The paper defines a new type of product of association schemes (and of the related objects, permutation groups and orthogonal block structures), which generalizes the direct and wreath products (which are referred to as ‘crossing’ and ‘nesting’ in the statistical literature). Given two association schemes $\mathcal{Q}_r$ for $r\,{=}\,1,2$, each having an inherent partition $F_r$ (that is, a partition whose equivalence relation is a union of adjacency relations in the association scheme), a product of the two schemes is defined, which reduces to the direct product if $F_1=U_1$ or $F_2=E_2$, and to the wreath product if $F_1 = E_1$ and $F_2 = U_2$, where $E_r$ and $U_r$ are the relation of equality and the universal relation on $\mathcal{Q}_r$. The character table of the crested product is calculated, and it is shown that, if the two schemes $\mathcal{Q}_1$ and $\mathcal{Q}_2$ have formal duals, then so does their crested product (and a simple description of this dual is given). An analogous definition for permutation groups with intransitive normal subgroups is created, and it is shown that the constructions for association schemes and permutation groups are related in a natural way.

The definition can be generalized to association schemes with families of inherent partitions, or permutation groups with families of intransitive normal subgroups. This time the correspondence is not so straightforward, and it works as expected only if the inherent partitions (or orbit partitions) form a distributive lattice.

The paper concludes with some open problems.

It is proved that all correlations of the sequence of Farey fractions exist and explicit formulas are provided for the correlation measures.

Explicit bounds are given for the residues at $s\,{=}\,1$ of the Dedekind zeta functions of number fields. As a consequence, a simple proof of the Brauer-Siegel theorem and explicit lower bounds for class numbers of number fields are obtained. Compared with Stark's original approach, the paper is explicit and more satisfactory for number fields containing quadratic subfields. Examples are given of fully explicit lower bounds for class numbers of various types of number fields, for example normal and non-normal number fields of odd degree, with an emphasis on cubic fields, real cyclic quartic number fields, and non-normal quartic number fields containing an imaginary quadratic subfield.

Finitely generated modules with finite $F$-representation type over Noetherian (local) rings of prime characteristic $p$ are studied. If a ring $R$ has finite $F$-representation type or, more generally, if a faithful $R$-module has finite $F$-representation type, then tight closure commutes with localizations over $R$. $F$-contributors are also defined, and they are used as an effective way of characterizing tight closure. Then it is shown that $\lim_{e \to \infty} ({\#(\e M,M_i)}/ {(ap^d)^e})$ always exists under the assumption that $(R,\m)$ satisfies the Krull–Schmidt condition and $M$ has finite $F$-representation type by $\{ M_1, M_2, \dots, M_s \}$, in which all the $M_i$ are indecomposable $R$-modules that belong to distinct isomorphism classes and $a=[R/\m:(R/\m)^p]$.

The defining ideal $I_{\bb X}$ of a set of points $\bb{X}$ in $\bb{P}^n_1 x \ldots x \bb{P}n_k$ is investigated with a special emphasis on the case when $\bb{X}$ is in generic position, that is, $\bb{X}$ has the maximal Hilbert function. When $\bb{X}$ is in generic position, the degrees of the generators of the associated ideal $I_{\bb X}$ are determined. $\nu(I_{\bb X})$ denotes the minimal number of generators of $I_{\bb X}$, and this description of the degrees is used to construct a function $v(s;n_1,\ldots,n_k)$ with the property that $\nu(I_{\bb X})\,{\geq}\, v(s;n_1,\ldots,n_k)$ always holds for $s$ points in generic position in $\bb{P}n_1 x \ldots x \bb{P}^n_k$. When $k\,{=}\,1$, $v(s;n_1)$ equals the expected value for $\nu(I_{\bb X})$ as predicted by the ideal generation conjecture. If $k\,{\geq}\, 2$, it is shown that there are cases with $\nu(I_{\bb X}) > v(s;n_1,\ldots,n_k$). However, computational evidence suggests that in many cases $\nu(I_{\bb X})\,{=}\, v(s;n_1,\ldots,n_k$).

Homogeneous fragmentations describe the evolution of a unit mass that breaks down randomly into pieces as time passes. They can be thought of as continuous time analogues of a certain type of branching random walk, which suggests the use of time-discretization to shift known results from the theory of branching random walks to the fragmentation setting. In particular, this yields interesting information about the asymptotic behaviour of fragmentations.

On the other hand, homogeneous fragmentations can also be investigated using a powerful technique of discretization of space due to Kingman, namely, the theory of exchangeable partitions of $\mathbb{N}$. Spatial discretization is especially well suited to the direct development for continuous times of the conceptual method of probability tilting of Lyons, Pemantle and Peres.

It is proved that the associated variety of a Poisson prime ideal of the centre of a symplectic reflection algebra at parameter $t \,{=}\, 0$ is irreducible.

We present a method for computing the 3-point genus zero Gromov–Witten invariants of the complex flag manifold $G/B$ from the relations of the small quantum cohomology algebra $QH^\ast G/B$ ($G$ is a complex semisimple Lie group and $B$ is a Borel subgroup). In [3] and [9], at least in the case $G={\rm GL}_n{\bf C}$, two algebraic/combinatoric methods have been proposed, based on suitably designed axioms. Our method is quite different, being differential geometric in nature; it is based on the approach to quantum cohomology described in [7], which is in turn based on the integrable systems point of view of Dubrovin and Givental.

The Kronecker quiver $K$ is considered, and the relations for the specialisation at $q=0$ of the generic composition algebra are given, as well as those for Reineke's composition monoid. As a corollary, it is deduced that the composition monoid is a proper factor of the specialisation of the composition algebra. A normal form is also obtained for the varieties occurring in the composition monoid in terms of Schur roots.

A Blaschke product $B$ with zero-sequence $(a_n)$ is called almost interpolating if the inequality $\liminf_n(1-|a_n|^2)|B'(a_n)|\geqslant \d>0$ holds. The sets $U$ for which there exists a Blaschke product $B$ such that $(a-B)/(1-\ov a B)$ is almost interpolating if and only if $a \in U$ are studied. Examples of such sets include open sets, containing the origin, and whose complement is the closure of an arbitrary set of concentric open arcs around the origin or open sets whose complement is of zero logarithmic capacity. Results on the range of interpolating Blaschke product s on the set of trivial points in the spectrum of $H^{\infty}$ are deduced.

The paper examines a higher-order ordinary differential equation of the form $\mathcal{P}[u]\,{:=} \sum_{j,k=0}^{m}D^{j}a_{jk}D^{k}u \,{=}\, \lambda{u}, x\in[0,b)$, where $D\,{=}\,i({d}/{dx})$, and where the coefficients $a_{jk}$, $j,k\in[0,m]$, with $a_{mm}\,{=}\,1$, satisfy certain regularity conditions and are chosen so that the matrix $(a_{jk})$ is hermitean. It is also assumed that $m\,{>}\,1$. More precisely, it is proved, using Paley–Wiener methods, that the corresponding spectral measure determines the equation up to conjugation by a function of modulus 1. The paper also discusses under which additional conditions the spectral measure uniquely determines the coefficients $a_{jk}$, $j,k\in[0,m]$, $j+k\neq{2m}$, as well as $b$ and the boundary conditions at 0 and at $b$ (if any).

Schottky uniformizations are provided of every closed Riemann surface $S$ of genus $g \in \{3,4\}$ admitting the symmetric group ${\mathcal S}_{4}$ as group of conformal automorphisms. These Schottky uniformizations reflect the group ${\mathcal S}_{4}$ and permit concrete representations of ${\mathcal S}_{4}$ to be obtained in the respective symplectic group $\mbox{Sp}_{g}({\mathbb Z})$. Their corresponding fixed points, in the Siegel space, give principally polarized Abelian varieties of dimension $g$. For $g=3$ and for some cases of $g=4$ they turn out to be holomorphically equivalent to the product of elliptic curves.

To gain understanding of the deformations of determinants and Pfaffians resulting from deformations of matrices, the deformation theory of composites $f\,{\circ}\,F$ with isolated singularities is studied, where $f\,{:}\,Y\,{\longrightarrow}\,\mathbb{C}$ is a function with (possibly non-isolated) singularity and $F:X\,{\longrightarrow}\, Y$ is a map into the domain of $f$, and $F$ only is deformed. The corresponding $T^1(F)$ is identified as (something like) the cohomology of a derived functor, and a canonical long exact sequence is constructed from which it follows that \[\tau=\mu(f\circ F)-\beta_0+\beta_1\], where $\tau$ is the length of $T^1(F)$ and $\beta_i$ is the length of $\mbox{Tor}_i^{\cal{O}_Y}({\cal O}_Y{/}J_f,\cal{O}_X)$. This explains numerical coincidences observed in lists of simple matrix singularities due to Bruce, Tari, Goryunov, Zakalyukin and Haslinger. When $f$ has Cohen–Macaulay singular locus (for example when $f$ is the determinant function), relations between $\tau$ and the rank of the vanishing homology of the zero locus of $f\circ F$ are obtained.

Consider the image of the Monge–Ampère operator acting on bounded functions, defined on a compact Kähler manifold, whose sum with the local Kähler potential is plurisubharmonic. It is shown that a nonnegative Borel measure belongs to this image if and only if it belongs to the image locally. In particular, those measures form a convex set.

Let $\{ \vf_k(x),\,k=1,2,\ldots\}$ be an arbitrary orthonormal system on [0,1] that is uniformly bounded by a constant $M$. Let $T$ be a subset of [0,1]$^2$ such that the Fourier series of all Lebesgue integrable functions on [0,1]$^2$ with respect to the product system $\{ \vf_k(x) \vf_l(y),\,k, l=1,2,\ldots\}$ converge in measure by squares on $T$. The following problem is studied. How large may the measure of $T$ be?

A theorem is proved that implies that for each such system, there is $\mu_2T \leq 1-M^{-4} $ (for the $d$-fold product systems, $\mu_dT \,{\leq}\, 1-M^{-2d}$, d\,{\geq}\, 2$). This estimate is sharp in the class of all such product systems.

The paper shows the existence of several new families of noncompact constant mean curvature surfaces: (i) singly punctured surfaces of arbitrary genus $g\,{\geq}\,1$, (ii) doubly punctured tori, and (iii) doubly periodic surfaces with Delaunay ends.