Finite p-groups all of whose maximal abelian subgroups are soft

A subgroup A of a p-group G is said to be soft in G if CG(A) = A and |NG(A)/A| = p. In this paper we determined finite p-groups all of whose maximal abelian subgroups are soft; see Theorem A and Proposition 2.4.     

真商群为s-自对偶群的有限p-群

如果有限群G的每个子群与G的某个商群同构,则称群G为s-自对偶群.如果s-自对偶群G的每个商群与G的某个子群同构,则称群G为自对偶群.本文分类了每个真商群均为s-自对偶群的有限p-群.作为推论,本文还分类了每个真截段均为s-自对偶群的有限p-群,每个真商群均为自对偶群的有限p-群,以及每个真截段均为自对偶群的有限p-群...     

The parity of the class number of the cyclotomic fields of prime conductor

Using a duality result for cyclotomic units proved by G.Gras, we derive a relation between the vanishing of some -components of the ideal class groups of abelian fields of prime conductor (Theorem 1). As a consequence, we obtain a criterion for the parity of the class number of any abelian number field of prime conductor.

     

任意两个非交换元均生成 p3阶子群的有限 p -群

该文分类了任意两个非交换元均生成 p³阶子群的有限 p -群.作为推论, 完全解决了文献[1]中提出的第237个问题: 对于所有的 x,y∈ G, 研究满足条件( x, y) |≤p³的 p -群 G.     

亚循环的Capable p-群

给定了一个群G,若存在另外的一个群H,能够使得H/Z(H)≌G,则称G是capable群.对cable群进行研究在p-群分类问题的研究中起着相当重要的作用.完全决定了亚循环的capable p-群G.     

关于有限p-群的正规闭包的一些条件

Berkovich 提出了研究满足下列条件的有限p-群G, 对于G的每一个非正规子群H满足(N₁)exp(H)=exp(H^G); (N₂) |H^G:H|≤ p; (N₃) H^G=HG''. 本文首先研究满足条件(N₃) 的有限p-群,然后讨论满足条件(N₁); (N₂) 和(N₃) 的有限p-群.     

On the normal subgroup lattice of a hypercentral p-group

We describe all hypercentral p-groups G whose lattice of normal subgroups n(G) is isomorphic to n(H) for a group H with hypercenwal derived subgroup and H not a pgroup.     

Asymptotics of Analytic Semigroups,II

Let $T(\cdot)$ be an analytic $C_0$-semigroup of operators in a sector $S_{\theta}$, such that $||T(\cdot)||$ is bounded in each proper subsector $S_{\theta_0}$. Let $A$ be its generator, and let $D^{\infty}(A)$ be its set of $C^{\infty}$-vectors. It is observed that the (general) Cauchy integral formula implies the following extension of Theorem 5.3 in [1] and Theorem 1 in [4]: for each proper subsector $S_{\theta_0}$, there exist positive constants $M,\,\delta$ depending only on $\theta_0$, such that $(\delta^n/n!)||z^nA^nT(z)x||\leq M\,||x||$ for all $n\in\Bbb N,\, z\in S_{\theta_0}$, and $x\in D^{\infty}(A)$. It follows in particular that the vectors $T(z)x$ (with $z\in S_{\theta}$ and $x\in D^{\infty}(A)$) are analytic vectors for $A$ (hence $A$ has a dense set of analytic vectors).     

有限伪反射群的相对不变式的 Poincar\'e级数

Let F be a field with characteristic 0, V = Fn the n-dimensional vector space over F and let G be a finite pseudo-reflection group which acts on V . Let χ ： G→ F＊ be a 1- dimensional representation of G. In this article we show that χ（g） = （detg）α（0 ≤ α ≤ r - 1）, where g ∈ G and r is the order of g. In addition, we characterize the relation between the relative invariants and the invariants of the group G, and then we use Molien’s Theorem of invariants to compute the Poincar′e series of relative invariants.     

混合图的Hermitian邻接矩阵的零维数

A mixed graph means a graph containing both oriented edges and undirected edges. The nullity of the Hermitian-adjacency matrix of a mixed graph G, denoted by ηH(G),is referred to as the multiplicity of the eigenvalue zero. In this paper, for a mixed unicyclic graph G with given order and matching number, we give a formula on ηH(G), which combines the cases of undirected and oriented unicyclic graphs and also corrects an error in Theorem 4.2 of [Xueliang LI, Guihai YU. The skew-rank of oriented graphs. Sci. Sin. Math., 2015, 45:93-104(in Chinese)]. In addition, we characterize all the n-vertex mixed graphs with nullity n-3, which are determined by the spectrum of their Hermitian-adjacency matrices.     

有限超特殊p-群的一个注记

本文获得了以下结果：设G为有限超特殊P-群．则下列条件等价：（1）G的非平凡特征子群的阶相同；（2）G的非平凡特征子群唯一；（3）当p〉2时，exp G=p；当P=2时，G≠D8．     

ON GROUPS OF ODD ORDER AND RANK $\[ \le 2\]$

The least commonmultiple of dimensions of chief factors of a finite solvable group is called the rank of the group. The groups of odd order with trivial Fratttini subgroups all of whose subgroups but itself have ranks are completely determined (Theorem A). For an odd number a necessary and sufficient condition for the groups of order n all to be of rank $\[ \le 2\]$ is obtained (Theorem B).     

素数幂、双素数幂阶元的共轭类长的个数为4的 有限群的结构

设群G为一个有限群.如果群G中素数幂、双素幂阶元的共轭类长的集合为{1,p~a,m,p~bm},那么群G是可解的,其中ab为正整数,p为素数且与m互素.进一步,给出了群G/Z(G)的结构,这是对文"Chen R F,Zhao X H.A criterion for a group to have nilpotent p-complements[J].Monatsh Math,2016,179(2):221-225"中定理A主要结论的一个推广.     

On Groups with a CC(n)-Subgroup

A subgroup H of G is a CC(n)-subgroup of G if |G: H| >n and |C_G(x): C_H(x)| ≤n for each element x ∈ H ? {1}. In this article, we study the finite p-groups with a nontrivial CC(p)-subgroup, and the locally nilpotent groups with a nontrivial CC(n)-subgroup.     

A Non-Linear Filtering Problem and Its Applications

For partially observed process in n-dimensional space $\[\left\{ \begin{array}{l} {\beta _i} = {\beta _0} + \int\limits_0^t {{A_1}(s,{\beta _t})ds + \int\limits_0^t {{B_1}(s,{\beta _s})d{w_s}^{(1)},} } \{\xi _i} = \int\limits_0^t {B(s,{\xi _s}){B^*}(s,{\xi _s})\varphi (s,{\beta _s})ds + \int\limits_0^t {B(s,{\xi _s})d{w_s},} } \end{array} \right.\]$ under non-Lipschitz (even discontinuous) condition, a Bayes formula different from [1] is derived (Theorem 1). By means of this formula the innovation problem for the above process under rather weak condition is solved (Theorem 2) .Then the existence of an optimal pathwise Bang-Bang control for a partially observed process with bounded controls is obtained (Theorem 4).     

estimates for a class of oscillatory integrals

We obtain a simpler proof of Theorem 3.1 of The complete mapping properties of some oscillatory integrals in several dimensions, by G. Sampson and P. Szeptycki (Canad. Math. J. 53 (5) (2001), 1031-1056).

     

Normal bases of ray class fields over imaginary quadratic fields

We develop a criterion for a normal basis (Theorem 2.4), and prove that the singular values of certain Siegel functions form normal bases of ray class fields over imaginary quadratic fields other than ${\mathbb{Q}(\sqrt{-1})}$ and ${\mathbb{Q}(\sqrt{-3})}$ (Theorem 4.2). This result would be an answer for the Lang-Schertz conjecture on a ray class field with modulus generated by an integer (≥2) (Remark 4.3).     

Limit Boundary Value Problems of a class of Nonlinear Differential Equations of the Second order

In this paper, the existence and uniqueness of solution of the limit boundary value problem $\[\ddot x = f(t,x)g(\dot x)\]$(F) $\[a\dot x(0) + bx(0) = c\]$(A) $\[x( + \infty ) = 0\]$(B) is considered, where $\[f(t,x),g(\dot x)\]$ are continuous functions on $\[\{ t \ge 0, - \infty < x,\dot x < + \infty \} \]$ such that the uniqueness of solution together with thier continuous dependence on initial value are ensured, and assume: 1)$\[f(t,0) \equiv 0,f(t,x)/x > 0(x \ne 0);\]$; 2) f(t,x)/x is nondecreasing in x>0 for fixed t and non-increasing in x<0 for fixed t, 3)$\[g(\dot x) > 0\]$, In theorem 1, farther assume: 4) $\[\int\limits_0^{ \pm \infty } {dy/g(y) = \pm \infty } \]$ Condition (A) may be discussed in the following three cases $x(0)=p(p \neq 0)$(A_1) $\[x(0) = q(q \ne 0)\]$(A_2) $\[x(0) = kx(0) + r{\rm{ }}(k > 0,r \ne 0)\]$(A_3) The notation $\[f(t,x) \in {I_\infty }\]$ will refer to the function f(t,x) satisfying $\[\int_0^{ + \infty } {\alpha tf(t,\alpha )dt = + \infty } \]$ for each $\alpha \neq 0$, Theorem. 1. For each $p \neq 0$, the boundary value problem (F), (A_1), (B) has a solution if and only if $f(t,x) \in I_{\infty}$ Theorem 2. For each$q \neq 0$, the boundary value problem (F), (A_2), (B) has a solution if and only if $f(t, x) \in I_{\infty}$. Theorem 3. For each k>0 and $r \neq 0$, the boundary value problem (F), (A_3), (B) has a solution if and only if f(t, x) \in I_{\infty}, Theorem 4. The boundary value problem (F), (A_j), (B) has at most one solution for j=l, 2, 3. .     

The Generalized Riemann-Hilbert problem For a Multi-connected Region of Second Order Non-linear Elliptic Equations

In this paper, we consider the generalized Riemann-Hilbert problem for second order non-linear elliptic complex equation $\frac{\partial ^2 w}{\partial \bar z ^2}=F(z,w,\frac{\partial w}{\partial \bar z},\frac{\partial w}{\partial z},\frac{\partial ^2 w}{\partial z \partial \bar z}),z\in G$(1) with the boundary condition $Re[z^-n_1e^-\pii\alpha_1(z)w]=r_1(z),Re[z^-n_2e^\pi i \alpha_2(z) \frac{\partial w}{\partial \bar z}]=r_2(z),z\in \Gamma$ where $\Gamma=\Gamma_0+\Gamma_1+\cdots+\Gamma_m$ is the smooth boundary of a multi-connected region G,$n_i(i=1,2)$ are called the indices of the boundary value problem. we also obtain the following existence theorem of generalized solution. Theorem, suppose that the indices $n_i>m-1$, the coefficients of the complex equation (1) and the boundary condition (2) satisftes the condition (c),and q^0 is sufficiently small, then the seneralized Riemann-Hilbert problem.(1), (2)is solvable and the solution has theexpression (7).