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.     

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     

Wavelet-type Transform and Bessel Potentials Associated with the Generalized Translation

Wavelet–type transform associated with singular Laplace–Bessel differential operator is introduced and the relevant Calderón–type reproducing formula is established. Representations of the generalized Bessel potentials 0)$ " align="middle" border="0"> and their inverses via the wavelet–type transform are obtained.     

The Maximum Size Of Graphs With A Unique<Emphasis Type="Italic">k</Emphasis>- Factor

For suitable positive integers n and k let m(n, k) denote the maximum number of edges in a graph of order n which has a unique k-factor. In 1964, Hetyei and in 1984, Hendry proved for even n and , respectively. Recently, Johann confirmed the following conjectures of Hendry: for and kn even and for n = 2kq, where q is a positive integer. In this paper we prove for and kn even, and we determine m(n, 3).     

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     

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     

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     

Wirtinger numbers and holomorphic symplectic immersions

For any subvariety of a compact holomorphic symplectic Kähler manifold, we define the symplectic Wirtinger number W(X). We show that and the equality is reached if and only if the subvariety is trianalytic, i.e. compatible with the hyperkähler structure on M. For a sequence of immersions of simple holomorphic symplectic manifolds, we show that      

Rigidity Theorem of Hypersurfaces with Constant Scalar Curvature in a Unit Sphere

In this paper, we give a characterization of tori S^1 （ √ nr＋2-n/nr）×S^n-1（√ n-2/nr） and S^m （ √n/m ） ×S^n-m （√n-m/n）. Our result extends the result due to Li （1996）on the condition that M is an n-dimensional complete hypersurface in Sn＋1 with two distinct principal curvatures. Keywords principal curvature, Clifford torus, Gauss equations     

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 .     

Ample vector bundles with zero loci of sectional genus two

Let X be a smooth complex projective variety and let be a smooth submanifold of dimension , which is the zero locus of a section of an ample vector bundle of rank on X. Let H be an ample line bundle on X, whose restriction H_Z to Z is generated by global sections. Triplets as above are classified under the assumption that is a polarized manifold of sectional genus 2. This can be regarded as a step towards the classification of ample vector bundles of corank one and curve genus two. Received: 6 June 2003     

Matching Polynomials And Duality

Let G be a simple graph on n vertices. An r-matching in G is a set of r independent edges. The number of r-matchings in G will be denoted by p(G, r). We set p(G, 0) = 1 and define the matching polynomial of G by and the signless matching polynomial of G by .It is classical that the matching polynomials of a graph G determine the matching polynomials of its complement . We make this statement more explicit by proving new duality theorems by the generating function method for set functions. In particular, we show that the matching functions and are, up to a sign, real Fourier transforms of each other.Moreover, we generalize Foatas combinatorial proof of the Mehler formula for Hermite polynomials to matching polynomials. This provides a new short proof of the classical fact that all zeros of µ(G, x) are real. The same statement is also proved for a common generalization of the matching polynomial and the rook polynomial.     

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     

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     

Set Systems with few Disjoint Pairs

Let X = {1, . . . , n}, and let be a family of subsets of X. Given the size of , at least how many pairs of elements of must be disjoint? In this paper we give a lower bound for the number of disjoint pairs in . The bound we obtain is essentially best possible. In particular, we give a new proof of a result of Frankl and of Ahlswede, that if satisfies then contains at least as many disjoint pairs as X^(r).The situation is rather different if we restrict our attention to : then we are asking for the minimum number of edges spanned by a subset of the Kneser graph of given size. We make a conjecture on this lower bound, and disprove a related conjecture of Poljak and Tuza on the largest bipartite subgraph of the Kneser graph.* Research partially supported by NSF grant DMS-9971788     

Stability of the magnetic Couette-Taylor flow

In this paper we consider the magnetic Couette-Taylor problem, that is, a conducting fluid between two infinite rotating cylinders, subject to a magnetic field parallel to the rotation axis. This configuration admits an equilibrium solution of the form It is shown that this equilibrium is Ljapounov stable under small perturbations in where provided that the parameters a, b, , are small. The methods of proof are a combination of an energy method, based on Bloch space analysis and small data techniques.Received: February 5, 2003; revised: September 29, 2003Dedicated to Prof. H. Amann on the occasion of his 65. birthday     

Stability of the Cauchy congruence on restricted domain

Let S be a real interval with , and be a function satisfying We show that if h is Lebesgue or Baire measurable, then there exists such that That result is motivated by a question of E. Manstaviius. Received: 11 February 2003     

Spectral triangles of Schrödinger operators with complex potentials

Consider the Schrödinger operator with a complex-valued potential v of period Let and be the eigenvalues of L that are close to respectively, with periodic (for n even), antiperiodic (for n odd), and Dirichelet boundary conditions on [0,1], and let be the diameter of the spectral triangle with vertices We prove the following statement: If then v(x) is a Gevrey function, and moreover