Let $X$ be a closed, symplectic 4-manifold. Suppose that there is either a symplectic or an anti-symplectic involution $\sigma : X\,{\to}\, X$ with a 2-dimensional compact, oriented submanifold $\Sigma$ as a fixed point set.

If $\sigma$ is a symplectic involution then the quotient $X/\sigma$ with $b_2^+(X/\sigma)\,{\ge}\, 1$ is a symplectic 4-manifold.

If $\sigma$ is an anti-symplectic involution and $\Sigma$ has genus greater than 1 representing non-trivial homology class, we prove a vanishing theorem on Seiberg-Witten invariants of the quotient $X/\sigma$ with $b_2^+(X/\sigma)\,{ >}\,1.$

If $\Sigma$ is a torus with self-intersection number 0, we get a relation between the Seiberg-Witten invariants on $X$ and those of $X/\sigma$ with $b_2^+(X), b_2^+(X/\sigma)\,{ >}\,2$ which was obtained in [21] when the genus $g(\Sigma)\,{ >}\,1$ and $\Sigma\cdot\Sigma\,{=}\,0$.This work was supported by a Korea Research Foundation Grant (No KRF-2002-072-C00010).

In this paper, we study Lagrangian submanifolds $M$ of the nearly Kähler 6-sphere $S^6(1)$. It is well known that such submanifolds, which are 3-dimensional, are always minimal and admit a symmetric cubic form. Following an idea of Bryant, developed in the study of Lagrangian submanifolds of $\mathbb C^3$, we then investigate those Lagrangian submanifolds for which at each point the tangent space admits an isometry preserving this cubic form. We obtain that all such Lagrangian submanifolds can be obtained starting from complex curves in $S^6(1)$ or from holomorphic curves in $\mathbb CP^2(4)$. In the final section we classify the Lagrangian submanifolds which admit a Sasakian structure that is compatible with the induced metric. This last result generalizes theorems obtained by Deshmukh and ElHadi. Note that in this case, the condition that $M$ admits a Sasakian structure implies that $M$ admits a pointwise isometry of the tangent space.

Signed digit representations with base $q$ and digits $-\frac q2,\dots,\frac q2$ (and uniqueness being enforced by applying a special rule which decides whether $-q/2$ or $q/2$ should be taken) are considered with respect to counting the occurrences of a given (contiguous) subblock of length $r$. The average number of occurrences amongst the numbers $0,\dots,n-1$ turns out to be const$\cdot\log_qn+\delta(\log_qn)+\smallOh(1)$, with a constant and a periodic function of period one depending on the given subblock; they are explicitly described. Furthermore, we use probabilistic techniques to prove a central limit theorem for the number of occurrences of a given subblock.

We study soluble groups in which every subgroup lying between a characteristic subgroup and its derived subgroup is normal.

It is shown that each finite translation generalized quadrangle (TGQ) $\mathcal{S}$, which is the translation dual of the point-line dual of a flock generalized quadrangle, has a line $[\infty]$ each point of which is a translation point. This leads to the fact that the full group of automorphisms of $\mathcal{S}$ acts $2$-transitively on the points of $[\infty]$, and the observation applies to the point-line duals of the Kantor flock generalized quadrangles, the Roman generalized quadrangles and the recently discovered Penttila-Williams generalized quadrangle. Moreover, by previous work of the author, the non-classical generalized quadrangles (GQ's) which have two distinct translation points, are precisely the TGQ's of which the translation dual is the point-line dual of a non-classical flock GQ.

We emphasize that, for a long time, it has been thought that every non-classical TGQ which is the translation dual of the point-line dual of a flock GQ has only one translation point. There are important consequences for the theory of generalized ovoids (or eggs) in PG$(4n - 1,q)$, the study of span-symmetric generalized quadrangles, derivation of flocks of the quadratic cone in PG$(3,q)$, subtended ovoids in generalized quadrangles, and the understanding of automorphism groups of certain generalized quadrangles. Several problems on these topics will be solved completely.

Given a contact form $\eta$, there is a one-to-one correspondence between the Riemannian structures $(\eta,g)$ and the CR-structures $(\eta,L)$. It is interesting to study the interaction between the two associated structures. We approach the geometry of contact Riemannian manifolds in connection with their associated CR-structures. In this context, for a contact Riemannian manifold $(M;\eta,g)$ we consider the Jacobi-type operator $R_{\dot\gamma}=R(\cdot,\dot\gamma)\dot\gamma$ along a self-parallel curve $\gamma$ with respect to the (generalized) Tanaka connection $\hatbnabla$.This work was financially supported by Chonnam National University in the program, 2001.

For a Banach space $X$ we consider three ways in which a subspace of $X^*$ can represent locally the whole dual space $X^*$. We obtain characterizations in terms of ultrapowers and we study the relationship between the subspaces of $X^*$ and the subspaces of the dual of an ultrapower of $X$.

The main problem investigated in this paper is the following. Assume that we are given a convergent projective system of topological measure spaces ordered by ordinals. When does there exist a consistent system of liftings (densities, linear liftings) on the projective system converging to a lifting (density, linear lifting) on the limit space. We look mainly for strong or strong completion Baire liftings. We reduce the problem to the question about the existence of strong liftings being inverse images of other strong liftings under measure preserving mappings (Proposition 2.4) and then we adapt a condition applied earlier by A. and C. Ionescu Tulcea [14] to get a strong lifting for an arbitrary measure on a product space (Theorem 2.7). In this way we get some results (see Theorems 2.7, 5.3, 5.7, 6.4 and 6.5) extending the well known achievements of A. and C. Ionescu Tulcea [14] and Fremlin [9].

The application of projective limits allows us to carry over results obtained earlier only for product spaces (see e.g. [23], [18], [19], [20], [21]) to more general classes of topological probability spaces. In particular, we can extend the class of spaces for which there is a positive answer to a problem of J. Kupka [17] concerning the permanence of the strong lifting property under the formation of products (see Theorem 6.5).

The automorphism group of a virtually polycyclic group $G$ is either virtually polycyclic or it contains a non-abelian free subgroup. We describe conditions on the structure of $G$ to decide which of the two alternatives occurs for $Aut(G).$

A well-known theorem of P. Hall says that if a group $G$ contains a normal nilpotent subgroup $N$ such that $G/N'$ is nilpotent then $G$ is nilpotent. We give a similar sufficient condition for a group $G$ to be an extension of a group of finite exponent by a nilpotent group.Supported by CNPq-Brazil.

Let $M$ be a subset of $\Bbb R$ with the following two invariance properties: (1) $M+k\subseteq M$ for all integers $k$, and (2) there exists a positive integer $l\ge 2$ such that $\frac{1}{l}M\subseteq M$. (For example, the set of Liouville numbers and the Besicovitch-Eggleston set of non-normal numbers satisfy these conditions.) We prove that if $h$ is a dimension function that is strongly concave at $0$, then the $h$-dimensional Hausdorff measure $\cal H^{h}(M)$ of $M$ equals $0$ or infinity.

We prove some inequalities for the spectral radius of positive operators on Banach function spaces. In particular, we prove the following extension of Levinger's theorem. Let $K$ be a positive compact kernel operator on $L^2(X, \mu)$ with the spectral radius $r(K)$. Then the function $\phi$ defined by $\phi(t) = r(t K + (1-t) K^*)$ is non-decreasing on $[0, \frac{1}{2}]$. We also prove that $\| A + B^* \| \ge 2 \cdot \sqrt{r(A B)}$ for any positive operators $A$ and $B$ on $L^2(X, \mu)$.

For a linearly ordered set $X$ we consider the relative rank of the semigroup of all order preserving mappings $\mathcal{O}_{X}$ on $X$ modulo the full transformation semigroup $\mathcal{T}_{X}$. In other words, we ask what is the smallest cardinality of a set $A$ of mappings such that $\genset{\mathcal{O}_{X}\cup A}=\mathcal{T}_{X}$. When $X$ is countably infinite or well-ordered (of arbitrary cardinality) we show that this number is one, while when $X=\mathbb{R}$ (the set of real numbers) it is uncountable.

A very important misprint occurs in the statement of the main theorem of the paper entitled ‘Weak behaviour of Fourier-Neumann series’ Glasgow Math. J.45 (2003), 97–104. The statement in lines 3–7 of p. 99 should be replaced by the following statement.

Born in London on 14 July 1947, Robert Winston Keith Odoni was the eldest of the five children of Walter Anthony and Lois Marie Theresa Odoni. The family name is Italian: grandfather Alfred Odoni had come to England from Switzerland in the 1920s and set up a manufacturing business in Birmingham, later transferring to London. Robert's father continued the business (then making cycle stands) until the 1970s. His mother was from Boulder in Colorado: they had met in America whilst he was on wartime naval service.