A logarithm type mean value theorem of the Riemann zeta function

For any integer K?2 and positive integer h, we investigate the mean value of |ζ(σ+it)|^2k×logh|ζ(σ+it)| for all real number 0<k<K and all σ>1−1/K. In case K=2, h=1, this has been studied by Wang in [F.T. Wang, A mean value theorem of the Riemann zeta function, Quart. J. Math. Oxford Ser. 18 (1947) 1-3]. In this note, we give a new brief proof of Wang's theorem, and, with this method, generalize it to the general case naturally.     

Moser iteration for (quasi)minimizers on metric spaces

We study regularity properties of quasiminimizers of the p-Dirichlet integral on metric measure spaces. We adapt the Moser iteration technique to this setting and show that it can be applied without an underlying differential equation. However, we have been able to run the Moser iteration fully only for minimizers. We prove Caccioppoli inequalities and local boundedness properties for quasisub- and quasisuperminimizers. This is done in metric spaces equipped with a doubling measure and supporting a weak (1, p)-Poincaré inequality. The metric space is not required to be complete. We also provide an example which shows that the dilation constant from the weak Poincaré inequality is essential in the condition on the balls in the Harnack inequality. This fact seems to have been overlooked in the earlier literature on nonlinear potential theory on metric spaces.     

The constrained degree and fixed-point index theory for set-valued maps

In the first part of the paper we give a construction of a topological degree theory for set-valued tangent vector fields with convex and nonconvex values defined on nonsmooth closed subsets of a Banach space. The obtained homotopy invariant is an extension of the classical degree for vector fields on manifolds. In the second part we propose a fixed-point index for inward maps on arbitrary closed convex sets.     

On solutions to the braid equation identified by Woronowicz

Solutions of the braid equation recently identified by Woronowicz are shown to be derived from subrepresentations of the regular representations of a quantum double Hopf algebra.     

Difference schrödinger operators with linear and exponential discrete spectra

Using the factorization method, we construct finite-difference Schrödinger operators (Jacobi matrices) whose discrete spectra are composed from independent arithmetic, or geometric series. Such systems originate from the periodic, orq-periodic closure of a chain of corresponding Darboux transformations. The Charlier, Krawtchouk, Meixner orthogonal polynomials, theirq-analogs, and some other classical polynomials appear as the simplest examples forN = 1 andN = 2 (N is the period of closure). A natural generalization involves discrete versions of the Painlevé transcendents.On leave of absence from the Institute for Nuclear Research, Russian Academy of Sciences, Moscow, Russia.     

Laser and fluorescence properties of dye-doped sol-gel silica from 400 nm to 800 nm

Lasing and fluorescence properties of ten dyes covering the spectral range from 400 nm to 800 nm were studied in a silica matrix prepared by the sol-gel technique. The dye-doped sol-gel silica samples fluoresced strongly under laser excitation. Many of the samples were also induced to superradiate. In comparison to dyes in alcohol solvents, significant red-shifting was observed in the sol-gel silica fluorescence spectra. Plausible causes of the red-shift are discussed.     

Upward planarity testing

Acyclic digraphs, such as the covering digraphs of ordered sets, are usually drawn upward, i.e., with the edges monotonically increasing in the vertical direction. A digraph is upward planar if it admits an upward planar drawing. In this survey paper, we overview the literature on the problem of upward planarity testing. We present several characterizations of upward planarity and describe upward planarity testing algorithms for special classes of digraphs, such as embedded digraphs and single-source digraphs. We also sketch the proof of NP-completeness of upward planarity testing.Research supported in part by the National Science Foundation under grant CCR-9423847.     

Discrepancy and the Error in Integration

 This paper is devoted to an estimation of the error of integration with respect to arbitrary unit measures μ and ν on only in terms of continuity or smoothness properties of the function f and the discrepancy . Here, stands for certain classes of (Borel-) test sets. The proofs are in part based on a continuous wavelet analysis of the integrated function by means of Haar-type wavelets. Received 26 January 2001; in revised form 23 September 2001     

Approximation Methods by Regular Functions

This paper is a survey of approximation results and methods by smooth functions in Banach spaces. The topics considered in the paper are the following: approximation by polynomials by C^k-functions using the method of smooth partitions of unity, approximation by the fine topology, analytic approximation and regularization in Banach spaces using the infimal convolution method.     

Dynamics of Thermally-Insulated Nonequilibrium Stefan Problem

We study a two-phase Stefan problem with kinetics. Here we prove existence of a finite-dimensional attractor for the problem without heat losses. Fot the most part we use a more elegant technique of energetic type estimates in appropriately defined weighted Sobolev spaces as opposite to the parabolic potentials of [9]. We demonstrate existence of compact attractors in the Sobolev spaces and prove that the attractor consists of sufficiently regular functions. This allows us to show that the Hausdorff dimension of the attractor is finite.