Scientific Notes of

Taurida National V.I. Vernadsky University

Taurida National V.I. Vernadsky University

Series

Physical and Mathematical Sciences

**Vol. 27 (66), No. 1. 2014****Physics and Mathematics Sciences.**

I**I****.****V****.**** Baran**

Mean value theorem and Taylor formula for symmetric derivatives and symmetric K-subdifferentials

**Y****.****V****.**** Bogdanski, Ya.Yu. Sanzharevskiy**

Laplacian on measure and the Dirichlet problem

*It was proposed**Laplace operator version on functions on a separable real Hilbert space H**(dimH 6 ∞) that is generated by the (non-negative finite Borel) measure μ**defined on H. It was studied both of existence and uniqueness of solutions**(including "weak" ones) of the Dirichlet peoblem for the elliptic equation in**a region G that is agreed with an initial measure μ. It was given an example of**agreeing of a measure μ with a region G.*

**E.V. Bozhonok, E.M. Kuz'menko**

Classes of variational functionals having nonlocal *K*–extremum in *W*1*,p*(*D*) on multi-dimensional domain

*In this paper the investigation scheme of nonlocal compact extremum at zero for**variational functional in Sobolev space **W*1*,**p*(*D*)*, **p **∈ *N*, on multi-dimensional**compact domain **D **⊂ *R*N**, **N **∈ *N*, is derived. Some examples of variational**functionals having nonlocal **K**–extremum are considered.*

**E****.****L****.**** Haziyev**

Spectral problem with transmission conditions on curvilinear interface

*This paper deals with a spectral problem arising in a problem of small motions of**a system "ideal capillary fluid–gas" in a rectangular vessel with the solid walls.**We suppose that a gas density is exponentially stratified opposite to the direction**of gravitational forces and conjugation conditions for potential displacement are**formulated on the fluid surface which is not horizontal at rest. A projection**method for finding a generalized solution is offered.*

**N****.****D****.**** Kopachevskii, K****.****A****.**** Radomirskaya**

Abstract mixed boundary and spectral transmission problems

*On the basis of the abstract Green’s formula for the triple of Hilbert spaces**and the generalized Green’s formula for the Laplace operator we consider a new**class of mixed boundary value and spectral transmission problems.*

**I****.****V****.**** Melnikova, O****.****S****.**** Starkov**

Weak and generalized solutions of the abstract cauchy problem

*We**consider three types of solutions (weak, generalized with respect to **t **and with**respect to a random variable) for the infinite dimensional stochastic Cauchy**problem **X**′*(*t*) = *AX*(*t*) + *B*W(*t*)*, t **≥ *0*,X*(0) = *ζ, **with **A **being the generator**of a regularized semigroup in a Hilbert space **H **and a white noise *W *in another**Hilbert space *H*, **B **∈ L*(H*,H*)*. It is proved coincidence of the solutions under**the conditions they exist.*

**E****.****V****.**** Siomkin****а**

Spectral problem associated with a Cauchy problem on small motions of a dissipative dynamical system

**N****.****G****.**** Soldatov****а**

On the solution of three-criteria problem under uncertainty

*In this paper the definition of guaranteed solution for three-criteria problem**under uncertainty is introduced. New solution is based on the method of**decision-making in hierarchical two-level game. The conditions of existence are**formulated. The example is given.*

**F****.****S****.**** Stonyakin**

Sequential version of Uhl Theorem on convexity and compactness
for vector measure range.

**F****.****S****.**** Stonyakin, R****.****O****.**** Shpilev**

Analog Lyapunov convexity theorm for e-quasimeasures and its application to fair division problem.

**Z.I. Khalilov****а**

Extreme variational problems with subsmooth integrand

**A****.****V****.**** Tsygankova**

Elimination of Jacobi equation in variational problems with non-smooth integrand

**V. Zhukovsky, P****.****K****.**** Ahrameev**

Guaranteed on risk solution in problem of sum distribution into
three deposits (in rubles, dollars and euros)

*We are looking at problems**of sum distribution (in rubles) into three deposits (in rubles, dollars and euros)**in order to obtain the maximum annual income (in terms of rubles). The**decision maker (investor) does not know the real dollar and the euro rates at**the end of the year, and has to look at some possible rate boundaries or limits**for them. The solution of this problem depends on the readiness of the decisionmaker**to take risks. The contents of this article are the ways to construct the**decisions of guaranteed risks onto account. This paper is actually devoted to**constructing of a guaranteed on risk solution.*

**V. Zhukovsky, M****.****I****.**** Vysokos**

Guaranteed in outcomes and risks solution for single-criterion problem

*Concept of Pareto-guaranteed in outcomes and risks solution for singlecriterion**problem under uncertainty is proposed. It is based on a modification**of the maximin. Explicit solution for the problem of a single contribution to the**diversification of ruble and foreign currency deposit is found.*

**A. V. Dereza**

Definition of Time Component Petri net for Different ways of it Construction

**A. S. Gorbatov, V. I. Zhukovskiy**

About Deposit Diversification Problem

**О. S. Kisel, J. S. Pashkova**

Comparison of Orlicz, Lorentz and Orlicz- Lorentz Spaces

**V. I. Voytitsky, D. A. Zakora**

On the Spectral Properties of Some Auxiliary Boundary Value Problems from Theory of Metamaterials

**V. I. Zhukovskiy, S. N. Sachkov, L. V. Smirnova**

Existence of Berge Equilibrium in Mixed Strategies