Position-space renormalization for systems with weak long-range interactions and the breakdown of hyperscaling

Exact renormalization group equations are derived for a position-space renormalization of spin systems with weak long-range forces. It is shown how an apparent dependence of the critical exponents on the choice of the renormalization group can be resolved via the mechanism of dangerous irrelevant variables and that this same mechanism is responsible for the breakdown of hyperscaling. The dimensiond=4 can be seen to be a borderline dimension above which classical critical exponents are expected.     

An intermediate spherical model of a ferromagnet

A one-parameter family of partition functions is considered which for zero value of the parameter reduces to the spherical model of a ferromagnet. The model for > 0 is closer to the usual discrete lattice spin model of a ferromagnet than is the spherical model. The first four terms in of the limiting value of the partition function are calculated above and below the critical temperature for arbitrary interactions using the saddle point method to calculate certain correlation functions for the spherical model. These calculations indicate that the critical temperature is independent of for small and certain interactions.Part of this research appeared in the author's doctoral thesis.⁽³⁾     

Improved RGF method to find saddle points

The predictor-corrector method for following a reduced gradient (RGF) to determine saddle points [Quapp, W. et al., J Comput Chem 1998, 19, 1087] is further accelerated by a modification allowing an implied corrector step per predictor but almost without additional costs. The stability and robustness of the RGF method are improved, and the new version in addition reduces the number of gradient and Hessian calculations.     

Hölder Continuity of the Saddle Point Set for Real-Valued Functions

This paper is concerned with Hölder continuity of the solution to a saddle point problem. Some new su?cient conditions for the uniqueness and Hölder continuity of the solution for a perturbed saddle point problem are established. Applications of the result on Hölder continuity of the solution for perturbed constrained optimization problems are presented under mild conditions. Examples are given to illustrate the obtained results.     

Optimality for set functions with values in ordered vector spaces

Let (X, , ) be a finite atomless measure space,L a convex subfamily of , andY andZ locally convex Hausdorff topological vector spaces which are ordered by the conesC andD, respectively. LetF:LY beC-convex andG:LZ beD-convex set functions. Consider the following optimization problem (P): minimizeF(), subject to L andG() _D . The paper generalizes the Moreau-Rockafellar theorem with set functions. By applying this theorem, a Kuhn-Tucker type optimality condition and a Fritz John type optimality condition for problem (P) are established. The duality theorem for problem (P) is also studied.This work was partially supported by National Science Council, Taipei, Taiwan. This paper was written while the first author was visiting at the University of Iowa, 1987-88.The authors would like to express their gratitude to the two anonymous referees for their valuable comments. Also, they would like to thank Professor P. L. Yu for his encouragement and suggestions which improved the material presented here considerably.     

一类Hermitian鞍点矩阵的特征值估计

本文研究了一类大型稀疏Hermitian鞍点线性系统Az=(B E E* 0)(x y)=(f g)=b系数矩阵的特征值,其中B∈C~(p×p)是Hermitian正定阵矩阵,E∈C~(p×q)是列降秩.本文分别给出了该系数矩阵正特征值与负特征值界的一个估计式,同时通过数值算例验证本文所给出的特征值界的估计是合理且有效的.     

Limit cycle bifurcations near a double homoclinic loop with a nilpotent saddle of order m

In this paper, we study the explicit expansion of the first order Melnikov function near a double homoclinic loop passing through a nilpotent saddle of order m in a near-Hamiltonian system. For any positive integer $m(m\ge 1)$, we derive the formulas of the coefficients in the expansion, which can be used to study the limit cycle bifurcations for near-Hamiltonian systems. In particular, for $m=2$, we use the coefficients to consider the limit cycle bifurcations of general near-Hamiltonian systems and give the existence conditions for 10, 11, 13, 15 and 16 (11, 13 and 16, respectively) limit cycles in the case that the homoclinic loop is of cuspidal type (smooth type, respectively) and their distributions. As an application, we consider a near-Hamiltonian system with a nilpotent saddle of order 2 and obtain the lower bounds of the maximal number of limit cycles.     

Bifurcation analysis of a generalized Gause model with prey harvesting and a generalized Holling response function of type III

In this paper we study a generalized Gause model with prey harvesting and a generalized Holling response function of type III: . The goal of our study is to give the bifurcation diagram of the model. For this we need to study saddle-node bifurcations, Hopf bifurcation of codimension 1 and 2, heteroclinic bifurcation, and nilpotent saddle bifurcation of codimension 2 and 3. The nilpotent saddle of codimension 3 is the organizing center for the bifurcation diagram. The Hopf bifurcation is studied by means of a generalized Liénard system, and for b=0 we discuss the potential integrability of the system. The nilpotent point of multiplicity 3 occurs with an invariant line and can have a codimension up to 4. But because it occurs with an invariant line, the effective highest codimension is 3. We develop normal forms (in which the invariant line is preserved) for studying of the nilpotent saddle bifurcation. For b=0, the reversibility of the nilpotent saddle is discussed. We study the type of the heteroclinic loop and its cyclicity. The phase portraits of the bifurcations diagram (partially conjectured via the results obtained) allow us to give a biological interpretation of the behavior of the two species.     

Kronecker product approximation preconditioners for convection–diffusion model problems

We consider the iterative solution of linear systems arising from four convection–diffusion model problems: scalar convection–diffusion problem, Stokes problem, Oseen problem and Navier–Stokes problem. We design preconditioners for these model problems that are based on Kronecker product approximations (KPAs). For this we first identify explicit Kronecker product structure of the coefficient matrices, in particular for the convection term. For the latter three model cases, the coefficient matrices have a 2 × 2 block structure, where each block is a Kronecker product or a summation of several Kronecker products. We then use this structure to design a block diagonal preconditioner, a block triangular preconditioner and a constraint preconditioner. Numerical experiments show the efficiency of the three KPA preconditioners, and in particular of the constraint preconditioner that usually outperforms the other two. This can be explained by the relationship that exists between these three preconditioners: the constraint preconditioner can be regarded as a modification of the block triangular preconditioner, which at its turn is a modification of the block diagonal preconditioner based on the cell Reynolds number. Copyright © 2009 John Wiley & Sons, Ltd.     

Linear-quadratic two-person zero-sum differential games,necessary and sufficient conditions: Comment

We give a simpler, easier-to-check, version of the theorem of the paper referred to, i.e., a necessary and sufficient condition for the existence of a saddle point to the linear-quadratic two-person zero-sum perfect information differential game.