We deal with a class of $p$-Laplacian Dirichlet boundary-value problems where the combined effects of ‘sublinear’ and‘superlinear’ growths allow us to establish the existence of at least two positive solutions.

AMS 2000 Mathematics subject classification: Primary 35J25; 35J60. Secondary 35J65; 35J70

In this paper finite groups factorized as products of pairwise totally permutable subgroups are studied in theframework of Fitting classes.

AMS 2000 Mathematics subject classification: Primary 20D10; 20D40

Working on a suitable cone of continuous functions, we give new results for integral equations of the form$\lambda u(t)=\int_{G}k(t,s)f(s,u(s))\,\mathrm{d} s:=Tu(t)$, where $G$ is a compact set in $\mathbb{R}^{n}$ and $k$ is apossibly discontinuous function that is allowed to change sign. We apply our results to prove existence of eigenvaluesof some non-local boundary-value problems.

AMS 2000 Mathematics subject classification: Primary 34B10. Secondary 34B18; 47H10; 47H30

Montel introduced the concept of quasi-normal families $f:\varOmega\to\mathbb{C}$ in 1922: $\mathcal{F}$ is quasi-normal oforder $N$ if every sequence $\{f_n\}$ from $\mathcal{F}$ has a subsequence which converges uniformly on compact subsetsof $\varOmega\setminus Z^\dagger$, where $Z^\dagger\subset\varOmega$ contains at most $N\in\mathbb{N}$ elements. ($\mathcal{F}$is of order $N:=\infty$ if every such exceptional set $Z^\dagger$ is finite.) The problem is that $Z^\dagger$ normallydepends on the subsequence. So even if every sequence has a subsequence which converges to a given function $f$ in$\varOmega$ except at $N$ points, the sequence itself may not converge in any domain $D\subseteq\varOmega$.

In this paper we introduce the concept of general convergence. Indeed, $\{f_n\}$ above converges generally to $f$. We also introduce a related concept, restrained sequences, and study some of their properties. Thedefinitions extend earlier concepts introduced for sequences of linear fractional transformations.

AMS 2000 Mathematics subject classification: Primary 20H10; 30B70; 30D45; 30F35; 58F08

We consider the class of graph-directed constructions which are connected and have the property of finite ramification.By assuming the existence of a fixed point for a certain renormalization map, it is possible to construct a Laplaceoperator on fractals in this class via their Dirichlet forms. Our main aim is to consider the eigenvalues of theLaplace operator and provide a formula for the spectral dimension, the exponent determining the power-law scaling inthe eigenvalue counting function, and establish generic constancy for the counting-function asymptotics. In order to dothis we prove an extension of the multidimensional renewal theorem. As a result we show that it is possible for theeigenvalue counting function for fractals to require a logarithmic correction to the usual power-law growth.

AMS 2000 Mathematics subject classification: Primary 35P20; 58J50. Secondary 28A80; 60K05; 31C25

We have constructed Lie superalgebras $D(2,1;\alpha)$, $G(3)$ and $F(4)$ from $U(-1,-1)$-balanced Freudenthal–Kantortriple systems in a natural way.

AMS 2000 Mathematics subject classification: Primary 17A40; 17B60

We show that the metric version of Pansu’s differentiability result for Lipschitz maps fails; this illustrates aninteresting difference between Euclidean domains and domains that are non-abelian stratified groups.

AMS 2000 Mathematics subject classification: Primary 26B05. Secondary 22E25

Let $q$ be a positive integer, let $\mathcal{I}=\mathcal{I}(q)$ and $\mathcal{J}=\mathcal{J}(q)$ be subintervals of integers in $[1,q]$ and let $\mathcal{M}$ be theset of elements of $\mathcal{I}$ that are invertible modulo $q$ and whose inverses lie in $\mathcal{J}$. We show that when $q$ approachesinfinity through a sequence of values such that $\varphi(q)/q\rightarrow0$, the $r$-spacing distribution betweenconsecutive elements of $\mathcal{M}$ becomes exponential.

AMS 2000 Mathematics subject classification: Primary 11K06; 11B05; 11N69

In this paper we examine periodic problems driven by the scalar $p$-Laplacian. Using non-smooth critical-point theoryand a recent multiplicity result based on local linking (the original smooth version is due to Brezis and Nirenberg),we prove three multiplicity results, the third for semilinear problems with resonance at zero. We also study aquasilinear periodic eigenvalue problem with the parameter near resonance. We prove the existence of three distinctsolutions, extending in this way a semilinear and smooth result of Mawhin and Schmitt.

AMS 2000 Mathematics subject classification: Primary 34C25

A strip of radius $r$ in the hyperbolic plane is the set of points within distance $r$ of a given geodesic. Wedefine the density of a packing of strips of radius $r$ and prove that this density cannot exceed

$$ \mathcal{S}(r)=\frac{3}{\pi}\sinh r\mathrm{arccosh}\biggl(1+\frac{1}{2\sinh^2r}\biggr). $$

This bound is sharp for every value of $r$ and provides sharp bounds on collaring theorems for simple geodesics onsurfaces.

AMS 2000 Mathematics subject classification: Primary 51M09; 52C15. Secondary 51M04

We consider the higher-rank graphs introduced by Kumjian and Pask as models for higher-rank Cuntz–Krieger algebras. Wedescribe a variant of the Cuntz–Krieger relations which applies to graphs with sources, and describe a local convexitycondition which characterizes the higher-rank graphs that admit a non-trivial Cuntz–Krieger family. We then proveversions of the uniqueness theorems and classifications of ideals for the $C^*$-algebras generated by Cuntz–Krieger families.

AMS 2000 Mathematics subject classification: Primary 46L05

It was shown by Huynh and Rizvi that a ring $R$ is semisimple artinian if and only if every continuous right $R$-moduleis injective. However, a characterization of rings, over which every finitely generated continuous rightmodule is injective, has been left open. In this note we give a partial solution for this question. Namely, we showthat for a right semi-artinian ring $R$, every finitely generated continuous right $R$-module is injective if and onlyif all simple right $R$-modules are injective.

AMS 2000 Mathematics subject classification: Primary 16D50. Secondary 16P20; 16P60

We study characters of an $n$-fold cover $\widetilde{SL}(n,\mathbb{F})$ of $SL(n,\mathbb{F})$ over anon-Archimedean local field. We compute the character of an irreducible representation of$\widetilde{SL}(n,\mathbb{F})$ in terms of the character of an irreducible representation of a cover$\widetilde{GL}(n,\mathbb{F})$ of $GL(n,\mathbb{F})$. We define an analogue of L-packets for$\widetilde{SL}(n,\mathbb{F})$, such that the character of a linear combination of the representations in such a packet is computed in terms of the character of an irreducible representation of $PGL(n,\mathbb{F})$. This is analogous to stable endoscopic lifting for linear groups. We also prove an ‘inversion’ formula expressing the character of a genuine irreducible representation of $\widetilde{SL}(n,\mathbb{F})$ as a linear combination of virtual characters, each of which is obtained from $PGL(n,\mathbb{F})$.

AMS 2000 Mathematics subject classification: Primary 22E50. Secondary 11F70

Let $G$ be a compact $p$-valued $p$-adic Lie group, and let $\varLambda(G)$ be its Iwasawa algebra. The present paper establishes results about the structure theory of finitely generated torsion $\varLambda(G)$-modules, up to pseudo-isomorphism, which are largely parallel to the classical theory when $G$ is abelian (except for basic differences which occur for those torsion modules which do not possess a non-zero global annihilator). We illustrate our general theory by concrete examples of such modules arising from the Iwasawa theory of elliptic curves without complex multiplication over the field generated by all of their $p$-power torsion points.

AMS 2000 Mathematics subject classification: Primary 11G05; 11R23; 16D70; 16E65; 16W70

We show that if an automorphism of a non-abelian free group $F_n$ is irreducible with irreducible powers, it acts on the boundary of Culler–Vogtmann’s outer space with north–south dynamics: there are two fixed points, one attracting and one repelling, and orbits accumulate only on these points. The main new tool we use is the equivariant assignment of a point $Q(X)$ to any end $X\in\partial F_n$, given an action of $F_n$ on an $\bm{R}$-tree $T$ with trivial arc stabilizers; this point $Q(X)$ may be in $T$, or in its metric completion, or in its boundary.

AMS 2000 Mathematics subject classification: Primary 20F65; 20E05; 20E08

Nous conjecturons que la réduction modulo $p$ des représentations cristallines irréductibles de dimension 2 sur $\bar{\bm{Q}}_p$ de $\Gal(\bar{\bm{Q}}_p/\bm{Q}_p)$ peut être prédite par la réduction modulo $p$ de représentations $p$-adiques localement algébriques de $\GL_2(\bm{Q}_p)$. Nous explicitons quelques calculs de telles réductions confirmant cette conjecture. Cela suggère un lien arithmétique non trivial entre les deux types de représentations.

AMS 2000 Mathematics subject classification: Primary 11F33; 11F70; 11F80; 11F85

In this note we are studying the Lie algebras associated to non-abelian unipotent periods on $P^1_{\mathbb{Q}(\mu_n)}\setminus\{0,\mu_n,\infty\}$. Let $n$ be a prime number. We assume that for any $m\geq 1$ the numbers $Li_{m+1}(\xi_n^k)$ for $1\leq k\leq (n-1)/2$ are linearly independent over $\mathbb{Q}$ in $\mathbb{C}/(2\pi\ri)^{m+1}\mathbb{Q}$. Let $S=\{k_1,\cdots,k_q\}$ be a subset of $\{1,\dots,p-1\}$ such that if $k\in S$, then $p-k\in S$ and $(S+S)\cap S=\emptyset$ (the sum of two elements of $S$ is calculated $\mathrm{Mod}p$). Then we show that in the Lie algebra associated to non-abelian unipotent periods on $P^1_{\mathbb{Q}(\mu_n)}\setminus \{0,\mu_n,\infty\}$ there are derivations $D^{k_1}_{m+1},\dots,D^{k_q}_{m+1}$ in each degree $m+1$ and these derivations are free generators of a free Lie subalgebra of this Lie algebra.

AMS 2000 Mathematics subject classification: Primary 11G55