• Volume 110, Issue 2

May 2000,   pages  121-232

• How to Recover an 𝐿-series from its Values at almost all Positive Integers. Some Remarks on a Formula of Ramanujan

We define a class of analytic functions which can be obtained from their values at almost all positive integers by a canonical interpolation procedure. All the usual 𝐿-functions belong to this class which is interesting in view of the extensive investigations of special values of motivic 𝐿-series. A number of classical contour integral formulas appear as particular cases of the interpolation scheme. The paper is based on a formula of Ramanujan and results of Hardy. An approach to the problem via distributions is also presented.

• RRF Rings which are not LRF

Define a ring 𝐴 to be RRF (respectively LRF) if every right (respectively left) 𝐴-module is residually finite. We determine the necessary and sufficient conditions for a formal triangular matrix ring $$T=\begin{pmatrix}A & 0\\ M & B\end{pmatrix}$$ to be RRF (respectively LRF). Using this we give examples of RRF rings which are not LRF.

• Inequalities for a Polynomial and its Derivative

For an arbitrary entire function 𝑓 and any 𝑟 > 0, let $M(f, r):=\max_{|z|=r}|f(z)|$. It is known that if 𝑝 is a polynomial of degree 𝑛 having no zeros in the open unit disc, and $m:=\min_{|z|=1}|p(z)|$, then

$$M(p',1)≤\frac{n}{2}\{M(p,1)-m\},$$

$$M(p, R)≤\left(\frac{R^n+1}{2}\right)M(p, 1)-m\left(\frac{R^n-1}{2}\right), R>> 1.$$

It is also known that if 𝑝 has all its zeros in the closed unit disc, then

$$M(p', 1)≥\frac{n}{2}\{M(p, 1)+m\}.$$

The present paper contains certain generalizations of these inequalities

• Oscillations of First Order Difference Equations

The oscillatory and asymptotic behaviour of solutions of first order difference equations is studied.

• Continuous Rearrangement and Symmetry of Solutions of Elliptic Problems

This work presents new results and applications for the continuous Steiner symmetrization. There are proved some functional inequalities, e.g. for Dirichlet-type integrals and convolutions and also continuity properties in Sobolev spaces 𝑊1, 𝑝. Further it is shown that the local minimizers of some variational problems and the nonnegative solutions of some semilinear elliptic problems in symmetric domains satisfy a weak, `local' kind of symmetry.

• Diameter Preserving Linear Maps and Isometries, II

We study linear bijections of simplex spaces $\mathcal{A}(S)$ which preserve the diameter of the range, that is, the seminorm $\varrho(f)=\sup\{|f(x)-f(y)|: x, y\in S\}$.

• On Vector Equilibrium Problem

This paper presents some existence results of a vector equilibrium problem. The several important special cases of the vector equilibrium problem are also discussed.

• Existence of Solutions of Nonlinear Integrodifferential Equations of Sobolev Type with Nonlocal Condition in Banach Spaces

In this paper we prove the existence of mild and strong solutions of a nonlinear integrodifferential equation of Sobolev type with nonlocal condition. The results are obtained by using semigroup theory and the Schauder fixed point theorem.

• # Proceedings – Mathematical Sciences

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