The convergence of harmonic Ritz values, harmonic Ritz vectors, and refined harmonic Ritz vectors

This paper concerns a harmonic projection method for computing an approximation to an eigenpair of a large matrix . Given a target point and a subspace that contains an approximation to , the harmonic projection method returns an approximation to . Three convergence results are established as the deviation of from approaches zero. First, the harmonic Ritz value converges to if a certain Rayleigh quotient matrix is uniformly nonsingular. Second, the harmonic Ritz vector converges to if the Rayleigh quotient matrix is uniformly nonsingular and remains well separated from the other harmonic Ritz values. Third, better error bounds for the convergence of are derived when converges. However, we show that the harmonic projection method can fail to find the desired eigenvalue --in other words, the method can miss if it is very close to . To this end, we propose to compute the Rayleigh quotient of with respect to and take it as a new approximate eigenvalue. is shown to converge to once tends to , no matter how is close to . Finally, we show that if the Rayleigh quotient matrix is uniformly nonsingular, then the refined harmonic Ritz vector, or more generally the refined eigenvector approximation introduced by the author, converges. We construct examples to illustrate our theory.

     

The Loewner differential equation and slit mappings

We show that the Loewner equation generates slits if the driving term is Hölder continuous with exponent 1/2 and small norm and that this is best possible.

     

Nonexcellence of the Function Field of the Product of Two Conics

Let k₀ be a field, k₀ ≠ 2, and α, β 2-fold Pfister forms over k₀. Denote by [α], [β] the classes of the corresponding quaternion algebras in ₂Brk₀, and by X_α, X_β the corresponding projective k₀-conics. Suppose ([α] + [β]) = 4. We construct a field F over k₀ such that the field extension F(X_α × X_β)/F is not excellent. Moreover, we find a 2-fold Pfister form γ over F such that ([α ] +[β ] + [γ]) = 4 and the homology group of the complex

Comparison of Green Functions and Harmonic Measures for Parabolic Operators

In this paper, we prove two-sided pointwise estimates for the Green function of a parabolic operator with singular first order term on a C^1,1-cylindrical domain . Basing on these estimates, we establish the equivalence of the parabolic measure, the adjoint parabolic measure and the surface measure on the lateral boundary of . These results are first studied by some authors for certain elliptic and less general parabolic operators. Mathematics Subject Classifications (2000) 31B25, 35B05, 35K10, 58J35.     

Boundary behavior of Gleason''''s problem in hyperbolic harmonic Bergman spaces

Necessary and sufficient conditions are obtained for the boundedness of Berezin transformation on Lebesgue space Lp(B, dVβ) in the real unit ball B in Rn. As an application, we prove that Gleason type problem is solvable in hyperbolic harmonic Bergman spaces. Furthermore we investigate the boundary behavior of the solutions of Gleason type problem.     

A polynomial algorithm for finding (<Emphasis Type="Italic">g,f</Emphasis>)-colorings orthogonal to stars in bipartite graphs

Let G be a bipartite graph with vertex set V(G) and edge set E(G), and let g and f be two nonnegative integer-valued functions defined on V(G) such that g(x) ≤ f(x) for every vertex x of V(G). A (g, f)-coloring of G is a generalized edge-coloring in which each color appears at each vertex x at least g(x) and at most f(x) times. In this paper a polynomial algorithm to find a (g, f)-coloring of a bipartite graph with some constraints using the minimum number of colors is given. Furthermore, we show that the results in this paper are best possible.     

Weak axioms of choice for metric spaces

In the framework of ZF, i.e., Zermelo-Fraenkel set theory without the axiom of choice AC, we show that if the family of all non-empty, closed subsets of a metric space has a choice function, then so does the family of all non-empty, open subsets of . In addition, we establish that the converse is not provable in ZF.

We also show that the statement ``every subspace of the real line with the standard topology has a choice function for its family of all closed, non-empty subsets" is equivalent to the weak choice form ``every continuum sized family of non-empty subsets of reals has a choice function".

     

On the existence of positive solutions of second order differential equations

Using the fixed point method we prove an existence result for positive solutions of nonlinear second order ordinary differential equations. An application to semilinear Schrödinger equations in exterior domains is also presented. Mathematics Subject Classification (2000) 34C10     

Structural Formulas for Orthogonal Matrix Polynomials Satisfying Second-Order Differential Equations, I

We develop structural formulas satisfied by some families of orthogonal matrix polynomials of size $2\times 2$ satisfying second-order differential equations with polynomial coefficients. We consider here two one-parametric families of weight matrices, namely \[ H_{a,1}(t)\;=\;e^{-t^2} \left( \begin{array}{@{}cc@{}} 1+\vert a\vert ^2t^2 & at \\bar at & 1 \end{array} \right) \quad {\rm and} \quad H_{a,2}(t)\;=\;e^{-t^2} \left( \begin{array} {@{}cc@{}} 1+\vert a\vert ^2t^4 & at^2 \\bar at^2 & 1 \end{array} \right), \] $a\in \mbox{\bf C} $ and $t\in \mbox{\bf R} $, and their corresponding orthogonal polynomials.