Mathematical description of surface effects in deformable solid bodies

We consider the general balance equations for the interface surface of two media and the interface line of three media, and we analyze the generalized Laplace and Young equations and study the balance equations for surface dislocations and disclinations. We establish the relations between the densities and flows of surface defects at the junction of three grain boundaries.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 42–47.     

Continuity equations for a deformable solid

For three-dimensional problems of the mechanics of a deformable solid in a Cartesian coordinate system, we show that there are only three continuity equations in terms of deformations and not six, as was considered earlier. We obtain continuity equations of the integro-differential type. They are reduced to the corresponding three of the six well-known Saint-Venant continuity differential equations only if the necessary conditions of consistency between displacements and deformations at the domain boundary are satisfied. As a results, the problem of closedness of the equations of elasticity and thermoelasticity in terms of stresses for three-dimensional problems as well as the equations of the mechanics of a deformable solid is finally solved as a whole. Pidstryhach Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences, L'viv. Translated from Matermatychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 117–123, April–June, 1998.     

Poisson point processes with exclusion

Summary We give the definition of Poisson point processes with exclusion by their local conditional distributions, treat the existence and uniqueness problem and their applications in percolation theory.     

Sample-path analysis of processes with imbedded point processes

The purpose of this paper is to review, unify, and extend previous work on sample-path analysis of queues. Our main interest is in the asymptotic behavior of a discrete-state, continuous-time process with an imbedded point process. We present a sample-path analogue of the renewal-reward theorem, which we callY=X. We then applyY=X to derive several relations involving the transition rates and the asymptotic (long-run) state frequencies at an arbitrary point in time and at the points of the imbedded point process. Included are sample-path versions of the rate-conservation principle, the global-balance conditions, and the insensitivity of the asymptotic frequency distribution to the distribution of processing time in a LCFS-PR service facility. We also provide a natural sample-path characterization of the PASTA property.The research of this author was partially supported by the U.S. Army Research Office, Contract DAAG29-82-K-0151 at N.C. State University, and by the National Science Foundation, Grant No. ECS-8719825, at the University of North Carolina, Chapel Hill.The research of this author was partially supported by the U.S. Army Research Office, Contract DAAG29-82-K-0151 at N.C. State University.     

The conduction current in electrode coatings of deformable piezoceramic bodies

We develop a theory for computing the conduction currents in electrode coatings of deformable piezoceramic media in the form of a special type of boundary-value problem for the Poisson equation. It is shown that the curl of the conduction currents in the coatings is always zero. We consider the example of computing the conduction currents and the magnetic field in radially deformed piezoceramic disks. We find that the magnetic field is confined to the volume of the ceramic material of the disk.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 124–127.     

On coupling of stochastic processes with embedded point processes

Criteria for semi-, wide-sense-, traditional regeneration and a coupling construction of stochastic processes with embedded point processes are presented.     

The nonlocal theory of elasticity and its applications to the description of defects in solid bodies

The nonlocal theory of elasticity takes account of remote action forces between atoms. This causes the stresses to depend on the strains not only at an individual point under consideration, but at all points of the body. The stresses caused by defects in a nonlocally elastic medium have no nonphysical singularities, in contrast to the corresponding solutions obtained in the classical theory of elasticity. Translated fromMatematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 1, 1998, pp. 90–96.     

Thermoelastic contact interaction of bodies in the presence of surface thermophysical irregularities

We study the thermoelastic contact interaction (in the absence of friction) of half-spaces under conditions of planar deformation in the presence of thin surface thermophysical irregularities that are taken into account by means of generalized conditions of thermal contact with one another. The problem is reduced to solving a system of singular integrodifferential equations with respect to the jumps of temperature and heat flow on the boundary of a section. We analyze the influence of a nonuniform thermal resistance distributed periodically along the surface or localized in one region of it on the distribution of temperature and stresses in the bodies and on their boundary. Four figures.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 23–28.     

A study of regimes for convective heating of deformable bodies with heat sources

We discuss a numerical/analytic method of solving the problem of determining regimes for heating electrically conducting elastic bodies using convection and electromagnetic radiation so as to optimize energy consumption. We give the results of studies of the temperature fields and stresses, and also the optimal heating regimes for a hollow glass ball.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 3, 1997, pp. 160–165.     

Area-interaction point processes

We introduce a new Markov point process that exhibits a range of clustered, random, and ordered patterns according to the value of a scalar parameter. In contrast to pairwise interaction processes, this model has interaction terms of all orders. The likelihood is closely related to the empty space functionF, paralleling the relation between the Strauss process and Ripley'sK-function. We show that, in complete analogy with pairwise interaction processes, the pseudolikelihood equations for this model are a special case of the Takacs-Fiksel method, and our model is the limit of a sequence of auto-logistic lattice processes.     

Inference for thinned point processes, with application to Cox processes

Given i.i.d. point processes N₁, N₂,…, let the observations be p-thinnings N′₁, N′₂,…, where p is a function from the underlying space E (a compact metric space) to [0, 1], whose interpretation is that a point of N_i at x is retained with probability p(x) and deleted with probability 1−p(x). Strongly consistent estimators of the thinning function p and the Laplace functional L_N(f) = E[e^−N(f)] of the N_i are constructed; associated “central limit” properties are given. Tests are presented, for the case when the N_i and N′_i are both observable, of the hypothesis that the N′_i are p-thinnings of the N_i. State estimation techniques are developed for the case where the N_i are Cox processes directed by unobservable random measures M_i; these techniques yield minimum mean-squared error estimators, based on observation of only the thinned processes N′_i of the N_i and the directing measures M_i. Limit theorems for empirical Laplace functionals of point processes are given.     

Testing separability in marked multidimensional point processes with covariates

In modeling marked point processes, it is convenient to assume a separable or multiplicative form for the conditional intensity, as this assumption typically allows one to estimate each component of the model individually. Tests have been proposed in the simple marked point process case, to investigate whether the mark distribution is separable from the spatial–temporal characteristics of the point process. Here, we extend these tests to the case of a marked point process with covariates, and where one is interested in testing the separability of each of the covariates, as well as the mark and the coordinates of the point process. The extension is not at all trivial, and covariates must be treated in a fundamentally different way than marks and coordinates of the process, especially when the covariates are not uniformly distributed. An application is given to point process models for forecasting wildfire hazard in Los Angeles County, California, and solutions are proposed to the problem of how to proceed when the separability hypothesis is rejected.     

Mathematical modeling of wear-resistance processes of bodies with coatings

We study a mathematical formulation of problems of contact interaction of coated bodies when the material is worn. We state the physical relations that connect the intensity of wear with the characteristic physical quantities. For the case of a half-plane with a thin elastic coating the problem is reduced to singular integro-differential equations whose solution is obtained in the class of orthogonal polynomials. We give estimates of the influence of the characteristics of the coating and the base on the wear process. One figure, two tables.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 51–55.     

Neural fuzzy point processes

Synaptic events in neural systems were described as generated by an apparatus @ possessing memory and encoding a fuzzy point process (the presynaptic discharge) into another N (the postsynaptic discharge). @ was considered to be a fuzzy automata, for which state membership is dependent on input membership and distribution as well as on a control exercised by other neural structures. In such a device, irregular input distributions favour a direct monotonic codification, whereas regular ones induce discontinuous and inverse relations between both fuzzy point processes. Both behaviors favour analogic and membership relations between the fuzzy input and output. However, there exist intermediate grades of irregularities which result in a context-free encoding, where similitude and equivalence relations predominate. The importance of such findings to neurophysiology is discussed.     

Inclusion-exclusion and point processes

A combinatorial (inclusion-exclusion) approach to the construction of point processes starting from densities is proposed. A formal sufficient crifficient criterion is derived and then applied with positive results to systems of functions having a special product form. Thus, a new class of point processes is derived to play a role within classical Gibbs processes.     

Existence of Gibbsian point processes with geometry-dependent interactions

We establish the existence of stationary Gibbsian point processes for interactions that act on hyperedges between the points. For example, such interactions can depend on Delaunay edges or triangles, cliques of Voronoi cells or clusters of k-nearest neighbors. The classical case of pair interactions is also included. The basic tools are an entropy bound and stationarity.     

Two-dimensional Gibbsian point processes with continuous spin symmetries

The conservation of continuous symmetries in two-dimensional systems with interaction is a classical subject of statistical mechanics. So far, all results of this sort required some smoothness properties of the interaction. Only recently Ioffe et al. (Comm. Math. Phys. 226 (2002) 433) succeeded to treat the case of lattice systems with continuous, rather than smooth, interaction. Here we establish a similar result for Gibbsian systems of point particles with internal degrees of freedom.