We give a new characterization of Hardy martingale cotype property of complex quasi-Banach space by using the existence of a kind of plurisubharmonic functions. We also characterize the best constants of Hardy martingale inequalities with values in the complex quasi-Banach space.

In this paper we establish the ${{L}^{p}}$ boundedness of a class of singular integrals with rough kernels associated to polynomial mappings.

The usual constructions of classifying spaces for monoidal categories produce $\text{CW}$-complexes with many cells that,moreover, do not have any proper geometric meaning. However, geometric nerves of monoidal categories are very handy simplicial sets whose simplices have a pleasing geometric description: they are diagrams with the shape of the 2-skeleton of oriented standard simplices. The purpose of this paper is to prove that geometric realizations of geometric nerves are classifying spaces for monoidal categories.

Let $P$ be a transitive permutation group of order ${{p}^{m}},\,p$ an odd prime, containing a regular cyclic subgroup. The main result of this paper is a determination of the suborbits of $P$. The main result is used to give a simple proof of a recent result by J. Morris on Cayley digraph isomorphisms.

In 1988 Rieger exhibited a differentiable function having a zero at the golden ratio $(-1\,+\,\sqrt{5})/2$ for which when Newton's method for approximating roots is applied with an initial value ${{x}_{0}}\,=\,0$, all approximates are so-called “best rational approximates”—in this case, of the form ${{F}_{2n}}/{{F}_{2n+1}}$, where ${{F}_{n}}$ denotes the $n$-th Fibonacci number. Recently this observation was extended by Komatsu to the class of all quadratic irrationals whose continued fraction expansions have period length 2. Here we generalize these observations by producing an analogous result for all quadratic irrationals and thus provide an explanation for these phenomena.

Unlike the (classical) Kolakoski sequence on the alphabet {1, 2}, its analogue on {1, 3} can be related to a primitive substitution rule. Using this connection, we prove that the corresponding bi-infinite fixed point is a regular generic model set and thus has a pure point diffraction spectrum. The Kolakoski-(3, 1) sequence is then obtained as a deformation, without losing the pure point diffraction property.

Let $M$ be a complete flat Lorentz 3-manifold $M$ with purely hyperbolic holonomy $\Gamma $. Recurrent geodesic rays are completely classified when $\Gamma $ is cyclic. This implies that for any pair of periodic geodesics ${{\gamma }_{1}}$, ${{\gamma }_{2}}$, a unique geodesic forward spirals towards ${{\gamma }_{1}}$ and backward spirals towards ${{\gamma }_{2}}$.

The main purpose of this paper is to determine isotropic immersions of complex space forms into real space forms with low codimension. This is an improvement of a result of S. Maeda.

We show that there exists a singular inner function $S$ which is universal for noneuclidean translates; that is one for which the set $\{S(\frac{z\,+\,{{z}_{n}}}{1\,+\,{{{\bar{z}}}_{n}}z})\,:\,n\,\in \,\mathbb{N}\}$ is locally uniformly dense in the set of all zero-free holomorphic functions in $\mathbb{D}$ bounded by one.

The congruences of a finite sectionally complemented lattice $L$ are not necessarily uniform (any two congruence classes of a congruence are of the same size). To measure how far a congruence $\Theta $ of $L$ is from being uniform, we introduce Spec $\Theta $, the spectrum of $\Theta $, the family of cardinalities of the congruence classes of $\Theta $. A typical result of this paper characterizes the spectrum $S=({{m}_{j}}|j<n)$ of a nontrivial congruence $\Theta $ with the following two properties:

$$({{S}_{1}})\,\,\,\,2\le n\,\,\text{and }n\ne 3.\,\,\,\,$$

$$({{S}_{2}})\,\,\,2\le {{m}_{j}}\,\,\text{and}\,\,{{m}_{j}}\ne 3,\,\,\,\text{for}\,\text{all}\,j<n.$$

We prove an uniform Hölder continuity of the resolvent of the Laplace-Beltrami operator on the real axis for a class of asymptotically Euclidean Riemannian manifolds. As an application we extend a result of Burq on the behaviour of the local energy of solutions to the wave equation.

We construct new examples of non-nil algebras with any number of generators, which are direct sums of two locally nilpotent subalgebras. Like all previously known examples, our examples are contracted semigroup algebras and the underlying semigroups are unions of locally nilpotent subsemigroups. In our constructions we make more transparent than in the past the close relationship between the considered problem and combinatorics of words.

Using weighted Delsarte surfaces, we give examples of $K3$ surfaces in positive characteristic whose formal Brauer groups have height equal to 5, 8 or 9. These are among the four values of the height left open in the article of Yui [11].

Dans cet article, à partir de la notion d’enlacement introduite dans [7] entre des paires d’ensembles $(B,\,A)$ et $(Q,\,P)$, nous établissons l’existence d’un point critique d’une fonctionnelle continue sur un espace métrique lorsqu’une de ces paires enlace l’autre. Des renseignements sur la localisation du point critique sont aussi obtenus. Ces résultats conduisent à une généralisation du théorème des trois points critiques. Finalement, des applications à des problèmes aux limites pour une équation quasi-linéaire elliptique sont présentées.

We show that Poincaré inequalities with reverse doubling weights hold in a large class of irregular domains whenever the weights satisfy certain conditions. Examples of these domains are John domains.

We prove an integral formula on closed oriented manifolds equipped with a codimension two foliation whose leaves are compact.

Leray's self-similar solution of the Navier-Stokes equations is defined by

$$u(x,\,t)\,=\,U(y)/\sqrt{2\sigma ({{t}^{*}}\,-\,t)},$$

where $y\,=\,x/\sqrt{2\sigma ({{t}^{*}}\,-\,t)},\,\sigma \,>\,0$. Consider the equation for $U(y)$ in a smooth bounded domain $\mathcal{D}$ of ${{\mathbb{R}}^{3}}$ with non-zero boundary condition:

$$-v\,\Delta \,U\,+\,\sigma U\,+\,\sigma y\,\cdot \,\nabla U\,+\,U\,\cdot \,\nabla U\,+\,\nabla P\,=\,0,\,\,\,y\,\in \,\mathcal{D}\,$$

$$\nabla \,\cdot \,U\,=\,0,\,\,\,y\,\in \,\mathcal{D},$$

$$U\,=\,\mathcal{G}(y),\,\,\,y\,\in \,\partial \mathcal{D}.$$

We prove an existence theorem for the Dirichlet problem in Sobolev space ${{W}^{1,2}}(\mathcal{D})$. This implies the local existence of a self-similar solution of the Navier-Stokes equations which blows up at $t\,=\,{{t}^{*}}$ with ${{t}^{*}}\,<\,+\infty $, provided the function $\mathcal{G}(y)$ is permissible.

A discrete group $G$ is called identity excluding if the only irreducible unitary representation of $G$ which weakly contains the 1-dimensional identity representation is the 1-dimensional identity representation itself. Given a unitary representation $\pi $ of $G$ and a probability measure $\mu $ on $G$, let ${{P}_{\mu }}$ denote the $\mu $-average $\int{\pi (g)\mu (dg)}$. The goal of this article is twofold: (1) to study the asymptotic behaviour of the powers $P_{\mu }^{n}$, and (2) to provide a characterization of countable amenable identity excluding groups. We prove that for every adapted probability measure $\mu $ on an identity excluding group and every unitary representation $\pi $ there exists and orthogonal projection ${{E}_{\mu }}$ onto a $\pi $-invariant subspace such that $s-{{\lim }_{n\to \infty }}\,(P_{\mu }^{n}\,-\,\pi {{(a)}^{n}}\,{{E}_{\mu }})\,\,=\,0$ for every $a\,\in $ supp $\mu $. This also remains true for suitably defined identity excluding locally compact groups. We show that the class of countable amenable identity excluding groups coincides with the class of $\text{FC}$-hypercentral groups; in the finitely generated case this is precisely the class of groups of polynomial growth. We also establish that every adapted random walk on a countable amenable identity excluding group is ergodic.

If $A$ is a set of $M$ positive integers, let $G(A)$ be the maximum of ${{a}_{i}}/\,\gcd ({{a}_{i}},\,\,{{a}_{j}}\,)$ over ${{a}_{i}},\,{{a}_{j}}\,\,\in \,\,A.$ We show that if $G(A)$ is not too much larger than $M$, then $A$ must have a special structure.

Order components of a finite simple group were introduced in [4]. It was proved that some non-abelian simple groups are uniquely determined by their order components. As the main result of this paper, we show that groups $PS{{U}_{11}}(q)$ are also uniquely determined by their order components. As corollaries of this result, the validity of a conjecture of J. G. Thompson and a conjecture of W. Shi and J. Bi both on $PS{{U}_{11}}(q)$ are obtained.