Bivariate Free Knot Splines

We consider the least squares approximation of gridded 2D data by tensor product splines with free knots. The smoothing functional to be minimized—a generalization of the univariate Schoenberg functional—is chosen in such a way that the solution of the bivariate problem separates into the solution of a sequence of univariate problems in case of fixed knots. The resulting optimization problem is a constrained separable least squares problem with tensor product structure. Based on some ideas developed by the authors for the univariate case, an efficient method for solving the specially structured 2D problem is proposed, analyzed and tested on hand of some examples from the literature.     

2+1维SU(2)格点规范场胶球质量的七阶RPA近似

采用无规相近似(RPA)方法,用空心图作为试探波函数,利用Feymann?Hellman定理计算七阶2?+?1维SU(2?)格点规范场的胶球质量,在弱耦合区1/g2?=1.0?–?1.8胶球质量表现出良好的标度行为,基本趋于常数(m/e2?≈1.2?0±0?.0?1)?.     

Magnetic bubble refraction and quasibreathers in inhomogeneous antiferromagnets

We study the dynamics of magnetic bubble solitons in a two-dimensional isotropic antiferromagnetic spin lattice in the case where the exchange integral J(x, y) is position dependent. In the near-continuum regime, this system is described by the relativistic O(3) sigma model on a space-time with a spatially inhomogeneous metric determined by J. We use the geodesic approximation to describe the low-energy soliton dynamics in this system: the n-soliton motion is approximated by geodesic motion in the moduli space M _n of static n-solitons equipped with the L ² metric γ. We obtain explicit formulas for γ for various natural choices of J(x, y). Based on these, we show that single soliton trajectories are refracted with J⁻¹ being analogous to the refractive index and that this refraction effect allows constructing simple bubble lenses and bubble guides. We consider the case where J has a disk inhomogeneity (with the value J ₊ outside a disk and J ₋ < J ₊ inside) in detail. We argue that for sufficiently large J ₊/J ₋, this type of antiferromagnet supports approximate quasibreathers: two or more coincident bubbles confined within the disk spin internally while their shape oscillates with a generically incommensurate period. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 1, pp. 191–208, July, 2007.     

Banach-Saks property in Marcinkiewicz spaces

We catalogue all Marcinkiewicz function and sequence spaces with the Banach-Saks property and present necessary and sufficient conditions for a wide subclass of spaces to possess the p-Banach-Saks property, 1<p<∞. We apply our results to several open problems.     

The James constant, the Jordan-von Neumann constant, weak orthogonality, and fixed points for multivalued mappings

We give some sufficient conditions for the Domínguez-Lorenzo condition in terms of the James constant, the Jordan-von Neumann constant, and the coefficient of weak orthogonality. As a consequence, we obtain fixed point theorems for multivalued nonexpansive mappings.     

A note on the complex semi‐definite matrix Procrustes problem

This note outlines an algorithm for solving the complex ‘matrix Procrustes problem’. This is a least‐squares approximation over the cone of positive semi‐definite Hermitian matrices, which has a number of applications in the areas of Optimization, Signal Processing and Control. The work generalizes the method of Allwright (SIAM J. Control Optim. 1988; 26 (3):537–556), who obtained a numerical solution to the real‐valued version of the problem. It is shown that, subject to an appropriate rank assumption, the complex problem can be formulated in a real setting using a matrix‐dilation technique, for which the method of Allwright is applicable. However, this transformation results in an over‐parametrization of the problem and, therefore, convergence to the optimal solution is slow. Here, an alternative algorithm is developed for solving the complex problem, which exploits fully the special structure of the dilated matrix. The advantages of the modified algorithm are demonstrated via a numerical example. Copyright © 2007 John Wiley & Sons, Ltd.     

On the properties of solutions to the Goursat-Darboux problem with boundary and distributed controls

We consider a control system described by the Goursat-Darboux equation. The system is controlled by distributed and boundary controls. The controls are subject to the constraints given as multivalued mappings with closed, possibly nonconvex, values depending on the phase variable. Alongside the initial constraints, we consider the convexified constraints and the constraints whose values are the extreme points of the convexified constraints. We study the questions of existence of solutions and establish connections between the solutions under various constraints.     

An extension of gap functions for a system of vector equilibrium problems with applications to optimization problems

In this paper, the notion of gap functions is extended from scalar case to vector one. Then, gap functions and generalized functions for several kinds of vector equilibrium problems are shown. As an application, the dual problem of a class of optimization problems with a system of vector equilibrium constraints (in short, OP) is established, the concavity of the dual function, the weak duality of (OP) and the saddle point sufficient condition are derived by using generalized gap functions. This work was supported by the National Natural Science Foundation of China (10671135) and the Applied Research Project of Sichuan Province (05JY029-009-1).     

The combination technique and some generalisations

The combination technique has repeatedly been shown to be an effective tool for the approximation with sparse grid spaces. Little is known about the reasons of this effectiveness and in some cases the combination technique can even break down. It is known, however, that the combination technique produces an exact result in the case of a projection into a sparse grid space if the involved partial projections commute.

The performance of the combination technique is analysed using a projection framework and the C/S decomposition. Error bounds are given in terms of angles between the spanning subspaces or the projections onto these subspaces. Based on this analysis modified combination coefficients are derived which are optimal in a certain sense and which can substantially extend the applicability and performance of the combination technique.     

Numerical analysis for stochastic age-dependent population equations with Poisson jumps

In this paper, stochastic age-dependent population equations with Poisson jumps are considered. In general, most of stochastic age-dependent population equations with jumps do not have explicit solutions, thus numerical approximation schemes are invaluable tools for exploring their properties. The main purpose of this paper is to develop a numerical Euler scheme and show the convergence of the numerical approximation solution to the true solution.