The Heat Kernel on Hyperbolic Space

The purpose of this note is to provide a new proof for the explicitformulas of the heat kernel on hyperbolic space. By definition,the hyperbolic space Hⁿ is a (unique) simply connected completen-dimensional Riemannian manifold with a constant negative sectionalcurvature –1. 1991 Mathematics Subject Classification58G32, 58G11.     

Hardy Inequality and Heat Semigroup Estimates for Riemannian Manifolds with Singular Data

Upper bounds are obtained for the heat content of an open set D in a geodesically complete Riemannian manifold M with Dirichlet boundary condition on ?D, and non-negative initial condition. We show that these upper bounds are close to being sharp if (i) the Dirichlet-Laplace-Beltrami operator acting in L ²(D) satisfies a strong Hardy inequality with weight δ², (ii) the initial temperature distribution, and the specific heat of D are given by δ^?α and δ^?β respectively, where δ is the distance to ?D, and 1 < α <2, 1 < β <2.     

Application of Chromatography and Spectrometry to the Authentication of Alcoholic Beverages

The use of gas chromatography (GC), gas chromatography–mass spectrometry (GC–MS), and UV–VIS spectrophotometry for identifying the falsification of strong alcoholic beverages (vodka, gin, cognac, and whiskey) was considered. In the GC analysis of ethyl alcohol and vodkas based on it, the test alcohol was assigned to synthetic alcohol or to biochemically produced alcohol using a set of typical impurities, markers of the alcohol nature, which present in the test alcohol in a certain ratio and can be determined by GC or GC–MS analysis. The multicomponent analysis of cognacs and related liquors can reveal the replacement of cognac spirit with alcohol produced from nongrape raw materials, to determine whether the cognac spirit was in contact with oak wood and how long was the duration of its aging, and to detect the falsification of the age by adding certain ingredients. The limitations of chromatographic and spectrometric analytical techniques in the identification of adulterated alcoholic beverages was demonstrated. The validation criteria for testing the identification of alcoholic beverage components by chromatographic techniques received special attention.     

On Nonnegative Solutions of the Inequality Δ u + uσ ≤ 0 on Riemannian Manifolds

We study the uniqueness of a nonnegative solution of the differential inequality on a complete Riemannian manifold, where σ > 1 is a parameter. We prove that if, for some x₀ ? M and all large enough r where , and B(x,r) is a geodesic ball, then the only nonnegative solution of (*) is identically 0. We also show the sharpness of the above values of the exponents p,q. © 2014 Wiley Periodicals, Inc.     

Comparison inequalities for heat semigroups and heat kernels on metric measure spaces

We prove a certain inequality for a subsolution of the heat equation associated with a regular Dirichlet form. As a consequence of this inequality, we obtain various interesting comparison inequalities for heat semigroups and heat kernels, which can be used for obtaining pointwise estimates of heat kernels. As an example of application, we present a new method of deducing sub-Gaussian upper bounds of the heat kernel from on-diagonal bounds and tail estimates.     

Asymptotic separation for independent trajectories of Markov processes

We say that n independent trajectories ξ₁(t),…,ξ _n (t) of a stochastic process ξ(t)on a metric space are asymptotically separated if, for some ɛ > 0, the distance between ξ _i (t _i ) and ξ _j (t _j ) is at least ɛ, for some indices i, j and for all large enough t ₁,…,t _n , with probability 1. We prove sufficient conitions for asymptotic separationin terms of the Green function and the transition function, for a wide class of Markov processes. In particular,if ξ is the diffusion on a Riemannian manifold generated by the Laplace operator Δ, and the heat kernel p(t, x, y) satisfies the inequality p(t, x, x) ≤ Ct ^−ν/2 then n trajectories of ξ are asymptotically separated provided . Moreover, if for some α∈(0, 2)then n trajectories of ξ^(α) are asymptotically separated, where ξ^(α) is the α-process generated by −(−Δ)^α/2. Received: 10 June 1999 / Revised version: 20 April 2000 / Published online: 14 December 2000 RID="*" ID="*" Supported by the EPSRC Research Fellowship B/94/AF/1782 RID="**" ID="**" Partially supported by the EPSRC Visiting Fellowship GR/M61573