A NEW FAMILY OF FILTRATION S ＋ 5 IN THE STABLE HOMOTOPY GROUPS OF SPHERES

In this article, by the algebraic method, the author proves the existence of a new nontrivial family of filtration s ＋ 5 in the stable homotopy groups of spheres πrS,which is represented by 0 ≠γ^-s＋3hnhm∈Ext^s＋5,A ^t（Zp,Zp）in the Adams spectral sequence,where r=q（p^m＋p^n＋（s＋3）p^2＋（s＋2）p＋（s＋1））-5,t=p^mq＋p^nq＋（s＋3）p^2q＋（s＋2）pq＋（s＋1）q＋s,p≥7,m≥n＋2〉5,0≤s〈p-3,q=2（p-1）.     

A Nontrivial Product of Filtration s ＋ 5 in the Stable Homotopy of Spheres

In this paper, some groups Ext A^s.t （Zp, Zp） with specialized s and t are first computed by the May spectrM sequence. Then we make use of the Adams spectral sequence to prove the existence of a new nontrivial family of filtration s＋5 in the stable homotopy groups of spheres πpnq＋（s＋3）pq＋（s＋1）q-5S which is represented （up to a nonzero scalar） by β＋2bohh∈ExtA^s＋5,P^nq＋（n＋3）pq＋（n＋1）q＋s（Zp, Zp） in the Adams spectral sequence, where p ≥ 5 is a prime number, n ≥3, 0≤ s 〈 p - 3, q = 2（p - 1）.     

A NON-TRIVIAL PRODUCT OF FILTRATION s＋6 IN THE STABLE HOMOTOPY GROUPS OF SPHERES

By a method improving that of [1], the authors prove the existence of a non-trivial product of filtration, s ＋ 6, in the stable homotopy groups of sphere, πt-6S, which is represented up to non-zero scalar by β^-s＋2ho（hmbn-1 -hnbm-1） ∈ ExtA^s＋6,t＋s（Zp, Zp） in the Adams spectral sequence, where p ≥ 7, n ≥ m ＋ 2 ≥ 5, q = 2（p- 1）, 0 ≤ s 〈 p - 2, t= （s ＋ 2 ＋ （s ＋ 2）p ＋ p^m ＋ p^n）q. The advantage of this method is to extend the range of s without much complicated argument as in [1].     

On a family involving R.L. Cohen's ζ-element(II)

In 1981, Cohen constructed an infinite family of homotopy elements ζk∈π*(S) represented by h0bk∈ Ext3,2(p-1)(pk+1+1)A(Z/p, Z/p) in the Adams spectral sequence, where p 2 and k≥1. In this paper,we make use of the Adams spectral sequence and the May spectral sequence to prove that the composite map ζn-1β2γs+3is nontrivial in the stable homotopy groups of spheres πt(s,n)-s-8(S), where p≥7, n 3,0≤s p- 5 and t(s, n) = 2(p- 1)[pn+(s + 3)p2 +(s + 4)p +(s + 3)] + s.     

On the Convergence of Products γ^-sh1hn in the Adams Spectral Sequence

Abstract Let A be the mod p Steenrod algebra and S the sphere spectrum localized at p, where p is an odd prime. In 2001 Lin detected a new family in the stable homotopy of spheres which is represented by （b0hn-h1bn-1）∈ ExtA^3,（p^n＋p）q（Zp,Zp） in the Adams spectral sequence. At the same time, he proved that i.（hlhn） ∈ExtA^2,（p^n＋P）q（H^＊M, Zp） is a permanent cycle in the Adams spectral sequence and converges to a nontrivial element ξn∈π（p^n＋p）q-2M. In this paper, with Lin＇s results, we make use of the Adams spectral sequence and the May spectral sequence to detect a new nontrivial family of homotopy elements jj′j^-γsi^-i′ξn in the stable homotopy groups of spheres. The new one is of degree p^nq ＋ sp^2q ＋ spq ＋ （s - 2）q ＋ s - 6 and is represented up to a nonzero scalar by hlhnγ-s in the E2^s＋2,＊-term of the Adams spectral sequence, where p ≥ 7, q = 2（p - 1）, n ≥ 4 and 3 ≤ s 〈 p.     

A NEW FAMILY OF FILTRATION s＋6 IN THE STABLE HOMOTOPY GROUPS OF SPHERES

In the year 2002, Lin detected a nontrivial family in the stable homotopy groups of spheres πt-6S which is represented by hngor3 ∈ ExtA6,t(Zp,Zp) in the Adams spectral sequence, where t=2pn(p-1) 6(p2 p 1)(p-1) and p≥7 is a prime number.This article generalizes the result and proves the existence of a new nontrivial family of filtration s 6 in the stable homotopy groups of spheres πt1-s-6S which is represented by hngors 3 6 ExtAs 6,t1(Zp, Zp) in the Adams spectral sequence, where n≥4, 0≤s相似文献   

Detection of Some Elements in the Stable Homotopy Groups of Spheres

Let A be the mod p Steenrod algebra and S be the sphere spectrum localized at an odd prime p. To determine the stable homotopy groups of spheres π＊S is one of the central problems in homotopy theory. This paper constructs a new nontrivial family of homotopy elements in the stable homotopy groups of spheres πp^nq＋2pq＋q-3S which isof order p and is represented by kohn ∈ ExtA^3,P^nq＋2pq＋q（Zp,Zp） in the Adams spectral sequence, wherep 〉 5 is an odd prime, n ≥3 and q = 2（p-1）. In the course of the proof, a new family of homotopy elements in πp^nq＋（p＋1）q-1V（1） which is represented by β＊i＇＊i＊（hn） ∈ ExtA^2,pnq＋（p＋1）q＋1 （H^＊V（1）, Zp） in the Adams sequence is detected.     

SOME NEW FAMILIES OF FILTRATION FIVE IN THE STABLE HOMOTOPY OF SPHERES

This study proves a general result on convergence of α2x ∈ ExtA^s＋2.tq＋2q＋1 （Zp, Zp） in the Adams spectral sequence and as a consequence, the study detects some new families in the stable homotopy groups of spheres πtq＋2q-4S which is represented in the Adams spectral sequence by α2fn,α2fn,α2huhmhn ∈ ExtA^5,tq＋2q＋1（Zp,Zp） with tq=p^n＋1q＋2p^nq,2p^n＋1q_P^nq,p^uq＋p^mq＋p^nq,respectively, where α2∈Extα^2,2q＋1（Zp,Zp）,fn∈ExtA^3,p^n＋1q＋2p^nq（Zp,Zp）,fn∈ExtA^3,2p^n＋2q＋p^nq（Zp,Zp）,hn∈ExtA^1,p^nq（Zp,Zp）and p≥5 is a prime,q=2（p=-1）,n≥2.     

A pull back theorem in the Adams spectral sequence

This paper proves that, for any generator x∈ExtA^s,tq（Zp,Zp）, if （1L ∧i）＊Ф＊（x）∈ExtA^s＋1,tq＋2q（H＊L∧M, Zp） is a permanent cycle in the Adams spectral sequence （ASS）, then b0x ∈ExtA^s＋1,tq＋q（Zp, Zp） also is a permenent cycle in the ASS. As an application, the paper obtains that h0hnhm∈ExtA^3,pnq＋p^mq＋q（Zp, Zp） is a permanent cycle in the ASS and it converges to elements of order p in the stable homotopy groups of spheres πp^nq＋p^mq＋q-3S, where p ≥5 is a prime, s ≤ 4, n ≥m＋2≥4 and M is the Moore spectrum.     

Whiteman广义分圆序列的自相关值

Let p,q be distinct primes with gcd(p-1,q-1) = 4.In this paper,we calculate the autocorrelation values of binary sequences of generalized cyclotomic sets of order four in Z pq and get conditions of p and q such that they are four-valued.     

σ-相关同伦元素的非平凡性

本文中,通过几何方法证明了σ相关同伦元素在球面稳定同伦群π_mS中是非平凡的,其中m=p~(n+1)q+2p~nq+(s+3)p~2q+(s+3)pq+(s+3)q-8,p≥7是奇素数,n3,0≤sp-3,且q=2(p-1).该σ相关同伦元素在Adams谱序列的E_2-项中由■_s+3■_ng0表示.     

A Relation in the Stable Homotopy Groups of Spheres

Let p≥7 be an odd prime. Based on the Toda bracket α1βp-11, α1 β1, p, γs,the authors show that the relation α1βp-11h2,0 γs= βp/p-1γs holds. As a result, they can obtain α1βp1h2,0 γs = 0 ∈π*(S0) for 2≤s≤p- 2, even though α1h2,0γs and β1α1h2,0 γs are not trivial. They also prove that βp-11α1 h2,0 γ3 is nontrivial in π*(S0) and conjecture that βp-11α1 h2,0 γs is nontrivial in π*(S0) for 3≤s≤p- 2. Moreover, it is known thatβp/p-1γ3 = 0 ∈ Ext5,*BP*BP(BP*, BP*), but βp/p-1γ3 is nontrivial in π*(S0) and represents the element βp-11α1 h2,0 γ3.     

球面稳定同伦群中的$\xi_n$-相关元素的非平凡性

利用Adams谱序列与May谱序列, 发掘了球面稳定同伦群中一族$\xi_n$的相关元素. 这里$\xi_n\in\pi_* M$在Adams 谱序列中由$h_0h_n\in \ext_A^{2,p^n q+q}(H^* M,\zz_p)$所表示, 其中$p\geqslant 7,\ n>3,\ q=2(p-1).$     

关于同伦元素 $\alpha_{1}\beta_{1}\beta_{2}\gamma_{s}$

设 $p\geq 7$ 为任意奇素数. 证明了当 $3\leq s 相似文献   

有循环极大子群的素数幂阶群的作用是边传递的图(II)

假定Γ是一个有限的、单的、无向的且无孤立点的图,G是Aut(Γ)的一个子群.如果G在Γ的边集合上传递,则称Γ是G-边传递图.我们完全分类了当G为一个有循环的极大子群的素数幂阶群时的G-边传递图.结果为:设图Γ含有一个阶为pn(p是素数,n≥2)的自同构群,且G有一个极大子群循环,则Γ是G-边传递的,当且仅当Γ同构于下列图之一1)pmK1,pn-1-m,0≤m≤n-1;2)pmK1,pn-m,0≤m≤n;3)pmKp,pn-m-1,0≤m≤n-2;4)pn-mCpm,pm≥3,m＜n;5)2n-2K1,1;6)pn-1-mCpm,pm≥3,m≤n-1;7)2pn-mCpm,pm≥3,m≤n-1;8)2pn-mK1,pm,0≤m≤n;9)pn-mK1,2pm,0≤m≤n;10)pn-mK2,pm,0＜m≤n;11)C(2pn-m,1,pm);12)pkC(2pm-k,1,pn-m),0＜k＜m,0＜m≤n;13)(t-s,2m)C(2m 1/(t-s,2m),1,2n-1-m),其中0≤m≤n-1,2n-2(s-1)≡0(mod 2m),t≡1(mod 2),s(≠)t(mod 2m),1≤s≤2m,1≤t≤2n-1;14)∪p i=1 Ci p n-1,其中Ci p n-1=Ca1a1 [1 (i-1)pn-2]a 1 2[1 (i--1)p n-2]…a 1 (pn-1-1)[1 (i-1)p n-2]≌Cp n-1,i=1,2,…,p;15)∪2 i=1 Ci 2n-1,其中Ci 2n-1=Ca1a 1 [1 (i-1)(2n-2-1)]a1 2[1 (i-1)(2n-2-1)]…a1 (2n-1-1)[1 (i-1)(2n-2-1)]≌C2n-1,i=1,2.     

On some congruences of certain binomial sums

For any prime \(p>3,\) we prove that

$$\begin{aligned} \sum _{k=0}^{p-1}\frac{3k+1}{(-8)^k}{2k\atopwithdelims ()k}^3\equiv p\left( \frac{-1}{p}\right) +p^3E_{p-3}\pmod {p^4}, \end{aligned}$$

where \(E_{0},E_{1},E_{2},\ldots \) are Euler numbers and \(\left( \frac{\cdot }{p}\right) \) is the Legendre symbol. This result confirms a conjecture of Z.-W. Sun. We also re-prove that for any odd prime \(p,\)

$$\begin{aligned} \sum _{k=0}^{\frac{p-1}{2}}\frac{6k+1}{(-512)^k}{2k\atopwithdelims ()k}^3\equiv p\left( \frac{-2}{p}\right) \pmod {p^2} \end{aligned}$$

using WZ method.

     

Some combinatorics of binomial coefficients and the Bloch-Gieseker property for some homogeneous bundles

A vector bundle has the Bloch-Gieseker property if all its Chern classes are numerically positive. In this paper we show that the non-ample bundle has the Bloch-Gieseker property, except for two cases, in which the top Chern classes are trivial and the other Chern classes are positive. Our method is to reduce the problem to showing, e.g. the positivity of the coefficient of in the rational function (for even).

     

具$p$-Laplacian 算子的多点边值问题迭代解的存在性

利用单调迭代技巧和推广的Mawhin定理得到下述带有p-Laplacian算子的多点边值问题迭代解的存在性,{(Фp(u'))' f(t,u, Tu)=0, 0(≤)t(≤)1,u(0)=q-1∑i=1γiu(δi),u(1)=m-1∑i=1ηiu(ξi),其中Фp(s)=|s|p-2s,p＞1;0＜δi＜1,γi＞0,1(≤)i(≤)q-1;0＜ξi＜1,ηi(≥)0,1(≤)i(≤)m-1且q-1∑i=1γi＜1,m-1∑i=1ηi(≤)1;Tu(t)=∫t0k(t,s)u(s)ds,k(t,s)∈C(I×I,R ).