Remarks on products of generalized topologies

We study the relationship between the product and other basic operations (namely σ, π, α and β) of generalized topologies. Also we discuss the connectedness, generalized connectedness and compactness of products of generalized topologies. It is proved that the connectedness and compactness are preserved under the product of generalized topologies, which shows that the definition of product of generalized topologies is quite reasonable.     

Generalized hyperconnectedness

The main purpose of this paper is to introduce and study generalized hyperconnected spaces. Various characterizations of generalized hyperconnected spaces and preservation theorems are discussed.     

Remarks on a generalized beta function

We show that a certain generalized beta function B(x,y;b) which reduces to Euler's beta functions B(x,y) when its variable b vanishes and preserves symmetry in its parameters may be represented in terms of a finite number of well known higher transcendental functions except (possibly) in the case when one of its parameters is an integer and the other is not. In the latter case B(x,y;b) may be represented as an infinite series of either Wittaker functions or Laguerre polynomials. As a byproduct of this investigation we deduce representations for several infinite series containing Wittaker functions, Laguerre polynomials, and products of both.     

Remarks on enlargements of generalized topologies

The concept of enlargement of a generalized topology was introduced by á. Császár [4]. He also introduced the concepts of (κ,λ)-continuity and (κ _μ ,λ _ν)-continuity on enlargements. In this paper, we characterize the (κ,λ)-continuity and introduce the concept of strong (κ,λ)-continuity on enlargements. In particular, we study characterizations for the strong (κ,λ)-continuity and the relationships among (μ,ν)-continuity, (κ,λ)-continuity, strong (κ,λ)-continuity and (κ _μ ,λ _ν)-continuity.     

Remarks on the generalized Newton method

We give some convergence results for the generalized Newton method for the computation of zeros of nondifferentiable functions which we proposed in an earlier work. Our results show that the generalized method can converge quadratically when used to compute the zeros of the sum of a differentiable function and the (multivalued) subgradient of a lower semicontinuous proper convex function. The method is therefore effective for variational inequalities and can be used to find the minimum of a function which is the sum of a twice-differentiable convex function and a lower semicontinuous proper convex function. A numerical example is given.     

Remarks on balance in generalized Tate cohomology

We consider two pairs of complete hereditary cotorsion theories on the category of left R-modules, such that We prove that for any left R-modules M, N and for any n ≧ 1, the generalized Tate cohomology modules can be computed either using a left of M and a left of M or using a right a right of N. Received: 17 December 2004     

Remarks on the regularity criteria for generalized MHD equations

We study the Cauchy problem for the generalized MHD equations, and prove some regularity criteria involving the integrability of ∇u in the Morrey, multiplier spaces.     

Remarks about a generalized pseudo-relativistic Hartree equation

With appropriate hypotheses on the nonlinearity f, we prove the existence of a ground state solution u for the problem

${(?\mathrm{\Delta }+m^2)}^{\sigma }u+Vu=(W?F(u))f(u)\text{in }{\mathbb{R}}^N,$

where $0<\sigma <1$, V is a bounded continuous potential and F the primitive of f. We also show results about the regularity of any solution of this problem.     

Remarks on a unified theory for classical and generalized interpolation and extrapolation

A unified theory for generalized interpolation, as developed by Mühlbach, and classical polynomial interpolation is discussed. A fundamental theorem for generalized linear iterative interpolation is given and used to derive generalizations of the classical formulae due to Neville, Aitken and Lagrange. Using Mühlbach's definition of generalized divided differences, Newton's generalized interpolation formula, including an expression for the error term, is derived as a pure identity.     

Remarks on acyclic versions of generalized von Neumann and Nash equilibrium theorems

A fixed-point theorem on compact compositions of acyclic maps on admissible (in the sense of Klee) convex subset of a t.v.s. is applied to obtain a cyclic coincidence theorem for acyclic maps, generalized von Neumann type intersection theorems, the Nash type equilibrium theorems, and the von Neumann minimax theorem. Our new results generalize earlier works of Lassonde [1], Simons [2], and Park [3,4].     

关于弱拟第一可数空间的注记

给出一个弱拟第一可数空间成为弱第一可数空间的充要条件,证明了空间是弱第一可数空间当且仅当它是具有序列点Gδ性质的弱拟第一可数空间且不含Sw的闭拷贝.同时还证明了每一弱第一可数空间(弱拟第一可数空间)都是某个第一可数空间的商二到一映像(商可数到一映像),作为应用,部分回答了林寿(2007)的一个问题.     

Remarks on quasi-topologies

A quasi-topology is a generalized topology (cf. [3]) closed for finite intersections. The paper discusses the question whether statements valid for topologies can be generalized for quasi-topologies.     

Remarks on projectivities

Aequationes mathematicae -