Perturbation of the SVD in the presence of small singular values

This paper gives SVD perturbation bounds and expansions that are of use when an m × n, m ? n matrix A has small singular values. The first part of the paper gives subspace bounds that are closely related to those of Wedin but are stated so as to isolate the effect of any small singular values to the left singular subspace. In the second part first and second order approximations are given for perturbed singular values. The subspace bounds are used to show that all approximations retain accuracy when applied to small singular values. The paper concludes by deriving a subspace bound for multiplicative perturbations and using that bound to give a simple approximation to a singular value perturbed by a multiplicative perturbation.     

Efficient estimation of sparse Jacobian matrices by differences

In difference Newton-like methods for solving F(x)=0, the Jacobian matrix F′(x) is approximated by differences between values of F. If F′(x) is sparse, a consistent partition of its columns can be exploited to approximate F′(x) using relatively few values of F. We provide a local convergence theory for the resulting methods. A superlinearly convergent stable cyclic secant method, in which at each iteration two values of F are required and several columns of the Jacobian matrix approximation are updated simultaneously, is developed.     

Determination of algebraic relations among special zeta values in positive characteristic

As an analogue to special values at positive integers of the Riemann zeta function, we consider Carlitz zeta values ζC(n) at positive integers n. By constructing t-motives after Papanikolas, we prove that the only algebraic relations among these characteristic p zeta values are those coming from the Euler-Carlitz relations and the Frobenius pth power relations.     

Cubature formulas on a sphere that are invariant with respect to a group of dihedron rotations with inversion D 6h

An algorithm of searching for the best (in a sense) cubature formulas on a sphere that are invariant with respect to a group of dihedron rotations with inversion D _6h is developed. This algorithm is applied to find parameters of all the best cubature formulas of this group of symmetry up to the 23rd order of accuracy n. In the course of the study performed, exact values of parameters of the corresponding cubature formulas are found for n ≤ 11, and approximate values are obtained by numerical solving systems of nonlinear algebraic equations by a Newton-type method for other values of n. For the first time, ways of obtaining the best cubature formulas for a sphere are systematically investigated for the case of a group that is not a subgroup of the groups of symmetry of regular polyhedrons.     

Singular value inequalities for matrix sums and minors

This paper gives new proofs for certain inequalities previously established by the author involving sums of singular values of matrices A, B, C = A + B, and also sums of singular values of A, B, and C when A, B are complementary submatrices of C. Some new facts concerning these inequalities are also included.     

What students outside Asia value in effective mathematics lessons: a scoping study

This article reports on a scoping study conducted in Australia and Sweden to facilitate the understanding of the values that are associated with effective learning of mathematics in Asian classrooms. Nineteen ??Western?? values have been identified, of which three (explanation, sharing and fun) were commonly embraced in the two countries. It is also evident that traits that are valued can be and are demonstrated in different forms. The implications for improving mathematics education through the harnessing of values are discussed. The distinction made between what is valued and the forms the value can take should empower classroom teachers to incorporate values flexibly in different classroom contexts.     

Remarks on the set-valued integrals of Debreu and Aumann

Some recently obtained sufficient conditions for the weak compactness of subsets of L¹(m, X) are used to show that for functions whose values are compact, convex subsets of a Banach space the Debreu integral, when it exists, is the same as the Aumann integral. Here no assumption is made concerning the reflexivity of X. This result extends to functions whose values are weakly compact, convex subsets of Banach space.     

Numerical ranges of principal submatrices

It is shown that the ratio of the area of the convex hull of the fields of values of the (n?1)-by-(n?1) principal submatrices of an n-by-n matrix A to the area of the field of values of A is bounded below by a function of n which approaches 1 as n approaches ∞. Since this convex hull is necessarily contained in the field of values of A, an interpretation is that, asymptotically in the dimension, the field of any given matrix is “filled up” by the fields of the submatrices (collectively). Some new inequalities for the eigenvalues of principal submatrices of hermitian matrices, which are not implied by interlacing, are employed.     

The boundary Carathéodory-Fejér interpolation problem

We give a new solvability criterion for the boundary Carathéodory-Fejér problem: given a point x∈R and, a finite set of target values, to construct a function f in the Pick class such that the first few derivatives of f take on the prescribed target values at x. We also derive a linear fractional parametrization of the set of solutions of the interpolation problem with real target values. The proofs are based on a reduction method due to Julia and Nevanlinna.     

Solvability problems of bivariate interpolation I

This paper is devoted to bivariate interpolation. The problem is to find a polynomialP(x, y) whose values and the values of whose derivatives at given points match given data. Methods of Birkhoff interpolation are used throughout. We define interpolation matricesE, their regularity, their almost regularity, and finally the regularity of the pairE, Z for a given set of knotsZ. Many concrete examples and applications are possible.     

Mixed recurrence relations and interlacing of the zeros of some q-orthogonal polynomials from different sequences

We use the generating functions of some q-orthogonal polynomials to obtain mixed recurrence relations involving polynomials with shifted parameter values. These relations are used to prove interlacing results for the zeros of Al-Salam-Chihara, continuous q-ultraspherical, q-Meixner-Pollaczek and q-Laguerre polynomials of the same or adjacent degree as one of the parameters is shifted by integer values or continuously within a certain range. Numerical examples are given to illustrate situations where the zeros do not interlace.     

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.     

Stability and non-normality of the k-ε equations

The analytical and numerical solutions of the equations of the k-ε turbulence model are analyzed. Under certain conditions on the boundary values and the interior values of k and ε the analytical and numerical solutions are bounded. If the steady state solution is obtained numerically by a Runge-Kutta time-stepping method, then severe constraints on the time-step and the non-normality of the jacobian matrix make the convergence very slow. The simplifications and conclusions are supported by data from a numerical solution of flow over a flat plate.     

Arithmetic returns for investment performance measurement

This paper introduces new money-weighted metrics for investment performance analysis, based on arithmetic means of holding period rates weighted by the investment’s market values. This approach generates rates of return which measure a fund’s or portfolio’s performance and a fund manager’s performance. It also enables to show that the Internal Rate of Return (IRR) is a weighted mean of holding period rates associated with interim values which differ from market values, so that value additivity is violated. The manager’s Arithmetic Internal Rate of Return (AIRR) is shown to be the true period equivalent of the cumulative Time Weighted Rate of Return (TWRR), whereas the period TWRR (a geometric return) provides a different ranking. The method is easily generalized for coping with varying benchmark rates. We also cope with the practical problem of estimating interim values whenever they are not available.     

l k,s -Singular values and spectral radius of rectangular tensors

The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we study the singular values/vectors problem of real nonnegative partially symmetric rectangular tensors. We first introduce the concepts of l ^k,s -singular values/vectors of real partially symmetric rectangular tensors. Then, based upon the presented properties of l ^k,s -singular values /vectors, some properties of the related l ^k,s -spectral radius are discussed. Furthermore, we prove two analogs of Perron-Frobenius theorem and weak Perron-Frobenius theorem for real nonnegative partially symmetric rectangular tensors.     

Bilattices for deductions in multi-valued logic

In this exploratory paper we propose a framework for the deduction apparatus of multi-valued logics based on the idea that a deduction apparatus has to be a tool to manage information on truth values and not directly truth values of the formulas. This is obtained by embedding the algebraic structure V defined by the set of truth values into a bilattice B. The intended interpretation is that the elements of B are pieces of information on the elements of V. The resulting formalisms are particularized in the framework of fuzzy logic programming. Since we see fuzzy control as a chapter of multi-valued logic programming, this suggests a new and powerful approach to fuzzy control based on positive and negative conditions.     

Partitioning heuristics for two geometric maximization problems

Given n points randomly selected from a uniform distribution on the unit square, we describe linear-time partitioning heuristics which will construct a matching or a tour of these points. We show that the heuristics closely approximate the optimum values as n → ∞. Hence we show that the asymptotic values of the maximum matching and tour are about 0·3826n and twice this value respectively.     

On exceptional values of differential polynomials

Let ?(z) be a meromorphic function of finite non-zero orderρ. Assuming certain growth estimates on ? by comparing it withr ^p L (r) whereL (r) is a slowly changing function we have obtained the bounds on the zeros of ψ (z) —w where ψ (z) is a differential polynomial. These bounds are satisfied except for special values ofw. Examples are given to show that the exceptional values exist.