Powerful numbers in (1ℓ + qℓ)(2ℓ + qℓ)⋯(nℓ + qℓ)

Let q be a positive integer. Recently, Niu and Liu proved that, if $n\ge \max ?\{q,1198?q\}$, then the product $(1^3+q^3)(2^3+q^3)?(n^3+q^3)$ is not a powerful number. In this note, we prove (1) that, for any odd prime power ? and $n\ge \max ?\{q,11?q\}$, the product $(1^{?}+q^{?})(2^{?}+q^{?})?(n^{?}+q^{?})$ is not a powerful number, and (2) that, for any positive odd integer ?, there exists an integer $N_{q,?}$ such that, for any positive integer $n\ge N_{q,?}$, the product $(1^{?}+q^{?})(2^{?}+q^{?})?(n^{?}+q^{?})$ is not a powerful number.     

6q 1级2—传递置换群,q为素数

§1 引言 M.D.Atkinson在文[3]中提出一个猜想:若G是Ω上2—传递而非2—本原置换群,则G是下面四种群之一: ⅰ) 素数P级P(P-1)阶亚循环群; ⅱ) 对于素数P,是2~P级2~P(2~P-1)或2~P(2~P-1)P阶群; ⅲ) 是有λ=1的区组设计的自同构群; ⅳ) Sz(q)≤G≤Aut(Sz(q))     

6q+1级2—传递置换群,q为素数

In [ 3 ] M. D. Atkinson conjectured that if G is a doubly transitive but not doubly primitive permutation group on Ω, then G is of one of the following four types: i) Metacyclic groups of prime degree p and of order p(p -1); ii) Groups of degree 2^p and of order 2^p(2^p-1)or 2^p(2^p-l)p for some prime p;iii)Gr-oups of automorphisms of a block design with λ=1; iv) Sz(q)≤G≤Aut(Sz(g)).In this paper we proved this conjecture in a special case without using the result of classification of finte simple groups, Qur explicit result is as follows: Theorem. Let G be a doubly transitive group on set Ω,where |Ω|=6q+1 and q is a prime, then one of the following holds: i)G is doubly primitive on Ω;ii) G is sharply doubly transitive on Ω; iii) G is a groups of automorphisms of a block design with λ=1.     

(i1i)*(b12k0)及(i1i)*(b31k0)在π*V(1)中的收敛性

对连通有限型谱X,y,存在Adams谱序列{Es,tr,dr},满足(1)dr:Es,t,r→Es+r,t+r-1,r是谱序列的微分, (2)Es,t,2≌Exts,t,A(H*X,H*y),(3)收敛到[∑t-sY,X].当X分别是球谱S,Moore谱M,Toda-smith谱V(1)时,(πt-X)p分别是S,M,V(1)的稳定同伦群.利用Adams 谱序列,证明了(i1i)*(62,1k0)及(i1i)*(b3,1k0)是永久循环但不是边缘,因此收敛到π*V(1)中的非零元,其中P为奇索数,q=2p-2.     

Percolation in invariant Poisson graphs with?i.i.d.?degrees

Let each point of a homogeneous Poisson process in ℝ ^d independently be equipped with a random number of stubs (half-edges) according to a given probability distribution μ on the positive integers. We consider translation-invariant schemes for perfectly matching the stubs to obtain a simple graph with degree distribution μ. Leaving aside degenerate cases, we prove that for any μ there exist schemes that give only finite components as well as schemes that give infinite components. For a particular matching scheme which is a natural extension of Gale–Shapley stable marriage, we give sufficient conditions on μ for the absence and presence of infinite components.     

The Equivalence of Hypercontractivity and Logarithmic Sobolev Inequality for q ((1 ≤ q ≤ 1) -Ornstein-Uhlenbeck Semigroup

In this paper the author proves the equivalence of hypercontractivity and logarithmic Sobolev inequality for q-Ornstein-Uhlenbeck semigroup U_t~((q))= Γ_q(e~(-t)I)(-1 ≤q ≤ 1), where Γ_q is a q-Gaussian functor.     

On the construction of [q 4 + q 2 − q, 5,q 4 − q 3 + q 2 − 2q; q]-codes meeting the Griesmer bound

It is unknown whether or not there exists an [87, 5, 57; 3]-code. Such a code would meet the Griesmer bound. The purpose of this paper is to give a constructive proof of the existence of [q ⁴ + q ² – q, 5, q ⁴ – q ³ + q ² – 2q; q]-codes for any prime power q 3. As a special case, it is shown that there exists an [87, 5, 57; 3]-code with weight enumerator 1 + 156z ³⁷ + 82z ⁶⁰ + 2z ⁶³ + 2z ⁷⁸. The new construction settles an open problem due to Hill and Newton [10].     

A Note on the Difference Equation x n +1 = ∑ i =0 k α i x n − i p i

In this note we improve Theorem 2 in Ref. [3] , about the difference equation x n +1 = ~ i =0 k f i x n m i p i , n =0,1,2,..., where k is a positive integer, f i , p i ] (0, X ) for i =0,..., k , and the initial conditions x m k , x m k +1 ,..., x 0 are arbitrary positive numbers.     

A new proof of the existence of (q 2−q+1)-arcs inPG(2,q 2)

Journal of Geometry -     

一类 q=1的鞅 Φ-不等式

对于 q>1的鞅Φ-不等式,到目前为止已经作了较充分的研究.但是,对于q=1的鞅Φ-不等式,至今尚未见其研究成果.本文就一类满足一定条件的q=1的 Young 函数Φ,给出了与 q>1相类似的鞅Φ-不等式.从而把有关的结果推广到这一类 q=1的 Young 函数上去.     

一类 q=1的鞅 Φ-不等式

对于 q>1的鞅Φ-不等式,到目前为止已经作了较充分的研究.但是,对于q=1的鞅Φ-不等式,至今尚未见其研究成果.本文就一类满足一定条件的q=1的 Young 函数Φ,给出了与 q>1相类似的鞅Φ-不等式.从而把有关的结果推广到这一类 q=1的 Young 函数上去.     

The Existence of Complete Mappings of SL(2, q), q≡1 Modulo 4

In 1955, Hall and Paige conjectured that any finite group with a noncyclic Sylow 2-subgroup admits complete mappings. For the groups GL(2, q), SL(2, q), PSL(2, q), and PGL(2, q) this conjecture has been proved except for SL(2, q), q odd. We prove that SL(2, q), q≡1 modulo 4 admits complete mappings.     

不等式（n∑i=1ai^3）（n∑i=1bi^3）（n∑i=1ci^3）≥（n∑i=1aibici）^3的恒等式构造

文[1]为证明2001年第42届IMO第2题而通过独特的思路给出了一个恒等式；设实数ai，bi∈R，A^3=n∑i=1ai^3，B^3=n∑i=1bi^3，且AB≠0，则有恒等式     

Characterization of {(q + 1) + 2, 1;t,q}-min · hypers and {2(q + 1) + 2, 2; 2,q}-min · hypers in a Finite projective geometry

In this paper, we shall characterize all {(q + 1) + 2, 1;t, q}-min · hypers and all {2(q + 1) + 2, 2; 2,q}-min · hypers for any integert 2 and any prime powerq 3. In the next paper [8], we shall characterize all {2(q + 1) + 2, 2;t, q}-min · hypers for any integert 3 and any prime powerq 5 using the results in this paper.     

忽视“q=-1”导致的错误

在一些与等比数列有关的问题中,若忽视“q= -1”常会导致一些结论性的错误．题一已知两个等比数列的公比不等,第5项相等,这两个等比数列中除第5项外,还有序号与数值都相等的项吗?(人教B版普通高中课程标准实验教科书《数学》5第50页练习B第3题)     

Simple 3-designs and PSL(2, q) with q ≡ 1 (mod 4)

In this paper, we consider the action of (2, q) on the finite projective line for q ≡ 1 (mod 4) and construct several infinite families of simple 3-designs which admit PSL(2, q) as an automorphism group. Some of the designs are also minimal. We also indicate a general outline to obtain some other algebraic constructions of simple 3-designs.