Lame equation and asymptotic higher-order periodic solutions to nonlinear evolution equations

Based on an auxiliary Lame equation and the perturbation method, a direct method is proposed to construct asymptotic higher-order periodic solutions to some nonlinear evolution equations. It is shown that some asymptotic higher-order periodic solutions to some nonlinear evolution equations in terms of Jacobi elliptic functions are explicitly obtained with the aid of symbolic computation.     

Comment on “Analytical evaluation for two-center nuclear attraction integrals over Slater type orbitals by using Fourier transform method” by S. ?zcan and E. ?ztekin (J. Math. Chem. DOI 10.1007/s10910-008-9398-z)

S. ?zcan and E. ?ztekin, (J. Math. Chem. doi:) published formulas for evaluating the two-center nuclear attraction integrals over Slater type orbitals. It is shown that the analytical relations for these integrals through the expansion coefficients of the electron charge density for the one-center case and the overlap integrals presented in Sect. 3 of this work can easily be derived by means of a simple algebra from the formulas published in our papers (I.I. Guseinov, J Mol Struct (Theochem) 417:117, 1997; J Math Chem 42:415, 2007 and B.A. Mamedov, Chin J Chem 22:545, 2004). It should be noted that the formulas of overlap integrals presented by E. ?ztekin et al., in previous paper (E. ?ztekin, M. Yavuz, Ş. Atalay, J Mol Struct (Theochem) 544:69, 2001) for the calculation of two-center nuclear attraction integrals also are obtained from our papers (see Comment: I.I. Guseinov, J Mol Struct (Theochem) 638:235, 2003).     

Analytical evaluation for two-center nuclear attraction integrals over slater type orbitals by using Fourier transform method

In this study, we shall suggest analytical expressions for two-center nuclear attraction integrals over STO’s with a one-center charge distribution by using Fourier transform method. The derivation is based on partial-fraction decompositions and Taylor expansions of rational functions. Analytical expressions obtained by this method are expressed in terms of Gegenbauer, and binomial coefficients and linear combinations of STO’s. Finally, it is relatively easy to express the Fourier integral representations of two-center nuclear attraction integrals with a one-center charge distribution mentioned above as finite and infinite of series of STO’s and irregular solid harmonics which may be considered to be limiting cases of STO’s.     

Complex dynamics of a discrete fractional‐order Leslie‐Gower predator‐prey model

A proposed discretized form of fractional‐order prey‐predator model is investigated. A sufficient condition for the solution of the discrete system to exist and to be unique is determined. Jury stability test is applied for studying stability of equilibrium points of the discretized system. Then, the effects of varying fractional order and other parameters of the systems on its dynamics are examined. The system undergoes Neimark‐Sacker and flip bifurcation under certain conditions. We observe that the model exhibits chaotic dynamics following stable states as the memory parameter α decreases and step size h increases. Theoretical results illustrate the rich dynamics and complexity of the model. Numerical simulation validates theoretical results and demonstrates the presence of rich dynamical behaviors include S‐asymptotically bounded periodic orbits, quasi‐periodicity, and chaos. The system exhibits a wide range of dynamical behaviors for fractional‐order α key parameter.     

两个四硫富瓦烯分子正离子自由基TTF·^＋之间的共价吸引作用

使用MP2／6-311G方法得到了四硫富瓦烯自由基正离子二聚体（TTF·^＋ -TTF·^＋）能量极小点的结构．这显示这种正离子间的吸引作用存在．这种新的吸引作用是望远镜形状的20中心2电子分子间共价π/π键．这种共价π/π键的键能约为-21kcal·mol^-1,它被正离子间的库伦排斥作用掩盖．有负离子围绕的四硫富瓦烯自由基正离子二聚体（TTF·^＋ -TTF·^＋）体系是稳定的．     

Asymptotic Results for the Sum of Dependent Non-identically Distributed Random Variables

In this paper we extend some results about the probability that the sum of n dependent subexponential random variables exceeds a given threshold u. In particular, the case of non-identically distributed and not necessarily positive random variables is investigated. Furthermore we establish criteria how far the tail of the marginal distribution of an individual summand may deviate from the others so that it still influences the asymptotic behavior of the sum. Finally we explicitly construct a dependence structure for which, even for regularly varying marginal distributions, no asymptotic limit of the tail of the sum exists. Some explicit calculations for diagonal copulas and t-copulas are given. Dominik Kortschak was supported by the Austrian Science Fund Project P18392.     

Prey-predator systems with intertwined basins of attraction

In this article, we show the existence of intertwined basins of attraction for a class of prey-predator systems, which improves the extant results by deleting some crucial and hard testing conditions.     

Charged patch attraction in dispersion: effect of polystyrene sulphonate molecular weight or patch size

The molecular weight (Mw) effect of polystyrene sulphonate (PSS) on the yield stress of ZrO₂ dispersion was evaluated. Three “monodispersed” samples with Mw of 1430, 9960 and 130,000, and two broad distribution samples of Mw 70,000 and 1.0 million were investigated. Adsorbed PSS gave rise to charged patch attraction that enhanced the maximum yield stress, τ _{y max}, of dispersions at charge neutral condition [3]. As a PSS patch consists of only one molecule, the patch size should increase with increasing Mw. For PSS of Mw 1430 with just only seven repeating units the patch formed would be rectangular (linear) and small with a high charge density. For Mw of 9960 and higher, the patch would be irregular in shape with a smaller charge density. At a moderate surface coverage of ˜20%, the τ _{y max} increases quite sharply from Mw 9960 to 130,000 and very gradually from 130,000 to 1.0 million. From Mw 9960 to 130,000, the strength of the patch attraction increases quite strongly with increasing patch size. From Mw 130,000 to 1million, the small increase in τ _{y max} reflected a limiting patch size effect. At the limiting size, the attraction is at maximum strength. At the same coverage, PSS of Mw 1430 also displayed a strong charge patch attraction. This was attributed to a higher number of charged patch interactions at the closest point of separation and also to the high charge density of the PSS patch. The low surface coverage of 6% revealed no significant molecular weight effect on τ _{y max}. The network structure consisted of a mixture of charge patch and bare surface interactions. Received: 4 July 2000/Accepted: 18 July 2000