In 1951, Dowker proved that a space $X$ is countably paracompact and normal if and only if $X \times {\bf I}$ is normal. A normal space $X$ is called a Dowker space if $X \times {\bf I}$ is not normal. The main thrust of this article is to extend this work with regards $\alpha$-normality and $\beta$-normality. Characterizations are given for when the product of a space $X$ and $(\omega + 1)$ is $\alpha$-normal or $\beta$-normal. A new definition, $\alpha$-countably paracompact, illustrates what can be said if the product of $X$ with a compact metric space is $\beta$-normal. Several examples demonstrate that the product of a Dowker space and a compact metric space may or may not be $\alpha$-normal or $\beta$-normal. A collectionwise Hausdorff. Moore space constructed by M. Wage is shown to be $\alpha$-normal but not $\beta$-nornal.

Let $c$ be a constant in ${\bf R}^t$. For a plane algebraic curve $r^{2m-n} = 2c^n \cos n\theta$, which depends on $m$ and $n$ in ${\bf N}$, we show that the whole length of the curve are given by a value of a product formula defined by the Beta function and Gauss's hypergeometric function depending $m$ and $n$ in ${\bf N}$. Besides, we point out the fact to be a similar model and an expansion for the complete elliptic integral of the second kind. Last, we give a background for the fact explaining the special case $m = n$.

In this paper we show that the finite subgroup scheme Spec $A[X, Y]/(X^{p^l}, Y^{p^l})$ of $\mathscr{E}^{\lambda, \mu, D} \in {\rm Ext}^1(\mathscr{G}^{(\lambda)}, \mathscr{G}^{(\mu)})$ is a Cartier dual of a certain finite subgroup scheme of the fiber product $W_{l,A} \times_{{\rm Spec} A} W_{l,A}$ of Witt vectors of length $l$ in positive characteristic $p$. After this, we treat the kernel of the type $F^2 + [a]F + [b]: W_{l,A} \to W_{l,A}$, where $F$ is the Frobenius endomorphism and $[a]$ is the Teichmüller lifting of $a \in A$, respectively.

For a one-parameter exponential family of distributions, a method to find the uniformly minimum variance unbiased (UMVU) estimator based on the complete sufficient statistic is given in Jani and Dave [1] by change of the expression of the unbiasedness condition. But, it heavily depends on the concrete form of the distribution of the statistic in obtaining indeed the UMVU estimator. In this paper, from the different point of view, the construction of the UMVU estimator for a one-parameter exponential families of distributions and certain two-parameter family of distributions is discussed. Some examples are also given.

We compare the spaces of Hermitian Jacobi forms (HJF) of weight $k$ and indices 1, 2 with classical Jacobi forms (JF) of weight $k$ and indices 1, 2, 4. Upper bounds for the order of vanishing of HJF at the origin are obtained. We compute the rank of HJF as a module over elliptic modular forms and prove the algebraic independence of the generators in case of index 1. Some related questions are discussed.

In this paper, we classify quadratic algebras in two variables at two levels: (1) up to isomorphism of graded algebras, (2) up to graded Morita equivalence. In general, it is difficult to classify algebras by looking at generators and relations, so we take a geometric approach, namely, using point schemes defined by Artin, Tate and Van den Bergh, to complete the classification.

We find particular relations which we call “Bernoulli-type” in some noncommutative polynomial ring with a single nontrivial relation. More precisely, our ring is isomorphic to the universal enveloping algebra of a two-dimensional non-abelian Lie algebra. From these Bernoulli-type relations in our ring, we can obtain a representation on a certain left ideal with the Bernoulli numbers as structure constants.

We investigate some odd dimensional Rimannian submanifolds admitting the almost contact metric structure $(\phi, \xi, \eta, \langle , \rangle)$ of a certain Euclidean sphere from the viewpoint of the weakly $\phi$-invariance of the second fundamental form. The family of such submanifolds contains some homogeneous submanifolds of the ambient sphere. In the latter half of this paper, we caluculate the mean curvature and the length of the derivative of the mean curvature vector of these homogeneous submanifolds.

We prove and disprove some generalizations of a result about some similarities involving normal operators due to M. R. Embry in 1970. Some interesting consequences are also given.