Case Of Polynomials Research Articles

Dual combinatorics of zonal polynomials

In this paper we establish a new combinatorial formula for zonal polynomials in terms of power-sums. The proof relies on the sign-reversing involution principle. We deduce from it formulas for zonal characters, which are defined as suitably normalized coefficients in the expansion of zonal polynomials in terms of power-sum symmetric functions. These formulas are analogs of recent developments on irreducible character values of symmetric groups. The existence of such formulas could have been predicted from the work of M. Lassalle who formulated two positivity conjectures for Jack characters, which we prove in the special case of zonal polynomials. Dans cet article, nous établissons une nouvelle formule combinatoire pour les polynômes zonaux en fonction des fonctions puissance. La preuve utilise le principe de l'involution changeant les signes. Nous en déduisons des formules pour les caractères zonaux, qui sont définis comme les coefficients des polynômes zonaux écrits sur la base des fonctions puissance, normalisés de manière appropriée. Ces formules sont des analogues de développements récents sur les caractères du groupe symétrique. L'existence de telles formules aurait pu être prédite à partir des travaux de M. Lassalle, qui a proposé deux conjectures de positivité sur les caractères de Jack, que nous prouvons dans le cas particulier des polynômes zonaux.

Robust Stability via Sign-Definite Decomposition

This paper considers the problem of robust stability of a polynomial family whose coefficients are polynomial functions of the parameters of interests. The problem occurs in the design of a fixed order or fixed structure multivariable feedback controller , parametrized by a real design parameter vector , for a plant , containing a vector of uncertain parameters. The characteristic polynomials of such systems often contain coefficients which depend polynomially on and . Using results on sign-definite decomposition, a new stability test is developed that gives a sufficient condition for Hurwitz stability of the family of closed loop systems that result when and vary over prescribed boxes. This test is reminiscent of Kharitonov's Theorem, even though the family of polynomials considered here is certainly not restricted to be interval or even convex. Moreover the test does reduce to Kharitonov's Theorem for the special case of interval polynomials. Using this criterion recursively and modularly, sets of controllers that stabilize the family of uncertain plants are determined.

Noether forms for the study of non-composite rational functions and their spectrum

In this paper, the spectrum and the decomposability of a multivariate rational function are studied by means of the effective Noether's irreducibility theorem given by Ruppert. With this approach, some new effective results are obtained. In particular, we show that the reduction modulo p of the spectrum of a given integer multivariate rational function r coincides with the spectrum of the reduction of r modulo p for p a prime integer greater or equal to an explicit bound. This bound is given in terms of the degree, the height and the number of variables of r. With the same strategy, we also study the decomposability of r modulo p. Some similar explicit results are also provided for the case of polynomials with coefficients in a polynomial ring.

High Accuracy Calculation of GaussWeights for Optimization Method by a Jacobi Pseudospectral Method

This paper presents modification of the Gauss weights accuracy for an optimal control solver using a Jacobipseudospectral method. For the purpose of computing the integral cost term, the accurate Gauss weights at the Jacobi- Gauss-Lobatto points are required. In the case of the specific polynomial approximation, such as the Legendre-Gauss-Lobatto points and the Chebyshev-Gauss-Lobatto points, using their analytic algorithms are sufficient. The method based upon the Vandermonde matrix is suitable for the low-order approximation calculation in the general polynomial case. Due to singularity of the Vandermonde matrix, this method cannot be used in the high-order approximation case. In such case, the modified method which is based upon the High-order Gauss-Lobatto formulae yields high accuracy weights value. Numerical examples demonstrate that this idea yields accurate results.

Ergodic averages of commuting transformations with distinct degree polynomial iterates

We prove mean convergence, as $N\to\infty$, for the multiple ergodic averages $\frac{1}{N}\sum_{n=1}^N f_1(T_1^{p_1(n)}x)... f_\ell(T_\ell^{p_\ell(n)}x)$, where $p_1,...,p_\ell$ are integer polynomials with distinct degrees, and $T_1,...,T_\ell$ are commuting, invertible measure preserving transformations, acting on the same probability space. This establishes several cases of a conjecture of Bergelson and Leibman, that complement the case of linear polynomials, recently established by Tao. Furthermore, we show that, unlike the case of linear polynomials, for polynomials of distinct degrees, the corresponding characteristic factors are mixtures of inverse limits of nilsystems. We use this particular structure, together with some equidistribution results on nilmanifolds, to give an application to multiple recurrence and a corresponding one to combinatorics.

Spreading of Gas from a Line Source in the Flow over Two‐dimensional Hill

AbstractFlow around the two‐dimensional hill is of great interest in engineering application, namely in transport and dispersion of pollutants in atmosphere. A case of polynomial shaped 2D hill with the line source of tracer gas has been studied in detail. Experimental study of both, the time‐averaged velocity field and the binary‐mixture concentration field has been done. (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Some algebraic methods for solving multiobjective polynomial integer programs

Multiobjective discrete programming is a well-known family of optimization problems with a large spectrum of applications. The linear case has been tackled by many authors during the past few years. However, the polynomial case has not been studied in detail due to its theoretical and computational difficulties. This paper presents an algebraic approach for solving these problems. We propose a methodology based on transforming the polynomial optimization problem to the problem of solving one or more systems of polynomial equations and we use certain Gröbner bases to solve these systems. Different transformations give different methodologies that are theoretically stated and compared by some computational tests via the algorithms that they induce.

Hermitian quadratic matrix polynomials: Solvents and inverse problems

A monic quadratic Hermitian matrix polynomial L(λ) can be factorized into a product of two linear matrix polynomials, say L(λ)=(Iλ-S)(Iλ-A). For the inverse problem of finding a quadratic matrix polynomial with prescribed spectral data (eigenvalues and eigenvectors) it is natural to prescribe a right solvent A and then determine compatible left solvents S. This problem is explored in the present paper. The splitting of the spectrum between real eigenvalues and nonreal conjugate pairs plays an important role. Special attention is paid to the case of real-symmetric quadratic polynomials and the allocation of the canonical sign characteristics as well as the eigenvalues themselves.

Zero location and asymptotic behavior for extremal polynomials with non-diagonal Sobolev norms

In this paper we are going to study the zero location and asymptotic behavior of extremal polynomials with respect to a non-diagonal Sobolev norm in the worst case, i.e., when the quadratic form is allowed to degenerate. The orthogonal polynomials with respect to this Sobolev norm are a particular case of those extremal polynomials. The multiplication operator by the independent variable is the main tool in order to obtain our results.

Two-stage stochastic matching and spanning tree problems: Polynomial instances and approximation

This article deals with the two-stage stochastic model, which aims at explicitly taking into account uncertainty in optimization problems, that Kong and Schaefer have recently studied for the maximum weight matching problem [N. Kong, A.J. Schaefer, A factor 1/2 approximation algorithm for two-stage stochastic matching problems, European Journal of Operational Research 172(3) (2006) 740–746]. They have proved that the problem is NP-hard, and they have provided a factor 1 2 approximation algorithm. We further study this problem and strengthen the hardness results, slightly improve the approximation ratio and exhibit some polynomial cases. We similarly tackle the maximum weight spanning tree problem in the two-stage setting. Finally, we make numerical experiments on randomly generated instances to compare the quality of several interesting heuristics.

The Smallest Eigenvalue of Hankel Matrices

Let H_N=(s_{n+m}),n,m\le N denote the Hankel matrix of moments of a positive measure with moments of any order. We study the large N behaviour of the smallest eigenvalue lambda_N of H_N. It is proved that lambda_N has exponential decay to zero for any measure with compact support. For general determinate moment problems the decay to 0 of lambda_N can be arbitrarily slow or arbitrarily fast. In the indeterminate case, where lambda_N is known to be bounded below by a positive constant, we prove that the limit of the n'th smallest eigenvalue of H_N for N tending to infinity tends rapidly to infinity with n. The special case of the Stieltjes-Wigert polynomials is discussed.

Growth properties of Nevanlinna matrices for rational moment problems

We consider rational moment problems on the real line with their associated orthogonal rational functions. There exists a Nevanlinna-type parameterization relating to the problem, with associated Nevanlinna matrices of functions having singularities in the closure of the set of poles of the rational functions belonging to the problem. We prove results related to the growth at the singularities of the functions in a Nevanlinna matrix, and in particular provide bounds on the growth analogous to the corresponding result in the classical polynomial case, when the number of singularities is finite.

Igusa's Conjecture on Exponential Sums Modulo p and p2 and the Motivic Oscillation Index

We prove the modulo p and modulo p2 cases of Igusa's conjecture on exponential sums. This conjecture predicts specific uniform bounds in the homogeneous polynomial case of exponential sums modulo pm when p and m vary. We introduce the motivic oscillation index of a polynomial f and prove the stronger, analogue bounds for m = 1,2 using this index instead of the original bounds. The modulo p2 case of our bounds holds for all polynomials; the modulo p case holds for homogeneous polynomials and under extra conditions also for nonhomogeneous polynomials. We obtain natural lower bounds for the motivic oscillation index by using results of Segers. We also show that, for p big enough, Igusa's local zeta function has a nontrivial pole when there are Fp-rational singular points on f = 0. We introduce a new invariant of f, the flaw of f.

Approximation of Iteration Number for Gauss-Seidel Using Redlich-Kister Polynomial

Problem statement: Development of mathematical models based on set of observed data plays a crucial role to describe and predict any phenomena in science, engineering and economics. Therefore, the main purpose of this study was to compare the efficiency of Arithmetic Mean (AM), Geometric Mean (GM) and Explicit Group (EG) iterative methods to solve system of linear equations via estimation of unknown parameters in linear models. Approach: The system of linear equations for linear models generated by using least square method based on (m+1) set of observed data for number of Gauss-Seidel iteration from various grid sizes. Actually there were two types of linear models considered such as piece-wise linear polynomial and piece-wise Redlich-Kister polynomial. All unknown parameters of these models estimated and calculated by using three proposed iterative methods. Results: Thorough several implementations of numerical experiments, the accuracy for formulations of two proposed models had shown that the use of the third-order Redlich-Kister polynomial has high accuracy compared to linear polynomial case. Conclusion: The efficiency of AM and GM iterative methods based on the Redlich-Kister polynomial is superior as compared to EG iterative method.

On rational Minkowski Pythagorean hodograph curves

Minkowski Pythagorean hodograph curves are polynomial curves with polynomial speed, measured with respect to Minkowski norm. Curves of this special class are particularly well suited for representing medial axis transforms of planar domains. In the present paper we generalize this polynomial class to a rational class of curves in Minkowski 3-space. We show that any rational Minkowski Pythagorean hodograph curve can be obtained in terms of its associated planar rational Pythagorean hodograph curve and an additional rational function. Moreover, both in the original polynomial and new rational case, we investigate the close relationship between these associated curves in Euclidean plane and Minkowski space.

Polynomials with prescribed zeros on an analytic curve

Recently Totik and Varju [17] presented an estimate of the uniform norm of a monic polynomial with prescribed zeros on the unit circle. In this paper we improve their estimate and extend it to the case of polynomials with some zeros on an arbitrary analytic curve in the complex plane.

A method to compute recurrence relation coefficients for bivariate orthogonal polynomials by unitary matrix transformations

We present an algorithm computing recurrence relation coefficients for bivariate polynomials, orthonormal with respect to a discrete inner product. These polynomials make it possible to give the solution of a discrete least squares approximation problem. To compute these polynomials, we pose the inverse eigenvalue problem and solve it efficiently and in a stable way, using a sequence of Givens rotations. We also show how to generalize the algorithm for the case of polynomials in more variables. Several numerical experiments show the validity of the approach.

Matrix Sylvester equations in the theory of orthogonal polynomials on the unit circle

In this paper we characterize sequences of orthogonal polynomials on the unit circle whose Carathéodory function satisfies a Riccati differential equation with polynomial coefficients, in terms of matrix Sylvester differential equations. For the particular case of semi-classical orthogonal polynomials on the unit circle, it is derived a characterization in terms of first order linear differential systems.

Quantum communication through a spin chain with interaction determined by a Jacobi matrix

We obtain the time-dependent correlation function describing the evolution of a single spin excitation state in a linear spin chain with isotropic nearest-neighbour XY coupling, where the Hamiltonian is related to the Jacobi matrix of a set of orthogonal polynomials. For the Krawtchouk polynomial case, an arbitrary element of the correlation function is expressed in a simple closed form. Its asymptotic limit corresponds to the Jacobi matrix of the Charlier polynomial, and may be understood as a unitary evolution resulting from a Heisenberg group element. Correlation functions for Hamiltonians corresponding to Jacobi matrices for the Hahn, dual Hahn and Racah polynomials are also studied. For the Hahn polynomials we obtain the general correlation function, some of its special cases and the limit related to the Meixner polynomials, where the su(1, 1) algebra describes the underlying symmetry. For the cases of dual Hahn and Racah polynomials, the general expressions of the correlation functions contain summations which are not of hypergeometric type. Simplifications, however, occur in special cases.

The eigenvalue distribution of products of Toeplitz matrices – Clustering and attraction

We use a recent result concerning the eigenvalues of a generic (non-Hermitian) complex perturbation of a bounded Hermitian sequence of matrices to prove that the asymptotic spectrum of the product of Toeplitz sequences, whose symbols have a real-valued essentially bounded product h , is described by the function h in the “Szegö way”. Then, using Mergelyan’s theorem, we extend the result to the more general case where h belongs to the Tilli class. The same technique gives us the analogous result for sequences belonging to the algebra generated by Toeplitz sequences, if the symbols associated with the sequences are bounded and the global symbol h belongs to the Tilli class. A generalization to the case of multilevel matrix-valued symbols and a study of the case of Laurent polynomials not necessarily belonging to the Tilli class are also given.