MULTIPLICATIVE APPROXIMATION OF A RANDOM PROCESS

In this paper we consider the stochastic Ito differential equation in an infinite-dimensional real Hilbert space. Using the method of multiplicative representations of Daletsky - Trotter, its approximate solution is constructed. Under classical conditions on the coefficients, there is a single to the stochastic equivalence of solutions of the stochastic equation, which is a random process. This development generates an evolutionary family of resolving operators by the formula x(t)= S(t, Construct the division of the segment by the points. An equation with time-uniform coefficients is considered on each elementary segment . There is a single solution of this equation on the elementary segment, which generates the resolving operator by the formula The multiplicative expression is constructed. Using the method of Dalecki-Trotter multiplicative representations, it is proved that this multiplicative expression is stochastically equivalent to the representation generated by the solution of the original equation. This means that the specified multiplicative expression is respectively a representation of the solution of the original equation. That is, the probability of one coincides with the solution of the original stochastic equation. It should be noted that this is possible under additional conditions for the coefficients of the equation. These conditions are the time continuity of the coefficients of the equation. Thus, the constructed multiplicative representation can be interpreted as an approximate solution of the original equation. This method of multiplicative approximation makes it possible to simplify the study of the corresponding random process both at the elementary segment and as a whole. It is known, that the solution of a stochastic equation by a known formula generates a solution of the inverse Kolmogorov equation in the corresponding space. This scheme of multiplicative approximation can be transferred to the solution of the parabolic equation, which is the inverse Kolmogorov equation. Thus, the method of multiplicative approximation makes it possible to simplify the study of both stochastic equations and partial differential equations.

Parametric Study of the Elementary Segment of an Antiprismatic Pop-Up Deployable Lattice Column

In this article the primary segment of an antiprismatic pop-up mast is analyzed, that can be applied for largely flexible architectural designs, like deployable bridges or transportable look-out towers. This deployable column, consisting of rigid plates, rigid and elastic bars, is characterized by its selfdeploying behavior due to the energy accumulated from lengthening the elastic bars during packing. The main goal of this paper is to prepare the analysis of the complex structure by a herein detailed investigation of the behavior of one, basic element of the deployable mast. After the analytical examination of the general behavior of the basic segment a geometrically nonlinear finite element formulation is used to trace the force-displacement diagram. Besides the parametric study, approximations of main mechanical parameters are herein given for facilitating preliminary design of such deployable structures.