Computable bounds for eigenvalues and eigenfunctions of elliptic differential operators

Summary We are concerned with bounds for the error between given approximations and the exact eigenvalues and eigenfunctions of self-adjoint operators in Hilbert spaces. The case is included where the approximations of the eigenfunctions don't belong to the domain of definition of the operator. For the eigenvalue problem with symmetric elliptic differential operators these bounds cover the case where the trial functions don't satisfy the boundary conditions of the problem. The error bounds suggest a certain defectminization method for solving the eigenvalue problems. The method is applied to the membrane problem.     

A two-sided omega-theorem for an asymmetric divisor problem

In this article we study the arithmetic functiond _a,b (l,k;n) which is defined as the number of representationsn=v ^a w ^b withw lying in the residue classl modulok (a,b andl,k fixed positive integers). For the remainder term in the asymptotic formula for Σ_n≤xd_a,b(l,k,;n) we obtain an Ω_± (under a certain restriction onl andk) which is sharper than the known results for the corresponding “unrestricted” problem.     

Large eigenvalues of the laplacian

Let G be a simple graph on n vertices and let L=L(G) be the Laplacian matrix of G corresponding to some ordering of the vertices. It is known that λ≤n for any eigenvalue λ of L. In this note we characterize when n is an eigenvalue of L with multiplicity m.     

On a non-linear wave equation and the control of an elastic string from one equilibrium location to another

This paper concerns a non-linear system of wave equations describing the motion in space of an elastic string. We derive the equations, determine the equilibrium solutions and, using the methods of quasilinear hyperbolic systems, prove that the natural initial, boundary value problem has classical solutions existing in neighbourhoods of the “stretched” equilibrium solutions. We then prove that the positions of the endpoints of the string can be controlled in such a way that the string moves from an equilibrium in one location to an equilibrium in another location.     

Influence of organic solvents on cutinase stability and accessibility to polyamide fibers

The stability of cutinase was measured in the presence of dimethylacetamide (DMA) and benzyl alcohol (BA). These solvents are known swelling agents of polyamide fibers. We found that cutinase retains some stability (half‐life of 14 h), enough for an enzymatic treatment with a solvent composition of 1.5% DMA + 10% BA. Enzymatic treatment in the presence of these solvent compositions yields more soluble amino groups as well as more amino end groups at the fiber level. We have shown evidence that the presence of minor quantities of solvent can increase the accessibility of the enzyme towards the polyamide fiber. © 2005 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 43: 2749‐2753, 2005