The classical Bertrand-Darboux problem

The well-known problem of classical mechanics considered by Bertrand (1857) and Darboux (1901) is reviewed in the context of Cartan’s geometry. __________ Translated from Fundamentalnaya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 12, No. 7, pp. 231–250, 2006.     

The classical overdetermined Serrin problem

Abstract

In this survey, we consider the classical overdetermined problem which was studied by Serrin in 1971. The original proof relies on Alexandrov’s moving plane method, maximum principles, and a refinement of Hopf’s boundary point Lemma. Since then other approaches to the same problem have been devised. Among them we consider the one due to Weinberger which strikes for the elementary arguments used and became very popular. Then we discuss also a duality approach involving harmonic functions, a shape derivative approach and a purely integral approach, all of them not relying on maximum principle. For each one, we consider pros and cons as well as some generalizations.     

THE SPECTRALITY DECISION PROBLEM

An effcient algorithm for deciding whether a given integer vector is thespectrum of some Boolean function is presented.The algorithm performs astep-by-step spectral decomposition of the input vector and checks at eachstep a set of necessary conditions for spectrality for the resulting vectors.The algorithm concludes that the input vector cannot lead to a valid Booleanfunction as soon as a vector not satisfying the conditions is found,which,as proved in the paper,for almost all cases happens after the first step ofthe decomposition.     

The stechkin problem for partial derivation operators on classes of finitely smooth functions

In this paper we find the order of the best value of approximation by operators from one space of functions with integral norm to another such space on classes of finitely smooth functions of partial derivation operators with bounded norm. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 77–86, January, 2000.     

The structure of social decision functions

This paper investigates possible extensions of the Arrow Impossibility Theorem by relaxing the collective rationality condition from transitivity to acyclicity. It is shown that some elements of dictatorship must persist even when two of Arrow's conditions are dropped (namely the Pareto principle and transitivity of social preferences).     

The classical moment problem: Hilbertian proofs

In view of its connection with a host of important questions, the classical moment problem deserves a central place in analysis. It is usually treated by methods striking in their virtuosity, but difficult to motivate. Here we describe an elementary approach which establishes the structural results and exposes the Hilbert space origins of the arguments.     

On the problem of the ergodic constant in the theory of classical Green's functions

Moscow Institute of Electronic Engineering. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 84, No. 3, pp. 459–473, September, 1990.     

The completion problem for partial packings

Packings (resolutions) of designs have been of interest to combinatorialists in recent years as a way of creating new designs from old ones. Line packings of projective 3-space were the first packings studied, but it is still unknown when a partial packing can be completed to a packing in this case. In this paper we show that there is no guarantee from a combinatorial point of view of completing such a partial packing even when the deficiency is 2. In particular, we construct for every odd prime power q a set of 2(q ²+1) lines which doubly cover the points of PG(3,q) and yet cannot be partitioned into two spreads (resolution classes). The method is based on manipulations of primitive elements of finite fields.This work was supported in part by the National Research Council of Canada.     

The completeness problem in partial hyperclones

The composition-closed sets of partial multi-valued operations, called partial hyperclones, defined on the finite set are investigated. It is shown that the lattice of all partial hyperclones is dually atomic, i.e., any non-full partial hyperclone is contained in a maximal partial hyperclone. Based on it some completeness criteria in the full partial hyperclone are established. Next the total list of maximal restriction-closed partial hyperclones is obtained and, thus, the completeness problem with respect to compositions and restrictions of partial hyperoperations is solved.     

The numerical solution of one-phase classical Stefan problem

In this paper, variable space grid and boundary Immobilisation Techniques based on the explicit finite difference are applied to the one-phase classical Stefan problem. It is shown that all the results obtained by the two methods are in good agreement with the exact solution, and exhibit the expected convergence as the mesh size is refined.     

Toeplitz matrices and classical and q-Bessel functions

We use biinfinite Toeplitz matrix analogues of classical and q-binomial identities in a commutative Banach algebra setting to characterize classical and q-Bessel functions of integer order and to establish properties of these functions.     

The interpolation problem for ridge functions

It is shown that the interpolation problem for ridge functions can be solved if and only if the rank of a certain matrix A equals the number of interpolation points. The elements of the matrix A are either 0 or 1 and can be easilyfound from the arguments of the unknown functions. It is shown that Sun's Characteristic, or incidence matrix C is given by C = AA ^T . From this it follows that the rank condition is equivalent to Sun's positive definite C condition.     

The Dirichlet problem for Baire-one functions

Let X be a compact convex set and let ext X stand for the set of all extreme points of X. We characterize those bounded function defined on ext X which can be extended to an affine Baire-one function on the whole set X.