The class $P_n(I)$ of matrix monotone functions of order $n$, defined on an open interval $I$, is considered in this paper. Explicit examples are given to prove that if $I$ is different from the whole real line, then $P_{n+1}(I)$ is a proper subset of $P_n(I)$ for each natural number $n$. The subhomogeneous C*-algebras are then characterized in terms of matrix monotone functions.

We present a simple proof of the fact that every countable group ${\Gamma}$ is weak Rohlin, that is, there is in the Polish space $mathbb{A}_{\Gamma}$ of measure preserving ${\Gamma}$-actions an action $\mathbf{T}$ whose orbit in $\mathbb{A}_{\Gamma}$ under conjugations is dense. In conjunction with earlier results this in turn yields a new characterization of non-Kazhdan groups as those groups which admit such an action $\mathbf{T}$ which is also ergodic.

The theory of the spectral shift function is extended to the case where only the difference of some powers of the resolvents of self-adjoint operators belongs to the trace class. As an example, a pair of Dirac operators is considered.

We give an elementary proof for the fact that the Julia set of a rational or entire function is the closure of the repelling periodic points.

We prove that if H is a finite-dimensional semisimple Hopf algebra acting on a PI-algebra R of characteristic 0, and R is either affine or algebraic over k, then the Jacobson radical of R is H-stable. Under the same hypotheses, we show that the smash product algebra R#H is semiprimitive provided that R is H-semiprime. More generally we show that the ‘finite’ Jacobson radical is H-stable, and that R#H is semiprimitive provided that R is H-semiprimitive and all irreducible representations of R are finite-dimensional. We also consider R#H when R is an FCR-algebra. Finally, we prove a general relationship between stability of the radical and semiprimeness of R#H; in particular if for a given H, any action of H stabilizes the Jacobson radical, then also any action of H stabilizes the prime radical.

In this paper, branching rules for the fundamental representations of the symplectic groups in positive characteristic are found. The submodule structure of the restrictions of the fundamental modules for the group Sp2n(K) to the naturally embedded subgroup Sp2n−2(K) is determined. As a corollary, inductive systems of fundamental representations for Sp∞(K) are classified. The submodule structure of the fundamental Weyl modules is refined.

In this paper we show that, given 0[les ]m, n[les ]∞, any monoid can be embedded into a monoid with left cohomological dimension m and right cohomological dimension n. The embedding can be chosen to preserve various monoid-theoretical properties.

The set of vectors tangent to complete simple geodesics at a Weierstrass point is investigated, and interpreted as an infinitesimal version of the Birman–Series set. Three new identities for the lengths of simple geodesics are deduced. Fundamental use is made of the properties of the elliptic involution.

Algebraic solutions of certain Painlevé VI equations are produced by solving linear systems with monodromy contained in the octahedral subgroup of $\SO(3)$. The method uses the algebraic geometry of special plane curves, and makes contact with some classical geometrical problems.

Based on the classification of contact structures on the 3-torus and classical contact geometric methods, it is shown that there exist essential loops in the space of contact structures on torus bundles over the circle.

Two-sided bounds for the heat kernel of the Schrödinger operator with a singular potential are obtained with explicit constants.

Let T be the Volterra operator on L2[0, 1]

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where f∈L2[0, 1], 0[les ]x[les ]1. It is well known that ∥n!Tn∥=O(1/n!). In a recent paper [1], D. Kershaw has proved that

formula here

a result which was first conjectured by Lao and Whitley in [2]. It is easy to prove that

formula here

For completeness, we give the proof using the familiar Schmidt norm estimate for the norm of an integral operator (see Section 2 below). Kershaw proves that

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by analysing the special positivity preserving properties of T*T. He uses one of the many abstract theorems on eigenvalues and eigenfunctions of compact operators which preserve a cone. In this paper we shall reprove (1), giving a short and direct proof of (2).

Functions with bounded variation and with a (total) variation are examined within Bishop's constructive mathematics. It is shown that the property of having a variation is hereditary downward on compact intervals, and hence that a real-valued function f with a variation on a compact interval can be expressed as a difference of two increasing functions. Moreover, if f is sequentially continuous, then the corresponding variation function, and hence f itself, is uniformly continuous.

We show that if X is a complex hereditarily indecomposable space, then every operator from a subspace Y of X to X is of the form λI+S, where I is the inclusion map and S is strictly singular.

Let Λ and Γ be finite dimensional algebras. It is shown that any stable equivalence f [ratio ] mod Λ → mod Γ between the categories of finitely generated modules induces a bijection M [map ] Mf between the sets of isomorphism classes of generic modules over Λ and Γ such that the endolength of Mf is bounded by the endolength of M up to a scalar which depends only on f. Using Crawley-Boevey's characterization of tame representation type in terms of generic modules, one obtains as a consequence a new proof for the fact that a stable equivalence preserves tameness. This proof also shows that polynomial growth is preserved.

We prove that if A and B are semisimple Banach algebras, then the separating subspace of every Lie isomorphism from A onto B is contained in the centre of B.

1. Definition of the A-polynomial

The A-polynomial was introduced in [3] (see also [5]), and we present an alternative definition here. Let M be a compact 3-manifold with boundary a torus T. Pick a basis λ,μ of π1T, which we shall refer to as the longitude and meridian. Consider the subset RU of the affine algebraic variety R = Hom (π1M, SL2[Copf ]) having the property that ρ(λ) and ρ(μ) are upper triangular. This is an algebraic subset of R, since one just adds equations stating that the bottom-left entries in certain matrices are zero. There is a well-defined eigenvalue map

formula here

given by taking the top-left entries of ρ(λ) and ρ(μ).

This paper investigates the tightness property of the capacity induced by the reduction operator with respect to the resolvent of a right Markov process. Tightness is verified in two particular situations: under the ‘weak duality hypothesis’, and if a substitute for ‘the axiom of polarity for the dual theory’ holds. In the second context, the quasi-continuity property for the excessive functions is derived. These are extensions and improvements of results of Lyons and Röckner, Ma and Röckner, Le Jan, and Fitzsimmons, mainly obtained in the context of Dirichlet forms.

A famous result of Kostant's states that the universal enveloping algebra of a semisimple complex Lie algebra is a free module over its center. An analogue of this result is proved for the class of special filtered algebras. This is then applied to show that the restricted Yangian and the universal enveloping algebra of the restricted current algebra, associated with the general linear Lie algebra, are both free over their centers.

We give a survey on the use of Eva Kallin's lemma. This lemma gives a condition on two polynomially convex sets in [Copf ]n under which their union is polynomially convex. This result has proved to be a useful tool in different areas of complex function theory of several variables, for instance in the study of polynomial convexity of the union of totally real surfaces, and in approximation problems in function algebras.