarithmic functions using polynomials. Either as an alternative to or a precursor to Taylor polynomial approximations at the precalculus level, interpolating polynomials are considered. A measure of error is given and the behaviour of the error function is…
Heat polynomial analogs for higher order evolution equations
Directory of Open Access Journals (Sweden)
G. N. Hile
2001-05-01
Full Text Available Polynomial solutions analogous to the heat polynomials are demonstrated for higher order linear homogeneous evolution equations with coefficients depending on the time variable. Further parallels with the heat polynomials are established when the equation is parabolic with constant coefficients and only highest order terms.
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.
Exact Potts/Tutte polynomials for polygon chain graphs
Shrock, Robert
2011-04-01
We present exact calculations of Potts model partition functions and the equivalent Tutte polynomials for polygon chain graphs with open and cyclic boundary conditions. Special cases of the results that yield flow and reliability polynomials are discussed. We also analyze special cases of the Tutte polynomials that determine various quantities of graph-theoretic interest.
Tutte polynomial of a small-world Farey graph
Liao, Yunhua; Hou, Yaoping; Shen, Xiaoling
2013-11-01
In this paper, we find recursive formulas for the Tutte polynomials of a family of small-world Farey graphs, which are modular, and has an exponential degree hierarchy. As applications of the recursive formula, the exact expressions for the chromatic polynomial and the reliability polynomial of Fare graphs are derived and the number of connected spanning subgraphs is also obtained.
Approximating Exponential and Logarithmic Functions Using Polynomial Interpolation
Gordon, Sheldon P.; Yang, Yajun
2017-01-01
This article takes a closer look at the problem of approximating the exponential and logarithmic functions using polynomials. Either as an alternative to or a precursor to Taylor polynomial approximations at the precalculus level, interpolating polynomials are considered. A measure of error is given and the behaviour of the error function is…
T. Kim; Choi, J.; Kim, Y. H.; C. S. Ryoo
2010-01-01
In this paper, we give a fermionic p-adic integral representions of Benstein polynomials associated with Euler numbers and polynomials. Finally, we give some interesting identities for the Euler numbers by using the properties of our integral represention.
On Skew Triangular Matrix Rings
Institute of Scientific and Technical Information of China (English)
Wang Wei-liang; Wang Yao; Ren Yan-li
2016-01-01
Letαbe a nonzero endomorphism of a ring R, n be a positive integer and Tn(R,α) be the skew triangular matrix ring. We show that some properties related to nilpotent elements of R are inherited by Tn(R,α). Meanwhile, we determine the strongly prime radical, generalized prime radical and Behrens radical of the ring R[x;α]/(xn), where R[x;α] is the skew polynomial ring.
Discrimination Power of Polynomial-Based Descriptors for Graphs by Using Functional Matrices
Dehmer, Matthias; Emmert-Streib, Frank; Shi, Yongtang; Stefu, Monica; Tripathi, Shailesh
2015-01-01
In this paper, we study the discrimination power of graph measures that are based on graph-theoretical matrices. The paper generalizes the work of [M. Dehmer, M. Moosbrugger. Y. Shi, Encoding structural information uniquely with polynomial-based descriptors by employing the Randić matrix, Applied Mathematics and Computation, 268(2015), 164–168]. We demonstrate that by using the new functional matrix approach, exhaustively generated graphs can be discriminated more uniquely than shown in the mentioned previous work. PMID:26479495
Random Matrix Theory and Quantum Chromodynamics
Akemann, Gernot
2016-01-01
These notes are based on the lectures delivered at the Les Houches Summer School in July 2015. They are addressed at a mixed audience of physicists and mathematicians with some basic working knowledge of random matrix theory. The first part is devoted to the solution of the chiral Gaussian Unitary Ensemble in the presence of characteristic polynomials, using orthogonal polynomial techniques. This includes all eigenvalue density correlation functions, smallest eigenvalue distributions and their microscopic limit at the origin. These quantities are relevant for the description of the Dirac operator spectrum in Quantum Chromodynamics with three colours in four Euclidean space-time dimensions. In the second part these two theories are related based on symmetries, and the random matrix approximation is explained. In the last part recent developments are covered including the effect of finite chemical potential and finite space-time lattice spacing, and their corresponding orthogonal polynomials. We also give some ...
Random matrix model for disordered conductors
Indian Academy of Sciences (India)
Zafar Ahmed; Sudhir R Jain
2000-03-01
We present a random matrix ensemble where real, positive semi-deﬁnite matrix elements, , are log-normal distributed, $\\exp[-\\log^{2}(x)]$. We show that the level density varies with energy, , as 2/(1 + ) for large , in the unitary family, consistent with the expectation for disordered conductors. The two-level correlation function is studied for the unitary family and found to be largely of the universal form despite the fact that the level density has a non-compact support. The results are based on the method of orthogonal polynomials (the Stieltjes-Wigert polynomials here). An interesting random walk problem associated with the joint probability distribution of the ensuing ensemble is discussed and its connection with level dynamics is brought out. It is further proved that Dyson's Coulomb gas analogy breaks down whenever the conﬁning potential is given by a transcendental function for which there exist orthogonal polynomials.
On the Complex Zeros of Some Families of Orthogonal Polynomials
Directory of Open Access Journals (Sweden)
Eugenia N. Petropoulou
2010-01-01
Full Text Available The complex zeros of the orthogonal Laguerre polynomials (( for <−, ultraspherical polynomials (( for <−, Jacobi polynomials (,( for <−, <−, +<−2(+1, orthonormal Al-Salam-Carlitz II polynomials ((; for <0, 0<<1, and -Laguerre polynomials ((; for <−, 0<<1 are studied. Several inequalities regarding the real and imaginary properties of these zeros are given, which help locating their position. Moreover, a few limit relations regarding the asymptotic behavior of these zeros are proved. The method used is a functional analytic one. The obtained results complement and improve previously known results.
Modelling Trends in Ordered Correspondence Analysis Using Orthogonal Polynomials.
Lombardo, Rosaria; Beh, Eric J; Kroonenberg, Pieter M
2016-06-01
The core of the paper consists of the treatment of two special decompositions for correspondence analysis of two-way ordered contingency tables: the bivariate moment decomposition and the hybrid decomposition, both using orthogonal polynomials rather than the commonly used singular vectors. To this end, we will detail and explain the basic characteristics of a particular set of orthogonal polynomials, called Emerson polynomials. It is shown that such polynomials, when used as bases for the row and/or column spaces, can enhance the interpretations via linear, quadratic and higher-order moments of the ordered categories. To aid such interpretations, we propose a new type of graphical display-the polynomial biplot.
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.
Linear precoding based on polynomial expansion: reducing complexity in massive MIMO
Mueller, Axel
2016-02-29
Massive multiple-input multiple-output (MIMO) techniques have the potential to bring tremendous improvements in spectral efficiency to future communication systems. Counterintuitively, the practical issues of having uncertain channel knowledge, high propagation losses, and implementing optimal non-linear precoding are solved more or less automatically by enlarging system dimensions. However, the computational precoding complexity grows with the system dimensions. For example, the close-to-optimal and relatively “antenna-efficient” regularized zero-forcing (RZF) precoding is very complicated to implement in practice, since it requires fast inversions of large matrices in every coherence period. Motivated by the high performance of RZF, we propose to replace the matrix inversion and multiplication by a truncated polynomial expansion (TPE), thereby obtaining the new TPE precoding scheme which is more suitable for real-time hardware implementation and significantly reduces the delay to the first transmitted symbol. The degree of the matrix polynomial can be adapted to the available hardware resources and enables smooth transition between simple maximum ratio transmission and more advanced RZF. By deriving new random matrix results, we obtain a deterministic expression for the asymptotic signal-to-interference-and-noise ratio (SINR) achieved by TPE precoding in massive MIMO systems. Furthermore, we provide a closed-form expression for the polynomial coefficients that maximizes this SINR. To maintain a fixed per-user rate loss as compared to RZF, the polynomial degree does not need to scale with the system, but it should be increased with the quality of the channel knowledge and the signal-to-noise ratio.
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.
Parabolic refined invariants and Macdonald polynomials
Chuang, Wu-yen; Donagi, Ron; Pantev, Tony
2013-01-01
A string theoretic derivation is given for the conjecture of Hausel, Letellier, and Rodriguez-Villegas on the cohomology of character varieties with marked points. Their formula is identified with a refined BPS expansion in the stable pair theory of a local root stack, generalizing previous work of the first two authors in collaboration with G. Pan. Haiman's geometric construction for Macdonald polynomials is shown to emerge naturally in this context via geometric engineering. In particular this yields a new conjectural relation between Macdonald polynomials and refined local orbifold curve counting invariants. The string theoretic approach also leads to a new spectral cover construction for parabolic Higgs bundles in terms of holomorphic symplectic orbifolds.
Polynomial threshold functions and Boolean threshold circuits
DEFF Research Database (Denmark)
Hansen, Kristoffer Arnsfelt; Podolskii, Vladimir V.
2013-01-01
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...... bound for PTFs that holds regardless of degree, thereby extending known lower bounds for THRMAJ circuits. We generalize two-party unbounded error communication complexity to the multi-party number-on-the-forehead setting, and show that communication lower bounds for 3-player protocols would yield size...... lower bounds for THRTHR circuits. We obtain several other results about PTFs. These include relationships between weight and degree of PTFs, and a degree lower bound for PTFs of constant length. We also consider a variant of PTFs over the max-plus algebra. We show that they are connected to PTFs over...
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....
Quantum Hurwitz numbers and Macdonald polynomials
Harnad, J
2015-01-01
Parametric families in the centre ${\\bf Z}({\\bf C}[S_n])$ of the group algebra of the symmetric group are obtained by identifying the indeterminates in the generating function for Macdonald polynomials as commuting Jucys-Murphy elements. Their eigenvalues provide coefficients in the double Schur function expansion of 2D Toda $\\tau$-functions of hypergeometric type. Expressing these in the basis of products of power sum symmetric functions, the coefficients may be interpreted geometrically as parametric families of quantum Hurwitz numbers, enumerating weighted branched coverings of the Riemann sphere. Combinatorially, they give quantum weighted sums over paths in the Cayley graph of $S_n$ generated by transpositions. Dual pairs of bases for the algebra of symmetric functions with respect to the scalar product in which the Macdonald polynomials are orthogonal provide both the geometrical and combinatorial significance of these quantum weighted enumerative invariants.
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.
Characteristic polynomials in real Ginibre ensembles
Energy Technology Data Exchange (ETDEWEB)
Akemann, G; Phillips, M J [Department of Mathematical Sciences and BURSt Research Centre, Brunel University West London, UB8 3PH Uxbridge (United Kingdom); Sommers, H-J [Fachbereich Physik, Universitaet Duisburg-Essen, 47048 Duisburg (Germany)], E-mail: Gernot.Akemann@brunel.ac.uk, E-mail: Michael.Phillips@brunel.ac.uk, E-mail: H.J.Sommers@uni-due.de
2009-01-09
We calculate the average of two characteristic polynomials for the real Ginibre ensemble of asymmetric random matrices, and its chiral counterpart. Considered as quadratic forms they determine a skew-symmetric kernel from which all complex eigenvalue correlations can be derived. Our results are obtained in a very simple fashion without going to an eigenvalue representation, and are completely new in the chiral case. They hold for Gaussian ensembles which are partly symmetric, with kernels given in terms of Hermite and Laguerre polynomials respectively, depending on an asymmetry parameter. This allows us to interpolate between the maximally asymmetric real Ginibre and the Gaussian orthogonal ensemble, as well as their chiral counterparts. (fast track communication)
Polynomial chaos representation of databases on manifolds
Energy Technology Data Exchange (ETDEWEB)
Soize, C., E-mail: christian.soize@univ-paris-est.fr [Université Paris-Est, Laboratoire Modélisation et Simulation Multi-Echelle, MSME UMR 8208 CNRS, 5 bd Descartes, 77454 Marne-La-Vallée, Cedex 2 (France); Ghanem, R., E-mail: ghanem@usc.edu [University of Southern California, 210 KAP Hall, Los Angeles, CA 90089 (United States)
2017-04-15
Characterizing the polynomial chaos expansion (PCE) of a vector-valued random variable with probability distribution concentrated on a manifold is a relevant problem in data-driven settings. The probability distribution of such random vectors is multimodal in general, leading to potentially very slow convergence of the PCE. In this paper, we build on a recent development for estimating and sampling from probabilities concentrated on a diffusion manifold. The proposed methodology constructs a PCE of the random vector together with an associated generator that samples from the target probability distribution which is estimated from data concentrated in the neighborhood of the manifold. The method is robust and remains efficient for high dimension and large datasets. The resulting polynomial chaos construction on manifolds permits the adaptation of many uncertainty quantification and statistical tools to emerging questions motivated by data-driven queries.
Moments, positive polynomials and their applications
Lasserre, Jean Bernard
2009-01-01
Many important applications in global optimization, algebra, probability and statistics, applied mathematics, control theory, financial mathematics, inverse problems, etc. can be modeled as a particular instance of the Generalized Moment Problem (GMP) . This book introduces a new general methodology to solve the GMP when its data are polynomials and basic semi-algebraic sets. This methodology combines semidefinite programming with recent results from real algebraic geometry to provide a hierarchy of semidefinite relaxations converging to the desired optimal value. Applied on appropriate cones,
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.
Piecewise polynomial solutions to linear inverse problems
DEFF Research Database (Denmark)
Hansen, Per Christian; Mosegaard, K.
1996-01-01
We have presented a new algorithm PP-TSVD that computes piecewise polynomial solutions to ill-posed problems, without a priori knowledge about the positions of the break points. In particular, we can compute piecewise constant functions that describe layered models. Such solutions are useful, e.g.......g., in seismological problems, and the algorithm can also be used as a preprocessor for other methods where break points/discontinuities must be incorporated explicitly....
Sparse DOA estimation with polynomial rooting
Xenaki, Angeliki; Gerstoft, Peter; Fernandez Grande, Efren
2015-01-01
Direction-of-arrival (DOA) estimation involves the localization of a few sources from a limited number of observations on an array of sensors. Thus, DOA estimation can be formulated as a sparse signal reconstruction problem and solved efficiently with compressive sensing (CS) to achieve highresolution imaging. Utilizing the dual optimal variables of the CS optimization problem, it is shown with Monte Carlo simulations that the DOAs are accurately reconstructed through polynomial rooting (Root...
Time-reversal symmetry and random polynomials
Braun, D; Kus, M.; Zyczkowski, K.
1996-01-01
We analyze the density of roots of random polynomials where each complex coefficient is constructed of a random modulus and a fixed, deterministic phase. The density of roots is shown to possess a singular component only in the case for which the phases increase linearly with the index of coefficients. This means that, contrary to earlier belief, eigenvectors of a typical quantum chaotic system with some antiunitary symmetry will not display a clustering curve in the stellar representation. M...
Georeferencing CAMS data: Polynomial rectification and beyond
Yang, Xinghe
The Calibrated Airborne Multispectral Scanner (CAMS) is a sensor used in the commercial remote sensing program at NASA Stennis Space Center. In geographic applications of the CAMS data, accurate geometric rectification is essential for the analysis of the remotely sensed data and for the integration of the data into Geographic Information Systems (GIS). The commonly used rectification techniques such as the polynomial transformation and ortho rectification have been very successful in the field of remote sensing and GIS for most remote sensing data such as Landsat imagery, SPOT imagery and aerial photos. However, due to the geometric nature of the airborne line scanner which has high spatial frequency distortions, the polynomial model and the ortho rectification technique in current commercial software packages such as Erdas Imagine are not adequate for obtaining sufficient geometric accuracy. In this research, the geometric nature, especially the major distortions, of the CAMS data has been described. An analytical step-by-step geometric preprocessing has been utilized to deal with the potential high frequency distortions of the CAMS data. A generic sensor-independent photogrammetric model has been developed for the ortho-rectification of the CAMS data. Three generalized kernel classes and directional elliptical basis have been formulated into a rectification model of summation of multisurface functions, which is a significant extension to the traditional radial basis functions. The preprocessing mechanism has been fully incorporated into the polynomial, the triangle-based finite element analysis as well as the summation of multisurface functions. While the multisurface functions and the finite element analysis have the characteristics of localization, piecewise logic has been applied to the polynomial and photogrammetric methods, which can produce significant accuracy improvement over the global approach. A software module has been implemented with full
Linear algebra for skew-polynomial matrices
Abramov, Sergei; Bronstein, Manuel
2002-01-01
We describe an algorithm for transforming skew-polynomial matrices over an Ore domain in row-reduced form, and show that this algorithm can be used to perform the standard calculations of linear algebra on such matrices (ranks, kernels, linear dependences, inhomogeneous solving). The main application of our algorithm is to desingularize recurrences and to compute the rational solutions of a large class of linear functional systems. It also turns out to be efficient when applied to ordinary co...
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.
Real meromorphic functions and linear differential polynomials
Institute of Scientific and Technical Information of China (English)
LANGLEY; J.; K.
2010-01-01
We determine all real meromorphic functions f in the plane such that f has finitely many zeros, the poles of f have bounded multiplicities, and f and F have finitely many non-real zeros, where F is a linear differential polynomial given by F = f (k) +Σk-1j=0ajf(j) , in which k≥2 and the coefficients aj are real numbers with a0≠0.
On form factors and Macdonald polynomials
Lashkevich, Michael
2013-01-01
We are developing the algebraic construction for form factors of local operators in the sinh-Gordon theory proposed in [B.Feigin, M.Lashkeivch, 2008]. We show that the operators corresponding to the null vectors in this construction are given by the degenerate Macdonald polynomials with rectangular partitions and the parameters $t=-q$ on the unit circle. We obtain an integral representation for the null vectors and discuss its simple applications.
Normality and shared values concerning differential polynomials
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
Let F be a family of functions meromorphic in a domain D, let P be a polynomial with either deg P≥3 or deg P = 2 and P having only one distinct zero, and let b be a finite nonzero complex number. If, each pair of functions f and g in F, P (f)f and P (g)g share b in D, then F is normal in D.
Completeness of the ring of polynomials
DEFF Research Database (Denmark)
Thorup, Anders
2015-01-01
Consider the polynomial ring R:=k[X1,…,Xn]R:=k[X1,…,Xn] in n≥2n≥2 variables over an uncountable field k. We prove that R is complete in its adic topology, that is, the translation invariant topology in which the non-zero ideals form a fundamental system of neighborhoods of 0. In addition we pro...
Some Orthogonal Polynomials in Four Variables
Directory of Open Access Journals (Sweden)
Charles F. Dunkl
2008-11-01
Full Text Available The symmetric group on 4 letters has the reflection group $D_3$ as an isomorphic image. This fact follows from the coincidence of the root systems $A_3$ and $D_3$. The isomorphism is used to construct an orthogonal basis of polynomials of 4 variables with 2 parameters. There is an associated quantum Calogero-Sutherland model of 4 identical particles on the line.
Algebraic polynomials and moments of stochastic integrals
Langovoy, Mikhail A
2011-01-01
We propose an algebraic method for proving estimates on moments of stochastic integrals. The method uses qualitative properties of roots of algebraic polynomials from certain general classes. As an application, we give a new proof of a variation of the Burkholder-Davis-Gundy inequality for the case of stochastic integrals with respect to real locally square integrable martingales. Further possible applications and extensions of the method are outlined.
Periodicity in Delta-modulated feedback control
Institute of Scientific and Technical Information of China (English)
Xiaohua XIA; Guanrong CHEN; Rudong GAI; Alan S. I. ZINOBER
2008-01-01
The Delta-modulated feedback control of a linear system introduces nonlinearity into the system through switchings between two input values. It has been found that Delta-modulation gives rise to periodic orbits. The existence of periodic points of all orders of Sigma-Delta modulation with "leaky" integration is completely characterized by some interesting groups of polynomials with "sign" coefficients. The results are naturally generalized to Sigma-Delta modulations with multiple delays, Delta-modulations in the "downlink", unbalanced Delta-modulations and systems with two-level quantized feedback. Further extensions relate to the existence of periodic points arising from Delta-modulated feedback control of a stable linear system in an arbitrary direction, for which some necessary and sufficient conditions are given.
A dynamic coefficient polynomial predistorter based on direct learning architecture
Institute of Scientific and Technical Information of China (English)
Li Bo; Ge Jianhua; Ai Bo
2008-01-01
A dynamic coefficient polynomial predistorter based on direct learning architecture is proposed. Compared to the existing polynomial predistorter, on the one hand, the proposed predistorter based on the direct learning architecture is more robust to initial conditions of the tap coefficients than that based on indirect learning architecture; on the other hand, by using two polynomial coefficient combinations, different polynomial coefficient combination can be selected when the input signal amplitude changes, which effectively decreases the estimate error. This paper introduces the direct learning architecture and gives the dynamic coefficient polynomial expression. A simplified nonlinear recursive least-squares (RLS) algorithm for polynomial coefficient estimation is also derived in detail. Computer simulations show that the proposed predistorter can attain 31dB, 28dB and 40dB spectrum suppression gain when our method is applied to the traveling wave tube amplifier (TWTA), solid state power amplifier (SSPA) and polynomial power amplifier (PA) model, respectively.
The bivariate Rogers Szegö polynomials
Chen, William Y. C.; Saad, Husam L.; Sun, Lisa H.
2007-06-01
We present an operator approach to deriving Mehler's formula and the Rogers formula for the bivariate Rogers-Szegö polynomials hn(x, y|q). The proof of Mehler's formula can be considered as a new approach to the nonsymmetric Poisson kernel formula for the continuous big q-Hermite polynomials Hn(x; a|q) due to Askey, Rahman and Suslov. Mehler's formula for hn(x, y|q) involves a 3phi2 sum and the Rogers formula involves a 2phi1 sum. The proofs of these results are based on parameter augmentation with respect to the q-exponential operator and the homogeneous q-shift operator in two variables. By extending recent results on the Rogers-Szegö polynomials hn(x|q) due to Hou, Lascoux and Mu, we obtain another Rogers-type formula for hn(x, y|q). Finally, we give a change of base formula for Hn(x; a|q) which can be used to evaluate some integrals by using the Askey-Wilson integral.
The bivariate Rogers-Szegoe polynomials
Energy Technology Data Exchange (ETDEWEB)
Chen, William Y C [Center for Combinatorics, LPMC, Nankai University, Tianjin 300071 (China); Saad, Husam L [Center for Combinatorics, LPMC, Nankai University, Tianjin 300071 (China); Sun, Lisa H [Center for Combinatorics, LPMC, Nankai University, Tianjin 300071 (China)
2007-06-08
We present an operator approach to deriving Mehler's formula and the Rogers formula for the bivariate Rogers-Szegoe polynomials h{sub n}(x, y vertical bar q). The proof of Mehler's formula can be considered as a new approach to the nonsymmetric Poisson kernel formula for the continuous big q-Hermite polynomials H{sub n}(x; a vertical bar q) due to Askey, Rahman and Suslov. Mehler's formula for h{sub n}(x, y vertical bar q) involves a {sub 3}{phi}{sub 2} sum and the Rogers formula involves a {sub 2}{phi}{sub 1} sum. The proofs of these results are based on parameter augmentation with respect to the q-exponential operator and the homogeneous q-shift operator in two variables. By extending recent results on the Rogers-Szegoe polynomials h{sub n}(x vertical bar q) due to Hou, Lascoux and Mu, we obtain another Rogers-type formula for h{sub n}(x, y vertical bar q). Finally, we give a change of base formula for H{sub n}(x; a vertical bar q) which can be used to evaluate some integrals by using the Askey-Wilson integral.
Maximal aggregation of polynomial dynamical systems.
Cardelli, Luca; Tribastone, Mirco; Tschaikowski, Max; Vandin, Andrea
2017-09-19
Ordinary differential equations (ODEs) with polynomial derivatives are a fundamental tool for understanding the dynamics of systems across many branches of science, but our ability to gain mechanistic insight and effectively conduct numerical evaluations is critically hindered when dealing with large models. Here we propose an aggregation technique that rests on two notions of equivalence relating ODE variables whenever they have the same solution (backward criterion) or if a self-consistent system can be written for describing the evolution of sums of variables in the same equivalence class (forward criterion). A key feature of our proposal is to encode a polynomial ODE system into a finitary structure akin to a formal chemical reaction network. This enables the development of a discrete algorithm to efficiently compute the largest equivalence, building on approaches rooted in computer science to minimize basic models of computation through iterative partition refinements. The physical interpretability of the aggregation is shown on polynomial ODE systems for biochemical reaction networks, gene regulatory networks, and evolutionary game theory.
Nested Canalyzing, Unate Cascade, and Polynomial Functions.
Jarrah, Abdul Salam; Raposa, Blessilda; Laubenbacher, Reinhard
2007-09-15
This paper focuses on the study of certain classes of Boolean functions that have appeared in several different contexts. Nested canalyzing functions have been studied recently in the context of Boolean network models of gene regulatory networks. In the same context, polynomial functions over finite fields have been used to develop network inference methods for gene regulatory networks. Finally, unate cascade functions have been studied in the design of logic circuits and binary decision diagrams. This paper shows that the class of nested canalyzing functions is equal to that of unate cascade functions. Furthermore, it provides a description of nested canalyzing functions as a certain type of Boolean polynomial function. Using the polynomial framework one can show that the class of nested canalyzing functions, or, equivalently, the class of unate cascade functions, forms an algebraic variety which makes their analysis amenable to the use of techniques from algebraic geometry and computational algebra. As a corollary of the functional equivalence derived here, a formula in the literature for the number of unate cascade functions provides such a formula for the number of nested canalyzing functions.
Simplified Storm Surge Simulations Using Bernstein Polynomials
Beisiegel, Nicole; Behrens, Jörn
2016-04-01
Storm surge simulations are vital for forecasting, hazard assessment and eventually improving our understanding of Earth system processes. Discontinuous Galerkin (DG) methods have recently been explored in that context, because they are locally mass-conservative and in combination with suitable robust nodal filtering techniques (slope limiters) positivity-preserving and well-balanced for the still water state at rest. These filters manipulate interpolation point values in every time step in order to retain the desirable properties of the scheme. In particular, DG methods are able to represent prognostic variables such as the fluid height at high-order accuracy inside each element (triangle). For simulations that include wetting and drying, however, the high-order accuracy will destabilize the numerical model because point values on quadrature points may become negative during the computation if they do not coincide with interpolation points. This is why the model that we are presenting utilizes Bernstein polynomials as basis functions to model the wetting and drying. This has the advantage that negative pointvalues away from interpolation points are prevented, the model is stabilized and no additional time step restriction is introduced. Numerical tests show that the model is capable of simulating simplified storm surges. Furthermore, a comparison of model results with third-order Bernstein polynomials with results using traditional nodal Lagrange polynomials reveals an improvement in numerical convergence.
Deterministic Polynomial Factoring and Association Schemes
Arora, Manuel; Karpinski, Marek; Saxena, Nitin
2012-01-01
The problem of finding a nontrivial factor of a polynomial f(x) over a finite field F_q has many known efficient, but randomized, algorithms. The deterministic complexity of this problem is a famous open question even assuming the generalized Riemann hypothesis (GRH). In this work we improve the state of the art by focusing on prime degree polynomials; let n be the degree. If (n-1) has a `large' r-smooth divisor s, then we find a nontrivial factor of f(x) in deterministic poly(n^r,log q) time; assuming GRH and that s > sqrt{n/(2^r)}. Thus, for r = O(1) our algorithm is polynomial time. Further, for r > loglog n there are infinitely many prime degrees n for which our algorithm is applicable and better than the best known; assuming GRH. Our methods build on the algebraic-combinatorial framework of m-schemes initiated by Ivanyos, Karpinski and Saxena (ISSAC 2009). We show that the m-scheme on n points, implicitly appearing in our factoring algorithm, has an exceptional structure; leading us to the improved time ...
Polynomiality of Hurwitz numbers, Bouchard-Mari\\~no conjecture, and a new proof of the ELSV formula
Dunin-Barkowski, Petr; Orantin, Nicolas; Shadrin, Sergey; Spitz, Loek
2013-01-01
In this paper we give a new proof of the ELSV formula. First, we refine an argument of Okounkov and Pandharipande in order to prove (quasi-)polynomiality of Hurwitz numbers without using the ELSV formula (the only way to do that before used the ELSV formula). Then, using this polynomiality we give a new prove of the Bouchard-Mari\\~no conjecture. After that, using the correspondence between the Givental group action and the topological recursion coming from matrix models, we prove the equivalence of the Bouchard-Mari\\~no conjecture and the ELSV formula (it is a refinement of an argument by Eynard).
Directory of Open Access Journals (Sweden)
Michael Basin
2011-04-01
Full Text Available In this paper, the mean-square filtering problem for polynomial system states confused with white Poisson noises over polynomial observations is studied proceeding from the general expression for the stochastic Ito differentials of the mean-square estimate and the error variance. In contrast to the previously obtained results, the paper deals with the general case of nonlinear polynomial states and observations with white Poisson noises. As a result, the Ito differentials for the mean-square estimate and error variance corresponding to the stated filtering problem are first derived. The procedure for obtaining an approximate closed-form finite-dimensional system of the filtering equations for any polynomial state over observations with any polynomial drift is then established. In the example, the obtained closed-form filter is applied to solve the third order sensor filtering problem for a quadratic state, assuming a conditionally Poisson initial condition for the extended third order state vector. The simulation results show that the designed filter yields a reliable and rapidly converging estimate.
Plane geometry and convexity of polynomial stability regions
Henrion, Didier
2008-01-01
The set of controllers stabilizing a linear system is generally non-convex in the parameter space. In the case of two-parameter controller design (e.g. PI control or static output feedback with one input and two outputs), we observe however that quite often for benchmark problem instances, the set of stabilizing controllers seems to be convex. In this note we use elementary techniques from real algebraic geometry (resultants and Bezoutian matrices) to explain this phenomenon. As a byproduct, we derive a convex linear matrix inequality (LMI) formulation of two-parameter fixed-order controller design problem, when possible.
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.
Higher-order numerical methods derived from three-point polynomial interpolation
Rubin, S. G.; Khosla, P. K.
1976-01-01
Higher-order collocation procedures resulting in tridiagonal matrix systems are derived from polynomial spline interpolation and Hermitian finite-difference discretization. The equations generally apply for both uniform and variable meshes. Hybrid schemes resulting from different polynomial approximations for first and second derivatives lead to the nonuniform mesh extension of the so-called compact or Pade difference techniques. A variety of fourth-order methods are described and this concept is extended to sixth-order. Solutions with these procedures are presented for the similar and non-similar boundary layer equations with and without mass transfer, the Burgers equation, and the incompressible viscous flow in a driven cavity. Finally, the interpolation procedure is used to derive higher-order temporal integration schemes and results are shown for the diffusion equation.
A Polynomial Time Algorithm for a Special Case of Linear Integer Programming
Ghasemiesfeh, Golnaz; Tabesh, Yahya
2011-01-01
According to the wide use of integer programming in many fields, affords toward finding and solving sub classes of these problems which are solvable in polynomial time seems to be important and useful. Integer linear programming (ILP) problems have the general form: $Min \\{C^{T}x: Ax=b, x\\geq 0, x\\in Z^{n}\\}$ where $Z^{n}$ is the set of n-dimensional integer vectors. Algorithmic solution of ILP is at great interest, in this paper we have presented a polynomial algorithm for a special case of the ILP problems; we have used a graph theoretical formulation of the problem which leads to an $O[mn(m+n)]$ solution where $m$ and $n$ are dimensions of coefficient matrix $X$.
HOMFLY polynomials in representation [3,1] for 3-strand braids
Mironov, A; Morozov, An; Sleptsov, A
2016-01-01
This paper is a new step in the project of systematic description of colored knot polynomials started in arXiv:1506.00339. In this paper, we managed to explicitly find the inclusive Racah matrix, i.e. the whole set of mixing matrices in channels R^3->Q with all possible Q, for R=[3,1]. The calculation is made possible by the use of a newly-developed efficient highest-weight method, still it remains tedious. The result allows one to evaluate and investigate [3,1]-colored polynomials for arbitrary 3-strand knots, and this confirms many previous conjectures on various factorizations, universality, and differential expansions. We consider in some detail the next-to-twist-knots three-strand family (n,-1|1,-1) and deduce its colored HOMFLY. Also confirmed and clarified is the eigenvalue hypothesis for the Racah matrices, which promises to provide a shortcut to generic formulas for arbitrary representations.
The Lie Algebraic Structure of Differential Operators Admitting Invariant Spaces of Polynomials
Finkel, F; Finkel, Federico; Kamran, Niky
1996-01-01
We prove that the scalar and $2\\times 2$ matrix differential operators which preserve the simplest scalar and vector-valued polynomial modules in two variables have a fundamental Lie algebraic structure. Our approach is based on a general graphical method which does not require the modules to be irreducible under the action of the corresponding Lie (super)algebra. This method can be generalized to modules of polynomials in an arbitrary number of variables. We give generic examples of partially solvable differential operators which are not Lie algebraic. We show that certain vector-valued modules give rise to new realizations of finite-dimensional Lie superalgebras by first-order differential operators.
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.
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, 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.
A Matricial Algorithm for Polynomial Refinement
King, Emily J
2011-01-01
In order to have a multiresolution analysis, the scaling function must be refinable. That is, it must be the linear combination of 2-dilation, $\\mathbb{Z}$-translates of itself. Refinable functions used in connection with wavelets are typically compactly supported. In 2002, David Larson posed the question, "Are all polynomials (of a single variable) finitely refinable?" That summer the author proved that the answer indeed was true using basic linear algebra. The result was presented in a number of talks but had not been typed up until now. The purpose of this short note is to record that particular proof.
Time-reversal symmetry and random polynomials
Braun, D; Zyczkowski, K
1996-01-01
We analyze the density of roots of random polynomials where each complex coefficient is constructed of a random modulus and a fixed, deterministic phase. The density of roots is shown to possess a singular component only in the case for which the phases increase linearly with the index of coefficients. This means that, contrary to earlier belief, eigenvectors of a typical quantum chaotic system with some antiunitary symmetry will not display a clustering curve in the stellar representation. Moreover, a class of time-reverse invariant quantum systems is shown, for which spectra display fluctuations characteristic of orthogonal ensemble, while eigenvectors confer to predictions of unitary ensemble.