A SIMPLIFIED BRAUER＇S THEOREM ON MATRIX EIGENVALUES

Let A=(a _ij)∈C ^n×n and . Suppose that for each row of A there is at least one nonzero off-diagonal entry. It is proved that all eigenvalues of A are contained in . The result reduces the number of ovals in original Brauer’s theorem in many cases. Eigenvalues (and associated eigenvectors) that locate in the boundary of are discussed. The project is supported in part by Natural Science Foundation of Guangdong.     

Numerical bifurcation and stability analysis for steady-states of reaction diffusion equations

Solutions of a semilinear elliptic boundary value problem, (with bounded below) can be put into a one-to-one correspondence with zeros of a function . Often d is small. The function is called the bifurcation function. It can also be shown that the eigenvalues of the matrix characterize the stability properties of the solutions of the elliptic problem as rest points of . A finite element method that can be used for computing B and B _c has recently been proposed. An overview of these results and the finite element method is given. This revised version was published online in June 2006 with corrections to the Cover Date.     

Stable rational modification of a weight

Consider a weight and a rational modification of it is a polynomial. An algorithm is stabilized for the following problem: “Given the coefficients of the three-term recurrence equation satisfied by the orthogonal polynomials relative to , compute the coefficients of the recurrence equation satisfied by the orthogonal polynomials relative to .” This revised version was published online in June 2006 with corrections to the Cover Date.     

Lower bounds of some singular integrals and their applications

Let and . We are interested in the lower bounds of the integral:

Accelerating infinite products

Slowly convergent infinite products are considered, where is a sequence of numbers, or a sequence of linear operators. Using an asymptotic expansion for the “remainder” of the infinite product a method for convergence acceleration is suggested. The method is in the spirit of the d-transformation for series. It is very simple and efficient for some classes of sequences . For complicated sequences it involves the solution of some linear systems, but it is still effective. This revised version was published online in June 2006 with corrections to the Cover Date.     

Equational Theories for Classes of Finite Semigroups

It is proved that there exists an infinite sequence of finitely based semigroup varieties such that, for all i, an equational theory for and for the class of all finite semigroups in is undecidable while an equational theory for and for the class of all finite semigroups in is decidable. An infinite sequence of finitely based semigroup varieties is constructed so that, for all i, an equational theory for and for the class of all finite semigroups in is decidable whicle an equational theory for and for the class of all finite semigroups in is not.     

Asymptotic zero distribution of hypergeometric polynomials

We show that the zeros of the hypergeometric polynomials , , cluster on the loop of the lemniscate as . We also state the equations of the curves on which the zeros of , lie asymptotically as . Auxiliary results for the asymptotic zero distribution of other functions related to hypergeometric polynomials are proved, including Jacobi polynomials with varying parameters and associated Legendre functions. Graphical evidence is provided using Mathematica. This revised version was published online in June 2006 with corrections to the Cover Date.     

Exponential Stability of Semigroups Related to Operator Models in Mechanics

In this paper, we consider equations of the form , where is a function with values in the Hilbert space , the operator B is symmetric, and the operator A is uniformly positive and self-adjoint in . The linear operator generating the C ₀-semigroup in the energy space is associated with this equation. We prove that this semigroup is exponentially stable if the operator B is uniformly positive and the operator A dominates B in the sense of quadratic forms.     

The C r-fundamental splines of Clough–Tocher and Powell–Sabin types for Lagrange interpolation on a three direction mesh

Let Δ⁽¹⁾ be the uniform three direction mesh of the plane whose vertices are integer points of .Let (respectively of degree d=3r (respectively d=3r+1 ) for r odd (respectively even) on the triangulation , and of degree d=2r (respectively d=2r+1) for r odd (respectively even) on the triangulation . Using linear combinations of translates of these splines we obtain Lagrange interpolants whose corresponding order of approximation is optimal. This revised version was published online in June 2006 with corrections to the Cover Date.     

Strong Positivity in Right-Invariant Order on Braid Groups and Quasipositivity

Dehornoy constructed a right invariant order on the braid group B _n uniquely defined by the condition 1{\text{ if }}\beta _0 ,\beta _1$$ " align="middle" border="0"> are words in . A braid is called strongly positive if 1$$ " align="middle" border="0"> for any . In the present paper it is proved that the braid is strongly positive if the word does not contain . We also provide a geometric proof of the result by Burckel and Laver that the standard generators of a braid group are strongly positive. Finally, we discuss relations between the right invariant order and quasipositivity.     

Hankel determinants of some polynomials arising in combinatorial analysis

In this paper we investigate Hankel determinants of the form , where c _n (t) is one of a number of polynomials of combinatorial interest. We show how some results due to Radoux may be generalized, and also show how “stepped up” Hankel determinants of the form may be evaluated. This revised version was published online in June 2006 with corrections to the Cover Date.     

Parameterization of m‐channel orthogonal multifilter banks

A complete parameterization for the m‐channel FIR orthogonal multifilter banks is provided based on the lattice structure of the paraunitary systems. Two forms of complete factorization of the m‐channel FIR orthogonal multifilter banks for symmetric/antisymmetric scaling functions and multiwavelets with the same symmetric center (1 + γ + γ/(m - 1)) for some nonnegative integer γ are obtained. For the case of multiplicity 2 and dilation factor m = 2, the result of the factorization shows that if the scaling function Φ and multiwavelet Ψ are symmetric/antisymmetric about the same symmetric center γ + for some nonnegative integer γ, then one of the components of Φ (respectively Ψ) is symmetric and the other is antisymmetric. Two examples of the construction of symmetric/antisymmetric orthogonal multiwavelets of multiplicity 3 with dilation factor 2 and multiplicity 2 with dilation factor 3 are presented to demonstrate the use of these parameterizations of orthogonal multifilter banks. This revised version was published online in June 2006 with corrections to the Cover Date.     

Rational approximations of pre-filtered transfer functions via the Lanczos algorithm

Given a single-input single-output system {A,b,c} with strictly proper transfer function g(s), we derive a Lanczos-based method to construct a tridiagonal state-space model approximating the “pre-filtered” transfer function f(s)g(s), where f(s) is given in factored form . We also show how to apply this idea to the Arnoldi process and mention a few other extensions. This revised version was published online in June 2006 with corrections to the Cover Date.     

Quantum duality in quantum deformations

In accordance with the quantum duality principle, the twisted algebra is equivalent to the quantum group and has two preferred bases: one inherited from the universal enveloping algebra and the other generated by coordinate functions of the dual Lie group . We show howthe transformation can be explicitly obtained for any simple Lie algebra and a factorable chain of extended Jordanian twists. In the algebra , we introduce a natural vector grading , compatible with the adjoint representation of the algebra. Passing to the dual-group coordinates allows essentially simplifying the costructure of the deformed Hopf algebra , considered as a quantum group . The transformation can be used to construct new solutions of the twist equations. We construct a parameterized family of extended Jordanian deformations and study it in terms of ; we find new realizations of the parabolic twist. Dedicated to the birthday of my teacher, Yurii Novozhilov __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 1, pp. 112–125, July, 2006.     

Finite rank operators in Jacobson radical \mathcal{R}_{\mathcal{N} \otimes \mathcal{M}}

In this paper we investigate finite rank operators in the Jacobson radical of Alg( ), where are nests. Based on the concrete characterizations of rank one operators in Alg( ) and , we obtain that each finite rank operator in can be written as a finite sum of rank one operators in and the weak closure of equals Alg( ) if and only if at least one of is continuous.     

Metrics in the sphere of a C*-module

Given a unital C*-algebra and a right C*-module over , we consider the problem of finding short smooth curves in the sphere = {x ∈ : 〈x, x〉 = 1}. Curves in are measured considering the Finsler metric which consists of the norm of at each tangent space of . The initial value problem is solved, for the case when is a von Neumann algebra and is selfdual: for any element x ₀ ∈ and any tangent vector ν at x ₀, there exists a curve γ(t) = e ^tZ (x ₀), Z ∈ , Z* = −Z and ∥Z∥ ≤ π, such that γ(0) = x ₀ and (0) = ν, which is minimizing along its path for t ∈ [0, 1]. The existence of such Z is linked to the extension problem of selfadjoint operators. Such minimal curves need not be unique. Also we consider the boundary value problem: given x ₀, x ₁ ∈ , find a curve of minimal length which joins them. We give several partial answers to this question. For instance, let us denote by f ₀ the selfadjoint projection I − x ₀ ⊗ x ₀, if the algebra f ₀ f ₀ is finite dimensional, then there exists a curve γ joining x ₀ and x ₁, which is minimizing along its path.      

K-Theory for the Banach Algebra of Operators on James''s Quasi-reflexive Banach Spaces

We prove that the K-groups of the Banach algebra of bounded, linear operators on the pth James space , where 1 < p < , are given by and . Moreover, for each Banach space and each non-zero, closed ideal contained in the ideal of inessential operators, we show that and . This enables us to calculate the K-groups of for each Banach space which is a direct sum of finitely many James spaces and -spaces.     

Coupled Vandermonde matrices and the superfast computation of Toeplitz determinants

Let n be a positive integer, let be complex numbers and let be a nonsingular n × n complex Toeplitz matrix. We present a superfast algorithm for computing the determinant of T. Superfast means that the arithmetic complexity of our algorithm is , where N denotes the smallest power of 2 that is larger than or equal to n. We show that det T can be computed from the determinant of a certain coupled Vandermonde matrix. The latter matrix is related to a linearized rational interpolation problem at roots of unity and we show how its determinant can be calculated by multiplying the pivots that appear in the superfast interpolation algorithm that we presented in a previous publication. This revised version was published online in June 2006 with corrections to the Cover Date.     

Distances entre exponentielles et puissances d’éléments de certaines algèbres de Banach

Let A be a Banach algebra which does not contain any nonzero idempotent element, let γ > 0, and let . We show that if then . We also show, assuming a suitable spectral condition on x, that if , then Received: 12 July 2006 Revised: 31 January 2007     

Finite-dimensional approximation of the inverse frame operator

A frame in a Hilbert space allows every element in to be written as a linear combination of the frame elements, with coefficients called frame coefficients. Calculations of those coefficients and many other situations where frames occur, requires knowledge of the inverse frame operator. But usually it is hard to invert the frame operator if the underlying Hilbert space is infinite dimensional. In the present paper we introduce a method for approximation of the inverse frame operator using finite subsets of the frame. In particular this allows to approximate the frame coefficients (even inl ²) using finite-dimensional linear algebra. We show that the general method simplifies in the important cases of Weil-Heisenberg frames and wavelet frames.