Let $D$ be a family of $k$-subsets (called blocks) of a $v$-set $X\left( v \right)$. Then $D$ is a $\left( v,\,k,\,t \right)$ covering design or covering if every $t$-subset of $X\left( v \right)$ is contained in at least one block of $D$. The number of blocks is the size of the covering, and the minimum size of the covering is called the covering number. In this paper we consider the case $t\,=\,2$, and find several infinite classes of covering numbers. We also give upper bounds on other classes of covering numbers.

A binary structure $S$ has the pigeonhole property $\left( P \right)$ if every finite partition of $S$ induces a block isomorphic to $S$. We classify all countable tournaments with $\left( P \right)$; the class of orders with $\left( P \right)$ is completely classified.

Abstract. The weighted mean matrix ${{M}_{a}}$ is the triangular matrix $\left\{ {{a}_{k}}/{{A}_{n}} \right\}$, where ${{a}_{n}}\,>\,0$ and ${{A}_{n}}\,:=\,{{a}_{1}}\,+\,{{a}_{2}}\,+\cdots +\,{{a}_{n}}$. It is proved that, subject to ${{n}^{c}}{{a}_{n}}$ being eventually monotonic for each constant $c$ and to the existence of $\alpha \,:=\,\lim \,\frac{{{A}_{n}}}{n{{a}_{n}}},\,{{M}_{a}}\,\in \,B\left( {{l}_{p}} \right)$ for $1\,<\,p\,<\infty $ if and only if $\alpha \,<\,p$.

We construct a countably infinite family of pairwise non-isomorphic maximal $\mathbb{Q}\left[ X \right]$-orders such that the full 2 by 2 matrix rings over these orders are all isomorphic.

Let $\text{Z}$ be the unital simple nuclear infinite dimensional ${{C}^{*}}$-algebra which has the same Elliott invariant as $\mathbb{C}$, introduced in [9]. A ${{C}^{*}}$-algebra is called $\text{Z}$-stable if $A\,\cong \,A\,\otimes \,\text{Z}$. In this note we give some necessary conditions for a unital simple ${{C}^{*}}$-algebra to be $\text{Z}$-stable.

Travelling wave solutions to the vortex filament flow generated by elastica produce surfaces in ${{\mathbb{R}}^{3}}$ that carry mutually orthogonal foliations by geodesics and by helices. These surfaces are classified in the special cases where the helices are all congruent or are all generated by a single screw motion. The first case yields a new characterization for the Bäcklund transformation for constant torsion curves in ${{\mathbb{R}}^{3}}$, previously derived fromthe well-known transformation for pseudospherical surfaces. A similar investigation for surfaces in ${{H}^{3}}$ or ${{S}^{3}}$ leads to a new transformation for constant torsion curves in those spaces that is also derived from pseudospherical surfaces.

A new class of 3-dimensional contact metric manifolds is found. Moreover it is proved that there are no such manifolds in dimensions greater than 3.

Suppose $m$ and $n$ are integers such that $1\,\le \,m\,\le \,n$. For a subgroup $H$ of the symmetric group ${{S}_{m}}$ of degree $m$, consider the generalized matrix function on $m\,\times \,m$ matrices $B\,=\,\left( {{b}_{ij}} \right)$ defined by ${{d}^{H}}\left( B \right)\,=\,\sum{_{\sigma \in H}\,\prod{_{j=1}^{m}}\,{{b}_{j\sigma \left( j \right)}}}$ and the generalized numerical range of an $n\,\times \,n$ complex matrix $A$ associated with ${{d}^{H}}$ defined by

$${{W}^{H}}\left( A \right)\,=\,\left\{ {{d}^{H}}\left( {{X}^{*}}\,AX \right)\,:\,X\,is\,n\,\times \,m\,\text{such}\,\text{that}\,{{X}^{*}}X\,\text{=}\,{{\text{I}}_{m}} \right\}$$

It is known that ${{W}^{H}}\left( A \right)$ is convex if $m\,=\,1$ or if $m\,=\,n\,=\,2$. We show that there exist normal matrices $A$ for which ${{W}^{H}}\left( A \right)$ is not convex if $3\,\le \,m\,\le \,n$. Moreover, for $m\,=\,2\,<\,n$, we prove that a normal matrix $A$ with eigenvalues lying on a straight line has convex ${{W}^{H}}\left( A \right)$ if and only if $vA$ is Hermitian for some nonzero $v\,\in \,\mathbb{C}$. These results extend those of Hu, Hurley and Tam, who studied the special case when $2\,\le \,m\,\le \,3\,\le \,n$ and $H\,=\,{{S}_{m}}$.

We generalize to a field of characteristic zero certain properties of the invariant functions of the coadjoint representation of solvable Lie algebras with abelian nilradicals, previously obtained over the base field $\mathbb{C}$ of complex numbers. In particular we determine their number and the restricted type of variables on which they depend. We also determine an upper bound on the maximal number of functionally independent invariants for certain families of solvable Lie algebras with arbitrary nilradicals.

This paper contains a geometric proof of an estimate for a restricted x-ray transform. The result complements one of A. Greenleaf and A. Seeger.

Many classes of singularly perturbed reaction-diffusion equations possess localized solutions where the gradient of the solution is large only in the vicinity of certain points or interfaces in the domain. The problems of this type that are considered are an interface propagation model from materials science and an activator-inhibitor model of morphogenesis. These two models are formulated as nonlocal partial differential equations. Results concerning the existence of equilibria, their stability, and the dynamical behavior of localized structures in the interior and on the boundary of the domain are surveyed for these two models. By examining the spectrum associated with the linearization of these problems around certain canonical solutions, it is shown that the nonlocal term can lead to the existence of an exponentially small principal eigenvalue for the linearized problem. This eigenvalue is then responsible for an exponentially slow, or metastable, motion of the localized structure.

We extend Maxwell’s representation of harmonic polynomials to $h$-harmonics associated to a reflection invariant weight function ${{h}_{k}}$. Let ${{\mathcal{D}}_{i}},\,1\,\le \,i\,\le \,d$, be Dunkl’s operators associated with a reflection group. For any homogeneous polynomial $P$ of degree $n$,we prove the polynomial ${{\left| x \right|}^{2\gamma +d-2+2n}}P\left( \mathcal{D} \right)\left\{ 1/{{\left| x \right|}^{2\gamma +d-2}} \right\}$ is a $h$-harmonic polynomial of degree $n$, where $\gamma \,=\,\sum \,ki$ and $\mathcal{D}\,=\,\left( {{\mathcal{D}}_{1}},\ldots ,{{\mathcal{D}}_{d}} \right)$. The construction yields a basis for $h$-harmonics. We also discuss self-adjoint operators acting on the space of $h$-harmonics.