Given a modulus of continuity $\omega$ and $q\in [1,\infty [$ then $H_q^{\omega }$ denotes the space of all functions $f$ with the period $1$ on $ℝ$ that are locally integrable in power $q$ and whose integral modulus of continuity of power $q$ (see(1)) is majorized by a multiple of $\omega$. The moduli of continuity $\omega$ are characterized for which $H_q^{\omega }$ contains “many” functions with infinite “essential” variation on an interval of length $1$.

[For the entire collection see Zbl 0742.00067.]Let ${𝔤}_k$ be the Lie algebra $𝔤l(k,𝒞)$, and let $U_k$ be the universal enveloping algebra for ${𝔤}_k$. Let $Z_k$ be the center of $U_k$. The authors consider the chain of Lie algebras ${𝔤}_n\supset {𝔤}_{n-1}\supset \cdots \supset {𝔤}_1$. Then $Z=\langle Z_k\mid k=1,2,\cdots n\rangle$ is an associative algebra which is called the Gel’fand-Zetlin subalgebra of $U_n$. A ${𝔤}_n$ module $V$ is called a $GZ$-module if $V={\sum }_x\oplus V\left(x\right)$, where the summation is over the space of characters of $Z$ and $V\left(x\right)=\{v\in V\mid {(a-x\left(a\right))}^{m}v=0$, $m\in {𝒵}_{+}$, $a\in 𝒵\}$. The authors describe several properties of $GZ$- modules. For example, they prove that if $V\left(x\right)=0$...

n the present paper the authors study some families of functions from a complex linear space $X$ into a complex linear space $Y$. They introduce the notion of $(j,k)$-symmetrical function ($k=2,3,\cdots$; $j=0,1,\cdots ,k-1$) which is a generalization of the notions of even, odd and $k$-symmetrical functions. They generalize the well know result that each function defined on a symmetrical subset $U$ of $X$ can be uniquely represented as the sum of an even function and an odd function.

The integrodifferential system with aftereffect (“heredity” or “prehistory”) dx/dt=Ax+-t R(t,s)x(s,)ds, is considered; here $\epsilon$ is a positive small parameter, $A$ is a constant $n\times n$ matrix, $R(t,s)$ is the kernel of this system exponentially decreasing in norm as $t\to \infty$. It is proved, if matrix $A$ and kernel $R(t,s)$ satisfy some restrictions and $\epsilon$ does not exceed some bound ${\epsilon }_{*}$, then the $n$-dimensional set of the so-called principal two-sided solutions $\tilde{x}(t,\epsilon )$ approximates in asymptotic sense the infinite-dimensional...

The concept of a $$-closed subset was introduced in [1] for an algebraic structure $=(A,F,R)$ of type $$ and a set $$ of open formulas of the first order language $L\left(\right)$. The set $C_{(})$ of all $$-closed subsets of $$ forms a complete lattice whose properties were investigated in [1] and [2]. An algebraic structure $$ is called $$- hamiltonian, if every non-empty $$-closed subset of $$ is a class (block) of some congruence on $$; $$ is called $$- regular, if $=𝔽$ for every two $$, $𝔽\in$ whenever they have a congruence class...

A compact set $K\subset {ℂ}^N$ satisfies Łojasiewicz-Siciak condition if it is polynomially convex and there exist constants B,β > 0 such that $V_K\left(z\right)\ge B{\left(dist(z,K)\right)}^{\beta }$ if dist(z,K) ≤ 1. (LS) Here $V_K$ denotes the pluricomplex Green function of the set K. We cite theorems where this condition is necessary in the assumptions and list known facts about sets satisfying inequality (LS).