Limiting distributions and estimation in generalized Polya random walk scheme

Asymptotic behavior of distributions generated by Polya random walks is investigated. Unbiased estimators of these distributions are constructed for closed first-arrival plans.Translated from Statisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 20–36, 1986.     

Bounds on compound distributions and stop-loss premiums

In the actuarial literature a lot of attention is given to the approximation of aggregate claims distributions by compound Poisson and Polya distributions and their subsequent numerical evaluation. The present contribution derives bounds for the tail of compound distributions and stop-loss premiums. The bounds are obtained in an elementary manner based on a version of the Chebyshev inequality. The main point of this contribution consists in deriving bounds with explicit dependence on the distribution function itself as well as on some partial moments of it.     

Negative binomial distributions for point processes

Negative binomial point processes are defined for which all finite-dimensional distributions associated with disjoint bounded Borel sets are negative binomial in the usual sense. For these processes we study classical notions such as infinite divisibility, conditional distributions, Palm probabilities, convergence, etc. Negative binomial point processes appear to be of interest because they are mathematically tractable models which can be used in many situations. The general results throw some new light on some well-known special cases like the Polya process and the Yule process.     

Temperature field of a layer of a fibrous composite material under conditions of external thermal radiation

On the basis of an expression for the scattering phase function of an individual fiber, relations modeling radiative properties of fibrous composites are obtained. We calculate the temperature state of a layer of such a material under condition of external thermal radiation. Dependences of temperature distributions on the volume fraction of fibers in the composite, their reflectance, and size are investigated. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 185–193, January–March, 2008.     

Distribution of self-balanced stresses in a stratified composite material with warped structures

We use the model of a linear, piecewise homogeneous elastic body to study the distribution of selfbalanced normal and tangential stresses for a horizontal deformation of a stratified composite material with warped structures which is under the action at infinity of uniformly distributed normal stresses directed along the stratification.Translated from Matematicheskie Metody i Fiziko-mekhanicheskie Polya, No. 26, pp. 83–89, 1987.     

Distributional analysis of a generalization of the Polya process

A nonhomogeneous birth process generalizing the Polya process is analyzed, and the distribution of the transition probabilities is shown to be the convolution of a negative binomial distribution and a compound Poisson distribution, whose secondary distribution is a mixture of zero-truncated geometric distributions. A simplified form of the secondary distribution is obtained when the transition intensities have a particular structure, and may sometimes be expressed in terms of Stirling numbers and special functions such as the incomplete gamma function, the incomplete beta function, and the exponential integral. Conditions under which the compound Poisson form of the marginal distributions may be improved to a geometric mixture are also given.     

Semiscalar equivalence and the smith normal form of polynomial matrices

We give conditions under which a set of polynomial matrices over a finite field can be simultaneously reduced by means of semiscalar equivalent transformations to a special triangular form with invariant factors on the principal diagonals. We investigate multiplicative properties of the Smith normal form of polynomial matrices and in particular we identify a class of polynomial matrices for which the Smith normal form of the product matrix is equal to the product of the Smith normal forms of the factor matrices.Translated from Matematicheskie Metody i Fiziko-mekhanicheskie Polya, No. 26, pp. 13–16, 1987.     

Quasistatic thermoelasticity problems for shells heated by moving heart sources

Solutions of thermoelasticity problems for cylindrical and spherical shells heated by moving heat sources distributed according to a normal (Gaussian) law are obtained in a quasistatic formulation. The results of temperature and stress calculations for the case of a plate are given.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 25, pp. 40–44, 1987.     

Solvability and properties of solutions of a system of integral equations of the convolution type with power-difference kernels

We established the conditions of solvability of systems of integral equations of the second kind with polynomial-difference kernels of the convolution type in the normal and exceptional cases and study some properties of their solutions. In each case, we determine the spaces to which these solutions belong. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 42–52, January–March, 2008.     

Attenuation of normal waves in a cavity with a liquid in the presence of an elastic shell

The attenuation of normal waves is investigated in a cylindrical cavity of an unbounded viscoelastic medium, in which there is a thin elastic shell filled with a viscous compressible liquid. It is assumed that the absorption of waves in a liquid or solid is small. The effect of the shell thickness on the value of the absorption coefficient of the normal waves is studied.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 25, pp. 49–52, 1987.     

A new class of multivariate counting processes and its characterization

In this paper, we suggest a new class of multivariate counting processes which generalizes and extends the multivariate generalized Polya process recently studied in Cha and Giorgio [On a class of multivariate counting processes, Adv. Appl. Probab. 48 (2016), pp. 443–462]. Initially, we define this multivariate counting process by means of mixing. For further characterization of it, we suggest an alternative definition, which facilitates convenient characterization of the proposed process. We also discuss the dependence structure of the proposed multivariate counting process and other stochastic properties such as the joint distributions of the number of events in an arbitrary interval or disjoint intervals and the conditional joint distribution of the arrival times of different types of events given the number of events. The corresponding marginal processes are also characterized.     

Potential of a simple layer for a torus with density depending on the distance from a tangent plane to the center

Using a bulk potential, we construct a one-dimensional toroidal layer with corresponding density for the mass distribution. We determine the density of a simple layer, and also find the potential of a simple layer distributed along the surface of an elliptical torus of revolution with density depending on distance as measured along the normal between a tangent plane and the center of the torus.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 29, pp. 85–87, 1989.     

The Polya Tree Sampler: Towards Efficient and Automatic Independent Metropolis-Hastings Proposals

We present a simple, efficient, and computationally cheap sampling method for exploring an un-normalized multivariate density on ?(d), such as a posterior density, called the Polya tree sampler. The algorithm constructs an independent proposal based on an approximation of the target density. The approximation is built from a set of (initial) support points - data that act as parameters for the approximation - and the predictive density of a finite multivariate Polya tree. In an initial "warming-up" phase, the support points are iteratively relocated to regions of higher support under the target distribution to minimize the distance between the target distribution and the Polya tree predictive distribution. In the "sampling" phase, samples from the final approximating mixture of finite Polya trees are used as candidates which are accepted with a standard Metropolis-Hastings acceptance probability. Several illustrations are presented, including comparisons of the proposed approach to Metropolis-within-Gibbs and delayed rejection adaptive Metropolis algorithm.     

A variational method for the solution of biharmonic problems for a rectangular domain

We develop a variational method for the solution of biharmonic problems for a rectangular domain where, at one pair of its opposite sides, the unknown function and its normal derivative take zero values, and, at the other pair, certain inhomogeneous conditions are valid. The cases of semiinfinite and finite domain are considered. The method is based on the minimization of a quadratic functional determining the deviation of the solution from the given inhomogeneous conditions in the norm of L ₂. To solve this variational problem, we apply the expansion of the solution in the systems of complex biharmonic functions (the so-called Papkovich homogeneous solutions), which satisfy identically the given homogeneous conditions at the pair of opposite sides of the rectangle. This representation of the solution is somewhat different from that proposed earlier [V. F. Chekurin, “A variational method for the solution of direct and inverse problems of the theory of elasticity for a semiinfinite strip,” Izv. Ross. Akad. Nauk, Mekh. Tverdogo Tela, No. 2, 58–70 (1999)]. We consider several variants of inhomogeneous boundary conditions arising in the problems of the two-dimensional theory of elasticity. Finally, we give an example of applying the proposed method for the determination of stress distributions in a rectangular area one of whose sides is rigidly fastened and the opposite one is subjected to the action of normal forces. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 88–98, January–March, 2008.     

Temperature fields in laminar anisotropic shells

A numerical solution of the nonlinear heat-conduction problem is obtained for laminar anisotropic shells with temperature-dependent physicomechanical properties.Translated from Matematicheskie Metody i Fiziko-mekhanicheskie Polya, No. 26, pp. 98–101, 1987.     

Investigation of the effect of an electrical strap on the current distribution in a system of underground metallic commodity-carrying pipelines electrically protected from corrosion by the method of integral equations

tA method of determining the potential and current density in a system of parallel pipelines with allowance for assembly straps is addressed.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 25, pp. 95–100, 1987.     

Local Regularity of Nonsmooth Wavelet Expansions and Application to the Polya Function

We give criteria of pointwise regularity for expansions on Haar or Schauder basis (or spline-type wavelets) corresponding to large Hölder exponents. As an application, we determine the exact Hölder regularity of the Polya function at every point and show that it is multifractal.     

Boundary-value contact problem of thermoelastoplasticity for a two-layer eccentric cylindrical pipe

We formulate the plane two-dimensional static boundary-value contact problem of thermoelastoplasticity for a two-layer eccentric cylindrical pipe under the action of a temperature field and compressive normal stresses that are uniformly distributed on its lateral surfaces and present its approximate solution. We assume that the mechanical and thermophysical properties of the materials are temperature-independent, plastic strains arise on the interior lateral surface of the two-layer pipe and completely envelop it, and the material of the pipe is perfectly elastoplastic, incompressible in the domain of plasticity, and satisfies the Tresca-Saint-Venant plasticity condition. I. Franko L'viv State University, L'viv. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 57–66, April–June, 1998.     

Invariant interpolation formulas for higher-order derivatives of functions of a vector argument

Interpolation formulas for higher-order derivatives which guarantee the invariance of the results under changes of coordinates are obtained for functions of a vector argument.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 25, pp. 22–25, 1987.     

Robustifying Generalized Linear Mixed Models Using a New Class of Mixtures of Multivariate Polya Trees

In applied sciences, generalized linear mixed models have become one of the preferred tools to analyze a variety of longitudinal and clustered data. Due to software limitations, the analyses are often restricted to the setting in which the random effects terms follow a multivariate normal distribution. However, this assumption may be unrealistic, obscuring important features of among-unit variation. This work describes a widely applicable semiparametric Bayesian approach that relaxes the normality assumption by using a novel mixture of multivariate Polya trees prior to define a flexible nonparametric model for the random effects distribution. The nonparametric prior is centered on the commonly used parametric normal family. We allow this parametric family to hold only approximately, thereby providing a robust alternative for modeling. We discuss and implement practical procedures for addressing the computational challenges that arise under this approach. We illustrate the methodology by applying it to real-life examples.

Supplemental materials for this paper are available online.