On dynamics of vertices of locally connected polynomial Julia sets

Let be a polynomial whose Julia set is locally connected. Then a non-preperiodic non-precritical vertex of must have the limit set which coincides with the limit set of an appropriately chosen recurrent critical point of . In particular, if all critical points of are non-recurrent then all vertices of are preperiodic or precritical.

     

Positivity preserving finite element approximation

We consider finite element operators defined on ``rough' functions in a bounded polyhedron in . Insisting on preserving positivity in the approximations, we discover an intriguing and basic difference between approximating functions which vanish on the boundary of and approximating general functions which do not. We give impossibility results for approximation of general functions to more than first order accuracy at extreme points of . We also give impossibility results about invariance of positive operators on finite element functions. This is in striking contrast to the well-studied case without positivity.

     

Hausdorff Dimension of Sets of Generic Points for Gibbs Measures

For a translation invariant Gibbs measure on the configuration space X of a lattice finite spin system, we consider the set X of generic points. Using a Breiman type convergence theorem on the set X of generic points of an arbitrary translation invariant probability measure on X, we evaluate the Hausdorff dimension of the set X with respect to any metric out of a wide class of scale metrics on X (including Billingsley metrics generated by Gibbs measures).     

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.     

On the Stability of the O(N)-Invariant and the Cubic-Invariant Three-Dimensional N-Component Renormalization-Group Fixed Points in the Hierarchical Approximation

We compute renormalization-group fixed points and their spectrum in an ultralocal approximation. We study a case of two competing nontrivial fixed points for a three-dimensional real N-component field: the O(N)-invariant fixed point vs. the cubic-invariant fixed point. We compute the critical value N _c of the cubic ⁴-perturbation at the O(N)-fixed point. The O(N)-fixed point is stable under a cubic ⁴-perturbation below N _c; above N _c it is unstable. The Critical value comes out as 2.219435<N _c<2.219436 in the ultralocal approximation. We also compute the critical value of N at the cubic invariant fixed point. Within the accuracy of our computations, the two values coincide.     

On a Primal-Dual Analytic Center Cutting Plane Method for Variational Inequalities

We present an algorithm for variational inequalities VI( , Y) that uses a primal-dual version of the Analytic Center Cutting Plane Method. The point-to-set mapping is assumed to be monotone, or pseudomonotone. Each computation of a new analytic center requires at most four Newton iterations, in theory, and in practice one or sometimes two. Linear equalities that may be included in the definition of the set Y are taken explicitly into account.We report numerical experiments on several well—known variational inequality problems as well as on one where the functional results from the solution of large subproblems. The method is robust and competitive with algorithms which use the same information as this one.     

Numerical algorithms for uniform Airy-type asymptotic expansions

Airy-type asymptotic representations of a class of special functions are considered from a numerical point of view. It is well known that the evaluation of the coefficients of the asymptotic series near the transition point is a difficult problem. We discuss two methods for computing the asymptotic series. One method is based on expanding the coefficients of the asymptotic series in Maclaurin series. In the second method we consider auxiliary functions that can be computed more efficiently than the coefficients in the first method, and we do not need the tabulation of many coefficients. The methods are quite general, but the paper concentrates on Bessel functions, in particular on the differential equation of the Bessel functions, which has a turning point character when order and argument of the Bessel functions are equal.     

半无限规划的一个对偶问题

本文对半无限凸规划提出一个新的对偶问题，使用扰动函数、次微分和法锥，文中证明了相应的弱对偶性及强对偶性的充要条件．     

A NEW FIXED POINT THEOREM FOR CONE MAPS AND ITS APPLICATIONS

1IntroductionLetEbearealBanachspace.AnonemptyconvexclosedsetPCEiscalledaconeinEifitsatisfiesthefollowingtwoconditionsf(i)xEP,A20impliesAxEP,(n)xEP,--xEPimpliesx=0.AconePiscalledsolidifitcontainsinteriorpoints,i.e.,P/gi.TheconePinEdefinesapartialorderingx5…     

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.