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).

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.

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).

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.

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.

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.