Asymptotics of the heat kernel for nonminimal differential operators

A method of calculating the coefficients in the asymptotic expansion of the heat-conduction kernel is generalized to nonminimal differential operators. The first nontrivial coefficients of the expansion for second-order nonminimal operators on Riemannian manifolds of arbitrary dimension are calculated.Translated from Ukrainskii Maternaticheskii Zhurnal, Vol. 43, No. 11, pp. 1541–1551, November, 1991.     

Asymptotics of stationary measure under scaling in stochastic exchange processes

Asymptotic properties of an invariant distribution of exchange processes are studied on a two-dimensional lattice with fixed boundary conditions.     

Asymptotics for a nonlinear integral equation with a generalized heat kernel

This paper is concerned with a nonlinear integral equation $$(P)\qquad u(x, t)=\int_{{\bf R}^N}G(x-y, t)\varphi(y){d}y+\int_0^t \int_{{\bf R}^N}G(x-y, t-s)f(y, s:u){d}y{d}s, \quad $$ where N ≥  1, \({\varphi \in L^\infty({\bf R}^N) \cap L^1({\bf R}^N,(1+|x|^K){d}x)}\) for some K ≥  0. Here, G = G(x,t) is a generalization of the heat kernel. We are interested in the asymptotic expansions of the solution of (P) behaving like a multiple of the integral kernel G as \({t \to \infty}\) .     

Asymptotics of relative heat traces and determinants on open surfaces of finite area

The goal of this article is to prove that on surfaces with asymptotically cusp ends the relative determinant of pairs of Laplace operators is well defined. We consider a surface with cusps $(M,g)$ and a metric $h$ on the surface that is a conformal transformation of the initial metric $g$ . We prove the existence of the relative determinant of the pair $(\Delta _{h},\Delta _{g})$ under suitable conditions on the conformal factor. The core of the paper is the proof of the existence of an asymptotic expansion of the relative heat trace for small times. We find the decay of the conformal factor at infinity for which this asymptotic expansion exists and the relative determinant is defined. Following the paper by B. Osgood, R. Phillips, and P. Sarnak about extremal of determinants on compact surfaces, we prove Polyakov’s formula for the relative determinant and discuss the extremal problem inside a conformal class. We discuss necessary conditions for the existence of a maximizer.     

Elasto-plastic contact problems with heat exchange

This paper deals with a dynamical model for the motion of a one-dimensional visco-elasto-plastic body in contact with an elasto-plastic obstacle and undergoing thermal expansion. The problem for the unknown displacement and temperature is rewritten in accordance with the formalism of hysteresis operators as solution operators of the underlying variational inequalities. The system consists of the momentum balance and energy balance equations. The existence result for this thermodynamically consistent problem is obtained by using some a priori estimates established for the space discretized problem while the uniqueness result follows from the Lipschitz continuity of the nonlinearities.     

On heat exchange and heat transport in a geothermal plant

In the long term, the only way to address the challenging task of power supply, is to make renewable energy sources economically attractive and to use them efficiently. In particular, geothermal energy is promising to take over the base load of the power supply. Nevertheless, a lot of investigations needs to be made to use the almost inexhaustible source of thermal energy in the interior of the earth effectively. Starting from the initially isothermal state, a cold fluid is injected through a borehole into a rock. By the rising pressure gradient, the fluid flows through the porous rock and escapes through another borehole. While the fluid passes the micro cracks in the hot rock, the water is heated by the rock due to the heat exchange between the constituents. This process is simulated based on the Theory of Porous Media (TPM). The presented modelling approach of the heat transport and the flow processes in a fully saturated subsurface includes two non-isothermal constituents: an elastically deformable, materially incompressible solid skeleton where thermal expansion is neglected, and a viscous, materially incompressible fluid constituent. To solve the initial-boundary-value problem, the governing primary variables of the coupled model are spatially approximated by mixed finite elements, and the time-discretisation is carried out by an implicit Euler time-integration scheme. The aim of the presented numerical simulations is to study the heat transport and to evaluate the efficiency by varying flow rates. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Asymptotics of Drift

In this paper we study the drift of simple random walks on finitely generated groups. We show that there are infinitely many asymptotics of drift. Bibliography: 9 titles.     

Asymptotics of the Expected Posterior

Let (X₁,...,X,...,X_n) be independently and identically distributed observations from an exponential family p equipped with a smooth prior density w on a real d-dimensional parameter . We give conditions under which the expected value of the posterior density evaluated at the true value of the parameter, ₀, admits an asymptotic expansion in terms of the Fisher information I(₀), the prior w, and their first two derivatives. The leading term of the expansion is of the form n^d/2c₁(d, 0) and the second order term is of the form n^4/2-1c₂(d, ₀>, w), with an error term that is o(n^d/2-1). We identify the functions c₁ and c₂ explicitly. A modification of the proof of this expansion gives an analogous result for the expectation of the square of the posterior evaluated at ₀. As a consequence we can give a confidence band for the expected posterior, and we suggest a frequentist refinement for Bayesian testing.     

On the solution of heat conduction problems for thermosensitive bodies heated by convective heat exchange

We propose a method of solving heat conduction problems for thermosensitive bodies, in particular for a hollow cylinder from whose surfaces a convective heat exchange with the external environment occurs.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 28, 1988, pp. 83–86.     

Thermoelastic contact with Barber's heat exchange condition

We consider a nonlinear parabolic problem that models the evolution of a one-dimensional thermoelastic system that may come into contact with a rigid obstacle. The mathematical problem is reduced to solving a nonlocal heat equation with a nonlinear and nonlocal boundary condition. This boundary condition contains a heat-exchange coefficient that depends on the pressure when there is contact with the obstacle and on the size of the gap when there is no contact. We model the heat-exchange coefficient as both a single-valued function and as a measurable selection from a maximal monotone graph. Both of these models represent modified versions of so-called imperfect contact conditions found in the work of Barber. We show that strong solutions exist when the coefficient is taken to be a continuously differentiable function and that weak solutions exist when the coefficient is taken to be a measurable selection from a maximal monotone graph. The proofs of these results reveal an interesting interplay between the regularity of the initial condition and the behavior of the coefficient at infinity.     

Asymptotics for the heat equation in the exterior of a shrinking compact set in the plane via Brownian hitting times

Let and let be a continuous, nonincreasing function on satisfying . Consider the heat equation in the exterior of a time-dependent shrinking disk in the plane:

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If there exist constants and a constant 0$"> such that , for sufficiently large , then . The same result is also shown to hold when is replaced by , where . Also, a discrepancy is noted between the asymptotics for the above forward heat equation and the corresponding backward one. The method used is probabilistic.

     

Asymptotics for the Tukey Median

The asymptotic distribution of the Tukey median has recently been obtained by Nolan in a bivariate setting and by Bai and He in the general multivariate case. To establish their theorem, these authors made a strong symmetry hypothesis on the distribution and, in the case of Bai and He, assumed the existence of a density function with a gradient satisfying a finite-moment condition. This paper extends the above result by deriving the asymptotic distribution of the above location estimate without making any symmetry or differentiability assumption on the distribution.     

Large-Reynolds-Number Asymptotics of the Berman Problem

Similarity flow of a viscous fluid in a channel is considered, driven by uniform withdrawal of the fluid through the channel walls. The nonlinear ordinary differential boundary value problem that results has several branches of solutions; those of Types III, III₁, and I₁ are investigated here, in the limit of large wall-suction Reynolds number. This paper gives a markedly more accurate Type III asymptotic solution than previously available, and describes the true asymptotics of the other branches for the first time. The asymptotic structure of the Type III₁ solution is particularly subtle, requiring matching between seven different layers. Numerical solutions of the boundary value problem provide support for the asymptotic solutions obtained.     

Asymptotics of the Discrete Chebyshev Polynomials

The discrete Chebyshev polynomials are orthogonal with respect to a distribution, which is a step function with jumps one unit at the points , N being a fixed positive integer. By using a double integral representation, we have recently obtained asymptotic expansions for in the double scaling limit, namely, and , where and ; see [8]. In this paper, we continue to investigate the behavior of these polynomials when the parameter b approaches the endpoints of the interval (0, 1). While the case is relatively simple (because it is very much like the case when b is fixed), the case is quite complicated. The discussion of the latter case is divided into several subcases, depending on the quantities n, x, and , and different special functions have been used as approximants, including Airy, Bessel, and Kummer functions.     

Asymptotics of the Empirical Cross-over Function

We consider a combination of heavily trimmed sums and sample quantiles which arises when examining properties of clustering criteria and prove limit theorems. The object of interest, which we call the Empirical Cross-over Function, is an L-statistic whose weights do not comply with the requisite regularity conditions for usage of existing limit results. The law of large numbers, CLT and a functional CLT are proven.