We prove that, if $X$ is a Tychonoff connected space and $\chi (x, X) \leq \omega$ for some $x \in X$, then there exists a strictly stronger Tychonoff connected topology on the space $X$, i.e., the space $X$ is not maximal Tychonoff connected. We also establish that if X is locally connected or CT-compact or has pointwise countable type then $X$ cannot be maximal Tychonoff connected.

For a one-dimensional generalized diffusion process $\{X(t) : t \geq O\}$ on an interval $I$, we consider an expectation conditional on no hitting the end points of $I$. If the end points are not accessible, we take two sequences $\{ \xi_{n}\}$ and $\{ \eta_{n} \}$ which converge to the end points as $n \rightarrow \infty$, instead of end points. We obtain the asymptotic behavior of this conditional expectation as $t \rightarrow \infty$ and $n \rightarrow \infty$. As an application of our results, we discuss the asymptotic conditional distribution and related quantities in population genetics.

In this paper, we classify a Hopf hypersurface in a nonflat complex space form whose Levi-form is parallel with respect to the generalized Tanaka-Webster connection.

In this paper we establish, using Mahler's method, the algebraic independence of reciprocal sums of products of Fibonacci numbers including slowly increasing factors in their numerators (see Theorems 1, 5, and 6 below). Theorems 1 and 4 are proved by using Theorems 2 and 3 stating key formulas of this paper, which are deduced from the crucial Lemma 2. Theorems 5 and 6 are proved by using different technique. From Theorems 2 and 5 we deduce Corollary 2, the algebraic independence of the sum of a certain series and that of its subseries obtained by taking subscripts in a geometric progression.

In the paper a family of curvature conditions of pseudosymmetry type is determined. We show that the curvature tensor of some Reissner-Nordstr6m-de Sitter type spacetiroes satisfy these conditions.

In this article we study the products of subspaces of the Sorgenfrey line 9. Using an idea by D. K. Burke and J. T. Moore[2] we prove in particular the following: Let $X_{i}, i = 1, \ldots ,n, n \geq 1$, be subspaces of $\mathscr{S}$, where each $X_i$ is uncountable. Then $X_{1} \times \ldots \times X_{n} \times x \mathscr{Q}$ can be embedded in $\mathscr{S}^{n+1}$ but can not be embedded in $\mathscr{S}^{n}$, where $\mathscr{Q}$ is the space of rational numbers with the natural topology. This statement strengthens [2, Theorem 2.1].

We consider a second order weakly hyperbolic equation with coefficients depending both on time and space. We assume the coefficients of the equation to have some kind of Hölder behavior with respect to time, and we add for the coefficients of the lower order terms an appropriate Levi condition. We prove Gevrey well posedness of the Cauchy problem for this equation for a small enough Gevrey index.

In this paper, we study the theories of lexicographic products of ordered abelian groups.

In this paper, we study maximal, minimal, normal and average order of the function $$f(n) = \prod_{k=0}^{b} n/gcd(n,k!)$$ which is the cardinality of the set of polynomial maps from $\mathbb{Z}$ into $\mathbb{Z}_{n}$.