The noncommutative torus $C^*(\mathbb Z^n,\,\omega)$ of rank $n$ is realized as the $C^*$-algebra of sections of a locally trivial continuous $C^*$-algebra bundle over $\widehat{S_{\omega}}$ with fibres $C^*(\mathbb Z^n/S_{\omega},\, \omega_1)$ for some totally skew multiplier $\omega_1$ on $\mathbb Z^n/S_{\omega}$. It is shown that $C^*(\mathbb Z^n/S_{\omega},\,\omega_1)$ is isomorphic to $A_{\varphi}\otimes M_k(\mathbb C)$ for some completely irrational noncommutative torus $A_{\varphi}$ and some positive integer $k$, and that $A_{\omega} \otimes M_{l^{\infty}}$ has the trivial bundle structure if and only if the set of prime factors of $k$ is a subset of the set of prime factors of $l$. This is applied to understand the bundle structure of the tensor products of Cuntz algebras with noncommutative tori.

We study numerical semigroups of the form $(x+S)\cup\{0\}$ with $S$ a numerical semigroup and $x\in S\setminus\{0\}$. We pay special attention to the cases when $S$ is symmetric, pseudo-symmetric, Arf and saturated.

Let $\Omega$ be a bounded domain in $\mathbb R^N$ and $m_1$, $m_2$ two functions in $L^\infty(\Omega)$. In the present work, we study a new spectrum constitued by the set of pairs $(\alpha,\beta)$ of $\R ^2$ for which the problem $$ \left\{ \begin{array}{rcr} -\bigtriangleup u & = & \alpha m_1 u^+-\be m_2 u^- \quad\mbox{ in} \:\: \Omega,\\ u & = & 0 \quad \hspace{2,4cm}\quad\mbox{ on}\:\: \partial\Omega, \end{array} \right. $$ has a nontrivial solution, where $u^\pm=\di\max(0,\pm u)$. We study then the nonresonance with respect to this spectrum in a non autonomous problem.

In this paper we prove a common fixed point theorem of Greguš contraction type by using minimal type commutativity without continuity requirements. The theorem extends known results on compatible continuous maps of [1]-[3], [7], [8] and [9], to a larger class of mappings.

In this paper we consider a suitable $\mathbb R^d$-valued process $(Z_t)$ and a suitable family of nonempty subsets $(A(b):b>0)$ of $\mathbb R^d$ which, in some sense, decrease to empty set as $b\rightarrow \infty$. In general let $T_b$ be the first hitting time of $A(b)$ for the process $(Z_t)$. The main result relates the large deviations principle of $(\frac{T_b}{b})$ as $b\rightarrow \infty$ with a large deviations principle concerning $(Z_t)$ which agrees with a generalized version of Mogulskii Theorem. The proof has some analogies with the proof presented in \cite{DW} for a similar result concerning nondecreasing univariate processes and their inverses with general scaling function.

We are interested in the question whether every strongly locally $\varphi$-symmetric contact metric space is a $(\kappa,\mu)$-space. In this paper, we show that the answer is positive for left-invariant contact metric structures on Lie groups.

For a positive integer $s$ and for $0\le a< b,$ let $$K=K^s_{a,b}=\bigcup_{k=0}^{s-1}e^{2\pi i\frac{k}{s}}[a,b].$$ We find that the {\it capacity} $\mathrm{Cap} (K)$ of $K$ is $$\mathrm{Cap} (K) = \sqrt [s]{\frac{b^s-a^s}4}\cdot\eqno (1)$$ \par From this relation we derive several classical results, due to Akhiezer, Henrici, and Bartolomeo and He, on capacities of some sets in the complex plane. \par An extension relation (1) to more general sets in the complex plane, together with potential theoretic techniques, is then used to obtain {\it saturation} theorems pertaining to approximation by polynomials with integer coefficients.

We give a simple proof of the known fact that the symplectic quadrangle is self-dual if and only if the ground field is perfect of characteristic~2, and that a polarity exists exactly if there is a root of the Frobenius automorphism. Moreover, we determine all polarities, characterize the conjugacy classes of polarities, and use the results to give a simple proof that the centralizer of any polarity acts two-transitively on the ovoid of absolute points. The proofs use elementary calculations in solvable subgroups of the symplectic group.

In this paper we prove that, for any Galois finite field extension $F/K$ on which a separated group of operators $\Gamma$ is acting, there is an isomorphism between the group of equivariant isomorphism classes of finite dimensional central simple $K$-algebras endowed with a $\Gamma$-action and containing $F$ as an equivariant strictly maximal subfield and the second equivariant cohomology group of the Galois group of the extension.

We consider a metric $f$--structure on a manifold $M$ of dimension $2n+s$. We suppose that its kernel is paralellizable by global orthonormal vector fields $\xi_1,\dots,\xi_s$ and that the dual 1--forms satisfy $d\eta^k=F$ where $F$ is the associated Sasaki 2--form and $k=1,\dots,s$. We prove that if $n$ is greater than one then $M$ cannot be flat. This is a generalization of a result by D.E.Blair proved for contact metric manifolds. We also give a counterexample in the case $n=1$.