**Cover**/ Титул (UA) [ (PDF) ]**Editorial Board**/ Редакційна колегія (UA).*P. 2.*[ (PDF) ]**Contents / Зміст (UA)**.*P.3.*[ (PDF) ]**A.L. Vyshnevetskiy,**Random walks on finite groups with conjugate class probability: algebraic approach.*P. 4-9.*[ Abstract ] [ Full-text available (PDF) ]**D.V. Sysoev,**A сondition for the existence of a unique equilibrium position of the Cauchy problem for linear matrix differential-algebraic equations.*P. 10-17.*[ Abstract ] [ Full-text available (PDF) ]**A.Ya. Khrystiyanyn, Dz.V. Lukivska,**Some generalizations of $p$-loxodromic functions.*P. 18-25.*[ Abstract ] [ Full-text available (PDF) ]**B.O. Chornomaz,**Lower bound on the number of meet-irreducible elements in extremal lattices,*P. 26-48.*[ Abstract ] [ Full-text available (PDF) ]**Abdon E. Choque-Rivero,**The Kharitonov theorem and robust stabilization via orthogonal polynomials,*P. 49-68.*[ Abstract ] [ Full-text available (PDF) ]

Under well known conditions an $n$-fold convolution of probability on finite group $G$ converges to the uniform probability on $G$ ($n\rightarrow\infty$). A lot of works estimate a rate of that convergence. The aim of the article is to obtain estimates of the rate for the probabilities that are constant on classes of conjugate elements of finite groups.

*Keywords:* probability; finite group; convergency.

2010 Mathematics Subject Classification: 20D99; 60B15; 60B10. [ Full-text available (PDF) ] Top of the page.

Sufficient conditions for the existence of a unique equilibrium position of the Cauchy problem for differential-algebraic equations are proposed. The paper proposes a constructive scheme of the equilibrium position in the Cauchy problem in the general case, when a linear operator $L,$ corresponding to homogeneous of the equation, has no inverse.

*Keywords:* differential-algebraic matrix equation; pseudoinverse matrix.

2010 Mathematics Subject Classification: 15A24; 34В15; 34C25. [ Full-text available (PDF) ] Top of the page.

The functional equation of the form $f(qz) = p(z)f(z), z \in \mathbb{C} \backslash \{0\}, q \in \mathbb{C} \backslash \{0\}, |q|<1.$ is considered. For certain fixed elementary functions $p(z)$, meromorphic solutions of this equation are found. These solutions are some generalizations of $p$-loxodromic functions and can be represented via the Schottky-Klein prime function as well as classic $p$-loxodromic functions.

*Keywords:* loxodromic function; $p$-loxodromic function; the Schottky-Klein
prime function.

2010 Mathematics Subject Classification: 30D30. [ Full-text available (PDF) ] Top of the page.

Extremal lattices are lattices maximal in size with respect to the number $n$ of their join-irreducible elements with bounded Vapnik-Chervonekis dimension $k$. It is natural, however, to estimate the size of a lattice also with respect to the number of its meet-irreducible elements. Although this number may differ for nonequivalent $(n, k+1)$-extremal lattices, we show that each $(n,k+1)$-extremal lattice has $k$ disjoint chains of meet-irreducible elements, each of length $n-k+2$.

*Keywords:* Extremal lattices; Vapnik-Chervonekis dimension; meet-irreducible
elements.

2010 Mathematics Subject Classification: 06B05; 05D99. [ Full-text available (PDF) ] Top of the page.

Kharitonov's theorem for interval polynomials is given in terms of orthogonal polynomials on $[0,+\infty)$ and their second kind polynomials. A family of robust stabilizing controls for the canonical system is proposed.

*Keywords:* Kharitonov theorem; orthogonal polynomials; Hurwitz polynomials;
stabilization of control systems.

2010 Mathematics Subject Classification: 34D20; 42C05; 30E05; 93D21. [ Full-text available (PDF) ] Top of the page.

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