For an $l\times k$ matrix $A=(a_{ij})$ of integers, denote by $\mathcal L(A)$ the system of homogenous linear equations $a_{i1}x_1+\ldots+a_{ik}x_k=0$, $1\le i\le l$. We say that $A$ is {\it density regular} if every subset of $\bold N$ with positive density, contains a solution to $\mathcal L(A)$. For a density regular $l\times k$ matrix $A$, an integer $r$ and a set of integers $F$, we write $$F\to(A)_r$$ if for any partition $F=F_1\cup....\cup F_r$ there exists $i\in\{1,2,...,r\}$ and a column vector $\bold x$ such that $A\bold x=\bold 0$ and all entries of $\bold x$ belong to $F_i$. Let $[n]_N$ be a random $N$-element subset of $\{1,2,...,n\}$ chosen uniformly from among all such subsets. In this paper we determine for every density regular matrix $A$ a parameter $\alpha=\alpha(A)$ such that $\lim_{n\to\infty}\bold P([n]_N\to(A)_r)=0$ if $N=O(n^{\alpha})$ and 1 if $N=\Omega(n^{\alpha})$.

1991 Mathematics Subject Classification: 05D10, 11B25, 60C05

The general theory of locally coherent Grothendieck categories is presented. To each locally coherent Grothendieck category $\C$ a topological space, the Ziegler spectrum of $\C,$ is associated. It is proved that the open subsets of the Ziegler spectrum of $\C$ are in bijective correspondence with the Serre subcategories of $\coh \C,$ the subcategory of coherent objects of $\C.$ This is a Nullstellensatz for locally coherent Grothendieck categories. If $R$ is a ring, there is a canonical locally coherent Grothendieck category $\RC$ (respectively, $\CR$) used for the study of left (respectively, right) $R$-modules. This category contains the category of $R$-modules and its Ziegler spectrum is quasi-compact, a property used to construct large (not finitely generated) indecomposable modules over an artin algebra. Two kinds of examples of locally coherent Grothendieck categories are given: the abstract category theoretic examples arising from torsion and localization and the examples that arise from particular modules over the ring $R.$ The duality between $\coh (\RC)$ and $\coh \CR$ is shown to give an isomorphism between the topologies of the left and right Ziegler spectra of a ring $R.$ The Nullstellensatz is used to give a proof of the result of Crawley-Boevey that every character $\xi: K_0 (\coh \C) \to Z$ is uniquely expressible as a $Z$-linear combination of irreducible characters.

1991 Mathematics Subject Classification: 16D90, 18E15.

In order to apply the ideas of Iwasawa theory to the symmetric square of a newform, we need to be able to define non-archimedean analogues of its complex $L$-series. The interpolated $p$-adic $L$-function is closely connected via a “Main Conjecture” with certain Selmer groups over the cyclotomic $\Bbb Z_p$-extension of $\Bbb Q$. In the $p$-ordinary case these functions are well understood.

In this article we extend the interpolation to an arbitrary set $S$ of good primes (not necessarily satisfying ordinarity conditions). The corresponding $S$-adic functions can be characterised in terms of certain admissibility criteria. We also allow interpolation at particular primes dividing the level of the newform.

One interesting application is to the symmetric square of a modular elliptic curve $E$ defined over $\Bbb Q$. Our constructions yield $p$-adic $L$-functions at all primes of stable or semi-stable reduction. If $p$ is ordinary or multiplicative the corresponding analytic function is bounded; if $p$ is supersingular our function behaves like $\log^2(1+T)$.

1991 Mathematics Subject Classification: 11F67, 11F66, 11F33, 11F30

This paper gives one-term componentwise asymptotics for the $M$ and spectral matrices of a self-adjoint realisation of an even-order ordinary differential expression. The underlying interval is assumed to have at least one regular endpoint, and the boundary conditions are supposed to be separated. Furthermore, the weight function and the reciprocal of the highest-order coefficient are supposed to be of regular variation at the regular endpoint, in the sense of Bingham, Goldie and Teugels.

1991 Mathematics Subject Classification: 34B24, 34E05.

Semisimple Banach algebras are well-known to have a unique (complete) algebra norm topology, but such uniqueness may fail if the radical is even one-dimensional. We obtain a necessary condition for uniqueness of norm when the algebra has finite-dimensional radical. In the case where the Banach algebra is separable, the condition is shown to be also sufficient for a large class of algebras, and in particular under various hypotheses of commutativity. Examples are given to show the limitations of the various sufficiency results, and these also give a good indication of where the difficulties lie in general. We conjecture that, at least in the separable case, our condition is both necessary and sufficient.

1991 Mathematics Subject Classification: 46H20, 46H40.

Let $f$, $g$ : $({\Bbb R}^n,0)\rightarrow ({\Bbb R}^p,0)$ be two $C^\infty$ map-germs. Then $f$ and $g$ are $C^0$-equivalent if there exist homeomorphism-germs $h$ and $l$ of $({\Bbb R}^n,0)$ and $({\Bbb R}^p,0)$ respectively such that $g=l\circ f\circ h^{-1}$. Let $k$ be a positive integer. A germ $f$ is $k$-$C^0$-determined if every germ $g$ with $j^k g(0)=j^k f(0)$ is $C^0$-equivalent to $f$. Moreover, we say that $f$ is finitely topologically determined if $f$ is $k$-$C^0$-determined for some finite $k$. We prove a theorem giving a sufficient condition for a germ to be finitely topologically determined. We explain this condition below.

Let $N$ and $P$ be two $C^\infty$ manifolds. Consider the jet bundle $J^k(N,P)$ with fiber $J^k(n,p)$. Let $z\in J^k(n,p)$ and let $f$ be such that $z= j^kf(0)$. Define \[ \chi(f)=\dim_{\Bbb R} \frac{\theta (f)} {tf(\theta (n))+f^{\ast}(m_p)\theta(f)}. \] Whether $\chi(f)(k$ depends only on $z$, not on $f$ . We can therefore define the set $W^k= W^k(n,p)=\{z\in J^k(n,p)\vert \chi(f)\ge k$ for some representative $f$ of $z$\}. Let $W^k(N,P)$ be the subbundle of $J^k(N,P)$ with fiber $W^k(n,p)$. Mather has constructed a finite Whitney (b)-regular stratification ${\mathcal{S}}^k(n,p)$ of $J^k(n,p)-W^k(n,p)$ such that all strata are semialgebraic and $\mathcal K$-invariant, having the property that if ${\mathcal S}^k(N,P)$ denotes the corresponding stratification of $J^k(N,P)-W^k(N,P)$ and $f\in C^\infty (N,P)$ is a $C^\infty$ map such that $j^k f$ is multitransverse to ${\mathcal S}^k(N,P)$, $j^k f(N)\cap W^k(N,P)=\emptyset$ and $N$ is compact (or $f$ is proper), then $f$ is topologically stable.

For a map-germ $f:({\Bbb R}^n,0)\rightarrow ({\Bbb R}^p,0)$, we define a certain {\L}ojasiewicz inequality. The inequality implies that there exists a representative $f : U\rightarrow{\Bbb R}^p$ such that $j^kf(U-\{0\})\cap W^k({\Bbb R}^n,{\Bbb R}^p)=\emptyset$ and such that $j^kf$ is multitransverse to ${\mathcal S}^k({\Bbb R}^n,{\Bbb R}^p)$ at any finite set of points $S\subset U-\{0\}$. Moreover, the inequality controls the rate $j^k f$ becomes non-transverse as we approach 0. We show that if $f$ satisfies this inequality, then $f$ is finitely topologically determined.

1991 Mathematics Subject Classification: 58C27.

Let $\Omega$ be an open proper subset of $\Bbb R^n$. Its {\it skeleton} is the set of points with more than one nearest neighbour in the complement of $\Omega$; its {\it central set} is the set of centres in maximal open balls included in $\Omega$. Intuitively, if we think of $\Omega$ as a land mass in which height is proportional to distance from the sea, its skeleton and central set can be thought of as corresponding to ridges in the mountains of $\Omega$. In this note I discuss the metric and topological properties of such sets. I show that any skeleton in $\Bbb R^n$ is F$_{\sigma}$, and has dimension at most $n-1$, by any of the usual measures of dimension; that if $\Omega$ is bounded and connected, its skeleton and central set are connected; and that $\Omega$ separates $\Bbb R^n$ iff its skeleton does iff its central set does. Any central set in $\Bbb R^n$ is a G$_{\delta}$ set of topological dimension at most $n-1$. In the plane, I show that both skeletons and central sets are locally path-connected, and indeed include many paths of finite length. For any $\Omega$, its central set includes its skeleton; I give examples to show that the central set can be significantly larger than the skeleton.

1991 Mathematics Subject Classification: 54F99.