格点量子Zakharov方程组紧致核截面的存在性与熵的估计

讨论一类非自治带量子纠正的修正Zakharov方程组在无穷格点上解的渐近行为,应用截断函数的技巧证明该格点方程组紧致核截面的存在性,并给出核截面的Kolmogorov-ε熵的上界估计.     

Exponential Decay of Correlations for a Real-Valued Dynamical System Generated by a $k$ Dimensional System

As a first step towards modelling real time-series, we study a class of real-variable, bounded processes \(\{X_{n}, n\in \mathbb{N}\}\) defined by a deterministic \(k\)-term recurrence relation \(X_{n+k} = \varphi (X _{n}, \ldots , X_{n+k-1})\). These processes are noise-free. We immerse such a dynamical system into \(\mathbb{R}^{k}\) in a slightly distorted way, which allows us to apply the multidimensional techniques introduced by Saussol (Isr. J. Math. 116:223–248, 2000) for deterministic transformations. The hypotheses we need are, most of them, purely analytic and consist in estimates satisfied by the function \(\varphi \) and by products of its first-order partial derivatives. They ensure that the induced transformation \(T\) is dilating. Under these conditions, \(T\) admits a greatest absolutely continuous invariant measure (ACIM). This implies the existence of an invariant density for \(X_{n}\), satisfying integral compatibility conditions. Moreover, if \(T\) is mixing, one obtains the exponential decay of correlations.

     

Heat Kernel on Analytic Subvariety

In this paper, the author extends Peter Li and Tian Gang’s results on the heat kernel from projective varieties to analytic varieties. The author gets an upper bound of the heat kernel on analytic varieties and proves several properties. Moreover, the results are extended to vector bundles. The author also gets an upper bound of the heat operators of some Schr¨ondinger type operators on vector bundles. As a corollary, an upper bound of the trace of the heat operators is obtained.     

Exponential Decay of Correlations in Multi-Dimensional Dispersing Billiards

We prove exponential decay of correlations for a “reasonable” class of multi-dimensional dispersing billiards. The scatterers are required to be C ³ smooth, the horizon is finite, there are no corner points. In addition, we assume subexponential complexity of the singularity set. Submitted: November 10, 2007. Accepted: May 23, 2008.     

Poisson Kernel on the Quantum Lobachevsky Spaces

We define an analog of the Poisson transform on the quantum Lobachevsky spaces.     

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.     

Heat Kernel Estimates on Weighted Graphs

We prove upper and lower heat kernel bounds for the Laplacianon weighted graphs which include the case that the weights haveno strictly positive lower bound. Our estimates give rise toa very explicit probabilistic interpretation, and can be formulatedin terms of a weighted metric. Interestingly, this metric isnot equivalent to the intrinsic metric. 1991 Mathematics SubjectClassification 39A12.     

THE HEAT KERNEL OF A BALL IN C~n

By introducing the horosphere coordinate of a unit ball B~n in C~n and an integraltransformation formula of functions in such coordidates,the author constructs the heatkernel H_(B~n)(z,w,t)of the heat equation associated to the Bergman metric of B~n.That iswhere c_n is a well-defined constant and r(z,w)is the geodesic destance of two points zand w of B_n and t∈ R~+.Sincethenis the Green function of the topological product space B~m×B~n.     

Entropy and Decay of Correlations for Real Analytic Semi-Flows

We consider real analytic suspension semi-flows over uniformly expanding real-analytic map of the interval. We show that for any -invariant equilibrium measure related to an analytic potential g, there exists a Banach space of test functions such that for generic observables in , the corresponding correlation functions cannot decay faster than , where h_g is the measure theoretic entropy of . This statement implies the existence of essential spectrum for the Perron-Frobenius operator associated to the semi-flow, when acting on any reasonable Banach space. Submitted: September 16, 2008. Accepted: March 30, 2009.     

Decay of Solutions of a Nonlinear Hyperbolic System Describing Heat Propagation by Second Sound

In this work we establish a decay result for the solutions of a nonlinear hyperbolic system describing heat propagation, where the heat flux is given by Cattaneo's law.     

Non-Gaussian Aspects of Heat Kernel Behaviour

A large number of papers written over the last ten years haveconcerned the spectral theory of Laplace–Beltrami operatorson complete Riemannian manifolds, and of other self-adjointsecond order elliptic operators. Much of the interest has centredon the relationship between various types of Sobolev inequality,parabolic Harnack inequalities and the Liouville property onthe one hand, and Gaussian heat kernel bounds on the other.For manifolds of bounded geometry there is an important connectionbetween this problem and a corresponding one for discrete Laplacianson graphs. Standard references are [9, 37] and more recent literaturecan be traced via [5, 16, 32].     

完备流形上带位势的热核估计

在完备非紧流形上获得了关于带位势热方程正解的梯度估计；接着，利用测地线的技巧获得了Harnack不等式；进一步，建立了两个积分不等式，综合Harnack不等式获得了热核的上下界；最后，利用函数的结果来控制p形式的热核。     

Logarithmic Gradient of the Heat Kernel on Riemannian Manifolds

Mathematical Notes -     

Gradient Estimate of the Heat Kernel on Modified Graphs

We obtain a condition on the modification of graphs which guarantees the preservation of the Gaussian upper bound for the gradient of the heat kernel.      

论由周期核产生对应非周期核的几类方法

Given a non-periodic kernel, as well known, a corresponding periodic kernel can be constructed by Poisson's sum formula. In this paper we discuss how to construct the corresponding non-periodic kernel from a periodic kernel. To do this, multiplicator method and undetermined coefficient method are introduced; The exis-tence theorems are estallished also. In particular, Fejer-Коровкйн and Ghermanesco perturbed Fejer type non-periodic kernels are given out.     

The Heat Kernel on Constant Negative Curvature Space Form

Let M be a n-dimensional simply connected, complete Riemannian manifold with constant negative curvature. The heat kernel on M is denoted by H^M_t(x, y) = H^M_t(r(x, y)), where r(x, y) = dist(x, y). We have the explicit formula of H^M_t(x, y) for n=2, 3, and the induction formula of H^M_t(x, y) for n ≥ 4^{[-1]}. But the explicit formula is very complicated for n ≥ 4. ln this paper we give some simple and useful global estimates of H^M_t(x, y), and apply these estimates to the problem of eigenvalue.     

The Subelliptic Heat Kernel on the Anti-de Sitter Space

We study the subelliptic heat kernel of the sub-Laplacian on a 2n+1-dimensional anti-de Sitter space ?²ⁿ⁺¹ which also appears as a model space of a CR Sasakian manifold with constant negative sectional curvature. In particular we obtain an explicit and geometrically meaningful formula for the subelliptic heat kernel. The key idea is to work in a set of coordinates that reflects the symmetry coming from the Hopf fibration \(\mathbb{S}^{1}\to \mathbb{H}^{2n+1}\). A direct application is obtaining small time asymptotics of the subelliptic heat kernel. Also we derive an explicit formula for the sub-Riemannian distance on ?²ⁿ⁺¹.