1. Ordered spaces

Risking semantic incoherence, we propose in this paper to use the term ordered space to mean any totally ordered set X equipped with a topology [Tscr ], while retaining the term linearly ordered space for its usual role of denoting the special case in which [Tscr ] is the order topology: the topology generated by the order-open initial and final segments of X.

The categories that we shall consider are full subcategories of the category OrdTop of ordered spaces and order-preserving continuous maps. It is easy to see that — in contrast to the situation in the full subcategory LinOrdTop of linearly ordered spaces — every inverse spectrum in OrdTop has a limit and that OrdTop admits subspaces and quotient spaces. These limits, subspaces and quotient spaces are all constructed precisely as in the category Top and are given their naturally induced orderings: where, however, we note that, for the ordered space X to induce an ordering on a quotient space, the quotient space (construed as a decomposition) must partition X into intervals rather than into arbitrary subsets. (In the terminology of [1], these constructions define subspaces and quotient spaces in OrdTop as, respectively, the domains and codomains of extremal monomorphisms and epimorphisms.) The purpose of the paper is to identify the closures of LinOrdTop under these constructions.

This paper gives a necessary criterion in terms of Hodge theory for representability by quaternion algebras of certain 2-torsion classes in the unramified Brauer group of a complex function field. This criterion is used to give examples of threefolds with unramified Brauer group elements which are the classes of biquaternion division algebras.

Let $G$ be a compact Lie group, and let $\pi$ be any prime or set of primes. A ‘$\pi$-perfection map’ is constructed: that is, a continuous function from the space of conjugacy classes of all closed subgroups of $G$ to the space of conjugacy classes of $\pi$-perfect subgroups with finite index in their normalizer. This is used to show that the idempotent elements of the Burnside ring of $G$ localized at $\pi$ are in bijective correspondence with the open and closed subsets of the space of conjugacy classes of $\pi$-perfect subgroups of $G$ with finite index in their normalizer.

We prove that if n>66 and (n, 30) = 1, then there exist uncountably many infinite simple (2, 3, n)- groups, that is, groups generated by a pair of elements x, y, say, where the orders of x, y and xy are 2, 3 and n, respectively. This extends previous results of Schupp and the authors.

These results are used to prove the existence of subgroups of the modular group with special arithmetic properties.

Let f be an orientation-preserving univalent harmonic mapping of the unit disk U. Then f=h+[gmacron], where h and g are analytic in U. Furthermore, f satisfies the equation

formula here

in U, where a(z)=g′(z)/h′(z), and [mid ]a(z)[mid ]<1 in U. The function a(z) is the analytic dilatation of f.

In [2], Hengartner and Schober proved the following version of the Riemann mapping theorem for univalent harmonic mappings.

Let Γ be a discrete group acting on a compact manifold X, let V be a Γ-equivalent Hermitian vector bundle over X, and let D be a first-order elliptic self-adjoint Γ-equivalent differential operator acting on sections of V. This data is used to define Toeplitz operators with symbols in the transformation group C*-algebra C(X)[rtimes ]Γ, and it is shown that if the symbol of such a Toeplitz operator is invertible, then the operator is Fredholm. In the case where Γ is finite and acts freely on X, a geometric-topological formula for the index is stated that involves an explicitly constructed differential form associated to the symbol.

Let $C$ be an elliptic curve defined over $\mathbb{Q}$. We can associate two formal groups with $C$: the formal group $\^{C}(X,Y)$ determined by the formal completion of the Néron model of $C$ over $\mathbb{Z}$ along the zero section, and the formal group $F_L(X,Y)$ of the L-series attached to $l$-adic representations on $C$ of the absolute Galois group of $\mathbb{Q}$. Honda shows that $F_L(X, Y)$ is defined over $\mathbb{Z}$, and it is strongly isomorphic over $\mathbb{Z}$ to $\^{C}(X,Y)$. In this paper we give a generalization of the result of Honda to building blocks over finite abelian extensions of $\mathbb{Q}$. The difficulty is to define new matrix L-series of building blocks. Our generalization contains the generalization of Deninger and Nart to abelian varieties of $\rm{GL}_2$-type. It also contains the generalization of our previous paper to $\mathbb{Q}$-curves over quadratic fields.

a structure theorem is given for $n$-dimensional smooth subvarieties of the grassmannian $g(1,n)$, with $n\geq n+3$, that can be isomorphically projected to $g(1,n+1)$. a complete classification in the cases $n=2n+1$ and $n=2n$ follows, as a corollary.

We prove that Schanuel's conjecture for the reals is equivalent to a uniform version of itself.

In a recent paper, Külshammer, Olsson, and Robinson proved a deep generalization of the Nakayama conjecture for symmetric groups. We provide a similar but a shorter and relatively elementary proof of their result. Our method enables us to obtain a more general $H$-analogue of the Nakayama conjecture where $H$ is a set of positive integers.

Let G be a finite group acting faithfully and transitively on a set Ω of size m, and let E = {x1, …, xk} be a generating set for G with x1x2…xk = 1. If x∈G has cycles of length r1, …, rl in its action on Ω, define ind (x) = [sum ]l1(ri−1). Then the genusg = g(G, Ω, E) is defined by

formula here

In this paper, the strongly Cohen–Macaulay ideals of second analytic deviation one are characterized in terms of the depth properties of the powers of the ideal in the ‘standard range’. This provides an explanation of the behaviour of certain ideals that have appeared in the literature.

The ‘isomorphism problem for integral group rings’ (IP) is the question whether for two finite groups G and H, the existence of an isomorphism of the integral group rings ℤG and ℤH implies that G and H are isomorphic. Though (IP) is not true in general [4], it is still an interesting question for which classes of finite groups (IP) has a positive solution. In this note, we want to show that (IP) holds for finite groups of Lie type associated to Chevalley groups of universal type.

Note that if U(ℤG) denotes the units of ℤG, and V(ℤG) stands for the units with augmentation 1, then U(ℤG)≅V(ℤG)×U(ℤ), and we can always assume that H[les ]V(ℤG).

Let $(W,S)$ be a finite Coxeter system, and let $J\,{\subseteq}\, S$. Any $w\,{\in}\,W$ has a unique factorization $w\,{=}\,w^Jw_J$, where $w_J$ belongs to the parabolic subgroup $W_J$ generated by $J$, and $w^J$ is of minimal length in the coset $wW_J$. It is shown here that $w_I\,{=}\,w^J$ if and only if $w^I\,{=}\,w_J$, for all $I,J\,{\subseteq}\,S$. Furthermore, a similar symmetry property in arbitrary $(W_I,W_J)$-double cosets is conjectured, which links this result to the Solomon descent algebra of $W$.

In his curriculum vitae, John Hawkes lists his research interests as geometric measure theory, probability (Lévy processes), and potential theory (probabilistic). In fact, he made significant contributions to all three areas, and there are strong relationships between them. He used both geometric measure theory and potential theory as tools for his study of the trajectories of particular Lévy processes, but in many cases he needed to develop the tool before it was ready to be used. We will summarise his research later, but first we discuss what is known of his life history.

Consider the parabolic equation \[ \begin{array}{@{}l@{\qquad}l@{}} u_t=\Delta u+|u|^p& \text{in}\ R^d\times(0,T), u(x,0)=\phi(x)& \text{in}\ R^d, \end{array} \] up to a maximal time $T=T_\infty$, where $p>1$. Let $p^{\ast}=1+2/d$. It is a classical result that if $p\le p^{\ast}$, then there exist no non-negative, global solutions to the above equation for any choice of $\phi\geq0$; that is, necessarily $T_\infty<\infty$ and the solution blows up in some sense. On the other hand, if $p>p^{\ast}$, then there do exist non-negative, global solutions for appropriate choices of $\phi$. A recent paper by Zhang seems to be the first to consider the existence of global solutions for the above equation with sign-changing initial data. He proved that if $p\le p^{\ast}$ and $\int_{R^d}\phi(x)dx>0$, then the solution to the above equation is not global. In this paper it is shown that this result continues to hold if $\int_{R^d}\phi(x)dx=0$, and then it is shown that for each $p{>}1$ and each $l\in(-\infty,0)$, there exists a $\phi$ satisfying $\int_{R^d}\phi(x)dx=l$ and such that the corresponding solution to the above equation is global, and there exists a $\phi$ satisfying $\int_{R^d}\phi(x)dx=l$ and such that the corresponding solution is non-global.