Mordell–Tornheim multiple zeta values at non-positive integers

We study the Mordell–Tornheim multiple zeta function with all the same parameters. Its values at non-positive integers are evaluated explicitly.     

An algebraic identity and the Jacobi–Trudi formula

The first Jacobi–Trudi identity expresses Schur polynomials as determinants of matrices, the entries of which are complete homogeneous polynomials. The Schur polynomials were defined by Cauchy in 1815 as the quotients of determinants constructed from certain partitions. The Schur polynomials have become very important because of their close relationship with the irreducible characters of the symmetric groups and the general linear groups, as well as due to their numerous applications in combinatorics. The Jacobi–Trudi identity was first formulated by Jacobi in 1841 and proved by Nicola Trudi in 1864. Since then, this identity and its numerous generalizations have been the focus of much attention due to the important role which they play in various areas of mathematics, including mathematical physics, representation theory, and algebraic geometry. Various proofs of the Jacobi–Trudi identity, which are based on different ideas (in particular, a natural combinatorial proof using Young tableaux), have been found. The paper contains a short simple proof of the first Jacobi–Trudi identity and discusses its relationship with other well-known polynomial identities.     

A generalization of Ohnoʼs relation for multiple zeta values

In this paper, we prove that certain parametrized multiple series satisfy the same relation as Ohno?s relation for multiple zeta values. This result gives us a generalization of Ohno?s relation for multiple zeta values. By virtue of this generalization, we obtain a certain equivalence between the above relation among the parametrized multiple series and a subfamily of the relation. As applications of the above results, we obtain some results on multiple zeta values.     

Integrals over polytopes, multiple zeta values and polylogarithms, and Euler’s constant

Let T be the triangle with vertices (1, 0), (0, 1), (1, 1). We study certain integrals over T, one of which was computed by Euler. We give expressions for them both as linear combinations of multiple zeta values, and as polynomials in single zeta values. We obtain asymptotic expansions of the integrals, and of sums of certain multiple zeta values with constant weight. We also give related expressions for Euler’s constant, and study integrals, one of which is the iterated Chen (Drinfeld-Kontsevich) integral, over some polytopes that are higher-dimensional analogs of T. The latter leads to a relation between certain multiple polylogarithm values and multiple zeta values.     

Discrete-Time Dichotomous Well-Posed Linear Systems and Generalized Schur–Nevanlinna–Pick Interpolation

We introduce a class of matrix-valued functions W called “L²- regular”. In case W is J-inner, this class coincides with the class of “strongly regular J-inner” matrix functions in the sense of Arov–Dym. We show that the class of L²-regular matrix functions is exactly the class of transfer functions for a discrete-time dichotomous (possibly infinite-dimensional) input-state-output linear system having some additional stability properties. When applied to J-inner matrix functions, we obtain a state-space realization formula for the resolvent matrix associated with a generalized Schur–Nevanlinna–Pick interpolation problem. Communicated by Daniel Alpay Submitted: August 20, 2006; Accepted: September 13, 2006     

Generalized qd algorithm and Markov–Bernstein inequalities for Jacobi weight

The Markov–Bernstein inequalities for the Jacobi measure remained to be studied in detail. Indeed the tools used for obtaining lower and upper bounds of the constant which appear in these inequalities, did not work, since it is linked with the smallest eigenvalue of a five diagonal positive definite symmetric matrix. The aim of this paper is to generalize the qd algorithm for positive definite symmetric band matrices and to give the mean to expand the determinant of a five diagonal symmetric matrix. After that these new tools are applied to the problem to produce effective lower and upper bounds of the Markov–Bernstein constant in the Jacobi case. In the last part we com pare, in the particular case of the Gegenbauer measure, the lower and upper bounds which can be deduced from this paper, with those given in Draux and Elhami (Comput J Appl Math 106:203–243, 1999) and Draux (Numer Algor 24:31–58, 2000).      

Fractal Jacobi Systems and Convergence of Fourier–Jacobi Expansions of Fractal Interpolation Functions

The fractal interpolation function (FIF) is a special type of continuous function on a compact subset of \({\mathbb{R}}\) interpolating a given data set. They have been proved to be a very important tool in the study of irregular curves arising from financial series, electrocardiograms and bioelectric recording in general as an alternative to the classical methods. It is well known that Jacobi polynomials form an orthonormal system in \({\mathcal{L}^{2}(-1,1)}\) with respect to the weight function \({\rho^{(r,s)}(x)=(1-x)^{r} (1+x)^{s}}\), \({r > -1}\) and \({s > -1}\). In this paper, a fractal Jacobi system which is fractal analogous of Jacobi polynomials is defined. The Weierstrass type theorem providing an approximation for square integrable function in terms of \({\alpha}\)-fractal Jacobi sum is derived. A fractal basis for the space of weighted square integrable functions \({\mathcal{L}_{\rho}^{2}(-1,1)}\) is found. The Fourier–Jacobi expansion corresponding to an affine FIF (AFIF) interpolating certain data set is considered and its convergence in uniform norm and weighted-mean square norm is established. The closeness of the original function to the Fourier–Jacobi expansion of the AFIF is proved for certain scale vector. Finally, the Fourier–Jacobi expansion corresponding to a non-affine smooth FIF interpolating certain data set is considered and its convergence in uniform norm and weighted-mean square norm is investigated as well.     

Generalized Hopf Formulas for the Nonautonomous Hamilton–Jacobi Equation

Generalized Hopf formulas are provided for minimax (viscosity) solutions of Hamilton–Jacobi equations of the form V _t + H(t, D _x V) = 0 and V _t + H(t, V, D _x V) = 0 with the boundary condition V(T, x) = (x), where is a convex function. The bounds within which these formulas apply are elucidated.     

Hankel and Toeplitz–Schur multipliers

We study the problem of characterizing Hankel–Schur multipliers and Toeplitz–Schur multipliers of Schatten–von Neumann class for . We obtain various sharp necessary conditions and sufficient conditions for a Hankel matrix to be a Schur multiplier of . We also give a characterization of the Hankel–Schur multipliers of whos e symbols have lacunary power series. Then the results on Hankel–Schur multipliers are used to obtain a characterization of the Toeplitz–Schur multipliers of . Finally, we return to Hankel–Schur multipliers and obtain new results in the case when the symbol of the Hankel matrix is a complex measure on the unit circle. Received: 16 February 2001 / revised version: 2 December 2001 / Published online: 27 June 2002 The first author is partially supported by Grant 99-01-00103 of Russian Foundation of Fundamental Studies and by Grant 326.53 of Integration. The second author is partially supported by NSF grant DMS 9970561.     

The Bowman–Bradley theorem for multiple zeta-star values

TextThe Bowman–Bradley theorem asserts that the multiple zeta values at the sequences obtained by inserting a fixed number of twos between $3,1,\dots ,3,1$ add up to a rational multiple of a power of π. We establish its counterpart for multiple zeta-star values by showing an identity in a non-commutative polynomial algebra introduced by Hoffman.VideoFor a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=LpqA2OJ6vP8.     

Semigroups of Herz–Schur multipliers

In order to investigate the relationship between weak amenability and the Haagerup property for groups, we introduce the weak Haagerup property, and we prove that having this approximation property is equivalent to the existence of a semigroup of Herz–Schur multipliers generated by a proper function (see Theorem 1.2). It is then shown that a (not necessarily proper) generator of a semigroup of Herz–Schur multipliers splits into a positive definite kernel and a conditionally negative definite kernel. We also show that the generator has a particularly pleasant form if and only if the group is amenable. In the second half of the paper we study semigroups of radial Herz–Schur multipliers on free groups. We prove that a generator of such a semigroup is linearly bounded by the word length function (see Theorem 1.6).     

Fast and reliable high-accuracy computation of Gauss–Jacobi quadrature

Numerical Algorithms - There have been a couple of papers for the solution of the nonsingular symmetric saddle-point problem using three-parameter iterative methods. In most of them, regions of...     

Generalized Kojima–Functions and Lipschitz Stability of Critical Points

In this paper we consider systems of equations which are defined by nonsmooth functions of a special structure. Functions of this type are adapted from Kojima's form of the Karush–Kuhn–Tucker conditions for C²—optimization problems. We shall show that such systems often represent conditions for critical points of variational problems (nonlinear programs, complementarity problems, generalized equations, equilibrium problems and others). Our main purpose is to point out how different concepts of generalized derivatives lead to characterizations of different Lipschitz properties of the critical point or the stationary solution set maps.