We consider a Rayleigh beam equation with two dynamical boundary controls. First, by a multiplier method, we show that the smooth solution has a polynomial energy decay rate. Next, using a spectrum method, we justify that the polynomial energy decay rate is optimal.

A classification is given of all translation planes of order $q^{2}$ that admit a collineation group $G$ admitting a normal subgroup $N$ such that $% G/N $ is isomorphic to $PSL(2,q)$.

We give simple proofs of the representation theorems of Riesz and Schwartz.

This work considers the fundamental groups and diameters of positively Ricci curved Riemannian $n$-manifolds. By combining the results of equivarient Hausdorff convergence with the Ricci version of a splitting theorem, some new information on the topology of compact manifolds with positive Ricci curvature was discovered. Moreover, a weak Margulis's lemma was also obtained for Riemannian manifolds with a lower Ricci curvature bound.

In a recent preprint (see [C]), F. Calegari has shown that for $\ell = 2, 3, 5$ and $7$ there exist $2$-dimensional irreducible representations $\rho$ of Gal$(\bar{\Q}/\Q)$ with values in $\F_\ell$ coming from the $\ell$-torsion points of an elliptic curve defined over $\Q$, but not minimally, i.e., so that any elliptic curve giving rise to $\rho$ has prime-to-$\ell$ conductor greater than the (prime-to-$\ell$) conductor of $\rho$. In this brief note, we will show that the same is true for any prime $\ell >7$

We prove the non existence of real hypersurfaces in complex projective space whose structure Jacobi operator is parallel in any direction of the maximal holomorphic distribution.

Let $G$ be a permutation group on a set $\Omega$ with no fixed points in $\Omega$ and let $m$ be a positive integer. Then we define the movement of $G$ as, $m:=move(G):=sup_{\Gamma}\{|\Gamma^{g}\setminus\Gamma | | g\in G\}$. Let $p$ be a prime, $p\geq 5$, and let $move(G)=m$. We show that if $G$ is not a 2-group and $p$ is the least odd prime dividing $|G|$, then $n:=|\Omega|\leq 4m-p$. Moreover for an infinite family of groups the maximum bound $n=4m-p$ is attained.

This paper deals with a new Hilbert- type inequality by introducing a parameter and the Beta function. As applications, the equivalent form, the reversions and some particular results are considered. All the theorems provide some new estimates on this type of inequalities.

Beside the classical properties of monotonicity and additive homogeneity, the functions modelling the evolution of discrete event systems usually have a third property, which expresses the causal (i.e. non-anticipative) character of the transformation. These "causal" topical functions are always uniformly topical, in particular they have an asymptote and, for endofunctions, a spectral vector.

A version of the Kontorovich-Lebedev transformation, involving the Hankel function of second kind in its kernel and connected with the Helmholtz's equation, has been investigated from a classical point of view by D. S. Jones. The main objective of this work is to extend this transform to certain space of generalized functions, establishing the corresponding distributional inversion formula.

For a word in $n$ letters, in [1] the author introduced a notion: \emph{its standard exponent} and proved that the variety of residually finite groups defined by a word is almost nilpotent if and only if the standard exponent of this word is 1. In this paper we obtain the following result: let $\omega(x_1, \cdots, x_n)$ denote a word in $x_1, \cdots, x_n$. Then both $\omega(x_1, \cdots, x_n)$ and $\omega(x^{m_1}_1, \cdots, x^{m_n}_n)$, where $m_i$ are natural numbers, have the same standard exponents.

Nous démontrons qu'en l'absence de courbe minimisante singulière, la fonction distance sous-riemannienne, localement lipschitzienne hors de la diagonale, vérifie un théorème de Sard. On en déduit que les sphères sous-riemanniennes sont des hypersurfaces lipschitziennes pour presque tout rayon dans $d_{SR}(q_0,Q)$.

We prove that, in absence of singular minimizing curve, the sub-riemannian distance function is locally Lipschitz outside the diagonal and satisfies Sard's theorem. Hence we deduce that the spheres are Lipschitz hypersurfaces for almost every radius in $d_{SR}(q_0,Q)$.

In this paper, we study the regularity of solutions for a fourth order parabolic equation. Based on the Schauder type estimates and Campanato spaces, we prove the global existence of classical solutions.

Let $H(\mathrm{U})$ be the space of all analytic functions in the unit disk $\mathrm{U}$. For a given function $h\in\mathcal{A}$ we define the integral operator $\mathrm{I}_{h;\beta}:\mathcal{K}\rightarrow H(\mathrm{U})$, with $\mathcal K\subset H(\mathrm{U})$, by $$\mathrm{I}_{h;\beta}[f](z)=\left[\beta \int_0^zf^\beta(t)h^{-1}(t)h'(t)\operatorname{d}t\right]^{1/\beta},$$ where $\beta\in\mathbb{C}$ and all powers are the principal ones. We will determine sufficient conditions on $g_1$, $g_2$ and $\beta$ such that $$\left[\frac{zh'(z)}{h(z)}\right]^{1/\beta}g_1(z)\prec \left[\frac{zh'(z)}{h(z)}\right]^{1/\beta}f(z)\prec \left[\frac{zh'(z)}{h(z)}\right]^{1/\beta}g_2(z)$$ implies $$\mathrm{I}_{h;\beta}[g_1](z)\prec\mathrm{I}_{h;\beta}[f](z)\prec \mathrm{I}_{h;\beta}[g_2](z),$$ where the symbol ``$\prec$'' stands for subordination. We will call such a kind of result a {\em sandwich-type theorem}. In addition, $\displaystyle\mathrm{I}_{h;\beta}[g_1]$ will be the {\em largest} function and $\displaystyle\mathrm{I}_{h;\beta}[g_2]$ the {\em smallest} function so that the left-hand side, respectively the right-hand side of the above implication hold, for all $f$ functions satisfying the differential subordination, respectively the differential superordination of the assumption. We will give some particular cases of the main result obtained for appropriate choices of the $h$, that also generalize classic results of the theory of differential subordination and superordination. The concept of differential superordination was introduced by S. S. Miller and P. T. Mocanu like a dual problem of differential subordination

If $A$ is a (nonempty) bounded convex subset of an asymmetric normed linear space $(X,q),$ we define the closedness of $A$ as the set \textrm{cl}$% _{q}A\cap \mathrm{cl}_{q^{-1}}A,$ and denote by $CB_{0}(X)$ the collection of the closednesses of all (nonempty) bounded convex subsets of $(X,q).$ We show that $CB_{0}(X),$ endowed with the Hausdorff quasi-metric of $q,$ can be structured as a quasi-metric cone. Then, and extending a classical embedding theorem of L. H\"{o}rmander, we prove that there is an isometric isomorphism from this quasi-metric cone into the product of two asymmetric normed linear spaces of bounded continuous real functions equipped with the asymmetric norm of uniform convergence.