On the homotopy type of the classifying space of the exceptional Lie group F4

 Previous work of several authors shows that the exceptional Lie group of rank 4, F ₄, as a p-compact group, is determined up to isomorphism by the isomorphism type of its maximal torus normalizer for p > 2. This paper considers the case p= 2 proving that F ₄ as 2-compact group is also determined up to isomorphism by the isomorphism type of its maximal torus normalizer. This allows the authors to determine the integral homotopy type of F ₄ among connected finite loop spaces with maximal tori. Received: 21 June 2000 / Revised version: 4 December 2001     

（L^p,L^q） Estimates for Multilinear Oscillatory Singular Integralswith Smooth Phases

In this paper,the authors establish the weighted (L^p,L^q) estimates for a class of multilinear oscillatory singular integrals with smooth phases.Certain endpoint estimates are also considered.     

Some generalizations on the boundedness of bilinear operators

For 0<p<∞, let H^p(R ⁿ) denote the Lebesgue space for p>1 and the Hardy space for p ≤1. In this paper, the authors study H^p(R ⁿ)×H^q(R ⁿ)→H^r(R ⁿ) mapping properties of bilinear operators given by finite sums of the products of the standard fractional integrals or the standard fractional integral with the Calderón-Zygmund operator. The authors prove that such mapping properties hold if and only if these operators satisfy certain cancellation conditions. Supported by the NNSF and the National Education Comittee of China.     

The GL(3) mellin transform for twisted non-cuspidal forms of higher level

The main result in this paper is the explicit computation of the functional equation satisfied by the GL(3) Mellin transform of a twisted non-cuspidal metaplectic form of non-trivial level. For concreteness, we work with one particular metaplectic form, automorphic under Λ(3), although our methods extend without change to any form automorphic with respect to Λ(p),p an odd prime. We clearly show the computations one must undertake in order to determine the pole locations of this transform (which depend on the form in question), and carry out those straightforward computations in the case of our one specific form. This particular form, first considered by Bump and Hoffstein [BH], is the maximal parabolic Eisenstein series on the cubic cover of GL(3) (induced from the theta function on the cubic cover of GL(2)), which has the remarkable property that its Fourier coefficients are essentially Hecke cubicL-series. In joint work with Hoffstein, the authors have applied the main theorems in this paper to compute the average values of these Hecke cubicL-series, a result of great arithmetic interest. Both authors are partially supported by the National Science Foundation. Both authors wish to thank the Mathematical Sciences Research Institute for its hospitality during part of the time this research was conducted. In addition, the authors wish to thank Jeffrey Hoffstein for many helpful conversations on this and related problems, and the referee for several helpful comments.     

WeightedL p approximation by modified hermite interpolation of higher order

The authors establish necessary and sufficient conditions for the weightedL ^p convergence at given rates of Hermite interpolation of higher order based on Jacobi zeros plus the endpoints ±1. Theorems on simultaneous approximation are also proved.This material is based upon work supported by the Ministero della Università e della Ricerca Scientifica e Tecnologica (the first two authors), by the National Research Council (the second author) and by Hungarian National Foundation Grant No. 1910 (the third author).     

Commutative free loop space models at large primes

Let p be a prime number and a natural number. If E is a r-connected finite CW-complex of dimension at most pr, then E is an example of a p -Anick space. For p > 2 we construct a commutative cochain algebra over that is an -model of the free loop space on a p-Anick space, i.e., its cohomology algebra is isomorphic to the mod p cohomology of the free loop space. For p-Anick spaces that are p-formal, such as spheres and projective spaces, we define an even simpler commutative free loop space model that applies for all primes p. We then use the simplified model to compute the cohomology algebras of a number of free loop spaces explicitly. Received: 23 June 1999; in final form: 8 September 2000 // Published online: 7 April 2003     

The K-theory of p-compact groups

In this paper, we show that the p-adic K-theory of a connected p-compact is the ring of invariants of the Weyl group action on the K-theory of a maximal torus. We apply this result to show that a connected finite loop space admits a maximal torus if and only if its complex K-theory is -isomorphic to the K-theory of some BG, where G is a compact connected Lie group. Received: November 9, 1996     

On Connected Transversals to Dihedral Subgroups of Order 2p n

Let G be a group with a dihedral subgroup H of order 2pⁿ, where p is an odd prime. We show that if there exist H-connected transversals in G, then G is a solvable group. We apply this result to the loop theory and show that if the inner mapping group of a finite loop Q is dihedral of order 2pⁿ, then Q is a solvable loop.1991 Mathematics Subject Classification: 20D10, 20N05     

Quasinormal subgroups of order p 2

The location of quasinormal subgroups in a group is not particularly well known. Maximal ones always have to be normal, but little has been proved about the minimal ones. In finite groups, the difficulties arise in the p-groups. Here we prove that, for every odd prime p, a quasinormal subgroup of order p ² in a finite p-group G contains a quasinormal subgroup of G of order p. S. Stonehewer is grateful to the Australian National University for financial support during the preparation of this paper.     

扭曲退化同宿分支

§ 1　 HypothesesConsider the following system:z.=f(z) , (1 .1 )and its perturbed systemz.=f(z) +g(z,μ) (1 .2 )where z∈ Rm+n,μ∈ Rk,k≥ 3,0≤ |μ| 1 ,f,g∈ Cr,r≥ 4 ,g(z,0 ) =0 .For simplicity,we sup-pose thatf(p) =0 ,g(p,μ) =0 .Moreover,for(1 .1 ) we assume(H1 ) The stable manifold Wspand the unstable manifold Wupof z=p are m-dimension-al and n-dimensional,respectively.The linearization Df(p) atthe equilibrium z=p has realmultiple-2 eigenvaluesλ1 and -ρ1 ,such thatany remaining eige…     

Normal <Emphasis Type="Italic">π</Emphasis>-complements for finite groups

Let G be a finite group. For a finite p-group P the subgroup generated by all elements of order p is denoted by Ω₁(p). Zhang [5] proved that if P is a Sylow p-subgroup of G, Ω₁(P) ≦ Z(P) and N _G (Z(P)) has a normal p-complement, then G has a normal p-complement. The object of this paper is to generalize this result. This paper was partly supported by Hungarian National Foundation for Scientific Research Grant # T049841 and T038059.     

Majorants andL p norms

It is shown that ifG is an infinite compact abelian group, thenL ^p (G) has the upper majorant property only ifp is even orp=∞. The research for this paper was partially supported by National Research Council of Canada Operating Grant A-4822.     

An optimization problem on subsets of the symmetric positive-semidefinite matrices

Motivated by the metricsstress problem in multidimensional scaling, the authors consider the more general problem of minimizing a strictly convex function on a particular subset ofR ^{n × n} . The subset in question is the intersection of a linear subspace with the symmetric positive-semidefinite matrices of rank p. Because of the rank restriction, this subset is not convex. Several equivalent formulations of this problem are derived, and the advantages and disadvantages of each formulation are discussed.Part of this research was conducted while the authors were visitors at the Center for Research on Parallel Computation, Rice University, Houston, Texas. The first author was partially supported by National Science Foundation Grant RII-89-05080.     

Computational complexity of norm-maximization

This paper discusses the problem of maximizing a quasiconvex function over a convex polytopeP inn-space that is presented as the intersection of a finite number of halfspaces. The problem is known to beNP-hard (for variablen) when is thep ^th power of the classicalp-norm. The present reexamination of the problem establishesNP-hardness for a wider class of functions, and for thep-norm it proves theNP-hardness of maximization overn-dimensionalparallelotopes that are centered at the origin or have a vertex there. This in turn implies theNP-hardness of {–1, 1}-maximization and {0, 1}-maximization of a positive definite quadratic form. On the good side, there is an efficient algorithm for maximizing the Euclidean norm over an arbitraryrectangular parallelotope.The authors are indebted to J. O'Rourke, P, Pardalos and R. Freund for useful references. The second and third authors are indebted to the Institute for Mathematics and its Applications in Minneapolis, where much of this paper was written: they acknowledge additional support from the Alexander von Humboldt Stiftung and the National Science Foundation, respectively.     

Finite groups in which all p-subnormal subgroups form a lattice

O. Kegel, in 1962, introduced the concept of p-subnormal subgroups of a finite group as the subgroups whose intersections with all Sylow p-subgroups of the group are Sylow p-subgroups of the subgroup. The set of p-subnormal subgroup of a finite group is not a lattice in general. In this paper, the class of all finite groups in which all p-subnormal subgroups from a lattice is determined. This is the class of all finite p-soluble groups whose p-length and p′-length, both, are less or equal to 1. The join-semilattice case and the meet-semilattice case are analyzed separately. The authors are supported by Proyecto PB 94-1048 of DGICYT, Ministerio de Educación y Ciencia of Spain.     

A classification for 2-isometries of noncommutativeLp-spaces

In this paper we extend previous results of Banach, Lamperti and Yeadon on isometries ofL _p-spaces to the non-tracial case first introduced by Haagerup. Specifically, we use operator space techniques and an extrapolation argument to prove that every 2-isometryT:L _p(M) →L _p(N) between arbitrary noncommutativeL _p-spaces can always be written in the form Here π is a normal *-isomorphism fromM onto the von Neumann subalgebra π(M) ofN,w is a partial isometry inN, andE is a normal conditional expectation fromN onto π(M). As a consequence of this, any 2-isometry is automatically a complete isometry and has completely contractively complemented range. The first and second authors were partially supported by the National Science Foundation DMS-0088928 and DMS-0140067.     

On Connected Transversals to Subgroups Whose Order is a Product of Two Primes

We consider finite groups which have connected transversals to subgroups whose order is a product of two primespandq. We investigate those values ofpandqfor which the group is soluble. We can show that the solubility of the group follows ifq = 2 andp ≤ 61,q = 3 andp ≤ 31,q = 5 andp ≤ 11. We then apply our results on loop theory and we show that if the inner mapping group of a finite loop has orderpqwherepandqare as above then the loop is soluble.     

A Note on Linear Extension of Into-Isometries Between Two Unit Spheres of Atomic AL^p-Space （0 〈 p 〈∞）

In this paper, we shall present a short and simple proof on the isometric linear extension problem of into-isometries between two unit spheres of atomic abstract L^p-spaces （0 〈 p 〈 ∞）.     

On v-closed manis valution rings

ABSTRACT

The graded Lie algebra L associated to the Nottingham group with respect to its natural filtration is known to be a loop algebra of the first Witt algebra W ₁ . The fact that the Schur multiplier of W ₁ , in characteristic p > 3, is one-dimensional implies that L is not finitely presented. Consider the universal covering ? ₁ of W ₁ and the corresponding loop algebra M of ? ₁ . In this paper we prove that M itself is finitely presented for p > 3. In characteristic p >  11 the algebra M turns out to be presented by two relations.     

Remarks on the length of conjugacy classes of finite groups

Let G be a finite group. The question of how certain arithmetical conditions on the lengths of the conjugacy classes of G influence the group structure has been studied by several authors. In this paper we study restrictions on the structure of a finite group in which the lengths of conjugacy classes are not divisible by p ² for some prime p. We generalise and provide simplified proofs for some earlier results.