The invertibility of the product of unbounded Toeplitz operators

In this note I give necessary and sufficient conditions on outer functionsf andg for the operator to be bounded and invertible on H². I also discuss the relationship of this question to two open questions in operator theory and weighted norm inequalities.     

On nonnegative solvability of linear operator equations

LetE be a Banach lattice having order continuous norm. Suppose, moreover,T is a nonnegative reducible operator having a compact iterate and which mapsE into itself. The purpose of this work is to extend the previous results of the authors, concerning nonnegative solvability of (kernel) operator equations on generalL ^p-spaces. In particular, we provide necessary and sufficient conditions for the operator equation x=T x+y to possess a nonnegative solutionxE wherey is a given nonnegative and nontrivial element ofE and is any given positive parameter.     

Incomplete idempotent Schröder quasigroups and related packing designs

Summary In this paper it is proved that, for any positive integern 2, 3 (mod 4),n 7, there exists an incomplete idempotent Schröder quasigroup with one hole of size two IISQ(n, 2) except forn = 10. It is also proved that for any positive integern 0, 1 (mod 4), there exists an idempotent Schröder quasigroup ISQ(n) except forn = 5 and 9. These results completely determine the spectrum of ISQ(n) and provide an application to the packing of a class of edge-coloured block designs.Research supported by NSERC grant A-5320.Research supported by NSFC grant 19231060-2.     

Representation of elements of a sequence by sumsets

In this note the method of [5] and a result from [3] are combined to treat the following classical problem: Given a finite setA and an infinite sequenceS (both inZ), what is the minimal number of elements ofA whose sum lies inS? We obtain an upper bound depending only on the densities ofA andS (but not on their arithmetic nature).     

A note on “small representations of finite distributive lattices as congruence lattices”

G. Grätzer, H. Lakser, and E. T. Schmidt proved that every distributive lattice with n join-irreducible elements can be represented as the congruence lattice of a small lattice L, that is, a lattice L with O(n ² ) elements. G. Grätzer, I. Rival, and N. Zaguia proved that, for any <2, O(n ² ) can not be improved to O(n ). In this note we show that the theorem about small representation can be improved further to get a more delicate result.     

Finite energy sectors of the XY model

Continuing the earlier work on soliton sectors, we determine all finite energy representations of the XY model for almost all parameter values. In the interior of unique vacuum regions of parameters (i.e. the large external magnetic field region ||>1), the unique irreducible vacuum representation is the only finite energy representation.At the critical values of the parameters (||=1 as well as theXY symmetric case =0, ||1), there is an infinite number of mutually nonequivalent irreducible finite energy representations. Apart from the unique irreducible ground state representation and another associated irreducible representation, these infinite number of representations arise from an infinite number of nearly zero energy excitations of the ground state with a finite total energy and may be called infrared representations.In the remaining cases, as have been studied earlier, there are two additional irreducible finite energy representations besides two irreducible ground state representations and they are topological soliton sectors with different ground state limits in positive and negative spatial infinity. (For two exceptional values of parameters (, )=(0, ±1), they also become ground state representations.)     

用B样条曲线构造几何物体模型的方法

根据Ｂ样条曲线原理,提出了先用若干个截平面去切割几何物体,得到一组与几何物体轮廓线相交的点。根据这些点,再用Ｂ样条曲线去构造几何物体的近似轮廓线,从而得到几何物体模型。该方法方便、精确,便于在计算机几何造型系统中应用。     

基于神经网络的无监督自学习控制系统

所论系统的控制由两个神经网络完成,即对象放识器和系统控制器。并且提出了一种 新的学习算法,它能利用存贮于辨识网络的信息改进控制策略。该控制系统可以对非线性系统,不确定系统等进行无监督的学习控制。     

Exact solution of a vertex model ind dimensions

We consider a class of vertex models describing directed lines on a lattice in arbitraryd dimensions, and solve the model exactly for the Cartesian lattice and in the case that each loop of lines carries a fugacity - 1. Our analysis, which can be carried out for arbitrary lattices, is based on an equivalence of the vertex model with a dimer problem. The dimer problem is, in turn, solved using the method of Pfaffians. It is found that the system is frozen below a critical temperatureT _cwith the critical exponent = (3 –d)/2.     

On spectral properties ofq-oscillator operators

The property of self-adjointness of the operatorQ =a ₊ +a _- in three types ofq-oscillator algebras is considered. Spectral measures and generalized eigenfunctions ofQ are found in the cases when this operator is bounded. Generalized eigenvectors are expressed in terms ofq-Hermite polynomials. If the operatorQ is unbounded, then its closure is not self-adjoint. However, in this case, admits self-adjoint extensions. Deficiency subspaces are one-dimensional. These subspaces are explicitly found.