• Volume 94, Issue 2-3

December 1985,   pages  61-148

• On Extended Hermite-Fezér interpolation based on the zeros of Laguerre polynomials

Szász [2] proved the existence of an interpolatory polynomial which interpolates a given function at given data with some extended points. He also gave its explicit representation. We study the uniform convergence of the same, interpolating at the zeros of Laguerre polynomials with one point (as origin) extension.

• Quotient-bounded elements in locally convex algebras. II

Consideration of quotient-bounded elements in a locally convexGB*-algebra leads to the study of properGB*-algebras viz those that admit nontrivial quotient-bounded elements. The construction and structure of such algebras are discussed. A representation theorem for a properGB*-algebra representing it as an algebra of unbounded Hilbert space operators is obtained in a form that unifies the well-known Gelfand-Naimark representation theorem forC*-algebra and two other representation theorems forb*-algebras (also calledlmc*-algebras), one representinga b*-algebra as an algebra of quotient bounded operators and the other as a weakly unbounded operator algebra. A number of examples are discussed to illustrate quotient-bounded operators. An algebra of unbounded operators constructed out of noncommutativeLp-spaces on a regular probability gauge space and the convolution algebra of periodic distributions are analyzed in detail; whereas unbounded Hilbert algebras andLw-integral of a measurable field ofC*-algebras are discussed briefly.

• Rayleigh wave scattering due to a rigid barrier in a liquid halfspace

The paper presents a theoretical formulation for studying scattering of Rayleigh waves due to the presence of rigid barriers in oceanic waters. The Wiener-Hopf technique has been employed to solve the problem. Exact solution has been obtained in terms of Fourier integrals whose evaluation gives the reflected, transmitted and scattered waves. The scattered waves have the behaviour of cylindrical waves originating at the edge of the barrier. Numerical results for the amplitude of the scattered waves have been obtained for small depth of the barrier.

• Schiffer variation of complex structure and coordinates for Teichmüller spaces

Schiffer variation of complex structure on a Riemann surfaceX0 is achieved by punching out a parametric disc$$\bar D$$ fromX0 and replacing it by another Jordan domain whose boundary curve is a holomorphic image of$$\partial \bar D$$. This change of structure depends on a complex parameter ε which determines the holomorphic mapping function around$$\partial \bar D$$.

It is very natural to look for conditions under which these ε-parameters provide local coordinates for Teichmüller spaceT(X0), (or reduced Teichmüller spaceT#(X0)). For compactX0 this problem was first solved by Patt [8] using a complicated analysis of periods and Ahlfors' [2] τ-coordinates.

Using Gardiner's [6], [7] technique, (independently discovered by the present author), of interpreting Schiffer variation as a quasi conformal deformation of structure, we greatly simplify and generalize Patt's result. Theorems 1 and 2 below take care of all the finitedimensional Teichmüller spaces. In Theorem 3 we are able to analyse the situation for infinite dimensionalT(X0) also. Variational formulae for the dependence of classical moduli parameters on the ε's follow painlessly.

• Minimum degree of a graph and the existence ofk-factors

Let G be a graph andk a positive integer such that (i)k|V(G)| is even; (ii) δ(G) ≥1/2[|V(G)|], and (iii) |V(G)|≥4k-t. ThenG has ak-factor.

• Degenerated nonlinear hyperbolic equation with discontinuous coefficients

The partial differential equations with discontinuous coefficients have been extended in various directions by a number of authors [2], [3], [11], [13], [14], [16]. This paper deals with a mixed problem for a degenerated nonlinear hyperbolic equation with discontinuous coefficients.

• A three-layer asymptotic analysis of turbulent channel flow

In the classical theory for large-Reynolds number fully developed channel flow, the solutions obtained by asymptotic-expansion techniques for the outer Karman defect layer and the inner Prandtl wall layer are demonstrated to match through the introduction of an intermediate layer, based on a general intermediate limit. From an examination of the results for this general intermediate layer, the distinguished intermediate limit and the corresponding intermediate layer for which the turbulent and laminar contributions to the difference of the Reynolds stress from the wall stress are of the same order of magnitude are identified. The thickness of this distinguished intermediate layer is found to be of the order of the geometric mean of the thicknesses of the outer and inner layers

• # Proceedings – Mathematical Sciences

Current Issue
Volume 129 | Issue 3
June 2019