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.

     

Lifting of generating subgroups

Let be an epimorphism of finite groups. Suppose that is generated by its subgroups and that is generated by its subgroups . Furthermore, suppose that and are conjugate, . We prove that there exist such that generate and , .

     

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.

     

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 .

     

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 .

     

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.

     

A conformal differential invariant and the conformal rigidity of hypersurfaces

For a hypersurface of a conformal space, we introduce a conformal differential invariant , where and are the first and the second fundamental forms of connected by the apolarity condition. This invariant is called the conformal quadratic element of . The solution of the problem of conformal rigidity is presented in the framework of conformal differential geometry and connected with the conformal quadratic element of . The main theorem states:

Let , and let and be two nonisotropic hypersurfaces without umbilical points in a conformal space or a pseudoconformal space of signature . Suppose that there is a one-to-one correspondence between points of these hypersurfaces, and in the corresponding points of and the following condition holds: where is a mapping induced by the correspondence . Then the hypersurfaces and are conformally equivalent.

     

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 .

     

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 .

     

On reducibility of semigroups of compact quasinilpotent operators

The following generalization of Lomonosov's invariant subspace theorem is proved. Let be a multiplicative semigroup of compact operators on a Banach space such that for every finite subset of , where denotes the Rota-Strang spectral radius. Then is reducible.

This result implies that the following assertions are equivalent:

(A) For each infinite-dimensional complex Hilbert space , every semigroup of compact quasinilpotent operators on is reducible.

(B) For every complex Hilbert space , for every semigroup of compact quasinilpotent operators on , and for every finite subset of it holds that .

The question whether the assertion (A) is true was considered by Nordgren, Radjavi and Rosenthal in 1984, and it seems to be still open.

     

When is a -block?

Let and be distinct prime numbers and let be a finite group. If is a -block of and is a -block, we study when the set of ordinary irreducible characters in the blocks and coincide.

     

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 .

     

Existence of Bade functionals for complete Boolean algebras of projections in Fréchet spaces

A classical result of W. Bade states that if is any complete Boolean algebra of projections in an arbitrary Banach space then, for every there exists an element (called a Bade functional for with respect to in the dual space , with the following two properties: (i) is non-negative on and, (ii) whenever satisfies It is shown that a Fréchet space has this property if and only if it does not contain an isomorphic copy of the sequence space

     

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 .

     

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 .

     

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.

     

Mordell-Weil groups of the Jacobian of the 5-th Fermat curve

Let denote the Jacobian of the Fermat curve of exponent 5 and let . We compute the groups , , , where is the unique quadratic subfield of . As an application, we present a new proof that there are no -rational points on the 5-th Fermat curve, except the so called ``points at infinity".

     

Self-contragredient supercuspidal representations of

Let be a non-archimedean local field of residual characteristic . Then has tamely ramified self-contragredient supercuspidal representations if and only if or is even. When such representations exist, they do so in abundance.

     

A Strict Version of the Non-commutative Urysohn Lemma

Given a pair , of -commuting, hereditary -subalgebras of a unital -algebra , such that is -unital and , there is an element in , with , such that is strictly positive in and is strictly positive in in . Moreover, is strictly positive in in .