Infinite families of curves are constructed of genus 2 and 3 over Q whose jacobians have high rank over Q. More precisely, if [Escr ] is an elliptic curve with rank at least r over Q, an infinite family of curves are constructed of genus 2 whose jacobians have rank at least r+4 over Q, and, under certain conditions, an infinite family of curves are constructed of genus 3 whose jacobians have rank at least 2r over Q. On specialisation, a family of curves are obtained of genus 2 whose jacobians have rank at least 27 and a family of curves are obtained of genus 3 whose jacobians have rank at least 26; one of these has rank at least 42.

The relation between compactly generated groups of polynomial growth and FC−-groups is clarified. It follows that L1(G) is symmetric for G compactly generated and of polynomial growth.

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.

Infinite families of curves are constructed of genus 2 and 3 over Q whose jacobians have high rank over Q. More precisely, if [Escr ] is an elliptic curve with rank at least r over Q, an infinite family of curves are constructed of genus 2 whose jacobians have rank at least r+4 over Q, and, under certain conditions, an infinite family of curves are constructed of genus 3 whose jacobians have rank at least 2r over Q. On specialisation, a family of curves are obtained of genus 2 whose jacobians have rank at least 27 and a family of curves are obtained of genus 3 whose jacobians have rank at least 26; one of these has rank at least 42.

Let [gfr ] be a semisimple Lie algebra over an algebraically closed field [ ] of characteristic zero and G be its adjoint group. The notion of a principal nilpotent pair is a double counterpart of the notion of a regular (= principal) nilpotent element in [gfr ]. Roughly speaking, a principal nilpotent pair e = (e1, e2) consists of two commuting elements in [gfr ] that can independently be contracted to the origin and such that their simultaneous centralizer has the minimal possible dimension, that is, rk[gfr ]. The definition and the basic results are due to V. Ginzburg [3]. He showed that the theory of principal nilpotent pairs yields a refinement of well-known results by B. Kostant on regular nilpotent elements in [gfr ] and has interesting applications to representation theory. In particular, he proved that the number of G-orbits of principal nilpotent pairs is finite and gave a classification for [gfr ] = [sfr ][lfr ]([ ]). Trying to achieve a greater generality, Ginzburg also introduced a wider class of distinguished nilpotent pairs and, again, classified them for [sfr ][lfr ]([ ]). (The precise definitions for all notions related to nilpotent pairs are found in §1.)

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

The notion of ‘finite type’ iterated function systems of contractive similitudes is introduced, and a scheme for computing the exact Hausdorff dimension of their attractors in the absence of the open set condition is described. This method extends a previous one by Lalley, and applies not only to the classes of self-similar sets studied by Edgar, Lalley, Rao and Wen, and others, but also to some new classes that are not covered by the previous ones.

The paper studies the dynamics of rational maps with indifferent parabolic points by comparing their dynamical properties with those of their ‘jump transformation’ which is uniformly expanding on a non-compact set with infinite Markov partition. It establishes the spectral properties of a two-variables operator-valued function associated to the jump transformation and exploits their dynamical relevance to allow the analytic properties of the pressure, the escape rate from a neighbourhood of the Julia set and the asymptotic distribution of pre-images to be studied.

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.

Let [gfr ] be a semisimple Lie algebra over an algebraically closed field [ ] of characteristic zero and G be its adjoint group. The notion of a principal nilpotent pair is a double counterpart of the notion of a regular (= principal) nilpotent element in [gfr ]. Roughly speaking, a principal nilpotent pair e = (e1, e2) consists of two commuting elements in [gfr ] that can independently be contracted to the origin and such that their simultaneous centralizer has the minimal possible dimension, that is, rk[gfr ]. The definition and the basic results are due to V. Ginzburg [3]. He showed that the theory of principal nilpotent pairs yields a refinement of well-known results by B. Kostant on regular nilpotent elements in [gfr ] and has interesting applications to representation theory. In particular, he proved that the number of G-orbits of principal nilpotent pairs is finite and gave a classification for [gfr ] = [sfr ][lfr ]([ ]). Trying to achieve a greater generality, Ginzburg also introduced a wider class of distinguished nilpotent pairs and, again, classified them for [sfr ][lfr ]([ ]). (The precise definitions for all notions related to nilpotent pairs are found in §1.)

Let R be a commutative Noetherian ring. Let [Pscr ](R) (respectively, [Iscr ](R)) be the category of all finite R-modules of finite projective (respectively, injective) dimension. Sharp [9] constructed a category equivalence between [Iscr ](R) and [Pscr ](R) for certain Cohen–Macaulay local rings R. Thus many properties about finite modules of finite projective dimension can be connected with those of finite injective dimension through this equivalence.

The existence of positive solutions of a second order differential equation of the form

(formula here)

with the separated boundary conditions: αz(0) − βz′(0) = 0 and γz(1)+δz′(1) = 0 has proved to be important in physics and applied mathematics. For example, the Thomas–Fermi equation, where f = z3/2 and g = t−1/2 (see [12, 13, 24]), so g has a singularity at 0, was developed in studies of atomic structures (see for example, [24]) and atomic calculations [6]. The separated boundary conditions are obtained from the usual Thomas–Fermi boundary conditions by a change of variable and a normalization (see [22, 24]). The generalized Emden–Fowler equation, where f = zp, p > 0 and g is continuous (see [24, 28]) arises in the fields of gas dynamics, nuclear physics, chemically reacting systems [28] and in the study of multipole toroidal plasmas [4]. In most of these applications, the physical interest lies in the existence and uniqueness of positive solutions.

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.

Let R be a commutative Noetherian ring. Let [Pscr ](R) (respectively, [Iscr ](R)) be the category of all finite R-modules of finite projective (respectively, injective) dimension. Sharp [9] constructed a category equivalence between [Iscr ](R) and [Pscr ](R) for certain Cohen–Macaulay local rings R. Thus many properties about finite modules of finite projective dimension can be connected with those of finite injective dimension through this equivalence.

Let X be an infinite dimensional Banach space. The paper proves the non-coincidence of the vector-valued Hardy space Hp([ ], X) with neither the projective nor the injective tensor product of Hp([ ]) and X, for 1 < p < ∞. The same result is proved for some other subspaces of Lp. A characterization is given of when every approximable operator from X into a Banach space of measurable functions [Fscr ](S) is representable by a function F:S → X as x [map ] 〈F(·), x〉. As a consequence the existence is proved of compact operators from X into Hp([ ]) (1 [les ] p < ∞) which are not representable. An analytic Pettis integrable function F:[ ] → X is constructed whose Poisson integral does not converge pointwise.

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

formula here

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.

We consider the Dipper–James q-Schur algebra [Sscr ]q(n, r)k, defined over a field k and with parameter q ≠ 0. An understanding of the representation theory of this algebra is of considerable interest in the representation theory of finite groups of Lie type and quantum groups; see, for example, [6] and [11]. It is known that [Sscr ]q(n, r)k is a semisimple algebra if q is not a root of unity. Much more interesting is the case when [Sscr ]q(n, r)k is not semisimple. Then we have a corresponding decomposition matrix which records the multiplicities of the simple modules in certain ‘standard modules’ (or ‘Weyl modules’). A major unsolved problem is the explicit determination of these decomposition matrices.

The paper calculates the average density of the normalized Hausdorff measure on the fractal set generated by a conformal iterated function system. It equals almost everywhere a positive constant given by a truncated generalized s-energy integral, where s is the corresponding Hausdorff dimension. As a main tool a conditional Gibbs measure is determined. The appendix proves an appropriate extension of Birkhoff's ergodic theorem which is also of independent interest.

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.