Abstract:
By establishing two Lemmas, the exponential stability and the asymptotical stability for mild solution to the second-order neutral stochastic partial differential equations with infinite delay are obtained, respectively. Our results can generalize and improve some existing ones. Finally, an illustrative example is given to show the effectiveness of the obtained results. 1. Introduction The neutral stochastic differential equations can play an important role in describing many sophisticated dynamical systems in physical, biological, medical, chemical engineering, aero-elasticity, and social sciences [1–3], and the qualitative dynamics such as the existence and uniqueness, stability, and controllability for first-order neutral stochastic partial differential equations with delays have been extensively studied by many authors; see, for example, the existence and uniqueness for neutral stochastic partial differential equations under the non-Lipschitz conditions was investigated by using the successive approximation [4–6]; in [7], Caraballo et al. have considered the exponential stability of neutral stochastic delay partial differential equations by the Lyapunov functional approach; in [8], Dauer and Mahmudov have analyzed the existence of mild solutions to semilinear neutral evolution equations with nonlocal conditions by using the fractional power of operators and Krasnoselski-Schaefer-type fixed point theorem; in [9], Hu and Ren have established the existence results for impulsive neutral stochastic functional integrodifferential equations with infinite delays by means of the Krasnoselskii-Schaefer-type fixed point theorem; some sufficient conditions ensuring the controllability for neutral stochastic functional differential inclusions with infinite delay in the abstract space with the help of the Leray-Schauder nonlinear alterative have been given by Balasubramaniam and Muthukumar in [10]; Luo and Taniguchi, in [11], have studied the asymptotic stability for neutral stochastic partial differential equations with infinite delay by using the fixed point theorem. Very recently, in [12], the author has discussed the exponential stability for mild solution to neutral stochastic partial differential equations with delays by establishing an integral inequality. Although there are many valuable results about neutral stochastic partial differential equations, they are mainly concerned with the first-order case. In many cases, it is advantageous to treat the second-order stochastic differential equations directly rather than to convert them to first-order systems.

Abstract:
This paper is mainly concerned with the globally exponential stability in mean square of uncertain neutral stochastic systems with mixed delays and Markovian jumping parameters. The mixed delays are comprised of the discrete interval time-varying delays and the distributed time delays. Taking the stochastic perturbation and Markovian jumping parameters into account, some delay-dependent sufficient conditions for the globally exponential stability in mean square of such systems can be obtained by constructing an appropriate Lyapunov-Krasovskii functional, which are given in the form of linear matrix inequalities (LMIs). The derived criteria are dependent on the upper bound and the lower bound of the time-varying delay and the distributed delay and are therefore less conservative. Two numerical examples are given to illustrate the effectiveness and applicability of our obtained results.

Abstract:
The aim of this paper is devoted to obtain some sufficient conditions for the exponential stability in -moment as well as almost surely exponential stability for mild solution of neutral stochastic partial differential equations with delays by establishing an integral-inequality. Some well-known results are generalized and improved. Finally, an example is given to show the effectiveness of our results.

Abstract:
The aim of this paper is devoted to obtain some sufficient conditions for the exponential stability in p (p≥2)-moment as well as almost surely exponential stability for mild solution of neutral stochastic partial differential equations with delays by establishing an integral-inequality. Some well-known results are generalized and improved. Finally, an example is given to show the effectiveness of our results.

Abstract:
For triangulated surfaces, we introduce the combinatorial Calabi flow which is an analogue of smooth Calabi flow. We prove that the solution of combinatorial Calabi flow exists for all time. Moreover, the solution converges if and only if Thurston's circle packing exists. As a consequence, combinatorial Calabi flow provides a new algorithm to find circle packings with prescribed curvatures. The proofs rely on careful analysis of combinatorial Calabi energy, combinatorial Ricci potential and discrete dual-Laplacians.

Abstract:
Objective: To analyze the ultraviolet fingerprint of the root bark of Paeonia Suffruticosa Andr. and other medicinal crops of its kind by the method of sequential analysis of the double indexes of common and variant peak ratios in ultraviolet fingerprint. Method: In this article three solvents-Chloroform, absolute ethel alcohol and water are introduced to systematically extract the components of the root bark of Paeonia Suffruticosa Andr. and other medicinal plants of its kind in different polar zones, examine their ultraviolet fingerprints and study the differences of the samples. Result: This method is able to accurately identify the root bark of Paeonia Suffruticosa Andr. and other medicinal plants of its kind in detail. Conclusion: Tow or several samples can be reliably identified by the method of sequential analysis of the double indexes of common and variaut peak ratios.

Abstract:
We define Discrete Quasi-Einstein metrics (DQE-metrics) as the critical points of discrete total curvature functional on triangulated 3-manifolds. We study DQE-metrics by introducing some combinatorial curvature flows. We prove that these flows produce solutions which converge to discrete quasi-Einstein metrics when the initial energy is small enough. The proof relies on a careful analysis of discrete dual-Laplacians which we interpret as the Jacobian matrix of the curvature map. As a consequence, combinatorial curvature flow provides an algorithm to compute discrete sphere packing metrics with prescribed curvatures.

Abstract:
For triangulated surfaces locally embedded in the standard hyperbolic space, we introduce combinatorial Calabi flow as the negative gradient flow of combinatorial Calabi energy. We prove that the flow produces solutions which converge to ZCCP-metric (zero curvature circle packing metric) if the initial energy is small enough. Assuming the curvature has a uniform upper bound less than $2\pi$, we prove that combinatorial Calabi flow exists for all time. Moreover, it converges to ZCCP-metric if and only if ZCCP-metric exists.

Abstract:
In this paper, we generalize our results in \cite{GX3} to triangulated surfaces in hyperbolic background geometry, which means that all triangles can be embedded in the standard hyperbolic space. We introduce a new discrete Gaussian curvature by dividing the classical discrete Gauss curvature by an area element, which could be taken as the area of the hyperbolic disk packed at each vertex. We prove that the corresponding discrete Ricci flow converges if and only if there exists a circle packing metric with zero curvature. We also prove that the flow converges if the initial curvatures are all negative. Note that, this result does not require the existence of zero curvature metric or Thurston's combinatorial-topological condition. We further generalize the definition of combinatorial curvature to any given area element and prove the equivalence between the existence of zero curvature metric and the convergence of the corresponding flow.