On Martingale Approximation of Adapted Processes

We show that the existence of a martingale approximation of a stationary process depends on the choice of the filtration. There exists a stationary linear process which has a martingale approximation with respect to the natural filtration, but no approximation with respect to a larger filtration with respect to which it is adapted and regular. There exists a stationary process adapted, regular, and having a martingale approximation with respect to a given filtration but not (regular and having a martingale approximation) with respect to the natural filtration.     

Investigation of velocity dependence of rolling frictions as a means of studying adhesion

Investigation of the rolling of two cylinders on the device described in this paper has shown that a study of the rolling friction of bodies coated with polymer films offers a means of investigating adhesion processes. It has been found that the resistance to rolling of bodies coated with nonadhesive polymers has a tendency to increase with increase in velocity, in accordance with the electrical theory of adhesion. The resistance to rolling of bodies coated with adhesive polymers passes through a maximum as the velocity increases and then decreases, in accordance with the diffusion theory of adhesion.Mekhanika Polimerov, Vol. 1, No. 4 pp. 117–122, 1965     

Renormalization group study of the melting of a two-dimensional system of collapsing hard disks

We consider the melting of a two-dimensional system of collapsing hard disks (a system with a hard-disk potential to which a repulsive step is added) for different values of the repulsive-step width. We calculate the system phase diagram by the method of the density functional in crystallization theory using equations of the Berezinskii–Kosterlitz–Thouless–Halperin–Nelson–Young theory to determine the lines of stability with respect to the dissociation of dislocation pairs, which corresponds to the continuous transition from the solid to the hexatic phase. We show that the crystal phase can melt via a continuous transition at low densities (the transition to the hexatic phase) with a subsequent transition from the hexatic phase to the isotropic liquid and via a first-order transition. Using the solution of renormalization group equations with the presence of singular defects (dislocations) in the system taken into account, we consider the influence of the renormalization of the elastic moduli on the form of the phase diagram.     

The effect of discretization on the numerical simulation of the vibrations of the impacting cantilever beam

We study the dynamics of a cantilever beam with an unnegligible large mass and with a concentrated mass fixed at the end, which impacts on the base during motion. Generally to model such a system, the finite element method with appropriate number of degrees of freedom has to be employed. However, to analyse some selected aspects of its dynamic behaviour, particularly to predict if the motion with impacts will be periodic, lower-dimensional substitutive models with one degree or two degrees of freedom can be employed. The way to determine the parameters of such models and their applicability limits are discussed.     

The modulus of unfoldings of cusps in conformal geometry

In this paper we give the complete classification of generic 1-parameter unfoldings of germs of real analytic curves with a cuspidal point under conformal equivalence. A cusp is obtained by squaring an analytic curve having contact of order 1 with a line through the origin. We show that this point of view can be extended to the unfolding. This allows to reduce the classification of unfoldings of cusps to the classification of unfoldings of a pair of curves having a contact of order 1 at the origin, one being obtained from the other through a reflection with respect to the origin. This unfolding can be studied in the same way as an unfolding of a curvilinear angle with zero angle, called a horn. We then classify the unfoldings of the special horns corresponding to cusps by means of the associated diffeomorphisms. We interpret the results geometrically.     

The problem of the formalization of constructive topology

Formal topologies are today an established topic in the development of constructive mathematics. One of the main tools in formal topology is inductive generation since it allows to introduce inductive methods in topology. The problem of inductively generating formal topologies with a cover relation and a unary positivity predicate has been solved in [CSSV]. However, to deal both with open and closed subsets, a binary positivity predicate has to be considered. In this paper we will show how to adapt to this framework the method used to generate inductively formal topologies with a unary positivity predicate; the main problem that one has to face in such a new setting is that, as a consequence of the lack of a complete formalization, both the cover relation and the positivity predicate can have proper axioms.     

An algorithm of the method of difference potentials for domains with cuts

The method of Difference Potentials (DPM) is applied to solving a Dirichlet problem for the Laplace equation in a square with a cut. The DPM approach has been modified to achieve a more efficient numerical algorithm with respect to computational time. The considered problem can be a prototype for other problems formulated in domains with cuts including elastic problems related to cracks.     

Estimation of fractal dimension of random fields on the basis of variance analysis of increments

This paper deals with estimation of fractal dimension of realizations of random fields. Numerical methods are based on analysis of variance of increments. It is proposed to study fractal properties with the use of a specific characteristic of randomfields called a “variational dimension.” For a class of Gaussian fields with homogeneous increments the variational dimension converges to the Hausdorff dimension. Several examples are presented to illustrate that the concept of variational dimension can be used to construct effective computational methods.     

Duality of a Nonconvex Sum of Ratios

For mathematical programs with objective involving a sum of ratios of affine functions, there are few theoretical results due to the nonconvex nature of the program. In this paper, we derive a duality theory for these programs by establishing their connection with geometric programming. This connection allows one to bring to bear the powerful theory and computational algorithms associated with geometric programming.     

Application of a spectral method to numerical modeling of the propagation of seismic waves in porous media with energy dissipation

This paper presents an algorithm based on the spectral Laguerre method for approximation of time derivatives as applied to a problem of seismic wave propagation in porous media with energy dissipation. The initial system of equations is written as a first-order hyperbolic system in terms of velocities, stresses, and pore pressure. To numerically solve the problem, a combination of an analytical Laguerre transform and a finite-difference method is used. The method proposed in the paper is an analog of a well-known spectral method based on the Fourier transform. However, unlike the Fourier transform, the integral Laguerre transform with respect to time reduces the initial problem to a system of equations in which the expansion parameter is present only in the right-hand side of the equations as a recurrence relation. As compared to the finite-difference method, with an analytical transform in the spectral method it is possible to reduce the original problem to a system of differential equations having only derivatives with respect to the spatial coordinates. This allows using the known stable difference scheme for the recurrence solutions to similar systems. Such an approach is effective when solving dynamic problems for porous media. Because of the presence of a second longitudinal wave with low velocity, the use of difference schemes in all the coordinates to obtain stable solutions requires a small step consistent both with respect to time and space, which inevitably increases the execution time.     

Execution of active reduction of vibrations by means of kinematic excitations

The major objective of this study is to show how to carry out the process of structural and parametrical synthesis that is understood as a method intended to seek for the structure and parameters of a discrete model of the system with active suppression of vibrations. Active components are implemented as mechanical components with kinematic excitations. The other objective is to analyze how the active subsystem interacts with the overall investigated structure. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Deformations of algebras and cohomology of fixed point sets

In [13] it is shown that under certain conditions the cohomology algebra of the fixed point set of a space with group action is in an algebraic sense a deformation of the cohomology algebra of the space itself. Here we attempt to prove a converse of the above statement, i.e. we try to realize geometrically a given algebraic deformation of a (commutative) graded algebras as the cohomology algebra of the fixed point set of a suitable space with group action. The first part of this note in a sense reduces this realization problem in equivariant topology to a non-equivariant problem while the second part uses Sullivan's theory of minimal models to actually obtain a converse for S¹-actions, where cohomology is taken with rational coefficients.     

Estimation of statistical characteristics of attainability sets of controllable systems

We study an expansion of the notion of invariance for sets with respect to controllable systems and differential inclusions. Namely, we study statistically invariant sets and statistical characteristics of attainability sets of controllable systems. We obtain a lower bound for the lower relative frequency of the absorption of the attainability set of a system by a given set and establish new sufficient conditions of the statistical invariance of the set with respect to the controllable system. We give examples of the calculation of statistical characteristics for the linear Cauchy problem and a linear controllable system with almost periodic coefficients.     

Application of the method of direct cutting-out to the solution of the problem of longitudinal shear of a wedge with thin heterogeneities of arbitrary orientation

The method of direct cutting-out consists of modeling of a finite body, in particular, with thin heterogeneities, using a much simpler problem for a bounded or a partially bounded body with thin heterogeneities located in the same manner and the presence of additional cracks or absolutely rigid inclusions of fairy large length, which are modeled by the boundary conditions of a bounded body. The method is tested on the problems of antiplane deformation of a symmetrically loaded crack in a wedge with free faces and an absolutely rigid inclusion placed with some tension in a wedge with restrained faces. For an elastic inclusion, we construct generalized conditions of interaction, which enable us to unify the procedure of giving different boundary conditions in the case of using the method of direct cutting-out.     

Partial uniform boundedness of solutions of systems of differential equations with partly controlled initial conditions

We introduce the notion of partial uniform boundedness of solutions with partially controlled initial conditions in the general case, that is, the case in which the part of variables with respect to which the boundedness of solutions is studied is a subset of the part of variables with respect to which the initial conditions are controlled. We obtain a criterion for the partial uniform boundedness of solutions with partly controlled initial conditions. We introduce the notion of partial total boundedness of solutions with partly controlled initial conditions. We obtain a sufficient condition for the partial total boundedness of solutions with partly controlled initial conditions.     

Subspaces and quotients of Banach spaces with shrinking unconditional bases

The main result is that a separable Banach space with the weak* unconditional tree property is isomorphic to a subspace as well as a quotient of a Banach space with a shrinking unconditional basis. A consequence of this is that a Banach space is isomorphic to a subspace of a space with a shrinking unconditional basis if and only if it is isomorphic to a quotient of a space with a shrinking unconditional basis, which solves a problem dating to the 1970s. The proof of the main result also yields that a uniformly convex space with the unconditional tree property is isomorphic to a subspace as well as a quotient of a uniformly convex space with an unconditional finite dimensional decomposition.     

Stability of a linear difference equation and estimation of its fundamental solution

We consider a linear nonautonomous system of difference equations. Under natural restrictions we prove that the stability with respect to the initial function and that with respect to the right-hand side in spaces of bounded and summable functions are equivalent to the corresponding estimates for the fundamental solution. As a consequence, we obtain an effective stability condition for a system with a constant matrix.