Given a hyperbolic invariant set of a diffeomorphism on a surface, it is proved that, if the holonomies are sufficiently smooth, then the diffeomorphism on the hyperbolic invariant set is rigid in the sense that it is C$^{1+}$ conjugate to a hyperbolic affine model.

For a given integer n, all zero-mean cosine polynomials of order at most n which are non-negative on [0,(n/(n+1))π] are found, and it is shown that this is the longest interval [0,θ] on which such cosine polynomials exist. Also, the longest interval [0,θ] on which there is a non-negative zero-mean cosine polynomial with non-negative coefficients is found.

As an immediate consequence of these results, the corresponding problems of the longest intervals [θ,π] on which there are non-positive cosine polynomials of degree n are solved.

For both of these problems, all extremal polynomials are found. Applications of these polynomials to Diophantine approximation are suggested.

A spectral inclusion theorem and a spectral mapping theorem are proved for the functional calculus for sectorial operators. Most proofs are generic, so that similar results can be obtained for other functional calculi. Applications are a new proof for the spectral mapping theorem for fractional powers and the identity $\sigma(\log A)\,{=}\,\log(\sigma(A)\ohne\{0\})$ for any injective sectorial operator $A$.

We consider the cohomology of Mackey functors for infinite groups and define the Mackey-cohomological dimension ${\rm cd}_{{\mathfrak M}} G$ of a group $G$. We will relate this dimension to other cohomological dimensions such as the Bredon-cohomological dimension ${\rm cd}_{{\mathfrak F}}G$ and the relative cohomological dimension ${{\mathfrak F}}\text{-}{\rm cd} G$. In particular, we show that for virtually torsion free groups the Mackey-cohomological dimension is equal to both ${{\mathfrak F}}\text{-}{\rm cd} G$ and the virtual cohomological dimension.

The trace-norm convergence of the Trotter product formula is proved for nonself-adjoint Gibbs semigroups. For any m-sectorial generators A and B such that e−tReA is in the trace-class for t > 0, the Trotter product formula converges in the trace-norm. With smallness conditions on B with respect to A, we give error bound estimates of the convergence rate in this topology.

It is shown that D. Cohen's inequality bounding the isoperimetric function of a group by the double exponential of its isodiametric function is valid in the more general context of locally finite simply connected complexes. It is shown that in this context this bound is ‘best possible’. Also studied are second-dimensional isoperimetric functions for groups and complexes. It is shown that the second-dimensional isoperimetric function of a group is bounded by a recursive function. By a similar argument it is shown that the area distortion of a finitely presented subgroup of a finitely presented group is recursive. Cohen's inequality is extended to second-dimensional isoperimetric and isodiametric functions of 2-connected simplicial complexes.

The paper presents a construction of fibered links $(K, \Sigma)$ out of chord diagrams $\sL$. Let $\Gamma$ be the incidence graph of $\sL$. Under certain conditions on $\sL$ the symmetrized Seifert matrix of $(K, \Sigma)$ equals the bilinear form of the simply-laced Coxeter system $(W, S)$ associated to $\Gamma$; and the monodromy of $(K, \Sigma)$ equals minus the Coxeter element of $(W, S)$. Lehmer's problem is solved for the monodromy of these Coxeter links.

We render the cut locus more accessible to analysis by showing that each non-conjugate cut point has a neighbourhood on which the distance function is the minimum of finitely many smooth ‘radial functions’. This enables us to generalise to an arbitrary complete manifold a number of recent results, mainly from stochastic analysis, which were either limited or not valid on the cut locus because of the lack of differentiability there of the distance function.

Let $F$ be a germ of a holomorphic function at $0$ in ${\bb C}^{n+1}$ , having $0$ as a critical point not necessarily isolated, and let $\tilde{X}:= \sum^n_{j=0} X^j(\partial/\partial z_j)$ be a germ of a holomorphic vector field at $0$ in ${\bb C}^{n+1}$ with an isolated zero at $0$ , and tangent to $V := F^{-1}(0)$ . Consider the ${\cal O}_{V,0}$ -complex obtained by contracting the germs of Kähler differential forms of $V$ at $0$ \renewcommand{\theequation}{0.\arabic{equation}} \begin{equation} \Omega^i_{V,0}:=\frac{\Omega^i_{{\bb C}^{n+1},0}}{F\Omega^i_{{\bb C}^{n+1},0}+dF\wedge{\Omega^{i-1}}_{{\bb C}^{n+1}},0} \end{equation} with the vector field <formula form="inline" disc="math" id="frm14"><formtex notation="AMSTeX"> $X:=\tilde{X}|_V$ on $V$ : \begin{equation} 0\longleftarrow {\cal O}_{V,0} {\buildrel X\over\longleftarrow}\,\Omega_{V,0}^1\,{\buildrel X\over\longleftarrow}\, \cdots \,{\buildrel X\over\longleftarrow}\, \Omega_{V,0}^n\, {\buildrel X\over\longleftarrow}\, \Omega_{V,0}^{n+1}\longleftarrow 0. \end{equation}

The uniserial $p$-adic space groups of coclass $r$ play a central role in the classification of the finite $p$-groups of coclass $r$. A practical algorithm to determine up to isomorphism the uniserial $p$-adic space groups of coclass $r$, where $p$ is an odd prime, is described. As an application, these groups are constructed or counted for some values on $p$ and $r$. For example, it is observed that there are 137 299 953 383 uniserial 3-adic space groups of coclass 4.

Let G be a locally compact group not necessarily unimodular. Let μ be a regular and bounded measure on G. We study, in this paper, the following integral equation,

formula here

This equation generalizes the functional equation for spherical functions on a Gel'fand pair. We seek solutions ϕ in the space of continuous and bounded functions on G. If π is a continuous unitary representation of G such that π(μ) is of rank one, then tr(π(μ)π(x)) is a solution of [Escr ](μ). (Here, tr means trace). We give some conditions under which all solutions are of that form. We show that [Escr ](μ) has (bounded and) integrable solutions if and only if G admits integrable, irreducible and continuous unitary representations. We solve completely the problem when G is compact. This paper contains also a list of results dealing with general aspects of [Escr ](μ) and properties of its solutions. We treat examples and give some applications.

The integral cohomology rings of the configuration spaces of $n$-tuples of distinct points on arbitrary surfaces (not necessarily orientable, not necessarily compact and possibly with boundary) are studied. It is shown that for punctured surfaces the cohomology rings stabilize as the number of points tends to infinity, similarly to the case of configuration spaces on the plane studied by Arnold, and the Goryunov splitting formula relating the cohomology groups of configuration spaces on the plane and punctured plane to arbitrary punctured surfaces is generalized. Moreover, on the basis of explicit cellular decompositions generalizing the construction of Fuchs and Vainshtein, the first cohomology groups for surfaces of low genus are given.

In a 2004 paper, Totaro asked whether a G-torsor X that has a zero-cycle of degree $d>0$ will necessarily have a closed étale point of degree dividing d, where G is a connected algebraic group. This question is closely related to several conjectures regarding exceptional algebraic groups. Totaro gave a positive answer to his question in the following cases: G simple, split, and of type $G_2$, type $F_4$, or simply connected of type $E_6$. We extend the list of cases where the answer is ‘yes’ to all groups of type $G_2$ and some nonsplit groups of type $F_4$ and $E_6$. No assumption on the characteristic of the base field is made. The key tool is a lemma regarding linkage of Pfister forms.

Using properties of the Frobenius eigenvalues, we show that, in a precise sense, ‘most’ isomorphism classes of (principally polarized) simple abelian varieties over a finite field are characterized, up to isogeny, by the sequence of their division fields, and we show a similar result for ‘most’ isogeny classes. Some global cases are also treated.

Throughout this paper, $G$ is a connected semisimple algebraic group defined over an algebraically closed field ${\mathbb k}$ of characteristic zero, and ${\frak g}$ is its Lie algebra.

We investigate the problem of finding a set of prime divisors of the order of a finite group, such that no two irreducible characters are in the same p-block for all primes p in the set. Our main focus is on the simple and quasi-simple groups. For results on the alternating and symmetric groups and their double covers, some combinatorial results on the cores of partitions are proved, which may be of independent interest. We also study the problem for groups of Lie type. The sporadic groups (and their relatives) are checked using GAP.

We generalise group algebras to other algebraic objects with bounded Hilbert space representation theory; the generalised group algebras are called ‘host’ algebras. The main property of a host algebra is that its representation theory should be isomorphic (in the sense of the Gelfand–Raikov theorem) to a specified subset of representations of the algebraic object. Here we obtain both existence and uniqueness theorems for host algebras as well as general structure theorems for host algebras. Abstractly, this solves the question of when a set of Hilbert space representations is isomorphic to the representation theory of a C*-algebra. To make contact with harmonic analysis, we consider general convolution algebras associated to representation sets, and consider conditions for a convolution algebra to be a host algebra.

By an algebra Λ we mean an associative k-algebra with identity, where k is an algebraically closed field. All algebras are assumed to be finite dimensional over k (except the path algebra kQ). An algebra is said to be biserial if every indecomposable projective left or right Λ-module P contains uniserial submodules U and V such that U+V=Rad(P) and U∩V is either zero or simple. (Recall that a module is uniserial if it has a unique composition series, and the radical Rad(M) of a module M is the intersection of its maximal submodules.) Biserial algebras arose as a natural generalization of Nakayama's generalized uniserial algebras [2]. The condition first appeared in the work of Tachikawa [6, Proposition 2.7], and it was formalized by Fuller [1]. Examples include blocks of group algebras with cyclic defect group; finite dimensional quotients of the algebras (1)–(4) and (7)–(9) in Ringel's list of tame local algebras [4]; the special biserial algebras of [5, 8] and the regularly biserial algebras of [3]. An algebra Λ is basic if Λ/Rad(Λ) is a product of copies of k. This paper contains a natural alternative characterization of basic biserial algebras, the concept of a bisected presentation. Using this characterization we can prove a number of results about biserial algebras which were inaccessible before. In particular we can describe basic biserial algebras by means of quivers with relations.

A noninjective bounded-to-one factor map from an irreducible shift of finite type onto a sofic system can be factored as a composition of other such maps in only finitely many ways (up to isomorphism). This generalizes to factor maps from systems with canonical coordinates to finitely presented dynamical systems.

We develop a theory of harmonic analysis and duality for finite commutative hypergroups by considering somewhat more general objects called signed hypergroups. A notion of entropy is defined, and a Second Law of Thermodynamics is established. Applications to group theory and to the fusion rule algebras of conformal field theory are given.