Nontransitive quasi-uniformities in the Pervin quasi-proximity class

We show that each topological space that does not admit a unique quasi-uniformity possesses a Pervin quasi-proximity class containing at least nontransitive members.

     

Complex Hyperspherical Equations, Nodal-Partitioning, and First-Excited-State Theorems in ℝ n

Three problems related to the spherical quantum billiard in are considered. In the first, a compact form of the hyperspherical equations leads to their complex contracted representation. Employing these contracted equations, a proof is given of Courant's nodal-symmetry intersection theorem for diagonal eigenstates of spherical-like quantum billiards in . The second topic addresses the first-excited-state theorem for the spherical quantum billiard in . Wavefunctions for this system are given by the product form, ( )Z _q+()Y ⁽ⁿ⁾ , where is dimensionless displacement, is angular-momentum number, qis an integer function of dimension, Z() is either a spherical Bessel function (nodd) or a Bessel function of the first kind (neven) and represents (n– 1) independent angular components. Generalized spherical harmonics are written . It is found that the first excited state (i.e., the second eigenstate of the Laplacian) for the spherical quantum billiard in is n-fold degenerate and a first excited state for this quantum billiard exists which contains a nodal bisecting hypersurface of mirror symmetry. These findings establish the first-excited-state theorem for the spherical quantum billiard in . In a third study, an expression is derived for the dimension of the th irreducible representation (irrep) of the rotation group O(n) in by enumerating independent degenerate product eigenstates of the Laplacian.     

A formula related to irreducible diagram expansions

A formula is presented which can be regarded as the analytic basis for irreducible diagram expansions. It expresses the off-diagonal elements of the inverse of a matrix of operators by the off-diagonal elements and by diagonal elements of various inverses of the original operator. The formula can be obtained by purely analytic means without reference to statistical considerations. No infinite processes are involved if one deals with a finite matrix of operators.     

Homology of Nilpotent Subalgebras of the Lie Superalgebra K(1,1).3

We calculate the dimensions of the second homology groups with trivial coefficients of nilpotent subalgebras of the Lie superalgebra , which is the natural superanalog of the Witt algebra. The proof is based on direct calculations of the rank of the differential. As an application, we find deformations of the maximal nilpotent subalgebra in .     

nvolutorial Płonka sums

The construction of the sum of a direct (semilattice ordered) system of algebras introduced by J. Plonka – later known as the Plonka sum – is one of the most important methods of composition in universal algebra, having a number of applications in different algebraic theories, such as semigroup theory, semiring theory, etc. In this paper we present a more general way for constructing algebras with involution, that is, algebraic systems equipped with a unary involutorial operation which is at the same time an antiautomorphism of the underlying algebra. It is the sum – involutorial Plonka sum, as we call it – of an involution semilattice ordered system of algebras. We investigate its basic properties, as well as the problem of its subdirect decomposition.     

Geometric structure in stochastic approximation

1. IntroductionLet f: Re -- R be a differelltiable fUnction. f reaChes its extremes on the setJ = {x E R"lfx(x) = 0}, (1.1)where,x(X) = (V,..., V)". (1.2)If jx can be observed exactly at any x e R", then there are various numerical methods toconstruct {xh}, xk E Re such that the distance d(xk, J) between uk and J tends to zero ask -- co. However, in many application problems jx can only be observed with noise, i.e.,the observation at time k 1 isYk 1 = fi(~k) (k 1, (1'3)where xk is …     

Uniqueness and asymptotic behaviour for solutions of semilinear problems with boundary blow-up

In this paper we prove uniqueness of positive solutions to logistic singular problems , , 1$">, 0$"> in , where the main feature is the fact that . More importantly, we provide exact asymptotic estimates describing, in the form of a two-term expansion, the blow-up rate for the solutions near . This expansion involves both the distance function and the mean curvature of .

     

关于拓扑对策中的I~2(K)I(K)问题

证明了如果空间类κ为D-空间类或闭遗传不可约空间类,则I(κ)(∈)κ.这一结果对于κ为D空间类和闭遗传不可约空间类,肯定地回答了I(κ)是否包含I2(κ)问题.     

有限群的特征标的零点和可解φ-群

有两个对偶的问题如下：问题Ⅰ：将满足下述条件的有限群G分类：G的特征标表中，除一行外其余各行最多有一个零．问题Ⅱ：将满足下述条件的有限群G分类：G的特征标表中，除一列外其余各列最多有一个零．在这篇文章中，我们对于有限可解群解答上述两个问题，并确定和这两个问题密切相关的一类有限可解群的结构（这类可解群在本文中称之为可解φ-群）．附带我们还完全回答了[4]中的问题1，并说明[6，定理]的条件可以极大地减弱．