We provide a simplified proof of the following well-known theorem. The projection to Rn + 1of an n-dimensional rectifiable Legendrian subset of Rn+1 × Snis C2-rectifiable. Such a result generalizes the following simple fact. Let M be a hypersurface whose Gauss map has graph of class C1. Then M is of class C2.

Determining the distribution of the number of empty urns after a number of balls have been thrown randomly into the urns is a classical and well understood problem. We study a generalization: Given a finite alphabet of size σ and a word length q, what is the distribution of the number X of words (of length q) that do not occur in a random text of length n+q−1 over the given alphabet? For q=1, X is the number Y of empty urns with σ urns and n balls. For q[ges ]2, X is related to the number Y of empty urns with σq urns and n balls, but the law of X is more complicated because successive words in the text overlap. We show that, perhaps surprisingly, the laws of X and Y are not as different as one might expect, but some problems remain currently open.

The sizes of the cycles and unicyclic components in the random graph $G(n, n/2\pm s)$, where $n^{2/3}\ll s \ll n$, are studied using the language of point processes. This refines several earlier results by different authors. Asymptotic distributions of various random variables are given: these distributions include the gamma distributions with parameters 1/4, 1/2 and 3/4, as well as the Poisson–Dirichlet and GEM distributions with parameters 1/4 and 1/2.

The bond percolation critical probability of the Kagomé lattice is greater than 0.5209 and less than 0.5291. The proof of these bounds uses the substitution method, comparing the percolative behaviour of the Kagomé lattice bond model with that of the exactly solved hexagonal lattice bond model via stochastic ordering.

A proper vertex coloring of a graph is called equitable if the sizes of colour classes differ by at most 1. In this paper, we find the minimum number l=l(d, Δ) such that every d-degenerate graph with maximum degree at most Δ admits an equitable t-colouring for every t[ges ]l when Δ[ges ]27d.

In this paper we prove that, if a collection of sets [Ascr] is union-closed, then the average set size of $\cal A is at least $\frac{1}{2}\log_2 (\vert{\cal A}\vert)$.

A general spectral bound for the sizes of some vertex subsets, which are mutually at a given minimum distance in a graph, is derived. This unifies and improves some previous results. Some applications to the study of certain metric parameters of the graph are then discussed.

It has been shown [2] that if n is odd and m1,…,mt are integers with mi[ges ]3 and [sum ]i=1tmi=|E(Kn)| then Kn can be decomposed as an edge-disjoint union of closed trails of lengths m1,…,mt. This result was later generalized [3] to all sufficiently dense Eulerian graphs G in place of Kn. In this article we consider the corresponding questions for directed graphs. We show that the complete directed graph <?TeX \displaystyle{\mathop{K}^{\raise-2pt\hbox{$\scriptstyle\leftrightarrow$}}}_{n}?> can be decomposed as an edge-disjoint union of directed closed trails of lengths m1,…,mt whenever mi[ges ]2 and <?TeX \sum_{i=1}^t m_{i}=\vert E({\leftrightarrow}_{n})\vert ?>, except for the single case when n=6 and all mi=3. We also show that sufficiently dense Eulerian digraphs can be decomposed in a similar manner, and we prove corresponding results for (undirected) complete multigraphs.

We prove that, for all values of the edge probability $p(n)$, the largest eigenvalue of the random graph $G(n, p)$ satisfies almost surely $\lambda_1(G)=(1+o(1))\max\{\sqrt{\Delta}, np\}$, where Δ is the maximum degree of $G$, and the o(1) term tends to zero as $\max\{\sqrt{\Delta}, np\}$ tends to infinity.

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

AMS 2000 Mathematics subject classification: Primary 58J05

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

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 finitesequences in $\mathbb{N}$,then, except in trivial cases, there is a partition of $\mathbb{N}$into two cells, neither of which contains $\mathrm{MT}(\bm{a},\langlex_t\rangle_{t=1}^\infty)\cup \mathrm{MT}(\bm{b},\langle y_t\rangle_{t=1}^\infty)$ 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 sowith a proof which is significantly shorter and simpler than the original proof which applied only to positivecoefficients. We also derive some results concerning the existence of solutions of certain linear equations in$\beta\mathbb{Z}$. Inparticular, 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 findidempotents $p$ and $q$ in $\beta\mathbb{N}$ such that $a_1\cdot p+a_2\cdot p+\cdots+a_n\cdot p=b_1\cdot q+b_2\cdotq+\cdots+b_m\cdot q$, and thus determine exactly when the latter has a solution.

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

We show that for a non-flat bornological space there is always a bornological countable enlargement; moreover, when thespace 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, andevery quasibarrelled space that is not barrelled has a quasibarrelled countable enlargement.

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

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