Fixed points of the mapping class group in the moduli spaces

Let be a compact oriented surface with or without boundary components. In this note we prove that if then there exist infinitely many integers such that there is a point in the moduli space of irreducible flat connections on which is fixed by any orientation preserving diffeomorphism of . Secondly we prove that for each orientation preserving diffeomorphism of and each there is some such that has a fixed point in the moduli space of irreducible flat connections on . Thirdly we prove that for all there exists an integer such that the 'th power of any diffeomorphism fixes a certain point in the moduli space of irreducible flat connections on .

     

Commensurators of parabolic subgroups of Coxeter groups

Let be a Coxeter system, and let be a subset of . The subgroup of generated by is denoted by and is called a parabolic subgroup. We give the precise definition of the commensurator of a subgroup in a group. In particular, the commensurator of in is the subgroup of in such that has finite index in both and . The subgroup can be decomposed in the form where is finite and all the irreducible components of are infinite. Let be the set of in such that for all . We prove that the commensurator of is . In particular, the commensurator of a parabolic subgroup is a parabolic subgroup, and is its own commensurator if and only if .

     

Weakly coupled bound states in quantum waveguides

We study the eigenvalue spectrum of Dirichlet Laplacians which model quantum waveguides associated with tubular regions outside of a
bounded domain. Intuitively, our principal new result in two dimensions asserts that any domain obtained by adding an arbitrarily small ``bump' to the tube (i.e., , open and connected, outside a bounded region) produces at least one positive eigenvalue below the essential spectrum of the Dirichlet Laplacian . For sufficiently small ( abbreviating Lebesgue measure), we prove uniqueness of the ground state of and derive the ``weak coupling' result using a Birman-Schwinger-type analysis. As a corollary of these results we obtain the following surprising fact: Starting from the tube with Dirichlet boundary conditions at , replace the Dirichlet condition by a Neumann boundary condition on an arbitrarily small segment , , of . If denotes the resulting Laplace operator in , then has a discrete eigenvalue in no matter how small is.

     

Point spectrum and mixed spectral types for rank one perturbations

We consider examples of rank one perturbations with a cyclic vector for . We prove that for any bounded measurable set , an interval, there exist so that
eigenvalue agrees with up to sets of Lebesgue measure zero. We also show that there exist examples where has a.c. spectrum for all , and for sets of 's of positive Lebesgue measure, also has point spectrum in , and for a set of 's of positive Lebesgue measure, also has singular continuous spectrum in .

     

On solutions of real analytic equations

Analyticity of solutions , of systems of real analytic equations , is studied. Sufficient conditions for and power series solutions to be real analytic are given in terms of iterative Jacobian ideals of the analytic ideal generated by . In a special case when the 's are independent of , we prove that if a solution satisfies the condition , then is necessarily real analytic.

     

Poincaré flows

We study flows on a compact metric space with the property that corresponding to every non-zero element of there is either a cross section associated with or one associated with . We obtain necessary and sufficient conditions for this to hold; on the -dimensional torus these conditions take a classical form.

     

Hypersurfaces in a sphere with constant mean curvature

Let be a closed hypersurface of constant mean curvature immersed in the unit sphere . Denote by the square of the length of its second fundamental form. If , is a small hypersphere in . We also characterize all with .

     

Orbifolds with lower Ricci curvature bounds

We show that the first betti number of a compact Riemannian orbifold with Ricci curvature and diameter is bounded above by a constant , depending only on dimension, curvature and diameter. In the case when the orbifold has nonnegative Ricci curvature, we show that the is bounded above by the dimension , and that if, in addition, , then is a flat torus .

     

On Bernstein-Sato polynomials

We show that for fixed and the set of Bernstein-Sato polynomials of all the polynomials in at most variables of degrees at most is finite. As a corollary, we show that there exists an integer depending only on and such that generates as a module over the ring of the -linear differential operators of , where is an arbitrary field of characteristic 0, is the ring of polynomials in variables over and is an arbitrary non-zero polynomial of degree at most .

     

Hilbert Cmodules in which all closed submodules are complemented

Let be a Calgebra. If there exists a full Hilbert module such that for each closed submodule , then is *-isomorphic to a Calgebra of (not neccesarily all) compact operators on a Hilbert space.

     

Théorème de l'application spectrale pour le spectre essentiel quasi-Fredholm

In 1958, T. Kato proved that a closed semi-Fredholm operator in a Banach space can be written where is a nilpotent operator and is a regular one.

J. P. Labrousse studied and characterised this class of operators in the case of Hilbert spaces. He also defined a new spectrum named ``essential quasi-Fredholm spectrum' and denoted .

In this paper we prove that the essential quasi-Fredholm spectrum defined by J. P. Labrousse satisfies the mapping spectral theorem, i.e.: If is a bounded operator in a Hilbert space and an analytic function in a neighbourhood of the spectrum of , then .

RÉSUMÉ. En 1958, T. Kato a montré que si est un opérateur fermé dans un espace de Banach et semi-Fredholm, alors il existe tels que où est nilpotent et est régulier.

J. P. Labrousse a étudié et caractérisé cette classe d'opérateurs dans le cadre des espaces de Hilbert et a défini un nouveau spectre qu'on appelle ``spectre essentiel quasi-Fredholm' et noté par .

Dans ce travail nous allons démontrer que le spectre essentiel quasi-Fredholm défini par J. P. Labrousse vérifie le théorème de l'application spectrale, c'est à dire: Si est un opérateur bourné d'un espace de Hilbert dans lui même et une fonction analytique au voisinage du spectre de , alors .

     

A monotoneity property of the gamma function

In this paper we obtain a monotoneity property for the gamma function that yields sharp asymptotic estimates for as tends to , thus proving a conjecture about .

     

An improved estimate for the highest Lyapunov exponent in the method of freezing

Let and be the eigenvalues of the matrix . The main result of the Method of Freezing states that if , and , then

for the highest exponent of the system, where

The previous best known value and the substantially smaller values of are reduced to the still smaller value.

     

Coprimeness among irreducible character degrees of finite solvable groups

Given a finite solvable group , we say that has property if every set of distinct irreducible character degrees of is (setwise) relatively prime. Let be the smallest positive integer such that satisfies property . We derive a bound, which is quadratic in , for the total number of irreducible character degrees of . Three exceptional cases occur; examples are constructed which verify the sharpness of the bound in each of these special cases.

     

Finite linear groups and theorems of Minkowski and Schur

Let be a finite group with a faithful rational valued character of degree . A theorem of I. Schur gives a bound for the order of in terms of , generalizing an earlier result of H. Minkowski who showed that the same bound holds if . This note contains strengthened versions of these results which in particular show that a -subgroup of of maximum possible order contains a reflection.

     

Stable orders of stunted lens spaces mod

Let be the stunted lens space mod and its stable order. If , then was determined by H. Toda (1963). In this paper, we determine the number for .

     

Bilocal derivations of standard operator algebras

In this paper, we shall show the following two results: (1) Let be a standard operator algebra with , if is a linear mapping on which satisfies that maps into for all , then is of the form for some in . (2) Let be a Hilbert space, if is a norm-continuous linear mapping on which satisfies that maps into for all self-adjoint projection in , then is of the form for some in .

     

A note on GK dimension of skew polynomial extensions

Let be a finitely generated commutative domain over an algebraically closed field , an algebra endomorphism of , and a -derivation of . Then if and only if is locally algebraic in the sense that every finite dimensional subspace of is contained in a finite dimensional -stable subspace.

Similarly, if is a finitely generated field over , a -endomorphism of , and a -derivation of , then if and only if is an automorphism of finite order.