Let Mⁿ be an n-dimensional compact C^"-differentiable manifold, n >2, and let S be a C¹-differential system on Mⁿ. The system induces a one-parameter C¹. transformation group Ø_t(-" < t < ") over Mⁿ and, thus, naturally induces a one-parameter transformation group of the tangent bundle of Mⁿ. The aim of this paper, in essence, is to study certain ergodic properties of this latter transformation group.Among various results established in the paper, we mention here only the following, which might describe quite well the nature of our study.(A) Let M be the set of regular points in Mⁿ of the differential system S. With respect to a given C^" Riemannian metric of Mⁿ, we consider the bundle £^# of all (n-2) spheres Q_x^n-2, x"M, where Q_x^n-2 for each x consists of all unit tangent vectors of Mⁿ orthogonal to the trajectory through x. Then, the differential system S gives rise naturally to a one-parameter transformation group ψ_t^#(-" < t < ") of £^#. For an l-frame α= (u₁, u₂, ... , u_l) of Mⁿ at a point x in M, 1 > l > n-1, each u_i being in £^#, we shall denote the volume of the parallelotope in the tangent space of Mⁿ at x with edges u₁, u₂, ... , u_l by v(α), and let η^_*α(t) =v(ψ_t^#(u₁),ψ_t^#(u₂),...,ψ_t^#(u_l)).This is a continuous real function of t. Let α is said to be positively linearly independent of the mean if I₊*(α) > 0. Similarly, α is said to be negatively linearly independent of the mean if I_-*(α) > 0. A point x of M is said to possess positive generic index κ=κ₊*(x) if, at x, there is a κ-frame = (u₁, u₂, ... , u_κ) , u_i " £^#, of Mⁿ having the property of being positively linearly independent in the mean, but at x, every l-frame = (v₁, v₂, ... , v_l), v_i " £^#, of Mⁿ with l > κ does not have the same property. Similarly, we define the negative generic index κ_-*(x) of x. For a nonempty closed subset F of Mn consisting of regular points of S, invariant under Ø_t(-" < t < ") let the (positive and negative) generic indices of F be defined by Theorem κ₊*(F) = κ_-*(F). (B)We consider a nonempty compact metric space x and a one-parameter transformation group φ_t(-" < t < ") over X. For a given positive integer l "e 2, we assume that, to each x2X, there are associated l-positive real continuous functions h_x1(t), h_x2(t), ... , h_xl(t) of (-" < t < "). Assume further that these functions possess the following properties, namely, for each of k = 1, 2, ... , l, (i*) h_k(x, t) = h_xk(t) is a continuous function of the Cartesian product X×(-","). (ii*) h_φs(x)k(t)=

hxk(s+t)

hxk(s)

This paper presents a self-contained account concerning a dimensionfree Harnack inequality and its applications. This new type of inequality not only implies heat kernel bounds as the classical Li Yau s Harnack inequality did, but also provides a direct way to describe various dimension-free properties of finite and infinite-dimensional diffusion semigroups. The author starts with a standard weighted Laplace operator on a Riemannian manifold with curvature bounded from below, and then move further to the unbounded below curvature case and its infinite-dimensional settings.

This is a survey on the recent progresses in the study of branching processes with immigration, generalized Ornstein Uhlenbeck processes, and affine Markov processes.We mainly focus on the applications of skew convolution semigroups and the connections in those processes.

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Coupling method is used to obtain the explicit upper and lower bounds for convergence rates in strong ergodicity for Markov processes. For one-dimensional diffusion processes and birth death processes, these bounds are sharp in the sense that the upper one and the lower one are only different by a constant.

In this paper, the point-wise multipliers for the Herz-type Besov spaces are obtained, and the pseudo-differential operators are proved to be bounded in the Herz-type Besov spaces by using the point-wise multipliers.

In this paper, a class of biholomorphic mappings called complete quasiconvex mappings is introduced and studied in bounded convex Reinhardt domains of Cⁿ. Through a detailed analysis of the analytic characterization for this class of mappings, it is shown that this class of mappings contains the convex mappings and is also a subset of the class of starlike mappings. In the special case of the polydisc, a decomposition theorem is established for the complete quasiconvex mappings, which in turn is used to derive an improved sufficient condition for the convex mappings.

The purpose of this paper is to make clear the so-called Nomizu problem, whether it is possible to find examples of space-like isoparametric hypersurfaces in Hⁿ⁺¹₁ with more than two distinct principal curvatures. It is proved that a space-like isoparametric hypersurface in Hⁿ⁺¹₁ or testSⁿ⁺¹₁ can have at most two distinct principal curvatures. The authors present the classification and explicit analytic expressions of such type of isoparametric hypersurfaces.

Suppose that E and F are two Banach spaces and that B(E, F) is the space of all bounded linear operators from E to F. Let T₀∈B(E, F) with a generalized inverse T₀⁺ ∈B(E, F). This paper shows that, for every T∈B(E, F) with T₀+(T-T₀) <1, B(I + T₀⁺(T-T₀))-¹T₀⁺ is a generalized inverse of T if and only if (I-T₀⁺T₀)N(T) = N(T₀), where N( ") stands for the null space of the operator inside the parenthesis. This result improves a useful theorem of Nashed and Cheng and further shows that a lemma given by Nashed and Cheng is valid in the case where T₀ is a semi-Fredholm operator but not in general.

The purpose of this paper is to investigate the Cauchy problem of the Camassa Holm equation. By using the abstract method proposed and studied by T. Kato and priori estimates, the existence and uniqueness of the global solution to the Cauchy problem of the Camassa Holm equation in L^p frame under certain conditions are obtained. In addition, the continuous dependence of the solution of this equation on the linear dispersive coefficient contained in the equation is obtained.