Extensions and the Induced Characters of Cyclic p-Groups

For a prime p, we denote by B_n the cyclic group of order pⁿ. Let φ be a faithful irreducible character of B_n, where p is an odd prime. We study the p-group G containing B_n such that the induced character φ^G is also irreducible. The purpose of this article is to determine the subgroup N_G(N_G(B_n)) of G under the hypothesis [N_G(B_n):B_n]⁴ ≦ pⁿ.     

Classification of the factorial functions of Eulerian binomial and Sheffer posets

We give a complete classification of the factorial functions of Eulerian binomial posets. The factorial function B(n) either coincides with n!, the factorial function of the infinite Boolean algebra, or ²n−1, the factorial function of the infinite butterfly poset. We also classify the factorial functions for Eulerian Sheffer posets. An Eulerian Sheffer poset with binomial factorial function B(n)=n! has Sheffer factorial function D(n) identical to that of the infinite Boolean algebra, the infinite Boolean algebra with two new coatoms inserted, or the infinite cubical poset. Moreover, we are able to classify the Sheffer factorial functions of Eulerian Sheffer posets with binomial factorial function B(n)=²n−1 as the doubling of an upside-down tree with ranks 1 and 2 modified. When we impose the further condition that a given Eulerian binomial or Eulerian Sheffer poset is a lattice, this forces the poset to be the infinite Boolean algebra BX or the infinite cubical lattice . We also include several poset constructions that have the same factorial functions as the infinite cubical poset, demonstrating that classifying Eulerian Sheffer posets is a difficult problem.     

Integrability of rotationally symmetric n-harmonic maps

A rotationally symmetric n-harmonic map is a rotationally symmetric p-harmonic map between two n-dimensional model spaces such that p=n. We show that rotationally symmetric n-harmonic maps can be integrated and are n-harmonic diffeomorphism, and apply such results to investigate the asymptotic behaviors of these maps. We also derive this integrability using Lie theory.     

Computing Orbits of the Automorphism Group of the Subsequence Poset B m,n

Let B _m,n denote the set of all words obtained from the cyclic word of length n on an alphabet of m letters in by deleting on all possible ways and their natural order. In [Order, 16: 179–194, 1999] Burosch et al. computed the automorphism group of the poset B _m,n . In this paper, we apply this result to obtain all of orbits of the natural action of Aut(B _m,n ) on B _m,n .     

Randomized isomorphic Dvoretzky theoremVersion aléatoire isomorphique du théorème de Dvoretzky

Let K be a symmetric convex body in $\text{R}{}^{\mathrm{N}}$ for which B₂^N is the ellipsoid of minimal volume. We provide estimates for the geometric distance of a ‘typical’ rank n projection of K to B₂ⁿ, for 1?n<N. Known examples show that the resulting estimates are optimal (up to numerical constants) even for the Banach–Mazur distance. To cite this article: A. Litvak et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 345–350.     

The shape of extremal functions for Poincaré-Sobolev-type inequalities in a ball

We study extremal functions for a family of Poincaré-Sobolev-type inequalities. These functions minimize, for subcritical or critical p?2, the quotient ‖∇u_‖2/‖up_‖ among all u∈H¹(B)?{0} with B_∫u=0. Here B is the unit ball in RN. We show that the minimizers are axially symmetric with respect to a line passing through the origin. We also show that they are strictly monotone in the direction of this line. In particular, they take their maximum and minimum precisely at two antipodal points on the boundary of B. We also prove that, for p close to 2, minimizers are antisymmetric with respect to the hyperplane through the origin perpendicular to the symmetry axis, and that, once the symmetry axis is fixed, they are unique (up to multiplication by a constant). In space dimension two, we prove that minimizers are not antisymmetric for large p.     

On some classes of bounded univalent mappings in several complex variables

Let B ⁿ be the Euclidean unit ball in C ⁿ . In this paper, we study several properties of strongly starlike mappings of order α (0 < α < 1) and bounded convex mappings on B ⁿ . We prove that K-quasiregular strongly starlike mappings of order α on B ⁿ have continuous and univalent extensions to ${\overline{B}^n}$ . We show that bounded convex mappings on B ⁿ are strongly starlike of some order α. We give a coefficient estimate for K-quasiregular strongly starlike mappings of order α on B ⁿ . Finally, we give examples of strongly starlike mappings of order α and bounded convex mappings on B ⁿ .     

Hamiltonian Cycles and Symmetric Chains in Boolean Lattices

Let B(n) be the subset lattice of \({\{1,2,\dots, n\}.}\) Sperner’s theorem states that the width of B(n) is equal to the size of its biggest level. There have been several elegant proofs of this result, including an approach that shows that B(n) has a symmetric chain partition. Another famous result concerning B(n) is that its cover graph is hamiltonian. Motivated by these ideas and by the Middle Two Levels conjecture, we consider posets that have the Hamiltonian Cycle–Symmetric Chain Partition (HC-SCP) property. A poset of width w has this property if its cover graph has a hamiltonian cycle which parses into w symmetric chains. We show that the subset lattices have the HC-SCP property, and we obtain this result as a special case of a more general treatment.     

A Local-Global Principle for Macaulay Posets

We consider the shadow minimization problem (SMP) for Cartesian powers P ⁿ of a Macaulay poset P. Our main result is a local-global principle with respect to the lexicographic order L ⁿ . Namely, we show that under certain conditions the shadow of any initial segment of the order L ⁿ for n 3 is minimal iff it is so for n = 2. These conditions include such poset properties as additivity, shadow increasing, final shadow increasing and being rank-greedy. We also show that these conditions are essentially necessary for the lexicographic order to provide nestedness in the SMP.     

A generalization of N-matrices

Ky Fan defines an N-matrix to be a matrix of the form A = tI ? B, B ? 0, λ < t < ?(B), where ?(B) is the spectral radius of B and λ is the maximum of the spectral radii of all principal submatrices of B of order n ? 1. In this paper, we define the closure (N₀-matrices) of N-matrices by letting λ ? t. It is shown that if A ∈ Z and A^-1 < 0, then A ∈ N. Certain inequalities of N-matrices are shown to hold for N₀-matrices, and a method for constructing an N-matrix from an M-matrix is given.     

The ordinary quiver of a weight three block of the symmetric group is bipartite

Suppose F is a field of characteristic p?5, and that B is a p-block of the symmetric group Sn of defect 3. We prove that the Ext¹-quiver of B is bipartite, with the bipartition being described in a simple way using the leg-lengths of p-hooks of partitions.     

Approximate fibrations and bundle maps on Hilbert cube manifolds

This paper is concerned with parameterized families of approximate fibrations from a compact Hilbert cube manifold M to a compact polyhedron B. The main result shows how to straighten out certain of these families to be nearly like a product. As an application of this technique, it is shown that an approximate fibration p:M → B can be approximated arbitrarily closely by bundle maps if and only if p is homotopic via approximate fibrations to a bundle map. Another result is that the space of bundle maps from M to B is locally n-connected for each n ? 0.     

A Note on the Homology of Signed Posets

Let S be a signed poset in the sense of Reiner [4]. Fischer [2] defines the homology of S, in terms of a partial ordering P (S) associated to S, to be the homology of a certain subcomplex of the chain complex of P (S).In this paper we show that if P (S) is Cohen-Macaulay and S has rank n, then the homology of S vanishes for degrees outside the interval [n/2, n].Research partially supported by the National Science Foundation and the John Simon Guggenheim Foundation.     

Finite Commutative Chain Rings

A commutative ring with identity is called a chain ring if all its ideals form a chain under inclusion. A finite chain ring, roughly speaking, is an extension over a Galois ring of characteristic pⁿusing an Eisenstein polynomial of degree k. When p∤k, such rings were classified up to isomorphism by Clark and Liang. However, relatively little is known about finite chain rings when p∣k. In this paper, we allowed p∣k. When n=2 or when p∥k but (p−1)∤k, we classified all pure finite chain rings up to isomorphism. Under the assumption that (p−1)∤k, we also determined the structures of groups of units of all finite chain rings.     

The Automorphism Group of the Subsequence Poset B m,n

In this paper we are interested in the automorphism group of the poset B _{m, n} . B _{m, n} constitutes the words obtained from the cyclic word of length n on an alphabet of m letters in by deleting on all possible ways and their natural order. We prove: Résumé: Le but de ce papier est la détermination du groupe d"automorphismes des ordres B _{m, n} . Il s"agit des mots obtenus à partir du mot cyclique de longeur n sur un alphabet de m lettres par suppression successive de lettres et ordonnés naturellement. On prouve: AutB _{m, n} = {S _n for 1 n m, S ₂ S _2m-n for m + 1n 2m - 1, S ₂for 2m n.     

A note on transitive permutation groups of prime degree

LetG be a nonsolvable transitive permutation group of prime degreep. LetP be a Sylow-p-subgroup ofG and letq be a generator of the subgroup ofN _G(P) fixing one point. Assume that |N _G(P)|=p(p?1) and that there exists an elementj inG such thatj ^?1qj=q^(p+1)/2. We shall prove that a group that satisfies the above condition must be the symmetric group onp points, andp is of the form 4n+1.     

Effective equidistribution of eigenvalues of Hecke operators

In 1997, Serre proved an equidistribution theorem for eigenvalues of Hecke operators on the space S(N,k) of cusp forms of weight k and level N. In this paper, we derive an effective version of Serre's theorem. As a consequence, we estimate, for a given d and prime p coprime to N, the number of eigenvalues of the pth Hecke operator Tp acting on S(N,k) of degree less than or equal to d. This allows us to determine an effectively computable constant Bd such that if J₀(N) is isogenous to a product of Q-simple abelian varieties of dimensions less than or equal to d, then N?Bd. We also study the effective equidistribution of eigenvalues of Frobenius acting on a family of curves over a fixed finite field as well as the eigenvalue distribution of adjacency matrices of families of regular graphs. These results are derived from a general “all-purpose” equidistribution theorem.     

Hypersurfaces with isotropic para-Blaschke tensor

Let Mnbe an n-dimensional submanifold without umbilical points in the(n + 1)-dimensional unit sphere Sn+1.Four basic invariants of Mnunder the Moebius transformation group of Sn+1are a 1-form Φ called moebius form,a symmetric(0,2) tensor A called Blaschke tensor,a symmetric(0,2) tensor B called Moebius second fundamental form and a positive definite(0,2) tensor g called Moebius metric.A symmetric(0,2) tensor D = A + μB called para-Blaschke tensor,where μ is constant,is also an Moebius invariant.We call the para-Blaschke tensor is isotropic if there exists a function λ such that D = λg.One of the basic questions in Moebius geometry is to classify the hypersurfaces with isotropic para-Blaschke tensor.When λ is not constant,all hypersurfaces with isotropic para-Blaschke tensor are explicitly expressed in this paper.