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1.
The Heat Kernel on Hyperbolic Space 总被引:1,自引:0,他引:1
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. 相似文献
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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. 相似文献
3.
S. A. Savchuk V. N. Vlasov S. A. Appolonova V. N. Arbuzov A. N. Vedenin A. B. Mezinov B. R. Grigor'yan 《Journal of Analytical Chemistry》2001,56(3):214-231
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. 相似文献
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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. 相似文献
7.
Heat kernels on metric measure spaces and an application to semilinear elliptic equations 总被引:3,自引:0,他引:3
Alexander Grigor'yan Jiaxin Hu Ka-Sing Lau 《Transactions of the American Mathematical Society》2003,355(5):2065-2095
We consider a metric measure space and a heat kernel on satisfying certain upper and lower estimates, which depend on two parameters and . We show that under additional mild assumptions, these parameters are determined by the intrinsic properties of the space . Namely, is the Hausdorff dimension of this space, whereas , called the walk dimension, is determined via the properties of the family of Besov spaces on . Moreover, the parameters and are related by the inequalities .
where is the generator of the semigroup associated with .
We prove also the embedding theorems for the space , and use them to obtain the existence results for weak solutions to semilinear elliptic equations on of the form
where is the generator of the semigroup associated with .
The framework in this paper is applicable for a large class of fractal domains, including the generalized Sierpinski carpet in .
8.
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. 相似文献
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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 相似文献