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 show that for a non-flat bornological space there is always a bornological countable enlargement; moreover, when the space is non-flat and ultrabornological the countable enlargement may be chosen to be both bornological and barrelled. It is also shown that countable enlargements for barrelled or bornological spaces are always Mackey topologies, and every quasibarrelled space that is not barrelled has a quasibarrelled countable enlargement.

AMS 2000 Mathematics subject classification: Primary 46A08; 46A20

Given a finite sequence $\bm{a}=\langle a_i\rangle_{i=1}^n$ in $\mathbb{N}$ and a sequence $\langle x_t\rangle_{t=1}^\infty$ in $\mathbb{N}$, the Milliken–Taylor system generated by $\bm{a}$ and $\langle x_t\rangle_{t=1}^\infty$ is

\begin{multline*} \qquad \mathrm{MT}(\bm{a},\langle x_t\rangle_{t=1}^\infty)=\biggl\{\sum_{i=1}^na_i\cdot\sum_{t\in F_i}x_t:F_1,F_2,\dots,F_n\text{ are finite non-empty} \\[-8pt] \text{subsets of $\mathbb{N}$ with }\max F_i\lt\min F_{i+1}\text{ for }i\ltn\biggr\}.\qquad \end{multline*}

It is known that Milliken–Taylor systems are partition regular but not consistent. More precisely, if $\bm{a}$ and $\bm{b}$ are finite sequences in $\mathbb{N}$, then, except in trivial cases, there is a partition of $\mathbb{N}$ into two cells, neither of which contains $\mathrm{MT}(\bm{a},\langle x_t\rangle_{t=1}^\infty)\cup \mathrm{MT}(\bm{b},\langle y_t\rangle_{t=1}^\infty)$ for any sequences $\langle x_t\rangle_{t=1}^\infty$ and $\langle y_t\rangle_{t=1}^\infty$.

Our aim in this paper is to extend the above result to allow negative entries in $\bm{a}$ and $\bm{b}$. We do so with a proof which is significantly shorter and simpler than the original proof which applied only to positive coefficients. We also derive some results concerning the existence of solutions of certain linear equations in $\beta\mathbb{Z}$. In particular, we show that the ability to guarantee the existence of $\mathrm{MT}(\bm{a},\langle x_t\rangle_{t=1}^\infty)\cup \mathrm{MT}(\bm{b},\langle y_t\rangle_{t=1}^\infty)$ in one cell of a partition is equivalent to the ability to find idempotents $p$ and $q$ in $\beta\mathbb{N}$ such that $a_1\cdot p+a_2\cdot p+\cdots+a_n\cdot p=b_1\cdot q+b_2\cdot q+\cdots+b_m\cdot q$, and thus determine exactly when the latter has a solution.

AMS 2000 Mathematics subject classification: Primary 05D10. Secondary 22A15; 54H13

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

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

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

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

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

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

The best-constant problem for Nash and Sobolev inequalities on Riemannian manifolds has been intensively studied in the last few decades, especially in the compact case. We treat this problem here for a more general family of Gagliardo–Nirenberg inequalities including the Nash inequality and the limiting case of a particular logarithmic Sobolev inequality. From the latter, we deduce a sharp heat-kernel upper bound.

AMS 2000 Mathematics subject classification: Primary 58J05

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

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

Montel introduced the concept of quasi-normal families $f:\varOmega\to\mathbb{C}$ in 1922: $\mathcal{F}$ is quasi-normal of order $N$ if every sequence $\{f_n\}$ from $\mathcal{F}$ has a subsequence which converges uniformly on compact subsets of $\varOmega\setminus Z^\dagger$, 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 $Z^\dagger$ normally depends 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. The definitions extend earlier concepts introduced for sequences of linear fractional transformations.

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

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 the set of elements of $\mathcal{I}$ that are invertible modulo $q$ and whose inverses lie in $\mathcal{J}$. We show that when $q$ approaches infinity through a sequence of values such that $\varphi(q)/q\rightarrow0$, the $r$-spacing distribution between consecutive elements of $\mathcal{M}$ becomes exponential.

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

Recently, ratio-dependent predator–prey systems have been regarded by some researchers as being more appropriate forpredator–prey interactions where predation involves serious searching processes. Due to the fact that every populationgoes through some distinct life stages in real-life, one often introduces time delays in the variables being modelled.The presence of time delay often greatly complicates the analytical study of such models. In this paper, thequalitative behaviour of a class of ratio-dependent predator–prey systems with delay at the equilibrium in theinterior of the first quadrant is studied. It is shown that the interior equilibrium cannot be absolutely stable andthere exist non-trivial periodic solutions for the model. Moreover, by choosing delay $\tau$ as the bifurcationparameter we study the Hopf bifurcation and the stability of the periodic solutions.

AMS 2000 Mathematics subject classification: Primary 34C25; 92D25. Secondary 58F14

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

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

In this paper we examine periodic problems driven by the scalar $p$-Laplacian. Using non-smooth critical-point theory and 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 a quasilinear periodic eigenvalue problem with the parameter near resonance. We prove the existence of three distinct solutions, extending in this way a semilinear and smooth result of Mawhin and Schmitt.

AMS 2000 Mathematics subject classification: Primary 34C25