The authors prove in this paper that an involutory Hopf algebra with non-zero integral over a field of characteristic zero is cosemisimple.

Let (R, [mfr ]) be a local Noetherian ring. We show that if R is complete, then an R-module M satisfies local duality if and only if the Bass numbers μi([mfr ], M) are finite for all i. The class of modules with finite Bass numbers includes all finitely generated, all Artinian, and all Matlis reflexive R-modules. If the ring R is not complete, we show by example that modules with finite Bass numbers need not satisfy local duality. We prove that Matlis reflexive modules satisfy local duality in general, where R is any local ring with a dualizing complex.

The author considers globally defined $h$ Fourier integral operators $(h\, {\rm FIO})$ with complex-valued phase functions. Symbolic calculus of $h\, {\rm FIO}$ is considered and, using a new complex Gauss transform, the composition of $h$ pseudodifferential operators $(h\, {\rm PDO})$ and $h\, {\rm FIO}$ is considered. For a self-adjoint $h\, {\rm PDO}\, A(h)$ and $h\, {\rm PDO}\, P(h)$ and $Q(h)$ with compactly supported symbols, the results are applied to approximate the kernel of the operator \[ U_{P,Q}(t;h):=P(h)e^{-ih^{-1}tA(h)}Q(h)^{\ast},\quad t\in {\bb R},\;h>0, \] by a single, globally defined $h$ -oscillatory integral.

It has been known since the last century that a single quadratic form in at least five variables has a nontrivial zero in any p-adic field, but the analogous question for systems of quadratic forms remains unanswered. It is plausible that the number of variables required for solubility of a system of quadratic forms simply is proportional to the number of forms; however, the best result to date, from an elementary argument of Leep [6], is that the number of variables needed is at most a quadratic function of the number of forms. The purpose of this paper is to show how these elementary arguments can be used, in a certain class of fields including the p-adic fields, to refine the upper bound for the number of variables needed to guarantee solubility of systems of quadratic forms. This result partially addresses Problem 6 of Lewis' survey article [7] on Diophantine problems.

A lower and an upper bound are proved for the maximum size of a subset of $\mathbb{Z}/p\mathbb{Z}$ containing no solutions of a fixed linear equation.

John Arthur Todd was one of the last survivors of the school of classical algebraic geometry that flourished in Cambridge under H. F. Baker, FRS. But, unlike most of his contemporaries, with the notable exception of W. V. D. Hodge, FRS, Todd made seminal contributions to the more modern theory, and his name is now enshrined in the literature and widely known.

Let (G, X) be a locally compact transformation group in which G acts freely on X. We show that the associated transformation-group C*-algebra C0(X) [rtimes ] G is a Fell algebra if and only if X is a Cartan G-space.

Let $X$ and $Y$ be affine nonsingular real algebraic varieties. A general problem in real algebraic geometry is to try to decide when a continuous map $f : X \rightarrow Y$ can be approximated by regular maps in the space of ${\cal C}_0$ mappings from $X$ to $Y$ , equipped with the ${\cal C}_0$ topology. This paper solves this problem when $X$ is the connected component containing the origin of the real part of a complex Abelian variety and $Y$ is the standard 2-dimensional sphere.

Let G denote the p-adic group GL(n), and let S(G) denote the Schwartz algebra of G. We construct a Chern character from the K-theory of the reduced C*-algebra of G to the periodic cyclic homology of S(G) which becomes an isomorphism after tensoring over ℤ with [Copf ].

Throughout this paper, X will denote a Banach space, S=S(X) and B=B(X) will be the unit sphere and the closed unit ball of X, respectively, and [Gscr ]=[Gscr ](X) will stand for the group of all surjective linear isometries on X. Unless explicitly stated otherwise, all Banach spaces will be assumed to be real. Nevertheless, by passing to real structures, the results remain true for complex spaces.

This paper presents the rigidity of $\lambda_2$-eigenmaps from $S^{4}$ to itself, and the nonexistence of some $\lambda_2$-eigenmaps, as well as new examples of eigenmaps between spheres.

The descent algebra of the symmetric group, over a field of non-zero characteristic p, is studied. A homomorphism into the algebra of generalised p-modular characters of the symmetric group is defined. This is then used to determine the radical, and its nilpotency index. It also allows the irreducible representations of the descent algebra to be described.

Smale's mean value conjecture asserts that $\min_\theta |P(\theta)/\theta| \le K|P'(0)|$ for every polynomial $P$ of degree $d$ satisfying $P(0)\,{=}\,0$, where $K\,{=}\,(d-1)/d$ and the minimum is taken over all critical points $\theta$ of $P$. A stronger conjecture due to Tischler asserts that \[ \min_\theta\left|\frac12-\frac{P(\theta)}{\theta\cdot P'(0)}\right| \le K_1 \] with $K_1=\frac12-1/d$. Tischler's conjecture is known to be true: (i) for local perturbations of the extremum $P_0(z)=z^d-dz$, and (ii) for all polynomials of degree $d\le 4$. In this paper, Tischler's conjecture is verified for all local perturbations of the extremum $P_1(z)=(z-1)^d-(-1)^d$, but counterexamples to the conjecture are given in each degree $d\ge 5$. In addition, estimates for certain weighted $L^1$- and $L^2$-averages of the quantities $\frac12-{P(\theta)}/{\theta\cdot P'(0)}$ are established, which lead to the best currently known value for $K_1$ in the case $d=5$.

In this paper, Conway's group, $\Co_3$, is characterized among simple groups of even type with $e(G)=3$, by a restriction on the $p$-local structure for some odd prime $p$ for which $m_{2,p}(G) = 3$.

Cappell conjectured the following in ]2]; compare ]11, Problem 2].

Conjecture 0.1. Let π be the fundamental group of a knot complement in [ ]3. Then Lhn(π)≃ Lhn(ℤ), for all n, where Lhnis the Wall–Novikov surgery group for homotopy equivalence, and the isomorphism is induced from the abelianization homomorphism π→ℤ.

An extension is obtained to certain maps into full flag manifolds of compact simple Lie groups of the classical Plücker formulae for holomorphic curves in complex projective space.

The Stöhr–Voloch approach is used to obtain a new bound for the number of solutions in (Fq)2 of an equation f(X, Y) = 0, where f(X, Y) is an absolutely irreducible polynomial with coefficients in a finite field Fq.

We observe that the Bargmann transform arises from the polar decomposition of a simple restriction operator. By virtue of the Bargmann transform, the Wigner transform is imbedded in the geometric symmetry group of phase space. From this imbedding, we easily obtain that the spectrum of the Wigner transform consists of {eikπ/12[ratio ]k=0, 1, …, 23}.

Let M be a geometrically finite pinched negatively curved Riemannian manifold with at least one cusp. The asymptotics of the number of geodesics in M starting from and returning to a given cusp, and of the number of horoballs at parabolic fixed points in the universal cover of M, are studied in this paper. The case of SL(2, ℤ), and of Bianchi groups, is developed.

in two papers, littlewood studied seemingly unrelated constants: (i) the best $\alpha$ such that for any polynomial $f$, of degree $n$, the areal integral of its spherical derivative is at most $\const\cdot n^\alpha$, and (ii) the extremal growth rate $\beta$ of the length of green's equipotentials for simply connected domains. these two constants are shown to coincide, thus greatly improving known estimates on $\alpha$.