Groups G2(p) as quotients of (2,3,7;2p)

It is proved that for any prime the group is a quotient of      

Locally Finite Periodic Groups Saturated with Finite Simple Orthogonal Groups of Odd Dimension

Siberian Mathematical Journal - Suppose that $ n $ is an odd integer, $ n\geq 5 $ . We prove that a periodic group  $ G $ , saturated with finite simple orthogonal groups  $...     

一类精细化Eulerian多项式的若干性质

最近,孙华定义了一类新的精细化Eulerian多项式,即$$A_n(p,q)=\sum_{\pi\in \mathfrak{S}_n}p^{{\rm odes}(\pi)}q^{{\rm edes}(\pi)},\ \ n\ge 1,$$ 其中$S_n$表示$\{1,2,\ldots,n\}$上全体$n$阶排列的集合, odes$(\pi)$与edes$(\pi)$分别表示$S_n$中排列$\pi$的奇数位与偶数位上降位数的个数.本文利用经典的Eulerian多项式$A_n(q)$ 与Catalan 序列的生成函数$C(q)$,得到精细化Eulerian 多项式$A_n(p,q)$的指数型生成函数及$A_n(p,q)$的显示表达式.在一些特殊情形,本文建立了$A_n(p,q)$与$A_n(0,q)$或$A_n(p,0)$之间的联系,并利用Eulerian数表示多项式$A_n(0,q)$的系数.特别地,这些联系揭示了Euler数$E_n$与Eulerian数$A_{n,k}$之间的一种新的关系.     

NUMBERS OF CONJUGATE CLASSES OF SYMMETRIC AND ALTERNATING GROUPS

Let d(n)be the excess of the number of even conjugate classes of S_n over that of oddconjugate classes of S_n and q(n)the number of splitting classes of S_n.In this paper arecurrence formula for d(n)and one for q(n)are given.As a recurrence formula for thenumber p(n)of conjugate classes of S_n is known,one can make use of p(n),d(n)andq(n)to calculate the numbers of even(odd)conjugate classes of S_n and that of conjugateclasses of A_n.By means of a graphical method the author proves the identity d(n)=q(n)when n≥2,which seems to have been obtained first by Sylvester by use of generatingfunctions.     

量子外代数的Galois覆盖代数的Hochschild上同调

本文研究一类量子代数$\Lambda^n_q$的Hochschild上同调.量子代数$\Lambda^n_q$的极小投射双模分解被构造, $\Lambda^n_q$的各阶Hochschild上同调群的维数被清晰的给出.此外,对一些特殊的情况, $\Lambda^n_q$的上同调环也被清晰的刻画.     

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$p\geqslant 7,\ n>3,\ q=2(p-1).$     

边界条件含有特征参数的Sturm-Liouville算子的唯一性定理

建立了一类Sturm-Liouville问题的唯一性定理.对于固定的n∈Z,证明了该Sturm-Liouville问题的第n个特征值λ_n（q,a）关于a是严格单调的.对不同系数的a_k,如果能够测得第n个特征值的谱集合{λ_n（q,a_k）}_k=1^+∞,则谱集合{λ_n（q,a_k）}_k=1^+∞能够唯一确定[0,π]上的势函数q（x）.     

两类积图的邻点可区别全染色

本文.证明了,当n≥2时,Xat(K_n×K′_n)=2n;当p,q≥2时,Xat(C_(2p)×K_(2q))=2q 3,其中K_n×K′_n是两个不同标号完全图的积图,C_(2p)×K_(2q)是偶圈和偶阶完全图的积图.     

Homotopy Groups of Spheres and Lipschitz Homotopy Groups of Heisenberg Groups

We provide a sufficient condition for the nontriviality of the Lipschitz homotopy group of the Heisenberg group, ${\pi_m^{\rm Lip}(\mathbb{H}_n)}$ , in terms of properties of the classical homotopy group of the sphere, ${\pi_m(\mathbb{S}^n)}$ . As an application we provide a new simplified proof of the fact that ${\pi_n^{\rm Lip}(\mathbb{H}_n)\neq \{0\}, n=1,2,\ldots}$ , and we prove a new result that ${\pi_{4n-1}^{\rm Lip}(\mathbb{H}_{2n})\neq \{0\}}$ for n = 1,2,… The last result is based on a new generalization of the Hopf invariant. We also prove that Lipschitz mappings are not dense in the Sobolev space ${W^{1,p}(\mathcal{M},\mathbb{H}_{2n})}$ when ${\dim \mathcal{M} \geq 4n}$ and 4n?1 ≤  p < 4n.     

Recognition of Finite Simple Groups S 4(q) by Their Element Orders

It is proved that among simple groups $S_4(q)$ in the class of finite groups, only the groups $S_4(3^n)$, where $n$ is an odd number greater than unity, are recognizable by a set of their element orders. It is also shown that simple groups $U_3(9)$, ${^3D}_4(2)$, $G_2(4)$, $S_6(3)$, $F_4(2)$, and ${^2E}_6(2)$ are recognizable, but $L_3(3)$ is not.     

有限域上的本原最优正规元

Let q be a prime or prime power and Fqn the extension of q elements finite field Fq with degree n(n1).Davenport,Lenstra and Schoof proved that there exists a primitive element α∈ Fqn such that α generates a normal basis of Fqn over Fq.Later,Mullin,Gao and Lenstra,etc.,raised the definition of optimal normal bases and constructed such bases.In this paper,we determine all primitive type I optimal normal bases and all finite fields in which there exists a pair of reciprocal elements α and α-1 such that both of them generate optimal normal bases of Fqn over Fq.Furthermore,we obtain a sufficient condition for the existence of primitive type II optimal normal bases over finite fields and prove that all primitive optimal normal elements are conjugate to each other.     

一类奇型Sturm-Liouville算子的逆问题

研究了奇型Sturm-Liouville算子的逆问题.对于固定的n∈N,证明了Sturm-Liouville问题(1.3)-(1.5)的第n个特征值λ_n(q,H)关于H是严格单调增加的,及一组不同边界条件下的第n个特征值的谱集合{λ_n(q,H_k)}_(k=1)~(+∞)能够唯一确定(0,πr)上的势函数q(x).     

一类单圈图的拉普拉斯谱刻画

Let H（n; q, n1, n2, n3, n4） be a unicyclic graph with n vertices containing a cycle Cq and four hanging paths Ph1＋1, Pn2＋1, Pn3＋1 and Pn4＋1 attached at the same vertex of the cycle. In this paper, it is proved that all unicyclic graphs H （n; q, n1, n2, n3, n4） are determined by their Laplacian spectra.     

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.     

Cesăro summability of the character system of the p-series field in the Kaczmarz rearrangement

Let $G_p$ be the $p$-series field. In this paper we prove the a.e. convergence $\sigma_n f\to f$ $(n\to \infty)$ for an integrable function $f\in L^1(G_p)$, where $\sigma_nf$ is the $n$th $(C,1)$ mean of $f$ with respect to the character system in the Kaczmarz rearrangement. We define the maximal operator $\sigma^* $ by $\sigma^*f := \sup_n|\sigma_nf|$. We prove that $\sigma^*$ is of type $(q,q)$ for all $1相似文献   

Möbius Homogeneous Hypersurfaces with Three Distinct Principal Curvatures in Sn+1

Let x : M~n→ S~(n+1) be an immersed hypersurface in the(n + 1)-dimensional sphere S~(n+1). If, for any points p, q ∈ Mn, there exists a Mbius transformation φ :S~(n+1)→ S~(n+1) such that φox(Mn~) = x(M~n) and φ ox(p) = x(q), then the hypersurface is called a Mbius homogeneous hypersurface. In this paper, the Mbius homogeneous hypersurfaces with three distinct principal curvatures are classified completely up to a Mbius transformation.     

On the absence of positive eigenvalues of Schrödinger operators with rough potentials

We prove the absence of positive eigenvalues of Schrödinger operators $ H=-\Delta+V $ on Euclidean spaces $ \mathbb{R}^n $ for a certain class of rough potentials $V$. To describe our class of potentials fix an exponent $q\in[n/2,\infty]$ (or $q\in(1,\infty]$, if $n=2$) and let $\beta(q)=(2q-n)/(2q)$. For the potential $V$ we assume that $V\in L^{n/2}_{{\rm{loc}}}(\mathbb{R}^n)$ (or $V\in L^{r}_{{\rm{loc}}}(\mathbb{R}^n)$, $r>1$, if $n=2$) and$\begin{equation*}$$\lim_{R\to\infty}R^{\beta(q)}||V||_{L^q(R\leq |x|\leq 2R)}=0\,.$$\end{equation*}$Under these assumptions we prove that the operator $H$ does not admit positive eigenvalues. The case $q=\infty$ was considered by Kato [K]. The absence of positive eigenvalues follows from a uniform Carleman inequality of the form$\begin{equation*}$$||W_m u||_{l^a(L^{p(q)})(\mathbb R^n)}\leq C_q||W_m|x|^{\beta(q)}(\Delta+1)u||_{l^a(L^{p(q)})(\mathbb{R}^n)}$$\end{equation*}$for all smooth compactly supported functions $u$ and a suitable sequence of weights $W_m$, where $p(q)$ and $p(q)$ are dual exponents with the property that $1/p(q)-1/p(q)=1/q$.     

Line Transversals to Translates of Unit Discs

Let be a family of convex figures in the plane. We say that has property T if there exists a line intersecting every member of . Also, the family has property T(k) if every k-membered subfamily of has property T. Let B be the unit disc centered at the origin. In this paper we prove that if a finite family of translates of B has property T(4) then the family , where , has property T. We also give some results concerning families of translates of the unit disc which has either property T(3) or property T(5).     

Real Computational Universality: The Word Problem for a Class of Groups with Infinite Presentation

The word problem for discrete groups is well known to be undecidable by a Turing Machine; more precisely, it is reducible both to and from and thus equivalent to the discrete Halting Problem. The present work introduces and studies a real extension of the word problem for a certain class of groups which are presented as quotient groups of a free group and a normal subgroup. As a main difference to discrete groups these groups may be generated by uncountably many generators with index running over certain sets of real numbers. We study the word problem for such groups within the Blum–Shub–Smale (BSS) model of real number computation. The main result establishes the word problem to be computationally equivalent to the Halting Problem for such machines. It thus gives the first non-trivial example of a problem complete, that is, computationally universal for this model. M. Ziegler supported by (project Zi1009/1-2).     

循环群$Z_{2}$的模向量不变式

Let G be the finite cyclic group Z_2 and V be a vector space of dimension 2n with basis x_1,...,x_n,y_1,...,y_n over the field F with characteristic 2.If σ denotes a generator of G,we may assume that σ（x_i）= ayi,σ（y_i）= a～-1x_i,where a ∈ F.In this paper,we describe the explicit generator of the ring of modular vector invariants of F[V]～G.We prove that F[V]～G = F[l_i = x_i ＋ ay_i,q_i = x_iy_i,1 ≤ i ≤ n,M_I = X_I ＋ a～-I-Y_I],where I∈An = {1,2,...,n},2 ≤-I-≤ n.