For the classical groups, Kraft and Procesi have resolved the question of which nilpotent orbits have closures that are normal and which do not, with the exception of the very even orbits in $D_{2l}$ that have partitions of the form $(a^{2k}, b^2)$ for $a\,{>}\,b$ even natural numbers satisfying $a k\,{+}\,b\,{=}\,2 l$.

In this paper, these orbits are shown to have normal closure.

A famous result of Kostant's states that the universal enveloping algebra of a semisimple complex Lie algebra is a free module over its center. An analogue of this result is proved for the class of special filtered algebras. This is then applied to show that the restricted Yangian and the universal enveloping algebra of the restricted current algebra, associated with the general linear Lie algebra, are both free over their centers.

It is proved that, if $P$ is a polynomial with integer coefficients, having degree 2, and $1>\varepsilon>0$, then $n(n-1)\cdots(n-k+1)=P(m)$ has only finitely many natural solutions $(m,n,k)$, $n\ge k>n\varepsilon$, provided that the $abc$ conjecture is assumed to hold under Szpiro's formulation.

This paper proves the rationality of Poincaré series whose coefficients are defined using the number of conjugacy classes in a system of finite groups defined as finite images of linear groups over $\Bbb{Z}_{p}$ modulo natural congruence subgroups.

Three short results are given concerning the elliptic Harnack inequality, in the context of random walks on graphs. The first is that the elliptic Harnack inequality implies polynomial growth of the number of points in balls, and the second that the elliptic Harnack inequality is equivalent to an annulus-type Harnack inequality for Green's functions. The third result uses the lamplighter group to give a counter-example concerning the relation of coupling with the elliptic Harnack inequality

An attempt is made to generalise Bloch and Kato's calculation of some of the local Tamagawa factors appearing in their conjecture, from the crystalline case to the semi-stable case. Success is attained only under severe conditions, but the result applies to the symmetric powers of an elliptic curve with multiplicative reduction.

A classical theorem of Zimmermann describes the relation between almost split sequences in the category of finitely presented modules and those in the category of all modules over some fixed ring. An analogue of Auslander–Reiten triangles in triangulated categories is proved in this paper. This is used to explain the relation between different existence results for Auslander–Reiten triangles, which are based either on Brown's representability theorem, or on the existence of Serre functors.

a structure theorem is given for $n$-dimensional smooth subvarieties of the grassmannian $g(1,n)$, with $n\geq n+3$, that can be isomorphically projected to $g(1,n+1)$. a complete classification in the cases $n=2n+1$ and $n=2n$ follows, as a corollary.

As an application of the vector sieve and uniform estimates on the Fourier coefficients of cusp forms of half-integral weight, it is shown that any sufficiently large number $n\equiv 3$ (mod 24) with $5 \nmid n$ is expressible as a sum of three squares of integers having at most 521 prime factors.

It is proved that Epstein's zeta-function $\zeta_Q(s)$, related to a positive definite integral binary quadratic form, has a zero $ 1/2 +i\gamma $ with $T \leq \gamma \leq T+T^{5/11+\varepsilon }$ for sufficiently large positive numbers $T$. This improves a classical result of H. S. A. Potter and E. C. Titchmarsh (Proc. London Math. Soc. (2) 39 (1935) 372–384).

It is proved that each mod 2 homology class of a compact Nash manifold can be represented by a semialgebraic arc-symmetric subset.

Let V be a finite-dimensional vector space over a finite field. The general and special linear groups, GL(V) and SL(V), act on the exterior algebras $\Lambda^*V$ and $\Lambda^*V^*$ of V and its dual $V^*$, and on the symmetric algebra $\S^*V$. The subring of SL(V)-invariants of $\Lambda^*V\otimes\S^*V$ was determined by Dickson and Mui. This paper describes the equivalent, but simpler, calculation of the invariant subring of $\Lambda^*V^*\otimes\S^*V$ as a representation of GL(V)/SL(V).

Sufficient conditions are given for the global existence of solutions to the Rayleigh equation.

Shelah has recently proved that an uncountable free group cannot be the automorphism group of a countable structure. In fact, he proved a more general result: an uncountable free group cannot be a Polish group. A natural question is: can an uncountable $\aleph _{1}$-free group be a Polish group? A negative answer is given here; indeed, it is proved that an $\aleph _{1}$-free group cannot be a homomorphic image of a Polish group. In fact, a stronger result is proved, involving a non-commutative analogue of the notion of separable group.

In this paper, it is proved that if $B$ is a Brauer $p$-block of a $p$-solvable group, for some odd prime $p$, then the height of any ordinary character in $B$ is at most $2b$, where $p^b$ is the largest degree of the irreducible characters of the defect group of $B$. Some other results that relate the heights of characters with properties of the defect group are obtained.

let $r$ be a ring and $t$ a 1-tilting right $r$-module. then $t$ is of countable type. moreover, $t$ is of finite type in the case where $r$ is a prüfer domain.

Given a set $\pi$ of odd primes and a $\pi$-solvable finite group $G$, one can choose three (not necessarily distinct) Hall $\pi$-subgroups of $G$ whose intersection is $O_{\pi}(G)$. This statement relies on a result concerning long orbits in coprime actions of groups of odd order.

In the Kourovka notebook, Deaconescu asks whether $\gpord{\Aut G}\ge \phi(\gpord{G})$ for all finite groups $G$, where $\phi$ denotes the Euler totient function, and whether $G$ is cyclic whenever $\gpord{\Aut G}= \phi(\gpord{G})$. Both questions are answered in the negative in this paper. Moreover, $\gpord{\Aut G}/ \phi(\gpord{G)$ can be made arbitrarily small.

A proof is given that a permutation group in which different finite sets have different stabilizers cannot satisfy any group law. For locally compact topological groups with this property, almost all finite subsets of the group are shown to generate free subgroups. Consequences of these theorems are derived for: Thompson's group $F$, weakly branch groups, automorphism groups of regular trees, and profinite groups with alternating composition factors of unbounded degree.

It is proved that a ring (≠0) is projective-free precisely when it has invariant basis number and every idempotent matrix not equal to I is non-full.