The authors sharpen a result of Baker and Harman (1995), showing that [x, x + x0.525] contains prime numbers for large x. An important step in the proof is the application of a theorem of Watt (1995) on a mean value containing the fourth power of the zeta function. 2000 Mathematical Subject Classification: 11N05.

We study finitely generated modules $M$ over a ring $R$, noetherian on both sides. If $M$ has finite Gorenstein dimension $\mbox{G-dim}_RM$ in the sense of Auslander and Bridger, then it determines two other cohomology theories besides the one given by the absolute cohomology functors ${\rm Ext}^n_R(M,\ )$. Relative cohomology functors ${\rm Ext}^n_{\mathcal G}(M,\ )$ are defined for all non-negative integers $n$; they treat the modules of Gorenstein dimension $0$ as projectives and vanish for $n > \mbox{G-dim}_RM$. Tate cohomology functors $\widehat{\rm Ext}^n_R(M,\ )$ are defined for all integers $n$; all groups $\widehat{\rm Ext}^n_R(M,N)$ vanish if $M$ or $N$ has finite projective dimension. Comparison morphisms $\varepsilon_{\mathcal G}^n \colon {\rm Ext}^n_{\mathcal G}(M,\ ) \to {\rm Ext}^n_R(M,\ )$ and $\varepsilon_R^n \colon {\rm Ext}^n_R(M,\ ) \to \widehat{\rm Ext}^n_R(M,\ )$ link these functors. We give a self-contained treatment of modules of finite G-dimension, establish basic properties of relative and Tate cohomology, and embed the comparison morphisms into a canonical long exact sequence $0 \to {\rm Ext}^1_{\mathcal G}(M,\ ) \to \cdots \to {\rm Ext}^n_{\mathcal G}(M,\ ) \to {\rm Ext}^n_R(M,\ ) \to \widehat{\rm Ext}^n_R(M,\ ) \to {\rm Ext}^{n+1}_{\mathcal G}(M,\ ) \to \cdots$. We show that these results provide efficient tools for computing old and new numerical invariants of modules over commutative local rings.

2000 Mathematical Subject Classification: 16E05, 13H10, 18G25.

We give a new heuristic for all of the main terms in the integral moments of various families of primitive $L$-functions. The results agree with previous conjectures for the leading order terms. Our conjectures also have an almost identical form to exact expressions for the corresponding moments of the characteristic polynomials of either unitary, orthogonal, or symplectic matrices, where the moments are defined by the appropriate group averages. This lends support to the idea that arithmetical $L$-functions have a spectral interpretation, and that their value distributions can be modelled using Random Matrix Theory. Numerical examples show good agreement with our conjectures.

The Gauss circle problem and the Dirichlet divisor problem are special cases of the problem of counting the points of the integer lattice in a planar domain bounded by a piecewise smooth curve. In the Bombieri–Iwaniec–Mozzochi exponential sums method we must count the number of pairs of arcs of the boundary curve which can be brought into coincidence by an automorphism of the integer lattice. These coincidences are parametrised by integer points close to certain plane curves, the resonance curves.

This paper sets up an iteration step from a strong hypothesis about integer points close to curves to a bound for the discrepancy, the number of integer points minus the area, as in the latest work on single exponential sums. The Bombieri–Iwaniec–Mozzochi method itself gives bounds for the number of integer points close to a curve in part of the required range, and it can in principle be used iteratively. We use a bound obtained by Swinnerton-Dyer's approximation determinant method. In the discrepancy estimate $O(R^K (\log R)^{\Lambda })$ in terms of the maximum radius of curvature $R$, we reduce $K$ from 2/3 (classical) and 46/73 (paper II in this series) to 131/208. The corresponding exponent in the Dirichlet divisor problem becomes $K/2 = 131/416$.

Working in the category $\mathcal{T}$ of based spaces, we give the basic theory of diagram spaces and diagram spectra. These are functors $\mathcal{D}\longrightarrow \mathcal{T}$ for a suitable small topological category $\mathcal{D}$. When $\mathcal{D}$ is symmetric monoidal, there is a smash product that gives the category of $\mathcal{D}$-spaces a symmetric monoidal structure. Examples include \begin{enumerate} \item[] prespectra, as defined classically, \item[] symmetric spectra, as defined by Jeff Smith, \item[] orthogonal spectra, a coordinate-free analogue of symmetric spectra with symmetric groups replaced by orthogonal groups in the domain category, \item[] $\Gamma$-spaces, as defined by Graeme Segal, \item[] $\mathcal{W}$-spaces, an analogue of $\Gamma$-spaces with finite sets replaced by finite CW complexes in the domain category. \end{enumerate} We construct and compare model structures on these categories. With the caveat that $\Gamma$-spaces are always connective, these categories, and their simplicial analogues, are Quillen equivalent and their associated homotopy categories are equivalent to the classical stable homotopy category. Monoids in these categories are (strict) ring spectra. Often the subcategories of ring spectra, module spectra over a ring spectrum, and commutative ring spectra are also model categories. When this holds, the respective categories of ring and module spectra are Quillen equivalent and thus have equivalent homotopy categories. This allows interchangeable use of these categories in applications.

2000 Mathematics Subject Classification: primary 55P42; secondary 18A25, 18E30, 55U35.

When $m$ is odd, spreads in an orthogonal vector space of type $\Omega^+ (2m+2,2)$ are related to binary Kerdock codes and extremal line-sets in $\RR^{2^{m+1}}$ with prescribed angles. Spreads in a $2m$-dimensional binary symplectic vector space are related to Kerdock codes over $\ZZ_4$ and extremal line-sets in $\CC^{2^m}$ with prescribed angles. These connections involve binary, real and complex geometry associated with extraspecial 2-groups. A geometric map from symplectic to orthogonal spreads is shown to induce the Gray map from a corresponding $\ZZ_4$-Kerdock code to its binary image. These geometric considerations lead to the construction, for any odd composite $m$, of large numbers of $\ZZ_4$-Kerdock codes. They also produce new $\ZZ_4$-linear Kerdock and Preparata codes.

1991 Mathematics Subject Classification: primary 94B60; secondary 51M15, 20C99.

In recent years the theory of structured ring spectra (formerly known as A$_{\infty}$- and E$_{\infty}$-ring spectra) has been simplified by the discovery of categories of spectra with strictly associative and commutative smash products. Now a ring spectrum can simply be defined as a monoid with respect to the smash product in one of these new categories of spectra. In this paper we provide a general method for constructing model category structures for categories of ring, algebra, and module spectra. This provides the necessary input for obtaining model categories of symmetric ring spectra, functors with smash product, $\Gamma$-rings, and diagram ring spectra. Algebraic examples to which our methods apply include the stable module category over the group algebra of a finite group and unbounded chain complexes over a differential graded algebra. 1991 Mathematics Subject Classification: primary 55U35; secondary 18D10.

This paper follows the program of study initiated by S. Fomin and A. Zelevinsky, and demonstrates that the homogeneous coordinate ring of the Grassmannian $\mathbb{G}(k, n)$ is a {\it cluster algebra of geometric type}. Those Grassmannians that are of {\it finite cluster type} are identified and their cluster variables are interpreted geometrically in terms of configurations of points in $\mathbb{C}\mathbb{P}^2$.

It is known that a number of algebraic properties of the braid groups extend to arbitrary finite Coxeter-type Artin groups. Here we show how to extend the results to more general groups that we call Garside groups.

Define a Gaussian monoid to be a finitely generated cancellative monoid where the expressions of a given element have bounded lengths, and where left and right lowest common multiples exist. A Garside monoid is a Gaussian monoid in which the left and right lowest common multiples satisfy an additional symmetry condition. A Gaussian group is the group of fractions of a Gaussian monoid, and a Garside group is the group of fractions of a Garside monoid. Braid groups and, more generally, finite Coxeter-type Artin groups are Garside groups. We determine algorithmic criteria in terms of presentations for recognizing Gaussian and Garside monoids and groups, and exhibit infinite families of such groups. We describe simple algorithms that solve the word problem in a Gaussian group, show that these algorithms have a quadratic complexity if the group is a Garside group, and prove that Garside groups have quadratic isoperimetric inequalities. We construct normal forms for Gaussian groups, and prove that, in the case of a Garside group, the language of normal forms is regular, symmetric, and geodesic, has the 5-fellow traveller property, and has the uniqueness property. This shows in particular that Garside groups are geodesically fully biautomatic. Finally, we consider an automorphism of a finite Coxeter-type Artin group derived from an automorphism of its defining Coxeter graph, and prove that the subgroup of elements fixed by this automorphism is also a finite Coxeter-type Artin group that can be explicitly determined.

1991 Mathematics Subject Classification: primary 20F05, 20F36; secondary 20B40, 20M05.

Here, $F$ denotes a non-Archimedean local field (with finite residue field) and $G$ the group of $F$-points of a connected reductive algebraic group defined over $F$. Let $\frak R(G)$ denote the category of smooth, complex representations of $G$. Let ${\cal B}(G)$ be the set of pairs $(L,\sigma)$, where $L$ is an $F$-Levi subgroup of $G$ and $\sigma$ is an irreducible supercuspidal representation of $L$, taken modulo the equivalence relation generated by twisting with unramified quasi characters and $G$-conjugacy. To $\frak s\in {\cal B}(G)$, one can attach a full (abelian) sub-category $\frak R^\frak s(G)$ of $\frak R(G)$; the theory of the Bernstein centre shows that $\frak R(G)$ is the direct product of these $\frak R^\frak s(G)$. The object of the paper is to give a general method for describing these factor categories via representations of compact open subgroups within a uniform framework. Fix $\frak s\in {\cal B}(G)$. Let $K$ be a compact open subgroup of $G$ and $\rho$ an irreducible smooth representation of $K$. The pair $(K,\rho)$ is an $\frak s$-type if it has the following property: an irreducible representation $\pi$ of $G$ contains $\rho$ if and only if $\pi\in \frak R^\frak s(G)$. Let ${\cal H}(G,\rho)$ be the Hecke algebra of compactly supported $\rho$-spherical functions on $G$; if $(K,\rho)$ is an $\frak s$-type, then the category $\frak R^\frak s(G)$ is canonically equivalent to the category ${\cal H}(G,\rho) \text{-Mod}$ of ${\cal H}(G,\rho)$-modules. Let $M$ be a Levi subgroup of $G$; there is a canonical map ${\cal B}(M)\to {\cal B}(G)$. Take $\frak t\in {\cal B}(M)$ with image $\frak s\in {\cal B}(G)$. The choice of a parabolic subgroup of $G$ with Levi component $M$ gives functors of {\it parabolic induction\/} and {\it Jacquet restriction\/} connecting $\frak R^\frak t(M)$ with $\frak R^\frak s(G)$. We assume given a $\frak t$-type $(K_M,\rho_M)$ in $M$; the paper concerns a general method of constructing from this data an $\frak s$-type $(K,\rho)$ in $G$. One thus obtains a description of these induction and restriction functors in terms of an injective ring homomorphism ${\cal H}(M,\rho_M) \to {\cal H}(G,\rho)$. The method applies in a wide variety of cases, and subsumes much previous work. Under further conditions, observed in certain particularly interesting cases, one can go some distance to describing ${\cal H}(G,\rho)$ explicitly. This enables one to isolate cases in which the map on Hecke algebras is an isomorphism, and this in turn implies powerful intertwining theorems for the types.

1991 Mathematics Subject Classification: 22E50.

It is shown that the classical John ellipsoid, the Petty ellipsoid and a recently discovered ‘dual’ of the Legendre ellipsoid are all special cases ($p = \infty$, $p = 1$ and $p = 2$) of a family of $L_p$ ellipsoids which can be associated with a fixed convex body. This insight allows for a unified view of, alternative approaches to, and extensions of some basic results in convex geometry.

We prove that the $p$-Laplacian problem $-\Delta_p u = f(x, u)$, with $u \in W^{1, p}_0 (\Omega)$ on a bounded domain $\Omega \subset R^N$, with $p > 1$ arbitrary, has a nodal solution provided that $f : \Omega \times R \to R$ is subcritical, and $f(x, t) / |t|^{p - 2}$ is superlinear. Infinitely many nodal solutions are obtained if, in addition, $f(x, -t) = -f(x, t)$.

Asymptotic expansions are obtained for finite-dimensional symmetric stable distributions. They are used to prove the existence of continuous transition probability densities of stable and stable-like jump-diffusions, and to construct local multiplicative asymptotics and global two-sided estimates for these densities. As a consequence, the distribution of the first passage times for stable jump-diffusions is estimated and the integral test for the limsup behaviour of their sample paths as $t\to 0$ is provided. 1991 Mathematics Subject Classification: 60E07, 60G17, 60J35, 47D07.

A free semigroup algebra is the weak operator topology closed algebra generated by a set of isometries with pairwise orthogonal ranges. The most important example is the left regular free semigroup algebra generated by the left regular representation of the free semigroup on $n$ generators. This algebra is the appropriate non-commutative $n$-dimensional analogue of the analytic Toeplitz algebra. We develop a detailed picture of the invariant subspace structure analogous to Beurling's theorem and show that this algebra is hyper-reflexive with distance constant at most 51.

The free semigroup algebras, known as atomic, for which the range projections of words in the generators lie in an atomic masa are completely classified. This provides a complete classification for a large class of representations of the Cuntz C*-algebras $\mathcal{O}_n$. This allows us to describe completely the invariant subspace structure of these algebras, and thereby show that these algebras are all hyper-reflexive.

For a braided tensor category ${\cal C}$ and a subcategory ${\cal K}$ there is a notion of a centralizer $C_{\cal C}({\cal K})$, which is a full tensor subcategory of ${\cal C}$. A pre-modular tensor category is known to be modular in the sense of Turaev if and only if the center ${\cal Z}_2({\cal C})\equiv C_{\cal C}({\cal C})$ (not to be confused with the center ${\cal Z}_1$ of a tensor category, related to the quantum double) is trivial, that is, consists only of multiples of the tensor unit, and $\dim{\cal C}\ne 0$. Here $\dim{\cal C} = \sum_i d(X_i)^2$, the $X_i$ being the simple objects.

We prove several structural properties of modular categories. Our main technical tool is the following double centralizer theorem. Let ${\cal C}$ be a modular category and ${\cal K}$ a full tensor subcategory closed with respect to direct sums, subobjects and duals. Then $C_{\cal C}(C_{\cal C}({\cal K})) = {\cal K}$ and $\dim{\cal K}\cdot\dim C_{\cal C}({\cal K}) = \dim{\cal C}$.

We give several applications.

(1) If ${\cal C}$ is modular and ${\cal K}$ is a full modular subcategory, then ${\cal L}=C_{\cal C}({\cal K})$ is also modular and ${\cal C}$ is equivalent as a ribbon category to the direct product: ${\cal C}\simeq{\cal K}\boxtimes{\cal L}$. Thus every modular category factorizes (non-uniquely, in general) into prime modular categories. We study the prime factorizations of the categories $D(G)$-Mod, where $G$ is a finite abelian group.

(2) If ${\cal C}$ is a modular $*$-category and ${\cal K}$ is a full tensor subcategory then $\dim{\cal C}\ge\dim{\cal K}\cdot\dim {\cal Z}_2({\cal K})$. We give examples where the bound is attained and conjecture that every pre-modular ${\cal K}$ can be embedded fully into a modular category ${\cal C}$ with $\dim{\cal C}=\dim{\cal K}\cdot\dim {\cal Z}_2({\cal K})$.

(3) For every finite group $G$ there is a braided tensor $*$-category ${\cal C}$ such that ${\cal Z}_2({\cal C})\simeq\mbox{Rep}\,G$ and the modular closure/modularization $\overline{\cal C}$ is non-trivial.

The paper studies convex bodies and star bodies in $\Bbb R^n$ by using Radon transforms on Grassmann manifolds, $p$-cosine transforms on the unit sphere, and convolutions on the rotation group of $\Bbb R^n$. It presents dual mixed volume characterizations of $i$-intersection bodies and $L_p$-balls which are related to certain volume inequalities for cross sections of convex bodies. It considers approximations of special convex bodies by analytic bodies and various finite sums of ellipsoids which preserve special geometric properties. Convolution techniques are used to derive formulas for mixed volumes, mixed surface measures, and $p$-cosine transforms. They are also used to prove characterizations of geometric functionals, such as surface area and dual quermassintegrals.

This paper deals with non-constant positive steady-state solutions of a predator-prey system with non-monotonic functional response, also called Holling type-IV interaction terms, and diffusion under the homogeneous Neumann boundary condition. We first establish positive upper and lower bounds for such solutions, and then study their non-existence, global existence and bifurcation.

Bohr's inequality says that if $f(z) = \sum^{\infty}_{n = 0} a_n z^n$ is a bounded analytic function on the closed unit disc, then $\sum^{\infty}_{n = 0} \lvert a_n\rvert r^n \leq \Vert f\Vert_{\infty}$ for $0 \leq r \leq 1/3$ and that $1/3$ is sharp. In this paper we give an operator-theoretic proof of Bohr's inequality that is based on von Neumann's inequality. Since our proof is operator-theoretic, our methods extend to several complex variables and to non-commutative situations.

We obtain Bohr type inequalities for the algebras of bounded analytic functions and the multiplier algebras of reproducing kernel Hilbert spaces on various higher-dimensional domains, for the non-commutative disc algebra ${\mathcal A}_n$, and for the reduced (respectively full) group C*-algebra of the free group on $n$ generators.

We also include an application to Banach algebras. We prove that every Banach algebra has an equivalent norm in which it satisfies a non-unital version of von Neumann's inequality.

2000 Mathematical Subject Classification: 47A20, 47A56.

The local structure of homomorphisms of commutative noetherian rings is investigated from the point of view of dualizing complexes. A concept of finite Gorenstein dimension, which substantially weakens the notion of finite flat dimension, is introduced for homomorphisms. It is shown to impose structural constraints, due to a remarkable equivalence of subcategories of the derived category of all modules.

An essential part of this study is the development of relative notions of dualizing complexes and Bass numbers. It is proved that the Bass numbers of local homomorphisms are rigid, extending a known result for local rings. Quasi-Gorenstein homomorphisms are introduced as local homomorphisms that base-change a dualizing complex for the source ring into one for the target. They are shown to have the stability properties of the Gorenstein homomorphism that they generalize.

1991 Mathematics Subject Classification: primary 13H10, 13D23, 14E40; secondary 13C15.

We introduce the notion of tracial topological rank for ${\rm C}^*$-algebras. In the commutative case, this notion coincides with the covering dimension. Inductive limits of ${\rm C}^*$-algebras of the form $PM_n(C(X))P$, where $X$ is a compact metric space with ${\rm dim\,} X\le k$, and $P$ is a projection in $M_n(C(X))$, have tracial topological rank no more than $k$. Non-nuclear ${\rm C}^*$-algebras can have small tracial topological rank. It is shown that if $A$ is a simple unital ${\rm C}^*$-algebra with tracial topological rank $k$ ($<\infty$), then \begin{enumerate} \item[(i)] $A$ is quasidiagonal, \item[(ii)] $A$ has stable rank $1$, \item[(iii)] $A$ has weakly unperforated $K_0(A)$, \item[(iv)] $A$ has the following Fundamental Comparability of Blackadar: if $p,q\in A$ are two projections with $\tau(p)<\tau(q)$ for all tracial states $\tau$ on $A$, then $p\preceq q$. 2000 Mathematics Subject Classification: 46L05, 46L35.