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.

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 $\varepsilon, \gamma$ .

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.

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 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.

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.

Let $G$ be a finite group and suppose that $P$ is a soluble $\{2, 3\}^\prime$ -subgroup of $G$ . The reader will lose only a little by assuming that $P$ is a subgroup of prime order $p > 3$ . Define \[ \Sigma_G(P) = \{A \le G \mid A \hbox{ is soluble and } A = \langle P, P^a\rangle\hbox{ for some }a \in A\}. \] This set is partially ordered by inclusion and we let \[ \Sigma^*_G(P) \] denote the set of maximal members of $\Sigma_G(P)$ .

The $n$ th quantum derivative ${\cal D}_nf(x)$ of the real-valued function $f$ is defined for each real non-zero $x$ as \[ \lim_{q\to 1}\frac{\displaystyle\sum\limits^n_{k=0}(-1)^k\left[\begin{array}{c}n\\k\end{array}\right]_qq^{(k-1)k/2}f(q^{n-k}x)}{q^{(n-1)n/2}(q-1)^nx^n}, \] where $\left[\begin{array}{c}n\\k\end{array}\right]_q$ the $q$ -binomial coefficient. If the $n$ th Peano derivative exists at $x$ , which is to say that if $f$ can be approximated by an $n$ th degree polynomial at the point $x$ , then it is not hard to see that ${\cal D}_nf(x)$ must also exist at that point. Consideration of the function $|1-x|$ at $x=1$ shows that the second quantum derivative is more general than the second Peano derivative. However, it can be shown that the existence of the $n$ th quantum derivative at each point of a set necessarily implies the existence of the $n$ th Peano derivative at almost every point of that set.

Let the manifold $X$ parametrise a family of compact complex submanifolds of the complex (or CR) manifold $Z$ . Under mild conditions the Penrose transform typically provides isomorphisms between a cohomology group of a holomorphic vector bundle $V\longrightarrow Z$ and the kernel of a differential operator between sections of vector bundles over $X$ . When the spaces in question are homogeneous for a group $G$ the Penrose transform provides an intertwining operator between representations.

The paper develops a Penrose transform for compactly supported cohomology on $Z$ . It provides a number of examples where a compactly supported cohomology group is shown to be isomorphic to the cokernel of a differential operator between compactly supported sections of vector bundles over $X$ . It considers also how the Serre duality pairing carries through the transform.

Through the use of the so-called variational Lyapunov method, which is developed by combining the method of variation of parameters and the Lyapunov second method, stability and instability properties in terms of two measures for impulsive differential equations with variable moments of impulsive effects are discussed. Some stability and instability criteria are established. These theorems, together with an example, show that perturbation and impulsive effects may make a stable system uniformly asymptotically stable or unstable. These results significantly generalize the known ones.

A systematic treatment is given of several classes of singular integrals. Their $L^p$ boundedness is proved when their kernels are given by functions $\Omega$ in $L\log L({\bf S}^{n-1})$ .

Maximal accretive realizations and bound-preserving self-adjoint extensions are two fundamental problems in applications of semi-bounded operator theory to differential equations. On the basis of using differential operator theory in direct sum spaces and Phillips theory for maximal accretive extensions of accretive operators, a complete characterization of the set of maximal accretive boundary conditions for Sturm–Liouville differential operators is presented. As an application, all possible forms of bound-preserving self-adjoint extensions of regular Sturm–Liouville operators are also characterized via various explicit boundary conditions. The methodology can also be applied to dealing with general classes of semi-bounded symmetric differential operators.

A family of Hecke $C^*$ -algebras can be realised as crossed products by semigroups of endomorphisms. It is shown by dilating representations of the semigroup crossed product that the category of representations of the Hecke algebra is equivalent to the category of continuous unitary representations of a totally disconnected locally compact group.

In this paper we shall prove that the measure algebra $M(G)$ of a locally compact group $G$ is amenable as a Banach algebra if and only if $G$ is discrete and amenable as a group. Our contribution is to resolve a conjecture by proving that $M(G)$ is not amenable in the case where the group $G$ is not discrete. Indeed, we shall prove a much stronger result: the measure algebra of a non-discrete, locally compact group has a non-zero, continuous point derivation at a certain character on the algebra.

The paper studies the dual algebras of localization Roe algebras over proper metric spaces and develops a localization version of Paschke duality for $K$ -homology. It is shown that the localization $K$ -homology groups are isomorphic to Kasparov's $K$ -homology groups for the Rips complex of proper metric spaces with bounded geometry. It follows that the obstruction groups to the coarse Baum–Connes conjecture can also be derived from the dual localization algebras.

Iterative algorithms for nonexpansive mappings and maximal monotone operators are investigated. Strong convergence theorems are proved for nonexpansive mappings, including an improvement of a result of Lions. A modification of Rockafellar’s proximal point algorithm is obtained and proved to be always strongly convergent. The ideas of these algorithms are applied to solve a quadratic minimization problem.