The paper considers the possible cardinalities of maximal cofinitary groups and the cofinalities of permutation groups on the set of natural numbers. These two cardinals are compared with some other well-known cardinal invariants which are related to covering properties in several forcing models. An application of combinatorial group theory to a construction of a forcing partially ordered set is presented.

It is proved that a connected infinite graph has a normal spanning tree (the infinite analogue of a depth-first search tree) if and only if it has no minor obtained canonically from either an (ℵ0, ℵ1)-regular bipartite graph or an order-theoretic Aronszajn tree. This disproves Halin's conjecture that only the first of these obstructions was needed to characterize the graphs with normal spanning trees. As a corollary Halin's further conjecture is deduced, that a connected graph has a normal spanning tree if and only if all its minors have countable colouring number.

The precise classification of the (ℵ0, ℵ1)-regular bipartite graphs remains an open problem. One such class turns out to contain obvious infinite minor-antichains, so as an unexpected corollary Thomas's result that the infinite graphs are not well-quasi-ordered as minors is reobtained.

Asymptotic formulae for the number of rational points of bounded height on flag varieties have earlier been established. In the paper these asymptotic formulae are recovered by a new method for varieties in biprojective space defined over ℚ that are isomorphic to the flag variety of lines in hyperplanes.

The result is obtained by an application of Heath-Brown's new form of the circle method. It serves as a pointer to the investigation of rational points of bounded height on varieties in multiprojective space.

The j-invariants of the quadratic Q-curves without complex multiplication are studied. Some properties of the norms of these invariants are shown and a relationship between the field Q(j) and the degree of an isogeny of the Q-curve to its Galois conjugate is found. In the case when the degree of the isogeny is a prime p, some properties of the primes of potentially multiplicative reduction for the Q-curve and of the reduction of j modulo a prime [Pfr ] in Q(j) over p when the Q-curve has potentially good reduction at [Pfr ] are found.

If G is a finite group, then the usual version of the Green correspondence applies to finitely generated kG-modules when k is a field of characteristic p > 0 or a p-adic ring. The paper presents a categorical version of the Green correspondence and a version of the Burry–Carlson–Puig theorem that remain valid for arbitrary modules and coefficient rings. In this generality, however, it is not clear whether the Green correspondent of a finitely generated module is always finitely generated.

Let G be a non-compact, locally compact group. The minimal ideals of the group algebra L1(G), the measure algebra M(G), and other Banach algebras (usually larger than L1(G) and M(G)) such as the second dual, L1(G)**, of L1(G) with an Arens product, or LUC(G)* with an Arens-type product, are studied in the paper. Using integrable representations, which exist on some semisimple Lie groups, it is seen that the minimal left ideals can be of infinite dimension, and that the compactness of G is not necessary for these ideals to exist in L1(G) and M(G). It is shown also that, although the coefficients of integrable representations are minimal idempotents in LUC(G)* and L1(G)** they do not generate minimal right ideals in these algebras, and that for a large class of groups, they do not generate minimal left ideals either.

The paper considers the iterated function systems of similitudes which satisfy a separation condition weaker than the open set condition, in that it allows overlaps in the iteration. Such systems include the well-known Bernoulli convolutions associated with the PV numbers, and the contractive similitudes associated with integral matrices. The latter appears frequently in wavelet analysis and the theory of tilings. One of the basic questions is studied: the absolute continuity and singularity of the self-similar measures generated by such systems. Various conditions to determine the dichotomy are given.

Commuting maps on a class of algebras called triangular algebras are investigated. In particular, sufficient conditions are given such that every commuting map L on such an algebra is of the form L(a) = ax+h(a), where x lies in the center of the algebra and h is a linear map from the algebra to its center.

A general approach is proposed to prove that the combination of expansion with bounded distortion yields strong rigidity of conjugacies.

With probabilistic techniques, it is proved that an inner function in the little Bloch space can have every asymptotic radial behaviour on sets of Hausdorff dimension 1. This is applied to deduce some differentiability properties of positive singular smooth measures and a result on radial cluster set interpolation for inner functions.

Two classes of vector-valued BMOA spaces are defined, in the complex ball and on the complex sphere, respectively. In the case of the complex sphere, vector measures are involved, since the argument in the scalar setting is not appropriate. Several properties (the Lp-equivalent norm theorem, exponential decay, the Baernstein theorem, and so on) of BMOA in the complex ball are extended to the Banach space setting. The two classes of BMOA spaces are proved to be isomorphic; in particular, the corresponding John–Nirenberg exponential decay is shown. Finally, the vector-valued H1-BMOA duality theorem is proved.

The equality of the Milnor number and Tjurina number for functions on space curve singularities, as conjectured recently by V. Goryunov, is proved. As a consequence, the discriminant in such a situation is a free divisor.

First, two comparison theorems are established between the first order delay differential equation

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and two related second order ordinary differential equations in the case when

formula here

Next, by using these comparison theorems some new oscillation and non-oscillation criteria are given for (*) under condition (**). As a consequence, an open problem by Elbert and Stavroulakis is completely solved.

The Schwartz space of rapidly decaying test functions is characterized by the decay of the short-time Fourier transform or cross-Wigner distribution. Then a version of Hardy's theorem is proved for the short-time Fourier transform and for the Wigner distribution.

Let A be a Banach algebra, and let E be a Banach A-bimodule. A linear map S:A→E is intertwining if the bilinear map

formula here

is continuous, and a linear map D[ratio ]A→E is a derivation if δ1D=0, so that a derivation is an intertwining map. Derivations from A to E are not necessarily continuous.

The purpose of the present paper is to prove that the continuity of all intertwining maps from a Banach algebra A into each Banach A-bimodule follows from the fact that all derivations from A into each such bimodule are continuous; this resolves a question left open in [1, p. 36]. Indeed, we prove a somewhat stronger result involving left- (or right-) intertwining maps.

The classification of finite groups acting freely on a homology 3-sphere is still incomplete. However it is known that these groups have periodic cohomology and groups with periodic cohomology are classified by the Suzuki–Zassenhaus theorem. The paper considers, more generally, finite groups which may act on a homology 3-sphere possibly with fixed points and the natural generalization of the Suzuki–Zassenhaus theorem to this case is found. According to the Suzuki–Zassenhaus theorem a group with periodic cohomology is either solvable or has a unique composition factor which is not cyclic, isomorphic to PSL(2, q) for q an odd prime. The main point of the work is that this fact is still true in the more general situation.

The paper studies the connective complex K-theoretic Euler class of a certain bundle associated to a Euclidean immersion of the lens space L(2k)2n+1> to show that this manifold cannot be immersed in ℝ4n−2α(n) if k [ges ] α(n). The non-immersion is best possible in many cases. This suggests a close relationship between the immersion problem for complex projective spaces and that for ‘high’ 2-torsion lens spaces.