Multiserial rings

We introduce the notion of multiserial (n-serial) rings and study their properties. The second-order minors of such rings are investigated. We also find all possible forms of quivers for Noetherian and hereditaryn-serial rings and describe the structure of semiprime and hereditaryn-serial rings. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 48, No. 9, pp. 1223–1235, September, 1996.     

On the construction of CDN[]-groups with elementary commutant of rank two

We describe certain CDN-groups of order p ⁿ with elementary commutant of rank two. Scientific-Research Pedagogic Institute, Ukrainian Pedagogic Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 10, pp. 1396–1403, October, 1997     

Nonlinear d'alembert equation in the pseudo-euclidean spaceR 2,n and its solutions

We investigate the nonlinear D'Alembert equation in the pseudo-Euclidean spaceR _2,n and construct new exact solutions containing arbitrary functions. Ukrainian University of Food Technology, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 52, No. 6, pp. 820–827, June, 2000.     

On the structure of incoming and outgoing subspaces for a wave equations in ℝ n

We investigate the structure of incoming and outgoing subspaces in the Lax-Phillips scheme for the classic wave equation in ℝ ⁿ . Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 5, pp. 708–712, May, 1999.     

Stability of one difference equation with positive coefficients

We consider a linear homogeneous difference equation of ordern with positive coefficients, which is presented in the normal form. We obtain necessary and sufficient conditions for the stability and asymptotic stability. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 9, pp. 1276–1280, September, 1990.     

Convergence of distributions of integral functionals of extremal random functions

We study the convergence of distributions of integral functionals of random processes of the formU _n (t)=b _n (Z _n (t)-a _n G(t)),t⃛T, where {X=X(t), t⃛T} is a random process,X _n ,n≥1, are independent copies ofX, andZ _n (t)=max_1≤k≤n X _k (t). Ukrainian State Academy of Light Industry, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 9, pp. 1201–1209, September, 1999.     

A modular integer GCD algorithm

This paper describes the first algorithm to compute the greatest common divisor (GCD) of two n-bit integers using a modular representation for intermediate values U, V and also for the result. It is based on a reduction step, similar to one used in the accelerated algorithm [T. Jebelean, A generalization of the binary GCD algorithm, in: ISSAC '93: International Symposium on Symbolic and Algebraic Computation, Kiev, Ukraine, 1993, pp. 111–116; K. Weber, The accelerated integer GCD algorithm, ACM Trans. Math. Softw. 21 (1995) 111–122] when U and V are close to the same size, that replaces U by (U−bV)/p, where p is one of the prime moduli and b is the unique integer in the interval (−p/2,p/2) such that . When the algorithm is executed on a bit common CRCW PRAM with O(nlognlogloglogn) processors, it takes O(n) time in the worst case. A heuristic model of the average case yields O(n/logn) time on the same number of processors.     

On differentiability of mappings of finite-dimensional domains into Banach spaces

The well-known Stepanov criterion of the differentiability (approximate differentiability) of real functions is generalized to mappings of subsets ofR ⁿ into Banach spaces satisfying the Rieffel sharpness condition, in particular, reflexive Banach spaces. For Banach spaces that do not satisfy the Rieffel sharpness condition, this criterion is not true. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 1, pp. 3–11, January, 1999.     

Equilibrium in quantum systems of particles with magnetic interaction. Fermi and bose statistics

Quantum systems of particles interacting via an effective electromagnetic potential with zero electrostatic component are considered (magnetic interaction). It is assumed that the j th component of the effective potential for n particles equals the partial derivative with respect to the coordinate of the jth particle of “magnetic potential energy” of n particles almost everywhere. The reduced density matrices for small values of the activity are computed in the thermodynamic limit for d-dimensional systems with short-range pair magnetic potentials and for one-dimensional systems with long-range pair magnetic interaction, which is an analog of the interaction of three-dimensional Chern-Simons electrodynamics (“magnetic potential energy” coincides with the one-dimensional Coulomb (electrostatic) potential energy). Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Published in Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 5, pp. 691–698, May 1997.     

Investigation of one class of diophantine equations

We consider the problem of existence of solutions of the equation in natural numbers for differentm∈N. We prove that this equation possesses solutions in natural numbers form=a ²+5,a∈Z, and does not have solutions ifm=4p ²,p∈N, andp is not divisible by 3. We also prove that, forn≥12, the equation

Periodic solutions of a weakly nonlinear system of partial differential equations with pulse influence

We establish conditions of the existence of solutions periodic in t with period T for a weakly nonlinear system of partial differential equations with pulse influence. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 4, pp. 601–605, April, 1997.     

Interpolation of nonlinear functionals by integral continued fractions

We constructively prove the theorem of existence of an interpolation integral chain fraction for a nonlinear functionalF:Q[0,1]→R ¹. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 3, pp. 364–375, March, 1999.     

Singly generatedC *-algebras

We consider aC ^*-algebraA generated byk self-adjoint elements. We prove that, for , the algebraM _n (A) is singly generated, i.e., generated by one non-self-adjoint element. We present an example of algebraA for which the property thatM _n (A) is singly generated implies the relation . Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 8, pp. 1136–1141, August, 1999.     

On the theory of groups with generalized minimality condition for closed subgroups

We prove that a topological Abelian locally compact group with generalized minimality condition for closed subgroups is a group of one of the following types: 1) a group with minimality condition for closed subgroups, 2) an additive group of theJ _p -ring of integerp-adic numbers, 3) an additive groupR _p of the field ofp-adic numbers (p is a prime number). Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 3, pp. 398–409, March, 1999.     

Some properties of generalized associated Legendre functions of second kind

We investigate some new properties of generalized associated Legendre polynomials of the second kind, establish new relationships between these polynomials, construct differential operators with the functions P _k ^m,n (z), Q _k ^m,n (z), and describe some applications.Kiev Polytechnical Institute. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 68, pp. 34–45, 1989.     

Optimal stopping times for solutions of nonlinear stochastic differential equations and their application to one problem of financial mathematics

We solve the problem of finding the optimal switching time for two alternative strategies at the financial market in the case where a random processX _t ,t ∈ [0, T], describing an investor's assets satisfies a nonlinear stochastic differential equation. We determine this switching time τ∈[0,T] as the optimal stopping time for a certain processY _t generated by the processX _t so that the average investor's assets are maximized at the final time, i.e.,EX _T . Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 6, pp. 804–809, June, 1999.     

Distribution of the supremum of random processes from quasi-BanachK σ-spaces

We study random processes from quasi-BanachK _σ-spaces of random variables whose domain of definition is not necessarily a compact set. We establish conditions under which the supremum of a properly normalized process belongs to the same space as the process itself. We also obtain estimates for the norm of this supremum. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 7, pp. 918–930, July, 1999.     

On the best approximation of functions ofn variables

We propose a new approach to the solution of the problem of the best approximation, by a certain subspace for functions ofn variables determined by restrictions imposed on the modulus of, continuity of certain partial derivatives. This approach is based on the duality theorem and on the representation of a function as a countable sum of simple functions. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 10, pp. 1352–1359, October, 1999.     

Decompositions of direct products of groups

We propose a new method for the decomposition of direct products of groups into U subsets. By using this method, we prove the following generalization of the Comfort-van Mill theorem: An arbitrary nondiscrete topological Abelian group with a finite number of second-order elements can be decomposed into a countable number of dense subsets. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 10, pp. 1385–1395, October, 1997.     

Limiting process for integral functionals of a wiener process on a cylinder

We prove that integral functionals, whose integrands are bounded functions of a Wiener process on a cylinder, weakly converge to the processw ₁(τ(t)), τ(t) = β₁ t + (β₂ − β₁)mes {s:w ₂(s)≥0,s<t}, wherew ₁(t andw ₂(t) are independent one-dimensional Wiener processes, β₁ and β₂ are nonrandom values, and β₂≥β₁≥0. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 6, pp. 765–768, June, 1994.