On a problem of A. M. Vasil'ev

Vilnius Pedagogical University, Studentu 39, 2034 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 34, No. 3, pp. 388–407, July–September, 1994.     

On a problem of S.L. Sobolev

In his famous works of 1930 [1,2] Sergey L. Sobolev has proposed a construction of the solution of the Cauchy problem for the hyperbolic equation of the second order with variable coefficients in Rş. Although Sobolev did not construct the fundamental solution, his construction was modified later by Romanov (2002) and Smirnov (1964) to obtain the fundamental solution. However, these works impose a restrictive assumption of the regularity of geodesic lines in a large domain. In addition, it is unclear how to realize those methods numerically. In this paper a simple construction of a function, which is associated in a clear way with the fundamental solution of the acoustic equation with the variable speed in 3-d, is proposed. Conditions on geodesic lines are not imposed. An important feature of this construction is that it lends itself to effective computations.     

On a functional equation by Baak,Boo and Rassias

In Baak et al. (J Math Anal Appl 314(1):150–161, 2006) the authors considered the functional equation

$$\begin{aligned} r f\left( \frac{1}{r}\,\sum _{j=1}^{d}x_j\right)+ & {} \sum _{i(j)\in \{0,1\} \atop \sum _{1\le j\le d} i(j)=\ell }r f\left( \frac{1}{r}\,\sum _{j=1}^d (-1)^{i(j)}x_j\right) \\= & {} \left( {d-1\atopwithdelims ()\ell }-{d-1\atopwithdelims ()\ell -1} +1\right) \sum _{j=1}^{d} f(x_j) \end{aligned}$$

where \(d,\ell \in \mathbb {N}\), \(1<\ell <d/2\) and \(r\in \mathbb {Q}{\setminus }\{0\}\). The authors determined all odd solutions \(f:X\rightarrow Y\) for vector spaces X, Y over \(\mathbb {Q}\). In Oubbi (Can Math Bull 60:173–183, 2017) the author considered the same equation but now for arbitrary real \(r\not =0\) and real vector spaces X, Y. Generalizing similar results from (J Math Anal Appl 314(1):150–161, 2006) he additionally investigates certain stability questions for the equation above, but as for that equation itself for odd approximate solutions only. The present paper deals with the general solution of the equation and the corresponding stability inequality. In particular it is shown that under certain circumstances non-odd solutions of the equation exist.

     

On a problem of C. C. Chen and D. E. Daykin

Let α(k, p, h) be the maximum number of vertices a complete edge-colored graph may have with no color appearing more than k times at any vertex and not containing a complete subgraph on p vertices with no color appearing more than h times at any vertex. We prove that $\text{α(k, p, h) ≤ h + 1 + (k ? 1){(p ? h ? 1) × (h}{}^{\mathrm{p}}\text{+ 1)}}{}^{\mathrm{<mtext>1</mtext><mtext>h</mtext>}}$ and obtain a stronger upper bound for α(k, 3, 1). Further, we prove that a complete edge-colored graph with n vertices contains a complete subgraph on p vertices in which no two edges have the same color if

$\text{(}\text{n}\text{3}\text{)>(}\text{p}\text{3}\text{)}\text{Σ}\text{i=1}\text{t}\text{(}\text{e}{}_{\mathrm{i}}\text{2}\text{)}$

where e_i is the number of edges of color i, 1 ≤ i ≤ t.     

On a problem posed by S. Li and J. Liu

A subgroup H of a finite group G is said to be Hall normally embedded in G if there is a normal subgroup N of G such that H is a Hall subgroup of N. The aim of this note is to prove that a group G has a Hall normally embedded subgroup of order |B| for each subgroup B of G if and only if G is soluble with nilpotent residual cyclic of square-free order. This is the answer to a problem posed by Li and Liu (J. Algebra 388:1–9, 2013).     

On solving a D.C. programming problem by a sequence of linear programs

We are dealing with a numerical method for solving the problem of minimizing a difference of two convex functions (a d.c. function) over a closed convex set in ⁿ . This algorithm combines a new prismatic branch and bound technique with polyhedral outer approximation in such a way that only linear programming problems have to be solved.Parts of this research were accomplished while the third author was visiting the University of Trier, Germany, as a fellow of the Alexander von Humboldt foundation.     

On a problem of Sárközi

Periodica Mathematica Hungarica -     

Note on a problem of M. Talagrand

Answering a question of M. Talagrand, we show that there is a fixed L with the following property. For positive integers and , if is the set of subgraphs of K_n containing at least copies of K_k, then there is a set of subgraphs of K_n such that (i) each member of contains a member of and (ii) (where means number of edges). © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 663–668, 2015     

On M. G. Krein's contribution to the moment problem

This is a survey of M. G. Krein' s papers devoted to the moment problem.Translated from Ukrainskii Matematicheskii Zhumal, Vol. 46, Nos. 1–2, pp. 63–75, January–February, 1994.     

On M. V. Zaicev problem for Noetherian special Lie algebras

We solve the M. V. Zaicev problem in the following sense: Any Noetherian semiprime special Lie algebra is embedded into algebra of matrices over commutative ring which is the direct sum of fields.