On an adjoint functor to the Thom functor

We construct a right adjoint functor to the Thom functor, i.e., to the functor which assigns the Thom space to a vector bundle .

     

The differentiable chain functor is not homotopy equivalent to the continuous chain functor

Let S_∗ and be the functors of continuous and differentiable singular chains on the category of differentiable manifolds. We prove that the natural transformation , which induces homology equivalences over each manifold, is not a natural homotopy equivalence.     

On the tychonoff functor and w-compactness

The main purpose of this paper is to settle the following problem concerning a product formula for the Tychonoff functor τ, by introducing the notion of w-compact spaces: Characterize a topological space X such that τ(X×Y)=τ(X)×τ(Y) for any topological space Y. We also study the properties of w-compact spaces, and it is proved that, for any family {X_α} of w-compact spaces, the product ΠX_α is also w-compact and τ(ΠX_α)=Πτ(X_α).     

An extension of the Garrett-Stanojevic class

The extension is made for the theorem of [2], by considering the classes S_pr and C_r, 1<p≤2, r∈{0,1,2,…} instead of S_p and C. Namely, it is shown that the class S_pr is a subclass of C_r∩BV, 1<p≤2,r∈{0,1, 2,…}, where BV is the class of null sequences of bounded variation, and C_r, r∈{0,1,2…} is the extension of the Garrett-Stanojevci class.     

An extension of the Massera Theorem

This paper deals with the implications of the boundedness of a solution for the periodic dynamical system of second order.The Project supported by National Natural Science Foundation of China.     

An extension of the integral test

We identify a class of functions for which an extension of the well known integral test holds, namely, the series is Cesàro summable if and only if the integral is. We also give the corresponding multidimensional results.

     

An extension of the Lindeberg-Feller theorem

A new version is presented of the necessary and sufficient condition of convergence, to a normal law, of sums of independent variables in a nonclassical situation (i.e., absence of limiting negligibility of variables). The obtained condition differs from previously obtained conditions by the fact that it does not use Levy's metric and that is is closer to classical formulations. A similar condition is sufficient for the closeness of two convolutions when the number of components of the convolutions increases without bounds.     

An extension of the Dulmage-Mendelsohn theorem

We prove a local Dulmage-Mendelsohn theorem, a generalization of their Theorem 18[1] to matrices which are decomposable or have period greater than one. This can be used to estimate the index of a wide variety of matrices and has applications to finite automata theory, stochastic processes, graph theory, and other fields.