A problem on triples

If a system H of triples (3-uniform hypergraph) on n vertices has the following property: for every 3-coloring of the vertex-set there exists a 3-colored triple, what is the minimum size (S(n)) of H? The first values of S(n) are computed and the asymptotic behaviour of this function is studied.     

A problem on brachistochrone as invariant variational problem

Based on the modified theory of invariant variational problems developed by the author, we describe a theoretical group approach to the problem of brachistochrone.     

On finite-time blow-up for a nonlocal parabolic problem arising from shear bands in metals

Results on finite-time blow-up of solutions to the nonlocal parabolic problem

are established. They extend some known results to higher dimensions.

     

A note on Fermat's problem

The General Fermat Problem asks for the minimum of the weighted sum of distances fromm points inn-space. Dozens of papers have been written on variants of this problem and most of them have merely reproduced known results. This note calls attention to the work of Weiszfeld in 1937, who may have been the first to propose an iterative algorithm. Although the same algorithm has been rediscovered at least three times, there seems to be no completely correct treatment of its properties in the literature. Such a treatment, including a proof of convergence, is the sole object of this note. Other aspects of the problem are given scant attention.This paper was written while the author was Science Faculty Fellow of the National Science Foundation at the London School of Economics.     

A problem on convex functions

A conjecture is made on convex functions. It leads to the problem of characterizing a class of convex functions, which is of interest both from the theoretical point of view and in the field of minimization methods.     

A problem on random trees

The problem is considered of determining the distribution of the number of nodes in any given component of a forest obtained by removing edges from a random tree.     

A remark on Nehari's problem

An example of a strickly contractive Hankel matrix is given such that the central contractive extensionf _c of satisfies f _c =1. This way we answer a problem raised by Ciprian Foias.Partially supported by a Georgia State University Research Grant.     

A note on Chern-Yamabe problem

We propose a flow to study the Chern-Yamabe problem and discuss the long time existence of the flow. In the balanced case we show that the Chern-Yamabe problem is the Euler-Lagrange equation of some functional. The monotonicity of the functional along the flow is derived. We also show that the functional is not bounded from below.     

A problem on the conventional well-posedness of a multipoint problem

Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 30, No. 4, pp. 40–43, July–August, 1989.     

带簇及矩形带簇的推广

本将带簇和矩形带簇的概念推广到了n-带簇εn=[x^n 1=x^n]和n-矩形带簇εn=[x^nyx^n=x^n]，讨论了这两种半群簇之间的关系，特别对所有n-带簇的并集ε=∪εn和所有n-矩形带簇的并集Rε=∪Rεn进行了研究．     

A note on maximal subgroups of free idempotent generated semigroups over bands

We prove that all maximal subgroups of the free idempotent generated semigroup over a band B are free for all B belonging to a band variety V if and only if V consists either of left seminormal bands, or of right seminormal bands.