Vol 21, No 1 (2016)

We formulate and prove the theorem on well-posedness of abstract Volterra equations in metric spaces. We consider nonlinear nonlocal integral Volterra equation involving generalizing equations typically used in mathematical neuroscience. We investigate solutions that tend to zero at any moment with unbounded growth of the spatial variable. In the literature such solutions are called «spatially localized solutions» or «bumps». They correspond to normal brain functioning. For the main equation, we consider an impulsive control problem, where the control parameters are moments, when the solution discontinue, and corresponding jumps’ values. Such control systems model electrical stimulation used in the presence of various diseases of central nervous system. We define suitable complete metric (not linear) space. Using the aforementioned theorem, we obtain conditions for existence and uniqueness of solution to this equation and continuous dependence of this solution on the control.

The continuity of the measure Lagrange multiplier from the maximum principle for control problems with state constraints is investigated. It is proved, under certain regularity assumptions, that the distribution function of the measure is H¨older continuous with exponent 1/2.

This is a first paper in the series devoted to singularities of geodesic flows in generalized Finsler (pseudo-Finsler) spaces. Geodesics are defined as extremals of a certain auxiliary functional whose non-isotropic extremals coincide with extremals of the action functional. This allows to consider isotropic lines as (unparametrized) geodesics, similarly to pseudo-Riemannian metrics. In the next forthcoming paper we study generic singularities of sodefined geodesic flows in the case when the pseudo-Finsler metric is given by a generic 3-form on a two-dimensional manifold.

A stable solution to the problem of the continuation of potential fields with non-planar surfaces under non-periodic models was obtained.

The paper is devoted to transmutation theory for ordinary and partial differential operators. It has many applications to inverse problems, spectral theory, nonlinear and soliton problems, singular equations, generalized analytic functions, singular boundary-value problems, fractional integrals and function spaces embedding. In this survey paper the most important classes of transmutations in modern theory is introduced and the new method is proposed to find transmutations via integral transform compositions.

The simultaneous invertibility (well-posedness) of linear differential operators with partial derivatives in the Besov type functional spaces is studied.

The problems related to calculation of derivatives of interval-specified functions are considered. These problems are relevant in study of systems with any level of uncertainty (nondeterministic systems). Specifically we speak about simple systems described by elementary interval-specific functions. Accordingly we solved problem of calculating derivatives of elementary interval-specified functions. Previously obtained formulas and methods of finding derivatives of any intervally defined functions are used. Basic definitions related to derivatives of the interval-specified functions are given and formulas of two types that allow you to calculate interval derivatives are presented. The first type formulae express derivatives in the closed interval form, which requires to compute using the apparatus of interval mathematics. But formulae of the second type can express derivatives in the open interval form, i. e. in form of two formulas. Formulas above expresses the lower and the upper limits of the interval representing the derivative. Here finding the derivative of interval-defined function is reduced to computation of two ordinary certain functions. Using above mathematical apparatus we find the derivatives of all elementary interval functions: interval constant, interval power function, interval exponential function, interval logarithmic function, interval natural-logarithmic function, interval trigonometric functions (sine, cosine, tangent, cotangent), interval inverse trigonometric functions (arcsine, arccosine, arctangent and arccotangent). Formulae of all the derivatives are shown in form of an open interval. The difference between derivatives of interval elementary functions and the derivatives of normal (i.e. non-interval) elementary functions is discussed.

Data protection system Terminus, which was aimed at protection data from unsanctioned access is described. This protection includes mandate and discrete division.