On the purity of the free boundary condition Potts measure on random trees

We consider the free boundary condition Gibbs measure of the Potts model on a random tree. We provide an explicit temperature interval below the ferromagnetic transition temperature for which this measure is extremal, improving older bounds of Mossel and Peres. In information theoretic language extremality of the Gibbs measure corresponds to non-reconstructability for symmetric q-ary channels. The bounds for the corresponding threshold value of the inverse temperature are optimal for the Ising model and differ from the Kesten Stigum bound by only 1.50% in the case q=3 and 3.65% for q=4, independently of d. Our proof uses an iteration of random boundary entropies from the outside of the tree to the inside, along with a symmetrization argument.     

The temperature field of a multilayer cylinder in asymptotic thermal mode

We propose a method of determining one-dimensional temperature fields in a multilayer cylinder corresponding to asymptotic thermal mode. We give the temperature distribution when the temperature of the surrounding medium varies polynomially and harmonically with time and with a heat source density concentrated on one of the interfaces of adjacent layers.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 162–169.     

Asymptotic Localization of Energy in Nondisordered Oscillator Chains

We study two popular one‐dimensional chains of classical anharmonic oscillators: the rotor chain and a version of the discrete nonlinear Schrödinger chain. We assume that the interaction between neighboring oscillators, controlled by the parameter ? > 0, is small. We rigorously establish that the thermal conductivity of the chains has a nonperturbative origin with respect to the coupling constant ?, and we provide strong evidence that it decays faster than any power law in ? as ? → 0. The weak coupling regime also translates into a high‐temperature regime, suggesting that the conductivity vanishes faster than any power of the inverse temperature. To our knowledge, it is the first time that a clear connection has been established between KAM‐like phenomena and thermal conductivity. © 2015 Wiley Periodicals, Inc.     

Optimization of the two-dimensional temperature fields in inhomogeneous bodies with constraints on the phase coordinates

We state a problem of speed-optimal control of the two-dimensional nonstationary temperature regime in inhomogeneous bodies with constraints on the control (the temperature of the heating medium) and the phase coordinates (the temperature of the body, the temperature range, and the heat flux on the surface of the body), and we propose a method for solving the problem numerically. We give as an example the computation of the optimal control of heating of a hollow infinite cylinder under constraints on the temperature range. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 119–123.     

Heating of an electron gas by large temperature gradients and scale thermoelectric effects in semiconductors

We study the temperature fields and thermoelectric effects in semiconductors of monopolar conductivity under conditions of heating of an electron gas by large temperature gradients. We study the influence of surface cooling of the current carriers on the scale effects.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 31, 1990, pp. 16–21.     

The thermally stressed state of the glass shell of a kinescope under heating by electromagnetic radiation

We give the results of theoretical and experimental studies of the temperature and stresses in the glass shell of a kinescope under heat treatment using electromagnetic radiation. We propose a combined regime for thermal treatment.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 33, 1991, pp. 31–35.     

The influence of variable thermophysical characteristics on the temperature field in a ceramic plate

We discuss an analytic method of determining the temperature fields in a rectangular plate of ceramic base type or a plate without inclusions. To increase the accuracy of the results obtained we take account of the temperature dependence of the coefficient of thermal conductivity and the specific heat capacity Translated fromMatematicheskie Melody i Fiziko-Mekhanicheskie Polya, Issue 31, 1990, pp. 99–103.     

A method of determining the functional dependence between stress and deformation under isothermal loading of a reinforced plate

We propose an approximate method for specifying the functional dependence among the stress, deformation, temperature, and time under arbitrary laws of loading and heating of a reinforced plate in a uniaxially stressed state. The method is justified experimentally in tests of the uniaxial extension of models under isothermic and nonisothermic loading with stress and temperature varying at a constant rate.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 75–77.     

Positive definite temperature functions on the Heisenberg group

We characterize positive definite temperature functions, i.e., positive definite solutions of the heat equation, on the Heisenberg group in terms of the initial values. We also obtain an integral representation for positive definite and U(n)-invariant temperature functions with polynomial growth, where U(n) is the group of all n× n unitary matrices.     

Thermomechanical behavior of nonmetallic electrical conductors during high-temperature treatment

We give a mathematical statement and initial relations for the problem of determining the temperature fields and stresses in nonmetallic electrical conductors under induction heating to high temperatures. We study the influence of the temperature dependence of the characteristics of the material and the radiant heat transfer with the surrounding medium on the thermomechanical behavior of a cylinder. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 74–79.     

Rational regimes for induction heating of cylindrical articles for graphitation

We develop a method of computing regimes for induction heating of cylindrical carbon devices during graphitation. We study the electromagnetic, temperature, and mechanical fields as functions of the parameters of the heating regime.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 33, 1991, pp. 27–31.     

Solution of the two-dimensional integrodifferential equation of the heat conduction problem for an unbounded body with a thin thermally conducting inclusion

We determine the temperature field in an unbounded body with a thin heat-conducting foreign inclusion. We propose a method of solution of the corresponding two-dimensional singular integrodifferenlial equation of the problem based on the study of the behavior of the solution in a neighborhood of the boundary of the inclusion.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 28, 1988, pp. 78–83.     

Thermal Effects in Gravitational Hartree Systems

We consider the non-relativistic Hartree model in the gravitational case, i.e. with attractive Coulomb–Newton interaction. For a given mass M > 0, we construct stationary states with non-zero temperature T by minimizing the corresponding free energy functional. It is proved that minimizers exist if and only if the temperature of the system is below a certain threshold T* > 0 (possibly infinite), which itself depends on the specific choice of the entropy functional. We also investigate whether the corresponding minimizers are mixed or pure quantum states and characterize a critical temperature T_c ? (0, T*){T_c \in (0, T*)} above which mixed states appear.     

Generalized potentials and their application in solving problems of mathematical physics

We introduce the concept of potential for bounded bodies. We construct the potentials for a two-dimensional semi-infinite region. We solve the problems of determining the nonsteady state temperature fields in a semi-infinite plate with a straight-line cut situated perpendicularly to the boundary of the region under boundary conditions of the first or second kind on the boundary of the region and the edges of the cut.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 28, 1988, pp. 52–58.     

The heating of a multilayered half-space by friction

We propose an approach to determining the temperature caused by the action of friction forces in the functioning of braking systems. The nonsteady temperature field in a composite semi-infinite body being heated in a circular region of the boundary surface by a frictional heat flux is found on the basis of the model of homogenization of multilayered periodic media with microlocal parameters. We give a numerical analysis of the behavior of the temperature field as a function of the spatial coordinates and the parameters that characterize the braking process. Translated fromMatematichni Metodi i Fiziko-mekhanichni Polya, Vol. 40, No. 1, 1997, pp. 111–116.     

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.     

The kinetics equations in a multiplying medium with free boundaries

We study a mathematical model of neutron multiplication in a slab ??, by taking into account temperature feedback effects and considering one group of delayed neutrons. The thickness 2a of ?? is time dependent because of temperature variations due to the energy released by fissions. Starting from a quite detailed picture of the physical phenomena occurring in ??, we derive a system of three coupled ordinary differential equations for the total number of neutrons F? = F?(t), for the total number of precursors ? = ?(t), and for the half-thickness of ??, a = a(t). We finally examine some stability properties of such a system of ordinary differential equations.     

Optimization of the thermostressed state of piecewise homogeneous glass shells of revolution during thermal processing

We propose a physico-mathematical model and state a problem of stress-optimization of regimes for heating thin piecewise homogeneous glass shells of revolution with given ranges of admissible temperature and stress values by means of an optimal choice of temperature for the surrounding medium. To solve the problem we apply the method of local variations with a specially chosen initial value of the control function and numerical analytic solution of the corresponding direct problem of thermomechanics for the shell in question.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 36, 1992, pp. 52–56.     

A Blow-up Criterion of Strong Solutions to the Compressible Viscous Heat-Conductive Flows with Zero Heat Conductivity

We study the compressible Navier-Stokes equations of viscous heat-conductive fluids in a periodic domain \mathbbT³\mathbb{T}^{3} with zero heat conductivity k=0. We prove a blow-up criterion for the local strong solutions in terms of the temperature and positive density, similar to the Beale-Kato-Majda criterion for ideal incompressible flows.     

New approximate solutions of the problem of the thermomechanics of a hollow cylinder with a material of variable properties

We use the method of equivalent sources to obtain approximate solutions of an unrestricted quasistatic axisymmetric problem of the thermomechanics of a long hollow cylinder with ends fastened and materials whose properties depend on the temperature. We perform a comparative analysis of the results with the solutions obtained by the method of perturbations and the method of finite differences.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 71–74.