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 .

     

The boundary of a Busemann space

Let be a proper Busemann space. Then there is a well defined boundary, , for . Moreover, if is (Gromov) hyperbolic (resp. non-positively curved), then this boundary is homeomorphic to the hyperbolic (resp. non-positively curved) boundary.

     

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 .

     

An infinite series of Kronecker conjugate polynomials

Let be a field of characteristic , a transcendental over , and be the absolute Galois group of . Then two non-constant polynomials are said to be Kronecker conjugate if an element of fixes a root of if and only if it fixes a root of . If is a number field, and where is the ring of integers of , then and are Kronecker conjugate if and only if the value set equals modulo all but finitely many non-zero prime ideals of . In 1968 H. Davenport suggested the study of this latter arithmetic property. The main progress is due to M. Fried, who showed that under certain assumptions the polynomials and differ by a linear substitution. Further, he found non-trivial examples where Kronecker conjugacy holds. Until now there were only finitely many known such examples. This paper provides the first infinite series. The main part of the construction is group theoretic.

     

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 .

     

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.

     

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.

     

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 .

     

Inner derivations on ultraprime normed algebras

We show that, for every ultraprime Banach algebra , there exists a positive number satisfying for all in , where denotes the centre of and denotes the inner derivation on induced by . Moreover, the number depends only on the ``constant of ultraprimeness' of .

     

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 .

     

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 .

     

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.

     

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 .

     

Invariance of spectrum for representations of -algebras on Banach spaces

Let be a Banach space, a unital -algebra, and an injective, unital homomorphism. Suppose that there exists a function such that, for all , and all ,

(a) ,

(b) ,

(c) .
Then for all , the spectrum of in equals the spectrum of as a bounded linear operator on . If satisfies an additional requirement and is a -algebra, then the Taylor spectrum of a commuting -tuple of elements of equals the Taylor spectrum of the -tuple in the algebra of bounded operators on . Special cases of these results are (i) if is a closed subspace of a unital -algebra which contains as a unital -subalgebra such that , and only if , then for each , the spectrum of in is the same as the spectrum of left multiplication by on ; (ii) if is a unital -algebra and is an essential closed left ideal in , then an element of is invertible if and only if left multiplication by on is bijective; and (iii) if is a -algebra, is a Hilbert -module, and is an adjointable module map on , then the spectrum of in the -algebra of adjointable operators on is the same as the spectrum of as a bounded operator on . If the algebra of adjointable operators on is a -algebra, then the Taylor spectrum of a commuting -tuple of adjointable operators on is the same relative to the algebra of adjointable operators and relative to the algebra of all bounded operators on .

     

Wavelet bases in rearrangement invariant function spaces

We point out that the well known characterization of spaces () in terms of orthogonal wavelet bases extends to any separable rearrangement invariant Banach function space on (equipped with Lebesgue measure) with nontrivial Boyd's indices. Moreover we show that such bases are unconditional bases of .

     

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.

     

Fixed point property and normal structure for Banach spaces associated to locally compact groups

In this paper we investigate when various Banach spaces associated to a locally compact group have the fixed point property for nonexpansive mappings or normal structure. We give sufficient conditions and some necessary conditions about for the Fourier and Fourier-Stieltjes algebras to have the fixed point property. We also show that if a -algebra has the fixed point property then for any normal element of , the spectrum is countable and that the group -algebra has weak normal structure if and only if is finite.

     

An Engel condition with derivation for left ideals

We generalize a number of results in the literature by proving the following theorem: Let be a semiprime ring, a nonzero derivation of , a nonzero left ideal of , and let . If for some positive integers , and all , the identity holds, then either or else the ideal of generated by and is in the center of . In particular, when is a prime ring, is commutative.