A classical space and a nonstandard space are used as models for white noise. The expectation of the product of certain generalized functionals is first proved to be finite. Then the covariance and the correlation between Donsker's delta functions are calculated and given meaning.

The first nontrivial DCE (2-computably enumerable) Turing approximation to the class of computably enumerable degrees is obtained. This depends on the following extension of the splitting theorem for the DCE degrees. For any DCE degree ${\bf a}$ and any computably enumerable degree ${\bf b}$ , if ${\bf b} < {\bf a}$ , then there are DCE degrees ${\bf x_0}, {\bf x_1}$ such that ${\bf b} < {\bf x_0}, {\bf x_1} < {\bf a}$ and ${\bf a} = {\bf x_0} \lor {\bf x_1}$ . The construction is unusual in that it is incompatible with upper cone avoidance.

Throughout this paper, we shall suppose that all algebraic number fields, namely, all algebraic extensions over the rational field ${\bb Q}$ , are contained in the complex field ${\bb C}$ . Let $P$ be the set of all prime numbers. For any algebraic number field $F$ , let $C_F$ denote the ideal class group of $F$ and, writing $F^+$ for the maximal real subfield of $F$ , let $C^-_F$ denote the kernel of the norm map from $C_F$ to the ideal class group of $F^+$ ; for each $l \in P$ , let $C_F(l)$ denote the $l$ -class group of $F$ , that is, the $l$ -primary component of $C_F$ , and let $C^-_F(l)$ denote the $l$ -primary component of $C^-_F$ . Furthermore, for each $l \in P$ , we denote by ${\bb Z}_l$ the ring of $l$ -adic integers.

A complex valued function $g$ , defined on the positive integers, is multiplicative if it satisfies $g(ab) = g(a)g(b)$ whenever the integers $a$ and $b$ are mutually prime.

THEOREM 1. Let $D$ be an integer, $2 \le D \le x, \varepsilon > 0$ . Let $g$ be a multiplicative function with values in the complex unit disc.

There is a character $\chi_1({\rm mod}\, D)$ , real if $g$ is real, such that when $0 < \gamma < 1$ , \[ \sum_{\overset{n \le y}{n\equiv a({\rm mod}\, D)}} g(n)- \frac{1}{\phi(D)} \sum_{\overset{n\le y}{(n,D)=1)}} g(n)-\frac{\overline{\chi_1(a)}}{\phi(D)} \sum_{n\le y} g(n)\chi_1(n) \ll \frac{y}{\phi(D)} \left(\frac{\log D}{\log y}\right)^{1/4-\varepsilon} \] uniformly for $(a, D) = 1, D \le y, x^\gamma \le y \le x$ , the implied constant depending at most upon $\varepsilon, \gamma$ .

It is proved that all modules over an integral domain $R$ have strongly flat cover if and only if every flat $R$ -module is strongly flat. The domains satisfying this property are characterized by the property that all their proper quotients are perfect rings, and are called almost perfect. They are exactly the $h$ -local domains which are locally almost perfect. Various relevant classes of modules admitting or not admitting strongly flat cover are exhibited.

The object of the paper is a thorough analysis of the WP-Bailey tree, a recent extension of classical Bailey chains. The paper begins by observing how the WP-Bailey tree naturally requires a finite number of classical $q$ -hypergeometric transformation formulas. It then shows how to move beyond this closed set of results, and in the process, heretofore mysterious identities of Bressoud are explicated. Next, WP-Bailey pairs are used to provide a new proof of a recent formula of Kirillov. Finally, the relation between the approach in the paper and that of Burge is discussed.

For any pair $i,j\ge 0$ with $i+j=1$ let ${\mathbf Bad}(i,j)$ denote the set of pairs $(\alpha,\beta)\in {\bb R}^2$ for which $\max\{\|q\alpha\|^{1/i}\|q\beta\|^{1/j}\}>c/q$ for all $q\in {\bb N}$ . Here $c=c(\alpha,\beta)$ is a positive constant. If $i=0$ the set ${\mathbf Bad}(0, 1)$ is identified with ${\bb R}\times {\mathbf Bad}$ where ${\mathbf Bad}$ is the set of badly approximable numbers. That is, ${\mathbf Bad}(0, 1)$ consists of pairs $(\alpha, \beta)$ with $\alpha\in {\bb R}$ and $\beta\in {\mathbf Bad}$ If $j=0$ the roles of $\alpha$ and $\beta$ are reversed. It is proved that the set ${\mathbf Bad}(1,0)\cap {\mathbf Bad} (0,1)\cap {\mathbf Bad}(i,j)$ has Hausdorff dimension 2, that is, full dimension. The method easily generalizes to give analogous statements in higher dimensions.

The fact that smooth Schubert varieties in partial flag manifolds are iterated fiber bundles over Grassmannians is used to give a simple presentation for their integral cohomology ring, generalizing Borel's presentation for the cohomology of the partial flag manifold itself. More generally, such a presentation is shown to hold for a larger class of subvarieties of the partial flag manifolds (which are called subvarieties defined by inclusions). The Schubert varieties which lie within this larger class are characterized combinatorially by a pattern avoidance condition.

Let $F$ be an algebraically closed field of characteristic $0$ , and let $A$ be a $G$ -graded algebra over $F$ for some finite abelian group $G$ . Through $G$ being regarded as a group of automorphisms of $A$ , the duality between graded identities and $G$ -identities of $A$ is exploited. In this framework, the space of multilinear $G$ -polynomials is introduced, and the asymptotic behavior of the sequence of $G$ -codimensions of $A$ is studied.

Two characterizations are given of the ideal of $G$ -graded identities of such algebra in the case in which the sequence of $G$ -codimensions is polynomially bounded. While the first gives a list of $G$ -identities satisfied by $A$ , the second is expressed in the language of the representation theory of the wreath product $G \wr S_n$ , where $S_n$ is the symmetric group of degree $n$ .

As a consequence, it is proved that the sequence of $G$ -codimensions of an algebra satisfying a polynomial identity either is polynomially bounded or grows exponentially.

Let $\zeta(s, \alpha)$ be the Hurwitz zeta function with parameter $\alpha$ . Power mean values of the form $\sum^q_{a=1}\zeta(s,a/q)^h$ or $\sum^q_{a=1}|\zeta(s,a/q)|^{2h}$ are studied, where $q$ and $h$ are positive integers. These mean values can be written as linear combinations of $\sum^q_{a=1}\zeta_r(s_1,\ldots,s_r;a/q)$ , where $\zeta_r(s_1,\ldots,s_r;\alpha)$ is a generalization of Euler–Zagier multiple zeta sums. The Mellin–Barnes integral formula is used to prove an asymptotic expansion of $\sum^q_{a=1}\zeta_r(s_1,\ldots,s_r;a/q)$ , with respect to $q$ . Hence a general way of deducing asymptotic expansion formulas for $\sum^q_{a=1}\zeta(s,a/q)^h$ and $\sum^q_{a=1}|\zeta(s,a/q)|^{2h}$ is obtained. In particular, the asymptotic expansion of $\sum^q_{a=1}\zeta(1/2,a/q)^3$ with respect to $q$ is written down.

Let $G$ be a finite nonabelian simple group and let $\Gamma$ be a connected undirected Cayley graph for $G$ . The possible structures for the full automorphism group ${\rm Aut}\Gamma$ are specified. Then, for certain finite simple groups $G$ , a sufficient condition is given under which $G$ is a normal subgroup of ${\rm Aut}\Gamma$ . Finally, as an application of these results, several new half-transitive graphs are constructed. Some of these involve the sporadic simple groups $G = J_1, J_4$ , Ly and BM, while others fall into two infinite families and involve the Ree simple groups and alternating groups. The two infinite families contain examples of half-transitive graphs of arbitrarily large valency.

In 1998, Zalesskii proposed to classify all instances of irreducible characters of quasi- simple groups which are of prime power degree. In joint work, Malle and Zalesskii then dealt with all quasi-simple groups with the exception of the alternating groups and their double covers [5]. In an earlier article [1] we have classified all the irreducible characters of $S_n$ of prime power degree and have deduced from this also the corresponding classification for the alternating groups. In the present article we complete Zalesskii's programme by dealing with the final case left open in [5], the double covers of the alternating groups. We derive this result from a corresponding result on the double covers of the symmetric groups. If one is only interested in spin characters, one easily sees that only 2-powers can occur as prime power degrees. However from a combinatorial point of view it is natural to ask more generally: when is the number of shifted standard tableaux of a given shape a prime power? Since our method is independent of the prime, we will answer this question, showing that apart from a few accidental cases for small $n$ only the ‘obvious’ partitions satisfy the prime power condition. Thus in turn for the spin characters, apart from exceptions for small $n$ , the ‘obvious’ spin characters of 2-power degree are indeed the only ones (this confirms the conjecture stated in [5]).

Let $G$ be a metacyclic group of order $pq$ , where $p$ and $q$ are distinct odd prime numbers, let $N| k$ be a Galois extension whose Galois group $G(N| k)$ is isomorphic to $G$ . Let $R_N, R_k$ be the rings of integers of $N$ and $k$ . As $R_k$ -module $R_N$ is completely determined by $[N:k]$ and by its class in the class group of $R_k$ . The paper determines the classes realized by tame Galois extensions $N|k$ with $G(N|k)\cong G$ and proves that they form a group.

A transitive finite permutation group is called elusive if it contains no nontrivial semiregular subgroup. The purpose of the paper is to collect known information about elusive groups. The main results are recursive constructions of elusive permutation groups, using various product operations and affine group constructions. A brief historical introduction and a survey of known elusive groups are also included. In a sequel, Giudici has determined all the quasiprimitive elusive groups.

Part of the motivation for studying this class of groups was a conjecture due to Marušič, Jordan and Klin asserting that there is no elusive 2-closed permutation group. It is shown that the constructions given will not build counterexamples to this conjecture.

Kummer's incorrect conjectured asymptotic estimate for the size of the first factor of the class number of a cyclotomic field, $h_1(p)$ , is further examined. Whereas Kummer conjectured that $h_1(p) \sim G(p) := 2p(p/4\pi^2)^{(p-1)/4}$ it is shown, under certain plausible assumptions, that there exist constants $a_\alpha, b_\alpha$ such that $h_1(p) \sim \alpha G(p)$ for $\sim a_\alpha x/\log^{b_\alpha} x$ primes $p \le x$ whenever $\log \alpha$ is rational. On the other hand, there are $\ll_A x/\log^A x$ such primes when $\log \alpha$ is irrational. Under a weak assumption it is shown that there are roughly the conjectured number of prime pairs $p, mp\pm 1$ if and only if there are $\gg_m x/\log^2 x$ primes $p \le x$ for which $h_1(p) \sim e^{\pm 1/2m} G(p)$ .

A Coxeter group $W$ is said to be rigid if, given any two Coxeter systems $(W, S)$ and $(W, S^\prime)$ , there is an automorphism $\rho: W \longrightarrow W$ such that $\rho(S) = S^\prime$ . The class of Coxeter systems $(W, S)$ for which the Coxeter graph $\Gamma_S$ is complete and has only odd edge labels is considered. (Such a system is said to be of type $K_n$ .) It is shown that if $W$ has a type $K_n$ system, then any other system for $W$ is also type $K_n$ . Moreover, the multiset of edge labels on $\Gamma_S$ and $\Gamma_{S^\prime}$ agree. In particular, if all but one of the edge labels of $\Gamma_S$ are identical, then $W$ is rigid.

A criterion is given for showing that certain one-relator groups are residually finite. This is applied to a one-relator group with torsion $G = \langle a_1, \ldots, a_r \mid W^n\rangle$ . It is shown that $G$ is residually finite provided that $W$ is outside the commutator subgroup and $n$ is sufficiently large. An important ingredient in the proof is a criterion which implies that a subgroup of a group is malnormal. A graded small-cancellation criterion is developed which detects whether a map $A \rightarrow B$ between graphs induces a $\pi_1$ -injection, and whether $\pi_1 A$ maps to a malnormal subgroup of $\pi_1 B$ .

Let $\Lambda$ be a connected representation finite selfinjective algebra. According to $G$ . Zwara the partial orders $\le_{\rm ext}$ and $\le_{\rm deg}$ on the isomorphism classes of $d$ -dimensional $\Lambda$ -modules are equivalent if and only if the stable Auslander–Reiten quiver $\Gamma_{\Lambda}$ of $\Lambda$ is not isomorphic to ${\bb Z}D_{3m}/\tau^{2m-1}$ for all $m\ge 2$ . The paper describes all minimal degenerations $M\le_{\rm deg} N$ with $M\not\leq_{\rm ext} N$ in the case when $\Gamma_\Lambda\cong {\bb Z}D_{3m}/\tau^{2m-1}$ for some $m\ge 2$ .

A real algebraic, plane, projective curve $A$ of degree $m$ is given by a homogeneous polynomial of degree $m$ in three variables, with real coefficients, defined up to multiplication by a non-zero scalar. If $F$ is such a polynomial defining $A$ , we denote by ${\bb C} A$ and ${\bb R} A$ , respectively, the sets of solutions of the equation $F = 0$ in ${\bb C} P^2$ , respectively ${\bb R} P^2$ . We suppose that the curve $A$ is non-singular, that is, $F$ has no critical points in ${\bb C}^3\backslash 0$ . Then ${\bb C} A$ is a Riemannian surface of genus $g = (m-1)(m-2)/2$ , and ${\bb R} A$ is a collection of $L \le g+1$ circles embedded in ${\bb R} P^2$ . If $L = g+1$ , we say that $A$ is an $M$ -curve. A circle embedded in ${\bb R} P^2$ is called oval, or pseudo-line, depending on whether it realizes the class 0 or 1 of $H_1({\bb R} P^2)$ . If $m$ is even, the $L$ components of ${\bb R} A$ are ovals; if $m$ is odd, ${\bb R} A$ contains exactly one pseudo-line, which will be denoted by ${\cal J}$ . Note that an oval separates ${\bb R} P^2$ into two pieces, a Möbius band and a disc. The latter is called the interior of the oval. An oval of ${\bb R} A$ is said to be empty if its interior contains no other oval of ${\bb R} A$ . Two ovals form an injective pair if one of them lies in the interior of the other one.

A matrix $A$ with minimum polynomial $m_A$ and characteristic polynomial $c_A$ is said to be cyclic if $m_A = c_A$ , semisimple if $m_A$ has no repeated factors, and separable if it is both cyclic and semisimple. For any set $T$ of matrices, we write $C_T$ for the proportion of cyclic matrices in $T, SS_T$ for the proportion of semisimple matrices, and $S_T$ for the proportion of separable matrices. We will write $C_{{\rm GL}(\infty,q)}$ for $\lim_{d\rightarrow\infty}C_{{\rm GL}(d,q)}$ , and so on.