In our article Absolute norms on $C^n$,, we characterized absolute normalized norms on $\Bbb C^n$ by using a continuous convex function with some appropriate conditions on a certain convex subset of $\mathbb R^n$. In this paper, we introduce the notion of generalized $\ell_p$-spaces, that is, $\ell_{\psi}$-spaces, and study their structures.

In this paper we derive error bounds for asymptotic expansions of the hypergeometric functions ${}_1F_1(n; n+b; Z)$ and ${}_1F_1(n; n+b; -Z)$, where $Z$ is a $p \times p$ symmetric nonnegative definite matrix. The first result is applied for theoretical accuracy of approximating the moments of $\Lambda=|S_e|/|S_e+S_h|$, where $S_h$ and $S_e$ are independently distributed as a noncentral Wishart distribution $W_p(q, \Sigma, \Sigma^{1/2} \Omega \Sigma^{1/2})$ and a central Wishart distribution $W_p(n, \Sigma)$, respectively. The second result is applied for theoretical accuracy of approximating the probability density function of the maximum likelihood estimators of regression coefficients in the growth curve model.

Our main result states that two signed measures $\mu$ and $\nu$ with bounded support contained in the zero set of a polynomial $P(x)$ are equal if they coincide on the subspace of all polynomials of polyharmonic degree $N_{P}$ where the natural number $N_{P}$ is explicitly computed by the properties of the polynomial $P\left( x\right) $. The method of proof depends on a definition of a multivariate Markov transform which is another major objective of the present paper. The classical notion of orthogonal polynomial of second kind is generalized to the multivariate setting: it is a polyharmonic function which has similar features to those in the one-dimensional case.

Let $L^{n}(3)$ be the $(2n+1)$-dimensional standard lens space mod 3 and let $\nu$ denote the normal bundle associated to an immersion of $L^{n}(3)$ in the Euclidean $(4n+3)$-space. In this paper we obtain a theorem on stable unextendibility of $R$-vector bundles over $L^{n}(3)$ improving some results in Extendibility and stable extendibility of vector bundles over lens spaces and Stable extendibility of normal bundles associated to immersions of real projective spaces and lens spaces, and study relations between stable extendibility and span of vector bundles over $L^{n}(3)$. Furtheremore, we prove that $c\nu$ is extendible to $L^{m}(3)$ for every $m > n$ if and only if $0 \leq n \leq 5$, and prove that $c(\nu \otimes \nu)$ is extendible to $L^{m}(3)$ for every $m > n$ if and only if $0 \leq n \leq 13$ or $n = 15$, where $c$ stands for the complexification and $\otimes$ denotes the tensor product.

This paper gives the definition and some properties of $a$-minimally thin sets at $\infty$ in a cone. Our results are based on estimating Green potential with a positive measure by connecting with a kind of density of the modified measure.

We will give an explicit description of non-commutative extensions over an ${\F}_p$-algebra of the additive group scheme (resp. the additive formal group scheme) by the group scheme (resp. the formal group scheme) which gives a deformation of the additive group scheme to the multiplicative group scheme (resp. the additive formal group scheme to the multiplicative formal group scheme).

The purpose of this paper is to establish the formulas on the power of the normal bundle associated to an immersion of the real projective space $\RP^n$ in the Euclidean space $\R^{n+k}$, and apply them to the problem of extendibility and stable extendibility. Furthermore, we give an example of a $2$-dimensional $\R$-vector bundle over $\RP^2$ that is stably extendible to $\RP^3$ but is not extendible to $\RP^3$.

There are several nonlinear models for analyzing repeated measurements. The mean response for an individual depends on the regression parameter specific to that individual. One of the simple form is the sum of a vector of fixed parameters and a vector of random effect. In this paper, we give a confidence region of the fixed parameters approximately.

Under the assumption that the three-factor and higher-order interactions are negligible, we consider a partially balanced fractional $2^{m_1+m_2}$ factorial design derived from a simple partially balanced array such that the general mean, all the $m_1+m_2$ main effects, and some linear combinations of $\binom{m_1}{2}$ two-factor interactions, of the $\binom{m_2}{2}$ ones and of the $m_1m_2$ ones are estimable, where $2\leq m_k$ for $k=1,2$. This paper presents optimal designs with respect to the generalized A-optimality criterion when the number of assemblies is less than the number of non-negligible factorial effects, where $2\leq m_1, m_2 \leq 4$.