Let M be a Hamiltonian K-space with proper moment map $\mu$. The symplectic quotient $X=\mu^{-1}(0)/K$ is a singular stratified space with a symplectic structure on the strata. In this paper we generalise the Kirwan map, which maps the K equivariant cohomology of $\mu^{-1}(0)$ to the middle perversity intersection cohomology of X, to this symplectic setting.

The key technical results which allow us to do this are Meinrenken's and Sjamaar's partial desingularisation of singular symplectic quotients and a decomposition theorem, proved in Section 2 of this paper, exhibiting the intersection cohomology of a ‘symplectic blowup’ of the singular quotient X along a maximal depth stratum as a direct sum of terms including the intersection cohomology of X.

Extensions of the concept of asymptotic independence which are useful in the study of abelian C$^*$-algebras generated by the union of two of their C$^*$-subalgebras are examined. Non-trivial examples are also given in terms of classes of bounded continuous functions defined on a locally compact group.

Let $\Lambda$ be a connected representation finite selfinjective algebra. According to $G$ . Zwara the partial orders $\le_{\rm ext}$ and $\le_{\rm deg}$ on the isomorphism classes of $d$ -dimensional $\Lambda$ -modules are equivalent if and only if the stable Auslander–Reiten quiver $\Gamma_{\Lambda}$ of $\Lambda$ is not isomorphic to ${\bb Z}D_{3m}/\tau^{2m-1}$ for all $m\ge 2$ . The paper describes all minimal degenerations $M\le_{\rm deg} N$ with $M\not\leq_{\rm ext} N$ in the case when $\Gamma_\Lambda\cong {\bb Z}D_{3m}/\tau^{2m-1}$ for some $m\ge 2$ .

In this paper we study the abstract quasi-linear evolution equation of second order

formula here

in a general Banach space Z. It is well-known that the abstract quasi-linear theory due to Kato [10, 11] is widely applicable to quasi-linear partial differential equations of second order and that his theory is based on the theory of semigroups of class (C0). (For example, see the work of Hughes et al. [9] and Heard [8].) However, even in the special case where A(t, w, v) = A is independent of (t, w, v), it is found in [2] and [14] that there exist linear partial differential equations of second order for which Cauchy problems are not solvable by the theory of semigroups of class (C0) but fit into the mould of well-posed problems where the solution and its derivative depend continuously on the initial data if the initial condition is measured in the graph norm of a suitable power of A. (See also work by Krein and Khazan [13] and Fattorini [6, Chapter 8].) This kind of Cauchy problem has recently been studied extensively, using the theory of integrated semigroups or regularized semigroups. The theory of integrated semigroups was studied intensively by Arendt [1] and that of regularized semigroups was initiated by Da Prato [3] and renewed by Davies and Pang [4]. For the theory of regularized semigroups we refer the reader to [5] and [16].

We introduce a numerical radius operator space $(X, \mathcal{W}_n)$. The conditions to be a numerical radius operator space are weaker than Ruan's axiom for an operator space $(X, \mathcal{O}_n)$. Let $w(\cdot)$ be the numerical radius on $\mathbb{B}(\mathcal{H})$. It is shown that, if $X$ admits a norm $\mathcal{W}_n(\cdot)$ on the matrix space $\mathbb{M}_n(X)$ which satisfies the conditions, then there is a complete isometry, in the sense of the norms $\mathcal{W}_n(\cdot)$ and $w_n(\cdot)$, from $(X, \mathcal{W}_n)$ into $(\mathbb{B}(\mathcal{H}), w_n)$. We study the relationship between the operator space $(X, \mathcal{O}_n)$ and the numerical radius operator space $(X, \mathcal{W}_n)$. The category of operator spaces can be regarded as a subcategory of numerical radius operator spaces.

It is proved that the word problem of the direct product of two free groups of rank 2 can be recognised by a 2-tape real-time but not by a 1-tape real-time Turing machine. It is also proved that the Baumslag–Solitar groups $B$(1, $r$) have the 5-tape real-time word problem for all $r\neq 0$.

If the sum of a series of positive numbers is not convergent and the series satisfies certain regularity conditions, a probability measure can be found whose support contains no interval, yet whose Fourier series decreases faster than the given series.

Let $T=U(1)$ and $M$ be a Hamiltonian $T$-space with proper moment map $\mu\,{:}\,M\longrightarrow \rr$. When 0 is not a regular value of $\mu$, the symplectic quotient $X=\mu^{-1}(0)/T$ is a singular stratified space. A description is provided of the middle perversity intersection cohomology of $X$ as a subspace of the equivariant cohomology $H^*_T(\mu^{-1}(0))$. The approach is sheaf theoretic.

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 formula of Poisson summation type is established for sums involving relative norms in an extension of number fields.

The paper is concerned with the uniqueness and existence problem for a multidimensional reacting and convection system with the vanishing viscosity method. The uniqueness theorem is obtained from the stability with respect to the initial data. To solve the existence problem, the uniqueness and existence of solutions to a viscous system are first proved, and then the L1-modulus of continuity of solutions independent of the small viscosity is obtained.

Finite-dimensional algebras over an algebraically closed field are divided into two disjoint classes, called tame and wild respectively, by Drozd's tame and wild dichotomy (see [5] and [2]). A tame algebra, roughly speaking, has its n-dimensional indecomposable modules parametrized by finitely many one-parameter families, for all natural numbers n, but a wild algebra has more indecomposable modules and it is considered hopeless to classify them. In [2], Crawley-Boevey showed that all but finitely many n-dimensional indecomposable modules over a tame algebra are τ-invariant, for all natural numbers n, and conjectured that the converse would be true, where τ := DTr is the Auslander–Reiten translation (see [1]) and we call an indecomposable module X τ-invariant if X ≅ = τX.

The paper is concerned with rings of polynomial invariants of finite groups. In particular, it will be shown that these rings are isomorphic as modules over the Steenrod algebra [Pscr ]* if and only if the group representations are pointwise conjugate. An application to cohomology is the construction of classifying spaces of finite groups which are not homotopy equivalent, but where the cohomology rings are isomorphic as unstable modules over the (topological) Steenrod algebra.

Let $G$ be a reductive $p$-adic group. Consider the category of smooth (complex) representations of $G$ in which a (fixed) closed cocompact subgroup of the centre acts by a (fixed) character. It is well known that the supercuspidal representations in this category are both injective and projective. It is shown that, conversely, an admissible injective or projective object is necessarily supercuspidal.

The close relationship between the notions of positive forms and representations for a C*-algebra A is one of the most basic facts in the subject. In particular the weak containment of representations is well understood in terms of positive forms: given a representation π of A in a Hilbert space H and a positive form φ on A, its associated representation π φ is weakly contained in π (that is, ker π φ ⊃ ker π) if and only if φ belongs to the weak* closure of the cone of all finite sums of coefficients of π. Among the results on the subject, let us recall the following ones. Suppose that A is concretely represented in H. Then every positive form φ on A is the weak* limit of forms of the type x [map ] [sum ]ki=1 〈ξi, xξi〉 with the ξi in H; moreover if A is a von Neumann subalgebra of ℒ(H) and φ is normal, there exists a sequence (ξi)i [ges ] 1 in H such that φ (x) = [sum ]i [ges ] 1 〈ξi, xξi〉 for all x.

A version of interpolation inequalities for derivatives in logarithmic Orlicz spaces is obtained where the first gradient of $u$ is estimated in terms of $u$ and its second gradient. One of the Orlicz functions considered is supposed to be ${\backslash\lambda^p}$. The motivation, examples and applications are discussed.