We investigate the topology of convergence to limit sets for one-dimensional linear cellular automata using ${\mathbb{Z}}_p$-valued upper semi-continuous functions, where $p$ is a prime integer.

We studied the geodesics of a spacetime with the pseudo-Riemannian metric:

$$d{s}^2=\frac{1}{x_4{x}_4}\{{\displaystyle \sum }_{b,c=1}^3\left({\delta }_{bc}-\frac{a{x}_b{x}_c}{1+a{r}^2}\right)d{x}_{b}d{x}_c-\frac{1}{1+a{x}_4{x}_4}d{x}_{4}d{x}_4\}$$

on $R^3\times R_{+}$, where $r^2={\displaystyle \sum }_{b=1}^3{x}_b{x}_b$ and $a=\text{ constant}$, which are plane quadratic curves (in [12]). In this paper, we shall determine all the Killing vector fields of this spacetime and choose special pairs out of them with interesting properties for the case .

For let $F_{\beta }$ denote the Fréchet algebra of all holomorphic functions $f$ on the unit disk $\text{Δ}$ for which ${\lim }_{r\to 1}{(1-r)}^{\beta }{\log }^{+}\left[{\max }_{\left|z\right|\le r}\right|f\left(z\right)\left|\right]=0$. Given a holomorphic self-map $\varphi$ of $\text{Δ}$ define the composition operator $C_{\varphi }$ on $F_{\beta }$ by: $C_{\varphi }f=f\circ \varphi ,f\in F_{\beta }$. This note shows that $C_{\varphi }$ exists always as a continuous operator. Furthermore, this note points out that boundedness and compactness of $C_{\varphi }$ are not only the same, but also equivalent to ${\varphi }^n\text{exp}[r{n}^{\beta /(1+\beta )}]\to 0$ in $F_{\beta }$ for some .

We study the exact asymptotic behaviour of some special generalized renewal measures on the whole line $\mathbf{R}$ and, in particular, that of the ordinary renewal measure on $\mathbf{R}$. When the underlying distribution $F$ is concentrated on $\left[0,\infty \right)$, these measures are closely related to the higher renewal moments $EN{(t)}^n$, where $N\left(t\right)$ is the number of renewals up to time $t$. The tail of $F$ is assumed to possess the tail behaviour of a distribution from the class $S\left(\gamma \right)$ for an arbitrary .

We present a new conceptual framework for Hashiguchi-Ichijyō’s theorems concerning Wagner manifolds and prove them intrinsically.

The initial value problem for a system of PDEs of the form $(*)\frac{\partial u_i(t,x)}{\partial t}+\frac{\partial u_i\left(t,x\right)}{\partial x}{\lambda }_i\left(t,x,u\left(t,x\right)\right)=f_i\left(t,x,u\left(t,x\right)\right),1\le i\le N.$ is solved. In $(*)t$ is a real variable, $x$ is a variable moving in a real Banach space $X$. $u_1,\dots ,u_N$ are unknown functions whose values are in a real Banach space $G$. $u$ denotes the ‘vector’ $(u_1,\dots ,u_N)$. Differentiation in $x$ is taken in the sense of the Fréchet derivative, $f_1,\dots ,f_N$ are $G$-valued given functions and ${\lambda }_1,\dots ,{\lambda }_N$ are $X$-valued given functions.

Statistical structure of the fraction of symmetrical $s^m$ factorial designs is investigated in some detail. In this paper, we show that the information matrix of a fractional $s^m$ factorial design is determined completely by its characteristic vector. We also give an explicit expression of the elements of the infomation matrix of the design derived from $s$-symbol balanced arrays in terms of its indices.