Volume 110, Issue 2
May 2000, pages 121-232
pp 121-132
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.
pp 133-136
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.
pp 137-146
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
pp 147-155
Oscillations of First Order Difference Equations
The oscillatory and asymptotic behaviour of solutions of first order difference equations is studied.
pp 157-204
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 𝑊^{1, 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.
pp 205-211
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 $\varrho(f)=\sup\{|f(x)-f(y)|: x, y\in S\}$.
pp 213-223
This paper presents some existence results of a vector equilibrium problem. The several important special cases of the vector equilibrium problem are also discussed.
pp 225-232
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.
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