The exponential convergence rate in entropy is studied for symmetric forms, with a special attention to the Markov chain with a state space having two points only. Some upper and lower bounds of the rate are obtained and ?ve examples with precise or qualitatively exact estimates are presented.

By manipulating classical Fisher information and employing various derivatives of density operators, and using entirely intuitive and direct methods, we introduce two families of quantum extensions of Fisher information that include those de?ned via the symmetric logarithmic derivative, via the right logarithmic derivative, via the Bogoliubov-Kubo-Mori derivative, as well as via the derivative in terms of commutators, as special cases. Some fundamental properties of these quantum extensions of Fisher information are investigated, a multi-parameter quantum Cram?r-Rao inequality is established, and applications to characterizing quantum uncertainty are illustrated.

The Ginzburg-Landau-type complex equations are simpli?ed mathematical models for various pattern formation systems in mechanics, physics, and chemistry. In this paper, the derivative complex Ginzburg-Landau (DCGL) equation in an unbounded domain Ω? R² is studied. We extend the Gagliardo-Nirenberg inequality to the weighted Sobolev spaces introduced by S. V. Zelik. Applied this Gagliardo-Nirenberg inequality of the weighted Sobolev spaces and based on the technique for the semi-linear system of parabolic equations which has been developed by M. A. Efendiev and S. V. Zelik, the global attractor A in the corresponding phase space is constructed, the upper bound of its Kolmogorov s ε-entropy is obtained, and the spatial chaos of the attractor A for DCGL equation in R² is detailed studied.

In this paper, we give some rigidity theorems which concern with compact minimal coisotropic submanifolds in CPⁿ, compact minimal quaternionic coisotropic submanifolds in QPⁿ and compact minimal hypersurfaces in P² (Cay).

In terms of continuous decomposition and choosing an appropriate BMO function, the authors obtain a sharp necessary condition for L^p boundedness of the commutators generated by Bochner-Riesz operators below the critical index and BMO functions.

In this paper, we prove that a Cayley digraph Γ = Cay (G, S) is a nontrivial lexicographical product if and only if there is a nontrivial subgroup H of G such that S\H is a union of some double cosets of H in G.

Let H be a Hilbert space, A ∈ L(H), y ∈ R(A), and y ∈R(A). We study the behavior of the distance square between y and A(B_τ), defined as a functional F(τ), as the radius τ of the ball B_τ of H tends to ∞. This problem is important in estimating the approximation error in learning theory. Our main result is to estimate the asymptotic behavior of F(τ) without the compactness assumption on the operator A. We also consider the Peetre K-functional and its convergence rates.

In this paper, ruin problems in the risk model with stochastic premium incomes and stochastic return on investments are studied. The logarithm of the asset price process is assumed to be a Lévy process. An exact expression for expected discounted penalty function is established. Lower bounds and two kinds of upper bounds for expected discounted penalty function are obtained by inductive method and martingale approach. Integro-differential equations for the expected discounted penalty function are obtained when the L暍y process is a Brownian motion with positive drift and a compound Poisson process, respectively. Some analytical examples and numerical examples are given to illustrate the upper bounds and the applications of the integro-differential equations in this paper.

A series of admissible wavelets is ?xed, which forms an orthonormal basis for the Hilbert space of all the quaternion-valued admissible wavelets. It turns out that their corresponding admissible wavelet transforms give an orthogonal decomposition of L²(IG(2),H).