a cauchy problem for a nonlinear convection-diffusion equation with periodic rapidly oscillating coefficients is studied. under the assumption that the convection term is large, it is proved that the limit (homogenized) equation is a nonlinear diffusion equation which shows dispersion effects. the convergence of the homogenization procedure is justified by using a new version of a two-scale convergence technique adapted to rapidly moving coordinates.

The paper determines all permutation groups with a transitive minimal normal subgroup that have no fixed point free elements of prime order. All such groups are primitive and are wreath products in a product action involving $M_{11}$ in its action on 12 points. These groups are not 2-closed and so substantial progress is made towards asserting the truth of the polycirculant conjecture that every 2-closed transitive permutation group has a fixed point free element of prime order. All finite simple groups $T$ with a proper subgroup meeting every ${\rm Aut}(T)$ -conjugacy class of elements of $T$ of prime order are also determined.

Let $D$ be an open set in euclidean space ${\bb R}^m$ with non-empty boundary $\partial D$ , and let $p_D : D \times D \times; [0,\infty)) \longrightarrow {\bb R}$ be the Dirichlet heat kernel for the parabolic operator ${-}\Delta + \partial/\partial t$ , where ${-}\Delta$ is the Dirichlet laplacian on $L^2(D)$ . Since the Dirichlet heat kernel is non-negative, we may define the (open) set function \renewcommand{\theequation}{1.1} \begin{equation} P_D = \int\nolimits^{\infty}_0 \int\nolimits_D \int\nolimits_D p_D (x,y;t)\,dx\,dy\,dt. \end{equation} We say that $D$ has finite torsional rigidity if $P_D < \infty$ . It is well known that if $D$ has finite volume, then $D$ has finite torsional rigidity [11]. As we shall see, the converse is not true. The main purpose of this paper is to obtain necessary and sufficient conditions on the geometry of $D$ to guarantee finite torsional rigidity and to gain some understanding of the behaviour of the expected lifetime of brownian motion in a certain natural class of domains that do not have finite torsional rigidity.

It is proved that there does not exist a set of four positive integers with the property that the product of any two of its distinct elements plus their sum is a perfect square. This settles an old problem investigated by Diophantus and Euler.

Suppose that X1, X2, X3, … are independent random points in Rd with common density f, having compact support Ω with smooth boundary ∂Ω, with f[mid ]Ω continuous. Let Rni, k denote the distance from Xi to its kth nearest neighbour amongst the first n points, and let Mn, k = maxi[les ]nRni, k. Let θ denote the volume of the unit ball. Then as n → ∞,

formula here

If instead the points lie in a compact smooth d-dimensional Riemannian manifold K, then nθMdn, k/ log n → (minKf)−1, almost surely.

An integral version of a classical embedding theorem concerning quaternion algebras B over a number field k is proved. Assume that B satisfies the Eichler condition, that is, some infinite place of k is not ramified in B, and let Ω be an order in a quadratic extension of k. The maximal orders of B which admit an embedding of Ω are determined. Although most Ω embed into either all or none of the maximal orders of B, it turns out that some Ω are ‘selective’, in the sense that they embed into exactly half of the isomorphism types of maximal orders of B. As an application, the maximal arithmetic subgroups of B*/k* which contain a given element of B*/k* are determined.

The paper shows that the times spent in [0, +∞) by certain processes Y which are defined by perturbations of Brownian motion involving reflection at maxima and minima are beta distributed. This result relies heavily on Ray–Knight theorems for such perturbed Brownian motions.

It is shown, by a simple and direct proof, that if a bounded valuation on a monotone convergence space is the supremum of a directed family of simple valuations, then it has a unique extension to a Borel measure. In particular, this holds for any directed complete partial order with the Scott topology. It follows that every bounded and continuous valuation on a continuous directed complete partial order can be extended uniquely to a Borel measure. The last result also holds for σ-finite valuations, but fails for directed complete partial orders in general.

The paper derives a formula for the second variation of the displacement function for polynomial perturbations of Hamiltonian systems with elliptic or hyperelliptic Hamiltonians H(x, y)=½y2−U(x) in terms of the coefficients of the perturbation. As an application, the conjecture stated by C. Chicone that a specific cubic system appearing in a deformation of singularity with two zero eigenvalues has at most two limit cycles is proved.

The paper proves that any two dust-like invariant sets of conformal iterated function systems have the same Hausdorff dimension if and only if they are nearly Lipschitz equivalent.

Suppose [Cscr ] is a category with a symmetric monoidal structure, which we will refer to as the smash product. Then the Picard category is the full subcategory of objects which have an inverse under the smash product in [Cscr ], and the Picard group Pic([Cscr ]) is the collection of isomorphism classes of such invertible objects. The Picard group need not be a set in general, but if it is then it is an abelian group canonically associated with [Cscr ].

There are many examples of symmetric monoidal categories in stable homotopy theory. In particular, one could take the whole stable homotopy category [Sscr ]. In this case, it was proved by Hopkins that the Picard group is just Z, where a representative for n can be taken to be simply the n-sphere Sn [8, 19]. It is more interesting to consider Picard groups of the E-local category, for various spectra E (all of which will be p-local for some fixed prime p in this paper). Here the smash product of two E-local spectra need not be E-local, so one must relocalize the result by applying the Bousfield localization functor LE. The best-known case is E=K(n), the nth Morava K-theory, considered in [8].

In this paper we study the case E=E(n), where E(n) is the Johnson–Wilson spectrum. In this case the E-localization functor is universally denoted Ln, and we denote the category of E-local spectra by [Lscr ]. Our main theorem is the following result.

In 1989 Happel asked the question whether, for a finite-dimensional algebra $A$ over an algebraically closed field $k,\ \text{\rm gl.dim\,} A< \infty$ if and only if $\text{\rm hch.dim\,} A < \infty$. Here, the Hochschild cohomology dimension of $A$ is given by $\text{\rm hch.dim\,} A := \inf \{ n \in \mathbb{N}_0 \mid \dim \textit{HH}^i(A)=0 \text{ for }i > n \}$. Recently Buchweitz, Green, Madsen and Solberg gave a negative answer to Happel's question. They found a family of pathological algebras $A_q$ for which $\text{\rm gl.dim\,} A_q = \infty$ but $\text{\rm hch.dim\,} A_q=2$. These algebras are pathological in many aspects. However, their Hochschild homology behaviors are not pathological any more; indeed one has $\text{\rm hh.dim\,} A_q = \infty=\text{\rm gl.dim\,} A_q$. Here, the Hochschild homology dimension of $A$ is given by $\text{\rm hh.dim\,} A := \inf \{ n \in \mathbb{N}_0 \mid \dim \textit{HH}_i(A)=0 \text{ for } i > n\}$. This suggests posing a seemingly more reasonable conjecture by replacing the Hochschild cohomology dimension in Happel's question with the Hochschild homology dimension: $\text{\rm gl.dim\,} A < \infty$ if and only if $\text{\rm hh.dim\,} A < \infty$ if and only if $\text{\rm hh.dim\,} A = 0$. The conjecture holds for commutative algebras and monomial algebras. In the case where $A$ is a truncated quiver algebra, these conditions are equivalent to the condition that the quiver of $A$ has no oriented cycles. Moreover, an algorithm for computing the Hochschild homology of any monomial algebra is provided. Thus the cyclic homology of any monomial algebra can be read off when the underlying field is characteristic 0.

The class of co-context-free groups is studied. A co-context-free group is defined as one whose co-word problem (the complement of its word problem) is context-free. This class is larger than the subclass of context-free groups, being closed under the taking of finite direct products, restricted standard wreath products with context-free top groups, and passing to finitely generated subgroups and finite index overgroups. No other examples of co-context-free groups are known. It is proved that the only examples amongst polycyclic groups or the Baumslag–Solitar groups are virtually abelian. This is done by proving that languages with certain purely arithmetical properties cannot be context-free; this result may be of independent interest.

We determine which groups ${\mathbb{Z}}/M{\mathbb{Z}}\oplus{\mathbb{Z}}/N{\mathbb{Z}}$ occur infinitely often as torsion groups $E(K)_{\operatorname{tors}}$ when $K$ varies over all quartic number fields and $E$ varies over all elliptic curves over $K$.

Let $\mu$ be the self-similar measure for a linear function system $S_jx=\rho x+b_j$ ($j=1,2,\ldots,m$) on the real line with the probability weight $\{p_j\}_{j=1}^m$. Under the condition that $\{S_j\}_{j=1}^m$ satisfies the finite type condition, the $L^q$-spectrum $\tau(q)$ of $\mu$ is shown to be differentiable on $(0,\infty)$; as an application, $\mu$ is exact dimensional and satisfies the multifractal formalism.

New lower bounds are given for the size of a point set in a Desarguesian projective plane over a finite field that contains at least a prescribed number s of points on every line. These bounds are best possible when q is square and s is small compared with q. In this case the smallest set is shown to be the union of disjoint Baer subplanes. The results are based on new results on the structure of certain lacunary polynomials, which can be regarded as a generalization of Rédei's results in the case when the derivative of the polynomial vanishes.

The object of the paper is to show that if f is a univalent, harmonic mapping of the annulus A(r, 1) = {z : r < [mid ]z[mid ] < 1} onto the annulus A(R, 1), and if s is the length of the segment of the Grötzsch ring domain associated with A(r, 1), then R < s. This gives the first, quantitative upper bound of R, which relates to a question of J. C. C. Nitsche that he raised in 1962. The question of whether this bound is sharp remains open.

Some almost sharp sufficient conditions of oscillation and nonoscillation are obtained for the superlinear delay differential equation

A classical result of Kadison concerning the extension, via the Hahn–Banach theorem, of extremal states on unital self-adjoint linear manifolds (that is, operator systems) in C*-algebras is generalised to the setting of noncommutative convexity, where one has matrix states (that is, unital completely positive linear maps) and matrix convexity. It is shown that if ϕ is a matrix extreme point of the matrix state space of an operator system R in a unital C*-algebra A, then ϕ has a completely positive extension to a matrix extreme point Φ of the matrix state space of A. This result leads to a characterisation of extremal matrix states as pure completely positive maps and to a new proof of a decomposition of C*-extreme points.

In this paper we construct quasinilpotent operators without nontrivial invariant subspaces. As well as being interesting in its own right, we believe that this is a sensible (perhaps necessary) first step for addressing the difficult problem of constructing topologically simple Banach algebras.