Products of characters and finite <Emphasis Type="Italic">p</Emphasis>-groups II

Let p be a prime number. Let G be a finite p-group and . Denote by the complex conjugate of . Assume that . We show that the number of distinct irreducible constituents of the product is at least . Received: 17 March 2003     

Type A Hecke algebras and related algebras

Let be the Hecke algebra of the symmetric group over a field K of characteristic and a primitive -th root of one in K. We show that an -module is projective if and only if its restrictions to any -parabolic subalgebra of is projective. Moreover, we give a new construction of blocks of -parabolic subalgebras, in terms of skew group algebras over local commutative algebras. Received: 30 June 2003     

Critical eigenvalues for a nonlinear problem

We study the application, , where is the supremum of positive s such that the problem admits a solution. Where B ₁ is the unit ball in We show that is a decreasing function, with where is the unique solution of the problem . We also give the explicit solutions of the problem , when and show that . We show that the problem doesnt admit a solution. In the end, we give a numerical approximation of , when .     

On spherical ideals of Borel subalgebras

The goal of this paper is to extend some previous results on abelian ideals of Borel subalgebras to so-called spherical ideals of These are ideals of such that their G-saturation is a spherical G-variety. We classify all maximal spherical ideals of for all simple G.Received: 25 March 2004     

Finite splitting fields of normal subgroups

Let G be a finite group, a normal subgroup, p a prime, a finite splitting field of characteristic p for G and We prove that is a splitting field for N, using the action of the Galois group of the field extension on the irreducible representations of N. As is a splitting field for the symmetric group S_n we get as a corollary that is a splitting field for the alternating group A_n. Received: 31 July 2003     

Non-collision periodic solutions for singular Hamiltonian systems

We study the existence of classical (non-collision) T-periodic solutions of the Hamiltonian system where and is a T-periodic function in t which has a singularity at like Under suitable conditions on H, we prove that if then (HS) possesses at least one non-collision solution and if then the generalized solution of (HS) obtained in [5] has at most one time of collision in its period.     

Algebraic radicals and incidence algebras

In this note we prove the uniqueness of U in a group G with a spherical split-BN-pair of rank ,i.e., if G has such a BN-pair with a nilpotent normal subgroup of B, and , then and is a normal subgroup of G. Here is the corresponding group of Lie-type and the subgroup of generated by all root-subgroups corresponding to positive roots. Received: 19 May 2003     

Divisibility of zeta functions of curves in a covering

We prove, as an analogy of a conjecture of Artin, that if is a finite flat morphism between two singular reduced absolutely irreducible projective algebraic curves defined over a finite field, then the numerator of the zeta function of X divides that of Y in . Then, we give some interpretations of this result in terms of semi-abelian varieties. Received: 23 July 2001     

A cohomological property of <Emphasis Type="Italic">p</Emphasis>-groups

Let p be a prime, a finite p-group, any finite group with order divisible by p, and any action of on . We show that the cardinality of the set of all derivations with respect to this action is a multiple of p. This generalises theorems of Frobenius and Hall. Received: 16 June 2003     

Bounded tightness for weak topologies

In every Hausdorff locally convex space for which there exists a strictly finer topology than its weak topology but with the same bounded sets (like for instance, all infinite dimensional Banach spaces, the space of distributions or the space of analytic functions in an open set , etc.) there is a set A such that 0 is in the weak closure of A but 0 is not in the weak closure of any bounded subset B of A. A consequence of this is that a Banach space X is finite dimensional if, and only if, the following property [P] holds: for each set and each x in the weak closure of A there is a bounded set such that x belongs to the weak closure of B. More generally, a complete locally convex space X satisfies property [P] if, and only if, either X is finite dimensional or linearly topologically isomorphic to . Received: 11 June 2003     

Automorphisms of principal blocks stabilizing Sylow subgroups

Let p be a prime, G a finite group which has a normal p-subgroup containing its own centralizer in G, and R a commutative local ring with residue class field of characteristic p. In this paper, it is shown that if is an augmented automorphism of RG which fixes a Sylow p-subgroup P of G, there is such that for all and is an inner automorphism of RG. Received: 26 July 2000     

A problem of Kusner on equilateral sets

R. B. Kusner [R. Guy, Amer. Math. Monthly 90, 196-199 (1983)] asked whether a set of vectors in such that the distance between any pair is 1, has cardinality at most d + 1. We show that this is true for p = 4 and any , and false for all 1<p<2 with d sufficiently large, depending on p. More generally we show that the maximum cardinality is at most if p is an even integer, and at least if 1<p<2, where depends on p. Received: 5 May 2003     

On higher order Bessel potentials

The solutions of the equation in , where are investigated, Bessel potentials of higher order are defined, and recurrence relations between these solutions and these Bessel potentials are obtained. It is also proved that these solutions and the solutions of , under certain conditions, are identical. Received: 6 November 2002     

A new bound for an extension of Mason’s theorem for functions of several variables

In this paper, using the generalized Wronskian, we obtain a new sharp bound for the generalized Masons theorem [1] for functions of several variables. We also show that the Diophantine equation (The generalized Fermat-Catalan equation) where , such that k out of the n-polynomials are constant, and under certain conditions for has no non-constant solution. Received: 20 March 2003     

A Helly-type theorem for countable intersections of starshaped sets

Let k and d be fixed integers, 0kd, and let be a collection of sets in If every countable subfamily of has a starshaped intersection, then is (nonempty and) starshaped as well. Moreover, if every countable subfamily of has as its intersection a starshaped set whose kernel is at least k-dimensional, then the kernel of is at least k-dimensional, too. Finally, dual statements hold for unions of sets.Received: 3 April 2004     

Maximal regularity for evolution equations in weighted <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

Let X be a Banach space and let A be a closed linear operator on X. It is shown that the abstract Cauchy problem enjoys maximal regularity in weighted L _p -spaces with weights , where , if and only if it has the property of maximal L _p -regularity. Moreover, it is also shown that the derivation operator admits an -calculus in weighted L _p -spaces. Received: 26 February 2003