The norm of a bounded self-adjoint operator $T:$ $l^{2}\rightarrow $ $l^{2}$ is considered. As applications, a new bilinear inequality with a best constant factor and some Hilbert's type inequalities are built.

On considère différents aspects d'une formule dans les groupes de Coxeter finis.

Several aspects of a formula which counts full reflections in finite Coxeter groups are considered.

We prove the existence of at least three periodic solutions of a nonlinear differential equation involving $p$-Laplacian and convex-concave nonlinearities.

It is proven that for an octonion-valued monogenic function $f(\mathbf{x})$, $\mathbf{x} \in \mathbf{R}^8$, its powers $|f|^p$ are subharmonic for any $p\geq 6/7$. This implies, in particular, Hadamard's three circles and three lines theorems and a Phragmén-Lindelöf theorem for monogenic functions.

Let $X, Y$ be Banach modules over a $C^*$-algebra and let $r_1, \cdots, r_n \in (0, \infty)$ be given. We prove the Hyers-Ulam-Rassias stability of the following functional equation in Banach modules over a unital $C^*$-algebra: \begin{eqnarray} \sum_{i=1}^{n} r_i f \left( \sum_{j=1}^{n}r_j(x_i-x_j) \right) + \left(\sum_{i=1}^{n} r_i\right) f \left(\sum_{i=1}^{n} r_ix_i \right) =\left(\sum_{i=1}^{n}r_i\right) \sum_{i=1}^{n} r_i f(x_i) . \end{eqnarray} We show that if $r_1=\cdots=r_n = r$ and an odd mapping $f : X \rightarrow Y$ satisfies the functional equation {\rm (0.1)} then the odd mapping $f : X \rightarrow Y$ is Cauchy additive. As an application, we show that every almost linear bijection $h : A \rightarrow B$ of a unital $C^*$-algebra $A$ onto a unital $C^*$-algebra $B$ is a $C^*$-algebra isomorphism when $h((nr)^d u y) = h((nr)^d u) h(y)$ for all unitaries $u \in A$, all $y \in A$, and all $d \in {\bf Z}$.

In this note, we characterize the order of fitness functions in terms of their Walsh coefficients.

The paper presents new Birkhoff-Kellogg type theorems for maps in the S-KKM class. Best proximity pair theorems are also established for the admissible class ${\mathfrak A}_c^\kappa$ and the $PK$ class.

Let $G\subset C $ be a finite Jordan domain, $z_{0}\in G;$ $B\Subset G$ be an arbitrary closed disk with $z_{0}\in B,$ and $w=\varphi (z,z_{0})$ be the conformal mapping of $G$ onto a disk $\{w:\left| w\right| <r\}$ normalized by $\varphi (z_{0},z_{0})=0$, $\varphi ^{\prime }(z_{0},z_{0})=1$ . It is well known that the Bieberbach polynomials $\{\pi _{n}(z,z_{0})\}$ for the pair $(G,z_{0})$ converge uniformly to $\varphi (z,z_{0})$ on compact subsets of the Jordan domain $G.$ In this paper we study the speed of $\left\| \varphi -\pi _{n}\right\| _{C(B)}\rightarrow 0,$ $n\rightarrow \infty ,$ in domains of the complex plane with a complicated boundary structure.

We construct $(k \pm 1)$-regular graphs which provide sequences of expanders by adding or substracting appropriate 1-factors from given sequences of $k$-regular graphs. We compute numerical examples in a few cases for which the given sequences are from the work of Lubotzky, Phillips, and Sarnak (with $k-1$ the order of a finite field). If $k+1 = 7$, our construction results in a sequence of $7$-regular expanders with all spectral gaps at least $6 - 2\sqrt 5 \approx 1.52$; the corresponding minoration for a sequence of Ramanujan $7$-regular graphs (which is not known to exist) would be $7 - 2\sqrt 6 \approx 2.10$.

We show that any hermitian $^{\ast}$-$l.m.c.a.$, the set of positive elements of which is a locally bounded cone, is necessarily a $Q$-algebra (the converse is not true). We also obtain that the algebra of complex numbers is the unique locally $C^{\ast}$-algebra without zero-divisors.

In this paper we extend a classical result, namely, the one that states that the only doubly ruled surfaces in $\mathbb R^3$ are the hyperbolic paraboloid and the hyperboloid of one sheet, in three directions: for all space forms, for any dimensions of the rulings and manifold, and to the conformal realm. We show that all this can be reduced, with the help of quite natural constructions, to just one simple example, the rank one real matrices. We also give the affine classification in Euclidean~space. To deal with the conformal case, we make use of recent developments on Ribaucour transformations.

We define an algebraic approximation of the Lusternik-Schnirelmann category of a map and show that this invariant lies between A-category and M-category. We derive from this result a characterization of the Lusternik-Schnirelmann category of a rational space.

For a Noetherian local domain $A$, there exists an upper bound $N_\tau(A)$ on the minimal number of generators of any height two ideal $\mathfrak a$ for which $A/\mathfrak a$ is Cohen-Macaulay of type $\tau$. If $A$ contains an infinite field, then we may take $N_\tau(A):=(\tau+1)e_{\textup{hom}}(A)$, where $e_{\textup{hom}}(A)$ is the homological multiplicity of $A$.

The aim of this note is to discuss in characteristic $2$ some conditions under which an anisotropic quadratic form is a Pfister neighbor when its anisotropic part over its own function field is defined over the base field. In this sense, we give a result which reduces the situation to the characterization of Pfister neighbors having a $2$-dimensional nonsingular part.

We show that if $C_\varphi$ is a Hilbert-Schmidt composition operator on an appropriately weighted Hardy space, then there exists a capacity, associated to the weight sequence of the space, so that the set on which the radial limit of $\varphi$ is unimodular has capacity zero. This extends recent results by Gallardo-Gutiérrez and González.

We construct some measure $\Theta^{\alpha}$ such that if $0<\alpha\leq2n-2$, $\beta=n-\frac{2+\alpha}{2}$ and $E$ is a circular set of type $G_{\delta}$ such that $E\subset\partial\Bbb B^{n}$ and $\Theta^{\alpha}(E)=0$ then there exists $f\in\Bbb O(\Bbb B^{n})\cap L^{2}(\Bbb B^{n})$ such that \[ E=E^{\beta}(f):=\left\{ z\in \partial B^{n}:\:\int_{\Bbb Dz}\left|f\right|^{2}\chi_{\beta}d\mathfrak{L}^{2}=\infty\right\} \] where $\chi_{s}:\Bbb B^{n}\ni z\longrightarrow\chi_{s}(z)=(1-\left\Vert z\right\Vert ^{2})^{s}$ and $\Bbb D$ denotes the unit disc in $\Bbb C$.