1. Introduction. Let p be a prime number and ${\mathbb{Z}}_p$ the ring of p-adic integers. Let k be a finite extension of the rational number field ℚ, $k_{\infty }$ a ${\mathbb{Z}}_p$-extension of k, $k_n$ the nth layer of $k_{\infty }/k$, and $A_n$ the p-Sylow subgroup of the ideal class group of $k_n$. Iwasawa proved the following well-known theorem about the order $A_n$ of $A_n$: Theorem A (Iwasawa). Let $k_{\infty }/k$ be a ${\mathbb{Z}}_p$-extension and $A_n$ the p-Sylow subgroup of the ideal class group of $k_n$, where $k_n$ is the $n$th layer of $k_{\infty }/k$. Then there exist integers $\lambda =\lambda (k_{\infty }/k)\ge 0$, $\mu =\mu (k_{\infty }/k)\ge 0$, $\nu =\nu (k_{\infty }/k)$, and n₀ ≥ 0...

We discuss the propagation of electromagnetic waves of a special form through an inhomogeneous isotropic medium which has a cylindrical symmetry and a nonlinear dielectric response. For the case where this response is of self-focusing type the problem is treated in [1]. Here we continue this study by dealing with a defocusing dielectric response. This tends to inhibit the guidance properties of the medium and so guidance can only be expected provided that the cylindrical stratification...

In this paper we deal with the Cauchy problem for differential inclusions governed by $m$-accretive operators in general Banach spaces. We are interested in finding the sufficient conditions for the existence of integral solutions of the problem $x^{\text{'}}\left(t\right)\in -Ax\left(t\right)+f(t,x\left(t\right))$, $x\left(0\right)=x_0$, where $A$ is an $m$-accretive operator, and $f$ is a continuous, but non-compact perturbation, satisfying some additional conditions.

Let $L\left(X\right)$ be the space of all lower semi-continuous extended real-valued functions on a Hausdorff space $X$, where, by identifying each $f$ with the epi-graph $epi\left(f\right)$, $L\left(X\right)$ is regarded the subspace of the space ${Cld}_F^{*}(X\times ℝ)$ of all closed sets in $X\times ℝ$ with the Fell topology. Let $$LSC\left(X\right)=\{f\in L\left(X\right)\mid f\left(X\right)\cap ℝ\ne \varnothing ,f\left(X\right)\subset (-\infty ,\infty ]\}\text{and}{LSC}_B\left(X\right)=\{f\in L\left(X\right)\mid f\left(X\right)\text{is}\text{a}\text{bounded}\text{subset}\text{of}ℝ\}.$$ We show that $L\left(X\right)$ is homeomorphic to the Hilbert cube $Q={[-1,1]}^{\mathbb{N}}$ if and only if $X$ is second countable, locally compact and infinite. In this case, it is proved that $(L\left(X\right),LSC\left(X\right),{LSC}_B\left(X\right))$ is homeomorphic to $(ConeQ,Q\times (0,1),\Sigma \times (0,1\left)\right)$ (resp. $(Q,s,\Sigma )$) if $X$ is compact (resp. $X$ is non-compact), where $ConeQ=(Q\times 𝐈)/(Q\times \{1\left\}\right)$ is...

1. Introduction. The recent article [1] gives explicit evaluations for exponential sums of the form $S(a,p^{\alpha }+1):={\sum }_{x{\in }_q}\chi \left(a{x}^{p^{\alpha }+1}\right)$ where χ is a non-trivial additive character of the finite field ${}_q$, $q=p^e$ odd, and $a\in {*}_q$. In my dissertation [5], in particular in [4], I considered more generally the sums S(a,N) for all factors N of $p^{\alpha }+1$. The aim of the present note is to evaluate S(a,N) in a short way, following [4]. We note that our result is also valid for even q, and the technique used in our proof can also be used to evaluate...

 Let $F_i=1,...,N$ be affine mappings of ${ℝ}^n$. It is well known that if there exists j ≤ 1 such that for every ${\sigma }_1,...,{\sigma }_j\in 1,...,N$ the composition (1) $F_{\sigma 1}\circ ...\circ F_{{\sigma }_j}$ is a contraction, then for any infinite sequence ${\sigma }_1,{\sigma }_2,...\in 1,...,N$ and any $z\in {ℝ}^n$, the sequence (2)$F_{\sigma 1}\circ ...\circ F_{{\sigma }_n}\left(z\right)$ is convergent and the limit is independent of z. We prove the following converse result: If (2) is convergent for any $z\in {ℝ}^n$ and any $\sigma ={\sigma }_1,{\sigma }_2,...$ belonging to some subshift Σ of N symbols (and the limit is independent of z), then there exists j ≥ 1 such that for every $\sigma ={\sigma }_1,{\sigma }_2,...\in \Sigma$ the composition (1) is a contraction....