Delay-Dependent Exponential Stability for Nonlinear Reaction-Diffusion Uncertain Cohen-Grossberg Neural Networks with Partially Known Transition Rates via Hardy-Poincaré Inequality

In this paper, stochastic global exponential stability criteria for delayed impulsive Markovian jumping reaction-diffusion Cohen-Grossberg neural networks(CGNNs for short) are obtained by using a novel Lyapunov-Krasovskii functional approach, linear matrix inequalities(LMIs for short) technique, It formula, Poincar′e inequality and Hardy-Poincaré inequality, where the CGNNs involve uncertain parameters, partially unknown Markovian transition rates, and even nonlinear p-Laplace diffusion(p 1). It is worth mentioning that ellipsoid domains in Rm(m ≥ 3) can be considered in numerical simulations for the first time owing to the synthetic applications of Poincar′e inequality and Hardy-Poincar′e inequality. Moreover, the simulation numerical results show that even the corollaries of the obtained results are more feasible and effective than the main results of some recent related literatures in view of significant improvement in the allowable upper bounds of delays.     

Global exponential stability of Cohen-Grossberg neural networks with variable delays

A class of generalized Cohen-Grossberg neural networks（CGNNs） with variable de- lays are investigated. By introducing a new type of Lyapunov functional and applying the homeomorphism theory and inequality technique, some new conditions axe derived ensuring the existence and uniqueness of the equilibrium point and its global exponential stability for CGNNs. These results obtained are independent of delays, develop the existent outcome in the earlier literature and are very easily checked in practice.     

Fractional Sobolev-Poincaré Inequalities in Irregular Domains

This paper is devoted to the study of fractional(q, p)-Sobolev-Poincar′e inequalities in irregular domains. In particular, the author establishes(essentially) sharp fractional(q, p)-Sobolev-Poincar′e inequalities in s-John domains and in domains satisfying the quasihyperbolic boundary conditions. When the order of the fractional derivative tends to 1, our results tend to the results for the usual derivatives. Furthermore, the author verifies that those domains which support the fractional(q, p)-Sobolev-Poincar′e inequalities together with a separation property are s-diam John domains for certain s, depending only on the associated data. An inaccurate statement in [Buckley, S. and Koskela, P.,Sobolev-Poincar′e implies John, Math. Res. Lett., 2(5), 1995, 577–593] is also pointed out.     

本刊英文版Vol.31(2015),No.1论文摘要（英文）

<正>Functionals for Multilinear Fractional Embedding William BECKNERAbstract A novel representation is developed as a measure for multilinear fractional embedding.Corresponding extensions are given for the Bourgain-Brezis-Mironescu theorem and Pitt's inequality.New results are obtained for diagonal trace restriction on submanifolds as an application of the Hardy-Littlewood-Sobolev inequality.Smoothing estimates are used to provide new structural understanding for density functional theory,the Coulomb interaction energy and quantum mechanics of phase space.Intriguing connections are drawn that illustrate interplay among classical inequalities in Fourier analysis.     

DIFFERENTIAL INVERSE VARIATIONAL INEQUALITIES IN FINITE DIMENSIONAL SPACES

In this article,a new differential inverse variational inequality is introduced and studied in finite dimensional Euclidean spaces.Some results concerned with the linear growth of the solution set for the differential inverse variational inequalities are obtained under different conditions.Some existence theorems of Caratheodory weak solutions for the differential inverse variational inequality are also established under suitable conditions.An application to the time-dependent spatial price equilibrium control problem is also given.     

具有变时滞的离散切换系统的时滞依赖稳定性的新判据

This paper deals with the issues of robust stability for uncertain discrete-time switched systems with mode-dependent time delays.Based on a novel difference inequality and a switched Lyapunov function,new delay-dependent stability criteria are formulated in terms of linear matrix inequalities （LMIs） which are not contained in known literature.A numerical example is given to demonstrate that the proposed criteria improves some existing results significantly with much less computational effort.     

The Convergence of the Sums of Independent Random Variables Under the Sub-linear Expectations

Let {Xn;n≥1} be a sequence of independent random variables on a probability space(Ω,F,P) and Sn=∑k=1n Xk.It is well-known that the almost sure convergence,the convergence in probability and the convergence in distribution of Sn are equivalent.In this paper,we prove similar results for the independent random variables under the sub-linear expectations,and give a group of sufficient and necessary conditions for these convergence.For proving the results,the Levy and Kolmogorov maximal inequalities for independent random variables under the sub-linear expectation are established.As an application of the maximal inequalities,the sufficient and necessary conditions for the central limit theorem of independent and identically distributed random variables are also obtained.     

The Same Distribution of Limit Cycles in a Hamiltonian System with Nine Seven-order Perturbed Terms

Using qualitative analysis and numerical simulation, we investigate the number and distribution of limit cycles for a cubic Hamiltonian system with nine different seven-order perturbed terms. It is showed that these perturbed systems have the same distribution of limit cycles. Furthermore, these systems have 13, 11 and 9 limit cycles for some parameters, respectively. The accurate positions of the 13, 11 and 9 limit cycles are obtained by numerical exploration, respectively. Our results imply that these perturbed systems are equivalent in the sense of distribution of limit cycles. This is useful for studying limit cycles of perturbed systems.     

Criteria for Super-and Weak-Poincaré Inequalities of Ergodic Birth-Death Processes

Criteria for the super-Poincaré inequality and the weak-Poincaré inequality about ergodic birth-death processes are presented. Our work further completes ten criteria for birth-death processes presented in Table 1.4 (p. 15) of Prof. Mu-Fa Chen's book "Eigenvalues, Inequalities and Ergodic Theory" (Springer, London, 2005). As a byproduct, we conclude that only ergodic birth-death processes on finite state space satisfy the Nash inequality with index 0 ν≤ 2.     

Sign-based Test for Mean Vector in High-dimensional and Sparse Settings

In this article, we introduce a robust sparse test statistic which is based on the maximum type statistic. Both the limiting null distribution of the test statistic and the power of the test are analysed. It is shown that the test is particularly powerful against sparse alternatives. Numerical studies are carried out to examine the numerical performance of the test and to compare it with other tests available in the literature. The numerical results show that the test proposed significantly outperforms those tests in a range of settings, especially for sparse alternatives.     

Poincaré type inequalities on the discrete cube and in the CAR algebra

We prove L ^p Poincaré inequalities with suitable dimension free constants for functions on the discrete cube {?1, 1} ⁿ . As well known, such inequalities for p an even integer allow to recover an exponential inequality hence the concentration phenomenon first obtained by Bobkov and Götze. We also get inequalities between the L ^p norms of $ \left\vert \nabla f\right\vert We prove L ^p Poincaré inequalities with suitable dimension free constants for functions on the discrete cube {−1, 1} ⁿ . As well known, such inequalities for p an even integer allow to recover an exponential inequality hence the concentration phenomenon first obtained by Bobkov and G?tze. We also get inequalities between the L ^p norms of and moreover L ^p spaces may be replaced by more general ones. Similar results hold true, replacing functions on the cube by matrices in the *-algebra spanned by n fermions and the L ^p norm by the Schatten norm C _p .     

p-Poincaré inequality versus ∞-Poincaré inequality: some counterexamples

We point out some of the differences between the consequences of p-Poincaré inequality and that of ∞-Poincaré inequality in the setting of doubling metric measure spaces. Based on the geometric characterization of ∞-Poincaré inequality given in Durand-Cartagena et al. (Mich Math J 60, 2011), we obtain a geometric property implied by the support of a p-Poincaré inequality, and demonstrate by examples that an analogous geometric characterization for finite p is not possible. The examples we give are metric measure spaces which are doubling and support an ∞-Poincaré inequality, but support no finite p-Poincaré inequality. In particular, these examples show that one cannot expect a self-improving property for ∞-Poincaré inequality in the spirit of Keith–Zhong (Ann Math 167(2):575–599, 2008). We also show that the persistence of Poincaré inequality under measured Gromov–Hausdorff limits fails for ∞-Poincaré inequality.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> harmonic 1-form on submanifold with weighted Poincaré inequality

We deal with complete submanifolds with weighted Poincaré inequality. By assuming the submanifold is δ-stable or has sufficiently small total curvature, we establish two vanishing theorems for L ^p harmonic 1-forms, which are extensions of the results of Dung-Seo and Cavalcante-Mirandola-Vitório.     

Weak Poincaré Inequalities, Decay of Markov Semigroups and Concentration of Measures

For a reasonable class of weak Poincaré inequalities, the decay of the corresponding Markov semigroups obtained earlier by Röckner and the first named author is improved by removing an extra L ²–norm. Next, a concentration estimate of the reference measure is presented for the weak Poincaré inequality, which is sharp as illustrated by some examples of one–dimensional diffusion processes.     

A Poincar�� Inequality for Orlicz�CSobolev Functions with Zero Boundary Values on Metric Spaces

We prove a Poincaré inequality for Orlicz–Sobolev functions with zero boundary values in bounded open subsets of a metric measure space. This result generalizes the (p, p)-Poincaré inequality for Newtonian functions with zero boundary values in metric measure spaces, as well as a Poincaré inequality for Orlicz–Sobolev functions on a Euclidean space, proved by Fuchs and Osmolovski (J Anal Appl (Z.A.A.) 17(2):393–415, 1998). Using the Poincaré inequality for Orlicz–Sobolev functions with zero boundary values we prove the existence and uniqueness of a solution to an obstacle problem for a variational integral with nonstandard growth.     

Vanishing theorems for ach Kähler manifolds and harmonic maps

We discuss a class of complete Kähler manifolds which are asymptotically complex hyperbolic near infinity. The main result is vanishing theorems for the second L ² cohomology of such manifolds when it has positive spectrum. We also generalize the result to the weighted Poincaré inequality case and establish a vanishing theorem provided that the weighted function ρ is of sub-quadratic growth of the distance function. We also obtain a vanishing theorem of harmonic maps on manifolds which satisfies the weighted Poincaré inequality.     

Dirichlet problem for semilinear edge-degenerate elliptic equations with singular potential term

In this paper, we introduce weighted p-Sobolev spaces on manifolds with edge singularities. We give the proof for the corresponding edge type Sobolev inequality, Poincaré inequality and Hardy inequality. As an application of these inequalities, we prove the existence of nontrivial weak solutions for the Dirichlet problem of semilinear elliptic equations with singular potentials on manifolds with edge singularities.     

THE POINCARÉ INEQUALITY FOR A VECTOR FIELD WITH ZERO TANGENTIAL OR NORMAL COMPONENT ON THE BOUNDARY

ABSTRACT

Poincaré's inequality is well known: given a bounded domain G, ∥u∥_p ? c∥?u∥_p provided u(x) vanishes on the boundary ?G. The case where u(x) is a vector field u(x) that does not vanish on the boundary ?G is considered. It is shown that when either the tangential component or the normal component vanishes on the boundary ?G, then the Poincaré inequality is satisfied.     

Poincaré inequalities and Newtonian Sobolev functions on noncomplete metric spaces

Let X be a noncomplete metric measure space satisfying the usual (local) assumptions of a doubling property and a Poincaré inequality. We study extensions of Newtonian Sobolev functions to the completion $\stackrel{?}{X}$ of X and use them to obtain several results on X itself, in particular concerning minimal weak upper gradients, Lebesgue points, quasicontinuity, regularity properties of the capacity and better Poincaré inequalities. We also provide a discussion about possible applications of the completions and extension results to p-harmonic functions on noncomplete spaces and show by examples that this is a rather delicate issue opening for various interpretations and new investigations.     

On the Discrete Poincaré–Friedrichs Inequalities for Nonconforming Approximations of the Sobolev Space H 1

We present a direct proof of the discrete Poincaré–Friedrichs inequalities for a class of nonconforming approximations of the Sobolev space H ¹(Ω), indicate optimal values of the constants in these inequalities, and extend the discrete Friedrichs inequality onto domains only bounded in one direction. We consider a polygonal domain Ω in two or three space dimensions and its shape-regular simplicial triangulation. The nonconforming approximations of H ¹(Ω) consist of functions from H ¹ on each element such that the mean values of their traces on interelement boundaries coincide. The key idea is to extend the proof of the discrete Poincaré–Friedrichs inequalities for piecewise constant functions used in the finite volume method. The results have applications in the analysis of nonconforming numerical methods, such as nonconforming finite element or discontinuous Galerkin methods.