Chen's Theorem with Small Primes

Let N be a sufficiently large even integer. In this paper it is proved that the equation N = p+P ₂, p≤N ^0,95, is solvable, where p denotes a prime and P ₂ denotes an almost prime with at most two prime factors. Received April 13, 1999, Accepted December 18, 2000     

Hua's Theorem on Prime Squares in Short Intervals

It is proved that every large integer N≡ 5(mod24) can be written as N = p ² ₁ + ... + p ² ₅ with each prime p _j satisfying This gives a short interval version of Hua's theorem on the quadratic Waring-Goldbach problem. Received October 14, 1999, Accepted April 17, 2000     

Permutability of minimal subgroups and<Emphasis Type="Italic">p</Emphasis>-nilpotency of finite groups

In this paper it is proved that ifp is a prime dividing the order of a groupG with (|G|,p − 1) = 1 andP a Sylowp-subgroup ofG, thenG isp-nilpotent if every subgroup ofP ∩G ^N of orderp is permutable inN _G (P) and whenp = 2 either every cyclic subgroup ofP ∩G ^N of order 4 is permutable inN _G (P) orP is quaternion-free. Some applications of this result are given. The research of the first author is supported by a grant of Shanxi University and a research grant of Shanxi Province, PR China. The research of the second author is partially supported by a UGC(HK) grant #2160126 (1999/2000).     

On Powerful Normal Subgroups of a p-Group

 In this paper we investigate the action of a p-group G on its powerful normal subgroup N. We shall mainly be concerned to find conditions which guarantee that G acts uniserially on N. We shall also study what consequences uniserial action on N has on the structure of N. (Received 21 August 1998; in revised form 28 June 1999)     

Nonexistence results of solutions of semilinear differential inequalities on the Heisenberg group

Denoting Δ_? the Laplacian operator on the (2N+1)-dimensional Heisenberg group ? ^N , we prove some nonexistence results for solutions of inequalities of the three types

A note on Property Np

Let M be a very ample line bundle on a smooth complex projective variety Y and let ϕ _M :Y→P(H ⁰(Y, M)^*) be the map associated to M; we are concerned with the problem to see whether the syzygies of ϕ _M (Y) give information on the syzygies of ϕ _M ^s (Y). In particular we prove that if Y is a smooth complex projective variety and M is a line bundle on Y satisfying Property N _p , then M ^s satisfies Property N _p if s≥p. Received: 11 June 1999 / Revised version: 22 November 1999     

Definable one dimensional structures in o-minimal theories

Let N be a structure definable in an o-minimal structure M and p ∈ S _N (N), a complete N-1-type. If dim _M (p) = 1, then p supports a combinatorial pre-geometry. We prove a Zilber type trichotomy: Either p is trivial, or it is linear, in which case p is non-orthogonal to a generic type in an N-definable (possibly ordered) group whose structure is linear, or, if p is rich then p is non-orthogonal to a generic type of an N-definable real closed field.     

Mean value estimates of the error terms of Lehmer problem

Let p be an odd prime and a be an integer coprime with p. Denote by N(a, p) the number of pairs of integers b, c with bc ≡ a(mod p), 1 ≤ b, c < p and with b, c having different parity. The main purpose of this paper is to study the mean square value problem of (N(a, p) − 1/2 (p−1)) over interval (N, N + M] with M, N positive integers by using the analytic methods, and finally by obtaining a sharp asymptotic formula.     

Extended genus of torsion-free finitely generated nilpotent groups of class two

The extended genus of a nilpotent group N is the set of isomorphism classes of nilpotent groups M, not necessarily finitely generated, such that the p-localizations M _p , N _p are isomorphic for all primes p. In this article, for any torsion-free finitely generated nilpotent group N of nilpotency class 2, the extended genus of N is analyzed by assigning to each of its members a sequence of triads of matrices with rational entries, generalizing the sequential representation which has been exploited elsewhere in the case when N is abelian. This approach allows, among other things, to obtain examples of groups in the ordinary (Mislin) genus of N     

On the holomorphic extension of CR functions from non-generic CR submanifolds of ? n

We give a holomorphic extension result for continuous CR functions on a non-generic CR submanifold N of ℂ ⁿ to complex transversal wedges with edges containing N. We show that given any v∈ℂ ⁿ ∖(T _p N+iT _p N), there exists a wedge of direction v whose edge contains a neighborhood of p in N, such that any continuous CR function defined locally near p extends holomorphically to that wedge.     

When almost all sets are difference dominated

We investigate the relationship between the sizes of the sum and difference sets attached to a subset of {0,1,…,N}, chosen randomly according to a binomial model with parameter p(N), with N^?1 = o(p(N)). We show that the random subset is almost surely difference dominated, as N → ∞, for any choice of p(N) tending to zero, thus confirming a conjecture of Martin and O'Bryant. The proofs use recent strong concentration results. Furthermore, we exhibit a threshold phenomenon regarding the ratio of the size of the difference to the sumset. If p(N) = o(N^?1/2) then almost all sums and differences in the random subset are almost surely distinct and, in particular, the difference set is almost surely about twice as large as the sumset. If N^?1/2 = o(p(N)) then both the sum and difference sets almost surely have size (2N + 1) ? O(p(N)^?2), and so the ratio in question is almost surely very close to one. If p(N) = c · N^?1/2 then as c increases from zero to infinity (i.e., as the threshold is crossed), the same ratio almost surely decreases continuously from two to one according to an explicitly given function of c. We also extend our results to the comparison of the generalized difference sets attached to an arbitrary pair of binary linear forms. For certain pairs of forms f and g, we show that there in fact exists a sharp threshold at c_f,g · N^?1/2, for some computable constant c_f,g, such that one form almost surely dominates below the threshold and the other almost surely above it. The heart of our approach involves using different tools to obtain strong concentration of the sizes of the sum and difference sets about their mean values, for various ranges of the parameter p. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Analysis of a circulant based preconditioner for a class of lower rank extracted systems

This paper proposes and studies the performance of a preconditioner suitable for solving a class of symmetric positive definite systems, A_px=b, which we call lower rank extracted systems (LRES), by the preconditioned conjugate gradient method. These systems correspond to integral equations with convolution kernels defined on a union of many line segments in contrast to only one line segment in the case of Toeplitz systems. The p × p matrix, A_p, is shown to be a principal submatrix of a larger N × N Toeplitz matrix, A_N. The preconditioner is provided in terms of the inverse of a 2N × 2N circulant matrix constructed from the elements of A_N. The preconditioner is shown to yield clustering in the spectrum of the preconditioned matrix similar to the clustering results for iterative algorithms used to solve Toeplitz systems. The analysis also demonstrates that the computational expense to solve LRE systems is reduced to O(N log N). Copyright © 2004 John Wiley & Sons, Ltd.     

Coalescing times for IID random variables with applications to population biology

We consider a coalescing particle model where particles move in discrete time. At each time period, each remaining ball is independently put in one of n bins according to a probability distribution p = (p₁, …, p_n), and all balls put into the same bin merge into a single ball. Starting with k balls, we are interested in the properties of E[N( p , k)], the expected time until all balls merge into one. We derive both upper and lower bounds for E[N( p , k)], some asymptotic results, and show that P{N( p , k) > t}, and thus E[N( p , k)], is a Schur concave function of p . Applications to population biology are noted. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 155–166, 2003     

On thep-ranks of net graphs

Let be ak-net of ordern with line-point incidence matrixN and letA be the adjacency matrix of its collinearity graph. In this paper we study thep-ranks (that is, the rank over ) of the matrixA+kl withp a prime dividingn. SinceA+kI=N ^T N thesep-ranks are closely related to thep-ranks ofN. Using results of Moorhouse on thep-ranks ofN, we can determiner _p (A+kI) if is a 3-net (latin square) or a desarguesian net of prime order. On the other hand we show how results for thep-ranks ofA+kI can be used to get results for thep-ranks ofN, especially in connection with the Moorhouse conjecture. Finally we generalize the result of Moorhouse on thep-rank ofN for desarguesian nets of orderp a bit to special subnets of the desarguesian affine plane of orderp ^e .The author is financially supported by the Cooperation Centre Tilburg and Eindhoven Universities.     

Subspaces ofl p N of small codimension

In this paper the structure of subspaces and quotients ofl _p ^N of dimension very close toN is studied, for 1≤p≤∞. In particular, the maximal dimensionk=k(p, m, N) so that an arbitrarym-dimensional subspaceX ofl _p ^N contains a good copy ofl _p ^k , is investigated form=N−o(N). In several cases the obtained results are sharp.     

On p-Regular G-Conjugacy Classes and the p-Structure of Normal Subgroups

Let N be a p-solvable normal subgroup of a group G such that N contains a noncentral Sylow r (≠ p)-subgroup R of G. It is proved that the p-complements of N are nilpotent if |x ^G |=1 or m for every p-regular element x of N whose order is divisible by at most two distinct primes. Our result, therefore, gives some information concerning the nilpotence of some kind of subgroups of a group G.     

On the Modular Version of the Gluck-Wolf Theorem

Let N be a normal subgroup of a finite group G. Let ϕ be an irreducible Brauer character of N. Assume π is a set of primes and χ(1)/ϕ(1) is a π′-number of any χ∈IBr _p (G/ϕ). If p∤|G:N|, and N is p-solvable, then G/N has an abelian-by-metabelian Hall-π subgroup; If p∉π then G/N has a metabelian Hall-π subgroup. Received February 22, 2000, Accepted May 9, 2001     

Sampling expansions associated with Kamke problems

The present paper is devoted to the derivation of sampling expansions for entire functions which are represented as integral transforms where a differential operator is acting on the kernels. The situation generalizes the results obtained in sampling theory associated with boundary value problems to the case when the differential equation has the form where N and P are two differential expressions of orders n and p respectively, and is the eigenvalue parameter. Both self adjoint and non self adjoint cases will be considered with examples in which the boundary conditions are strongly regular. Received February 16, 1998; in final form March 15, 1999     

Finiteness conditions on subgroups of profinite p-Poincaré duality groups

For a prime number p let G be a profinite p-PD _n group with a closed normal subgroup N such that G/N is a profinite p-PD _m group and that H _i (V, $ \mathbb{F} $ \mathbb{F} _p ) is finite for every open subgroup V of N and all i ≤ [n/2]. Generalising [12, Thm. 3.7.4] we show that m ≤ n and N is a profinite p-PD _{n − m} group. In case that G is a pro-p PD _n group of Euler characteristic 0 with a closed normal subgroup N of type FP _{[n−1 / 2]} such that G/N is soluble-by-finite pro-p group of finite rank, we show that N is a pro-p PD _{n − m} group, where m = vcd _p (G/N). As a corollary we obtain that a pro-p PD ₃ group with infinite abelianization is either soluble or contains a free nonprocyclic pro-p subgroup.     

A note on the Lehmer problem over short intervals

Let p be an odd prime, c be an integer with (c, p) = 1, and let N be a positive integer with N ≤ p − 1. Denote by r(N, c; p) the number of integers a satisfying 1 ≤ a ≤ N and 2 ∤ a + ā, where ā is an integer with 1 ≤ ā ≤ p − 1, aā ≡ c (mod p). It is well known that r(N, c; p) = 1/2N + O(p ^1/2log² p). The main purpose of this paper is to give an asymptotic formula for Σ _c=1 ^p−1(r(N, c; p) − 1/2N)².