Twisted traces of singular values

Let p be a prime for which the congruence group Γ₀(p)^* is of genus zero, and j _p ^* be the corresponding Hauptmodul. We investigate the twisted traces of singular values of j _p ^* and construct infinite products related to them. This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2006-331-C00006).     

The singular values of and

We obtain several inequalities relating the singular values of A+B and those of A+iB when A and B are hermitian, and when one or both of them are positive semidefinite.     

On the sum of superoptimal singular values

In this paper, we study the following extremal problem and its relevance to the sum of the so-called superoptimal singular values of a matrix function: Given an m×n matrix function Φ, when is there a matrix function Ψ_∗ in the set such that

     

Problems for elliptic singular equations with a gradient term

We investigate the homogeneous Dirichlet problem for a class of second-order nonlinear elliptic partial differential equations with singular data. In particular, we study the asymptotic behaviour of the solution near the boundary up to the second order under various assumptions on the growth of the coefficients of the equation.     

A singular nonlinear elliptic equation with natural growth in the gradient

This paper is devoted to the existence and regularity of the homogenous Dirichlet boundary value problem for a singular nonlinear elliptic equation with natural growth in the gradient. By certain transformations, the problem can be transformed formally into either a Dirichlet problem or boundary blowup problems without gradient term, for which the corresponding existence results are also derived, which is a partial extension and supplement to the previous works. Copyright © 2008 John Wiley & Sons, Ltd.     

Low rank update of singular values

The notion of a low rank update arises in many important applications. This paper deals with the inverse problem of updating a rectangular matrix by additive low rank matrices so as to reposition the associated singular values. The setting is analogous to the classical pole assignment problem where eigenvalues of a square matrix are relocated. Precise and easy-to-check necessary and sufficient conditions under which the problem is solvable are completely characterized, generalizing some traditional Weyl inequalities for singular values. The constructive proof makes it possible to compute such a solution numerically. A pseudo algorithm is outlined.

     

A note on the perturbation of singular values

In this note a variant of the classical perturbation theorem for singular values is given. The bounds explain why perturbations will tend to increase rather than decrease singular values of the same order of magnitude as the perturbation.     

On the singular values of Weber modular functions

The minimal polynomials of the singular values of the classical Weber modular functions give far simpler defining polynomials for the class fields of imaginary quadratic fields than the minimal polynomials of singular moduli of level 1. We describe computations of these polynomials and give conjectural formulas describing the prime decomposition of their resultants and discriminants, extending the formulas of Gross-Zagier for the level 1 case.

     

On the distribution singular values of Toeplitz matrices

We prove a second order formula concerning distribution of singular values of Toeplitz matrices in some cases when conditions of the H. Widom Theorem are not satisfied.

     

On the Cauchy problem for the singular parabolic equations with gradient term

The Cauchy problem for a singular parabolic equation with gradient term of the form

ut−div(|Du|^p−2Du)=|Duqσ^|     

Entire positive solutions of the singular Emden-Fowler equation with nonlinear gradient term

Let p and q be locally Hölder functions in ?^N, p > 0 and q ≥ 0. We study the Emden-Fowler equation $-\triangle u+q(x)\mid \nabla u\mid^\alpha=p(x)u^{-\gamma}$ in ?^N, where α and γ are positive numbers. Our main result establishes that the above equation has a unique positive solutions decaying to zero at infinity. Our proof is elementary and it combines the maximum principle for elliptic equations with a theorem of Crandall, Rabinowitz and Tartar.     

Blaschke products and singular functions with prescribed boundary values

This paper shows that there exists a Blaschke product having a prescribed radial limit at each point of a prescribed finite subset of the unit circle. In addition, an analogue for singular inner functions is proved; and an extension dealing with tangential limits is established.     

Extreme determinants on the convex hull of matrices with prescribed singular values

In a recent paper[1] the author carried through a comprehesive analysis of the diagonal elements of matrices having prescribed singular values, and as an application of this analysis a characterization was obtained of matrices with prescribed singular values. In the present note we obtain the greatest and least values for the determinants of the matrices in such convex hulls.     

Extreme determinants on the convex hull of matrices with prescribed singular values

In a recent paper[1] the author carried through a comprehesive analysis of the diagonal elements of matrices having prescribed singular values, and as an application of this analysis a characterization was obtained of matrices with prescribed singular values. In the present note we obtain the greatest and least values for the determinants of the matrices in such convex hulls.     

Interlacing inequalities for generalized singular values

A generalization of the concept of singular value of a complex matrix introduced by J. F. Maitre and N. Huu Vinh is considered. We give the relation between the singular values of two matrices and the singular values of their sum, and we find Cauchy interlacing-type inequalities for some of these singular values.     

Problems for elliptic singular equations with a quadratic gradient term

We investigate the homogeneous Dirichlet problem for a class of second-order elliptic partial differential equations with a quadratic gradient term and singular data. In particular, we study the asymptotic behaviour of the solution near the boundary under suitable assumptions on the growth of the coefficients of the equation.