Fibonacci-type polynomial as a trajectory of a discrete dynamical system

Families of polynomials which obey the Fibonacci recursion relation can be generated by repeated iterations of a 2×2 matrix,Q ₂, acting on an initial value matrix,R ₂. One matrix fixes the recursion relation, while the other one distinguishes between the different polynomial families. Each family of polynomials can be considered as a single trajectory of a discrete dynamical system whose dynamics are determined byQ ₂. The starting point for each trajectory is fixed byR ₂(x). The forms of these matrices are studied, and some consequences for the properties of the corresponding polynomials are obtained. The main results generalize to the so-calledr-Bonacci polynomials.     

On Distinguished Solutions of Truncated Matricial Hamburger Moment Problems

We study two slightly different versions of the truncated matricial Hamburger moment problem. A central topic is the construction and investigation of distinguished solutions of both moment problems under consideration. These solutions turn out to be nonnegative Hermitian q × q Borel measures on the real axis which are concentrated on a finite number of points. These points and the corresponding masses will be explicitly described in terms of the given data. Furthermore, we investigate a particular class of sequences (s_j)^∞_{j = 0} of complex q × q matrices for which the corresponding infinite matricial Hamburger moment problem has a unique solution. Our approach is mainly algebraic. It is based on the use of particular matrix polynomials constructed from a nonnegative Hermitian block Hankel matrix. These matrix polynomials are immediate generalizations of the monic orthogonal matrix polynomials associated with a positive Hermitian block Hankel matrix. We generalize a classical theorem due to Kronecker on infinite Hankel matrices of finite rank to block Hankel matrices and discuss its consequences for the nonnegative Hermitian case.     

On the Laguerre-Hahn Intertwining Operator and Application to Connection Formulas

In this paper, we show that the lowering operator D _u indexed by a linear functional on polynomials u, introduced by F. Marcellán and R. Sfaxi, namely the Laguerre-Hahn derivative, is intertwining with the standard derivative D by a linear isomorphism S _u on polynomials. This allows us to establish an intertwining relation between the nonsingular Laguerre-Hahn polynomials of class zero of Hermite type and the Hermite polynomials, as well as some new connection formulas between such Laguerre-Hahn polynomials and the canonical basis.     

On 3-monotone approximation by piecewise polynomials

We consider 3-monotone approximation by piecewise polynomials with prescribed knots. A general theorem is proved, which reduces the problem of 3-monotone uniform approximation of a 3-monotone function, to convex local L₁ approximation of the derivative of the function. As the corollary we obtain Jackson-type estimates on the degree of 3-monotone approximation by piecewise polynomials with prescribed knots. Such estimates are well known for monotone and convex approximation, and to the contrary, they in general are not valid for higher orders of monotonicity. Also we show that any convex piecewise polynomial can be modified to be, in addition, interpolatory, while still preserving the degree of the uniform approximation. Alternatively, we show that we may smooth the approximating piecewise polynomials to be twice continuously differentiable, while still being 3-monotone and still keeping the same degree of approximation.     

Polynomials over division rings

LetD be a division ring with a centerC, andD[X ₁, …,X _N] the ring of polynomials inN commutative indeterminates overD. The maximum numberN for which this ring of polynomials is primitive is equal to the maximal transcendence degree overC of the commutative subfields of the matrix ringsM _n(D),n=1, 2, …. The ring of fractions of the Weyl algebras are examples where this numberN is finite. A tool in the proof is a non-commutative version of one of the forms of the “Nullstellensatz”, namely, simpleD[X ₁, …,X _m]-modules are finite-dimensionalD-spaces. This paper was written while the authors were Fellows of the Institute for Advanced Studies, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem, Israel.     

Pseudo-Orthogonal Polynomials

We consider the classical problem of transforming an orthogonality weight of polynomials by means of the space R _n . We describe systems of polynomials called pseudo-orthogonal on a finite set of n points. Like orthogonal polynomials, the polynomials of these systems are connected by three-term relations with tridiagonal matrix which is nondecomposable but does not enjoy the Jacobi property. Nevertheless these polynomials possess real roots of multiplicity one; moreover, almost all roots of two neighboring polynomials separate one another. The pseudo-orthogonality weights are partly negative. Another result is the analysis of relations between matrices of two different orthogonal systems which enables us to give explicit conditions for existence of pseudo-orthogonal polynomials.     

Upward Extension of the Jacobi Matrix for Orthogonal Polynomials

Orthogonal polynomials on the real line always satisfy a three-term recurrence relation. The recurrence coefficients determine a tridiagonal semi-infinite matrix (Jacobi matrix) which uniquely characterizes the orthogonal polynomials. We investigate new orthogonal polynomials by adding to the Jacobi matrixrnew rows and columns, so that the original Jacobi matrix is shifted downward. Thernew rows and columns contain 2rnew parameters and the newly obtained orthogonal polynomials thus correspond to an upward extension of the Jacobi matrix. We give an explicit expression of the new orthogonal polynomials in terms of the original orthogonal polynomials, their associated polynomials, and the 2rnew parameters, and we give a fourth order differential equation for these new polynomials when the original orthogonal polynomials are classical. Furthermore we show how the 1?orthogonalizing measure for these new orthogonal polynomials can be obtained and work out the details for a one-parameter family of Jacobi polynomials for which the associated polynomials are again Jacobi polynomials.     

Improving projection‐based eigensolvers via adaptive techniques

We consider subspace iteration (or projection‐based) algorithms for computing those eigenvalues (and associated eigenvectors) of a Hermitian matrix that lie in a prescribed interval. For the case that the projector is approximated with polynomials, we present an adaptive strategy for selecting the degree of these polynomials such that convergence is achieved with near‐to‐optimum overall work without detailed a priori knowledge about the eigenvalue distribution. The idea is then transferred to the approximation of the projector by numerical integration, which corresponds to FEAST algorithm proposed by E. Polizzi in 2009. [E. Polizzi: Density‐matrix‐based algorithm for solving eigenvalue problems. Phys. Rev. B 2009; 79 :115112]. Here, our adaptation controls the number of integration nodes. We also discuss the interaction of the method with search space reduction methods.     

The computation of orthogonal rational functions and their interpolating properties

We shall consider nested spacesl _n ,n = 0, 1, 21... of rational functions withn prescribed poles outside the unit disk of the complex plane. We study orthogonal basis functions of these spaces for a general positive measure on the unit circle. In the special case where all poles are placed at infinity,l _n = _n , the polynomials of degree at mostn. Thus the present paper is a study of orthogonal rational functions, which generalize the orthogonal Szegö polynomials. In this paper we shall concentrate on the functions of the second kind which are natural generalizations of the corresponding polynomials.The work of the first author is partially supported by a research grant from the Belgian National Fund for Scientific Research     

Conditions for Singular Incidence Matrices

Suppose one looks for a square integral matrix N, for which NN_T has a prescribed form. Then the Hasse-Minkowski invariants and the determinant of NN_T lead to necessary conditions for existence. The Bruck-Ryser-Chowla theorem gives a famous example of such conditions in case N is the incidence matrix of a square block design. This approach fails when N is singular. In this paper it is shown that in some cases conditions can still be obtained if the kernels of N and N_T are known, or known to be rationally equivalent. This leads for example to non-existence conditions for self-dual generalised polygons, semi-regular square divisible designs and distance-regular graphs.     

To solving multiparameter problems of algebra. 8. The <Emphasis Type="Italic">RP-q</Emphasis> method and its applications

A new method (the RP-q method) for factorizing scalar polynomials in q variables and q-parameter polynomial matrices (q ≥ 1) of full rank is suggested. Applications of the algorithm to solving systems of nonlinear algebraic equations and some spectral problems for a q-parameter polynomial matrix F (such as separation of the eigenspectrum and mixed spectrum of F, computation of bases with prescribed spectral properties of the null-space of polynomial solutions of F, and computation of the hereditary polynomials of F) are considered. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 149–164.     

The parity of the number of irreducible factors for some pentanomials

It is well known that the Stickelberger–Swan theorem is very important for determining the reducibility of polynomials over a binary field. Using this theorem the parity of the number of irreducible factors for some kinds of polynomials over a binary field, for instance, trinomials, tetranomials, self-reciprocal polynomials and so on was determined. We discuss this problem for Type II pentanomials, namely x^m+xⁿ⁺²+xⁿ⁺¹+xⁿ+1F₂[x] for even m. Such pentanomials can be used for the efficient implementation of multiplication in finite fields of characteristic two. Based on the computation of the discriminant of these pentanomials with integer coefficients, we will characterize the parity of the number of irreducible factors over F₂ and establish necessary conditions for the existence of this kind of irreducible pentanomials.Our results have been obtained in an experimental way by computing a significant number of values with Mathematica and extracting the relevant properties.     

Computation of the canonical polynomials and applications to some optimal control problems

The Tau method is a numerical technique that consists in constructing polynomial approximate solutions for ordinary differential equations. This method has two approaches: operational and recursive. The former converts the differential problem to a matrix problem and produces approximations in terms of a prescribed orthogonal polynomials basis. In the recursive approach, we construct approximate solutions in terms of a special set of polynomials {Q _k (t); k?=?0, 1, 2...} called canonical polynomials basis. In some cases, the Q _k ??s can be obtained explicitly through a recursive formula. But no analogous formulae are reported in the literature for the general cases. In this paper, utilizing the operational Tau method, we develop an algorithm that allows to generate those canonical polynomials iteratively and explicitly. In addition, we demonstrate the capability of the operational Tau method in treating quadratic optimal control problems governed by ordinary differential equations.     

Reverse generalized Bessel matrix differential equation,polynomial solutions,and their properties

This paper is devoted to the study of reverse generalized Bessel matrix polynomials (RGBMPs) within complex analysis. This study is assumed to be a generalization and improvement of the scalar case into the matrix setting. We give a definition of the reverse generalized Bessel matrix polynomials Θ_n(A; B; z), , for parameter (square) matrices A and B, and provide a second‐order matrix differential equations satisfied by these polynomials. Subsequently, a Rodrigues‐type formula, a matrix recurrence relationship, and a pseudo‐generating function are then developed for RGBMPs. © 2013 The Authors Mathematical Methods in the Applied Sciences Published by John Wiley & Sons, Ltd.     

Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory

We consider asymptotics for orthogonal polynomials with respect to varying exponential weights w_n(x)dx = e^−nV(x) dx on the line as n → ∞. The potentials V are assumed to be real analytic, with sufficient growth at infinity. The principle results concern Plancherel‐Rotach‐type asymptotics for the orthogonal polynomials down to the axis. Using these asymptotics, we then prove universality for a variety of statistical quantities arising in the theory of random matrix models, some of which have been considered recently in [31] and also in [4]. An additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials (see also [4]). The orthogonal polynomial problem is formulated as a Riemann‐Hilbert problem following [19, 20]. The Riemann‐Hilbert problem is analyzed in turn using the steepest‐descent method introduced in [12] and further developed in [11, 13]. A critical role in our method is played by the equilibrium measure dμ_V for V as analyzed in [8]. © 1999 John Wiley & Sons, Inc.     

The bilateral birth–death chain generated by the associated Jacobi polynomials

We give a probabilistic interpretation of the associated Jacobi polynomials, which can be constructed from the three-term recurrence relation for the classical Jacobi polynomials by shifting the integer index n by a real number t. Under certain restrictions, this will give rise to a doubly infinite tridiagonal stochastic matrix, which can be interpreted as the one-step transition probability matrix of a discrete-time bilateral birth–death chain with state space on $\mathbb{Z}$. We also study the unique UL and LU stochastic factorizations of the transition probability matrix, as well as the discrete Darboux transformations and corresponding spectral matrices. Finally, we use all these results to provide an urn model on the integers for the associated Jacobi polynomials.     

Estimating the dimension of a linear system

The problem considered is that of estimating the integer or integers that prescribe the dimension of a linear system. These could be the Kronecker indices. Though attention is concentrated on the order or McMillan degree, which specifies the dimension of a minimal state vector, the same results are available for other cases. A fairly complete theorem is proved relating to conditions under which strong or weak convergence will hold for an estimate of the McMillan degree when the estimation is based on minimisation of a criterion of the form log det( _n) + nC(T)/T, where _n, is the estimate of the prediction error covariance matrix and the McMillan degree is assumed to be n. The conditions relate to the prescribed sequence C(T).     

Grassmann algebras over finite fields

We study an analogue of a problem of procesi about matrices [9, page 185(e)]: are there non-trivial polynomials over Z which become identities over Z _p - for grassmann algebras E? when 1 ? E, we show that such polynomials do not exist, but when 1 ?,E such polynomials exist - also for matrices over E. these results are deduced from a careful study of the various codimensions of these algebras.     

Toward an arithmetic of polynomials

Summary ForR a commutative ring, which may have divisors of zero but which has no idempotents other than zero and one, we consider the problem of unique factorization of a polynomial with coefficients inR. We prove that, if the polynomial is separable, then such a unique factorization exists. We also define a Legendre symbol for a separable polynomial and a prime of commutative ring with exactly two idempotents in such a way that the symbols of classical number theory are subsumed. We calculate this symbol forR = Q in two cases where it has classically been of interest, namely quadratic extensions and cyclotomic extensions. We then calculate it in a situation which is new, namely the so called generalized cyclotomic extensions from a paper by S. Beale and D. K. Harrison. We study the Galois theory in the general ring situation and in particular define a category of separable polynomials (this is an extension of a paper by D. K. Harrison and M. Vitulli) and a cohomology theory of separable polynomials.