Some results on extending and sharpening the Weierstrass product inequalities

In this paper, we establish two extensions of Weierstrass's inequality involving symmetric functions by means of the theory of majorization, and give an interesting sharpness of Weierstrass's inequality by using the arithmetic-geometric mean inequality. Furthermore, we apply these results to improve a well-known inequality and deduce some new inequalities.     

An overview of some recent results on the Euler system of isentropic gas dynamics

This overview is concerned with the well-posedness problem for the isentropic compressible Euler equations of gas dynamics. The results we present are in line with the programof investigatingthe efficiency of different selection criteria proposed in the literature in order to weed out non-physical solutions to more-dimensional systems of conservation laws and they build upon the method of convex integration developed by De Lellis and Székelyhidi for the incompressible Euler equations. Mainly following [5], we investigate the role of the maximal dissipation criterion proposed by Dafermos in [6]: we prove how, for specific pressure laws, some non-standard (i.e. constructed via convex integration methods) solutions to the Riemann problem for the isentropic Euler system in two space dimensions have greater energy dissipation rate than the classical self-similar solution emanating from the same Riemann data. We therefore show that the maximal dissipation criterion proposed by Dafermos does not favour in general the self-similar solutions.     

The nonsymmetric case of the Keller-Segel model in chemotaxis: some recent results

Looking at the nonsymmetric case of a reaction-diffusion model known as the Keller-Segel model, we summarize known facts concerning (global in time) existence and prove new blowup results for solutions of this system of two strongly coupled parabolic partial differential equations. We show in Section 4, Theorem 4, that if the solution blows up under a condition on the initial data, blowup takes place at the boundary of a smooth domain . Using variational techniques we prove in Section 5 the existence of nontrivial stationary solutions in a special case of the system. Received April 2000     

Low-rank and sparse matrix recovery from noisy observations via 3-block ADMM algorithm

Recovering low-rank and sparse matrix from a given matrix arises in many applications, such as image processing, video background substraction, and so on. The 3-block alternating direction method of multipliers (ADMM) has been applied successfully to solve convex problems with 3-block variables. However, the existing sufficient conditions to guarantee the convergence of the 3-block ADMM usually require the penalty parameter $\gamma$ to satisfy a certain bound, which may affect the performance of solving the large scale problem in practice. In this paper, we propose the 3-block ADMM to recover low-rank and sparse matrix from noisy observations. In theory, we prove that the 3-block ADMM is convergent when the penalty parameters satisfy a certain condition and the objective function value sequences generated by 3-block ADMM converge to the optimal value. Numerical experiments verify that proposed method can achieve higher performance than existing methods in terms of both efficiency and accuracy.     

Heuristic and optimization approaches to extending the Kohonen self organizing algorithm

The Kohonen self organizing neural network has been applied to an increasingly wider range of application problems that traditionally have been the domain of statistical and operational research techniques, such as data clustering and classification, and optimization and control. This Kohonen network is bestowed with a number of unique strengths which are, unfortunately, matched by an equally formidable set of limitations due its learning algorithm. There have been extensive studies over the last decade to extend the Kohonen neural network using heuristic and optimization approaches. This paper provides a comprehensive survey of the research efforts directed to enhancing the Kohonen self organizing neural network and its learning algorithm. We also point out some research directions for pursuing further improvements.     

Some recent and new results on the maximum entropy distribution

This letter concerns the maximum entropy (ME) distribution proposed by Theil and Laitinen (1980); it summarizes some of the results obtained with this distribution and its explores the conditioning of the ME covariance matrix as the number of variables increases.     

A short proof of some recent results related to Cesàro function spaces

We give a short proof of the recent results that, for every 1≤p<∞$1\le p<\infty$, the Cesàro function space Ce_sp(I)$Ce{s}_p\left(I\right)$ is not a dual space, has the weak Banach–Saks property and does not have the Radon–Nikodym property.     

Stability of synchronized dynamics and pattern formation in coupled systems: Review of some recent results

In arbitrarily coupled dynamical systems (maps or ordinary differential equations), the stability of synchronized states (including equilibrium point, periodic orbit or chaotic attractor) and the formation of patterns from loss of stability of the synchronized states are two problems of current research interest. These two problems are often treated separately in the literature. Here, we present a unified framework in which we show that the eigenvalues of the coupling matrix determine the stability of the synchronized state, while the eigenvectors correspond to patterns emerging from desynchronization. Based on this simple framework three results are derived: First, general approaches are developed that yield constraints directly on the coupling strengths which ensure the stability of synchronized dynamics. Second, when the synchronized state becomes unstable spatial patterns can be selectively realized by varying the coupling strengths. Distinct temporal evolution of the spatial pattern can be obtained depending on the bifurcating synchronized state. Third, given a desired spatiotemporal pattern, one is able to design coupling schemes which give rise to that pattern as the coupled system evolves. Systems with specific coupling schemes are used as examples to illustrate the general methods.     

Remarks on some recent metrical common fixed point theorems

In an interesting article, Bouhadjera and Godet-Thobie (2009) [6] introduced notions of subcompatibility and subsequential continuity, and utilized them to prove several common fixed point theorems. The results of the aforementioned article contain flaws, and they are not correct in their present form. But these results can be recovered in two ways by strengthening either of the newly introduced definitions. The results, in their corrected form, still bring about noted improvements over a multitude of relevant fixed point theorems of the existing literature, substantiating the utility of these (two) new notions.     

Vizing's conjecture: a survey and recent results

Vizing's conjecture from 1968 asserts that the domination number of the Cartesian product of two graphs is at least as large as the product of their domination numbers. In this paper we survey the approaches to this central conjecture from domination theory and give some new results along the way. For instance, several new properties of a minimal counterexample to the conjecture are obtained and a lower bound for the domination number is proved for products of claw‐free graphs with arbitrary graphs. Open problems, questions and related conjectures are discussed throughout the paper. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 46–76, 2012     

Comments on some recent methods for the simultaneous determination of polynomial zeros

In this note we give some comments on the recent results concerning a simultaneous method of the fourth-order for finding complex zeros in circular interval arithmetic. The main discussion is directed to a rediscovered iterative formula and its modification, presented recently in (Sun and Kosmol (J. Comput. Appl. Math. 130 (2001) 293)). The presented comments include some critical parts of the papers (Jpn. J. Indust. Appl. Math. 15 (1998) 295) and (Sun and Kosmol, 2001) which treat the same subject.     

关于单叶函数中几个结果的注记

首先就Mbius变换在几何函数论上应用时存在的两个误区予以讨论,指出其存在的问题与如何改正;其次明确地指出用Baernstein方法估计m次对称单叶函数之逆函数系数时的不足之处;最后就S ingh与Puri一文中的一个结果给以改正,并用Ruschew eyh的共轭方法得到该结果成立的最佳范围.     

Remarks on some associated Laguerre integral results

Motivated essentially by their possible need in a fairly large number of physical and chemical contexts, Mavromatis and Alassar [1] derived several associated Laguerre integral results by eliminating an unnecessary constraint used in an earlier paper on the subject by Mavromatis [2]. The main object of the present sequel to these recent works is to investigate and apply much more general families of integral formulas, involving products of two or more Laguerre polynomials, which have been considered in the mathematical literature rather extensively.