Regularity of Positive Solutions for an Integral System on Heisenberg Group

In this paper, we are concerned with the properties of positive solutions of the following nonlinear integral systems on the Heisenberg group $\mathbb{H}^n$, \begin{equation} \left\{\begin{array}{ll} u(x)=\int_{\mathbb{H}^n}\frac{v^{q}(y)w^{r}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ v(x)=\int_{\mathbb{H}^n}\frac{u^{p}(y)w^{r}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ w(x)=\int_{\mathbb{H}^n}\frac{u^{p}(y)v^{q}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ \end{array}\right.\end{equation} for $x\in \mathbb{H}^n$, where $0<\alpha 1$ satisfying $\frac{1}{p+1} $+ $\frac{1}{q+1} + \frac{1}{r+1} = \frac{Q+α+β}{Q}.$ We show that positive solution triples $(u,v,w)\in L^{p+1}(\mathbb{H}^n)\times L^{q+1}(\mathbb{H}^n)\times L^{r+1}(\mathbb{H}^n)$ are bounded and they converge to zero when $|x|→∞.$     

有限群的ss-置换子群

设G为有限群,称G的子群H为ss-置换子群,如果存在G的次正规子群B使得G=HB,且H与B的任意Sylow子群可以交换,即对任意X∈Syl（B）有XH=HX.利用子群的ss-置换性来研究有限群的结构,得到有限群超可解的两个充分条件.     

2-(v,6,1)设计的可解区传递自同构群

设G是一个2-（v,6,1）设计的可解区传递自同构群,且G非旗传递,则：（1）v＝91,G=Z91×Zd,这里3|d|12;（2）v=pm,G≤AL（1,pm）,之一成立．其中p≠2．当p＝3时,4|m见且m＞4;当p＞5时,pm≡1（mod30）。     

Symmetry and regularity of solutions to a system with three-component integral equations

Consider the system with three-component integral equations u(x) = Rn |x y|α nw(y)rv(y)q dy,v(x) = Rn |x y|α nu(y)pw(y)rdy,w(x) = Rn |x y|α nv(y)q u(y)pdy,where 0 < α < n,n is a positive constant,p,q and r satisfy some suitable conditions.It is shown that every positive regular solution(u(x),v(x),w(x)) is radially symmetric and monotonic about some point by developing the moving plane method in an integral form.In addition,the regularity of the solutions is also proved by the contraction mapping principle.The conformal invariant property of the system is also investigated.     

没有5-圈的平面图的BB-染色

设H为G的一个生成子图,（G,H）的一个BB-k-染色是指一个映射f：V（G）→{1,2,…,k},当uv∈E（H）,｜f（u）-f（v）｜≥2;当uv∈E（G）／E（H）,｜f（u）-f（v）｜≥1.定义（G,H）的BB色数x_b（G,H）为最小的整数k,使得（G,H）是BB-k可染的.本文研究了对于任意的连通,非二部平面图G,且G没有5-圈,都存在一棵生成树T,使得x_b（G,T）=4.     

Pm×Kn的邻点可区别全色数

设G是简单图．设f是一个从V(G)∪E(G)到{1,2,…,k}的映射．对每个v∈V(G),令C_f(v)={f(v)}∪{f(vw)|w∈V(G),vw∈E(G)}．如果f是k-正常全染色,且对任意u,v∈V(G),uv∈E(G),有C_f(u)≠C_f(v),那么称f为图G的邻点可区别全染色(简称为k-AVDTC)．数x_(at)(G)=min{k|G有k-AVDTC}称为图G的邻点可区别全色数．本文给出路P_m和完全图K_n的Cartesion积的邻点可区别全色数．     

L~p-decay rates to nonlinear diffusion waves for p-system with nonlinear damping

In this paper, we study the Lp (2≤p≤ ∞) convergence rates of the solutions to the Cauchy problem of the so-called p-system with nonlinear damping. Precisely, we show that the corresponding Cauchy problem admits a unique global solution (v (x,t), u(x, t)) and such a solution tends time-asymptotically to the corresponding nonlinear diffusion wave ((v|-)(x,t),(u|-)(x,t)) governed by the classical Darcy's law provided that the corresponding prescribed initial error function lies in and is sufficiently small. Furthermore, the Lp (2≤p≤ ∞) convergence rates of the solutions are also obtained.     

Cocyclic Generalised Hadamard Matrices and Central Relative Difference Sets

Cocyclic matrices have the form where G is a finite group, C is a finite abelian group and : G × G C is a (two-dimensional) cocycle; that is,

可解群的若干充分条件

A subgroup if of a group G is called semipermutable if it is permutable with every subgroup K of G with (|H|, |K|) - 1, and s-semipermutable if it is permutable with every Sylow p-subgroup of G with (p, |H|) = 1. In this paper, some sufficient conditions for a group to be solvable are obtained in terms of s-semipermutability.     

s-半置换子群对群构造的影响

群G的一个子群H称为半置换的,若对G的任意子群K,只要(|H|,|K|)=1,就有HK=KH;H称为s-半置换的,若对G的任意Sylow p-子群P,只要(p,|H|)=1,就有HP=PH.本文研究极大子群和极小子群的某些子群的s-半置换性对群构造的影响,推广和改进了Asaad及王品超等人所得的结果.     

偶图的K3.3剖分

设G＝（X，Y；E）是一个偶图。如果|X|≥2|Y|-3且d(v)=3对任意v∈X,那么G含有K3.3的剖分。有例子表明|X|的下界在一定程度上是不可改进的。     

关于P_(r,(2s-1))的奇优美标号

对于简单图G=〈V,E〉,如果存在一个映射f:V(G)→{0,1,2,…,2 |E|-1}满足1)对任意的u,v∈V,若u≠v,则(u)≠f(v);2)max{f(v)|v∈V}=2|E|-1;3)对任意的e_1,e_2∈E,若e_1≠e_2,则g(e_1)≠g(e_2),此处g(e)=|f(u)+f(v)|,e=uv;4){g(e)|e∈E}={1,3,5,…,2|E|-1},则称G是奇优美图,f称为G的奇优美标号.Gnanajoethi提出了一个猜想:每棵树都是奇优美的.证明了图P_(r,(2s-1)是奇优美图.     

Some remarks on systems of elliptic equations doubly critical in the whole $${\mathbb{R}^N}$$

We study the existence of different types of positive solutions to problem

Regular Component Splittable Languages

Every infinite regular language contains a regular subset of the formuv+w for some words u, v, w, where v is not the empty word. The regularsubsets of the above form are called regular components. Some well-knowncontext-free languages, such as the Dyck language and the balanced language(over an alphabet X with |X| = 2), are decomposed as disjointunions of disjunctive languages. In this paper, we investigate thedecompositions of some of the regular languages and the context-freelanguages as disjoint unions of regular components. An infinite language iscalled regular component splittable if it can be expressed as a disjointunion of regular components and a finite set. We show that the Dycklanguage, the balanced language and some disjunctive context-free splittablelanguages are regular component splittable. We also present an example toshow that there is a disjunctive context-free language which is not regularcomponent splittable.     

Contractions Satisfying the Absolute Value Property

Let B(H) denote the algebra of operators on a complex Hilbert space H, and let U denote the class of operators which satisfy the absolute value condition . It is proved that if is a contraction, then either A has a nontrivial invariant subspace or A is a proper contraction and the nonnegative operator is strongly stable. A Putnam-Fuglede type commutativity theorem is proved for contractions A in , and it is shown that if normal subspaces of . It is proved that if are reducing, then every compact operator in the intersection of the weak closure of the range of the derivation with the commutant of A* is quasinilpotent.     

Two Kinds of Boundary Value Problems for the First Order Systems of Nonlinear Elliptic Partial Differential Equations

In this paper we establish the esxistence and uniqueness theorems of a solution of the problem P(|z|=1,Re[z^-nw_z]=0) and H(|z|=1,Re[z^-nw]=0) for the nonlinear system (A):w_{\bar z}=g(z,w,w_z),|g(z,w,w_z^1)-g(z,w,w_z^2)| \leq q_0|w_z^1-w_z^2|,q_0=const.<1,z \in G={|z|<1}.for the problem P,where w(z) \in C_\alpha^1(\bar G),let C_\alpha(g(z,w,t)-g(z,w_1,t_1),\bar G) \leq H_3C_\alpha(w-w_1,\bar G)+H_4C_\alpha(t-t_1,\bar G),H_3,H_4 and C_\alpha(g(z,0,0),\bar G)be the suitable small constants.For the problem H,where w(z) \inW_p^(1)(\bar G),p>2,let |g(z,w,0)| \leq A(z,w)|w|^m+B(z,w),m>1,A(z,w) \in L_p(\bar G) and ||B(z,w)||_{L_p(\bar G)} be the suitable small.     

关于亚纯代数体函数奇异方向的存在性

本文证明了如下结果:对于有穷正级亚纯代数体函数,一定存在一条奇异方向L∶arg z=θ0(0≤θ0＜2π),使得对于任意δ∈(0,π/2),在角域Δ(θ0,δ)内,对任意复数a,对任意ε＞0,有∑i1|zi(a;Δ(θ0,δ))|σ=∞(σ等于ρ或ρ-ε)至多有2v个例外a值.     

The Normalizer Property for Integral Group Rings of Holomorphs of Finite Groups with Trivial Center

Let G=Hol(H) be the holomorph of a finite group H. If there is a prime q dividing |H| such that every q-central automorphism of H is inner and Z(H)=1, then every Coleman automorphism of G is inner. In particular, the normalizer property holds for G.     

Weighted Inequalities for Certain Maximal Functions in Orlicz Spaces

Let M_g be the maximal operator defined by $$M_g f\left( x \right) = \sup \frac{{\int_a^b {f\left( y \right)g\left( y \right){\text{d}}y} }}{{\int_a^b {g\left( y \right){\text{d}}y} }}$$ , where g is a positive locally integrable function on R and the supremum is taken over all intervals [a,b] such that 0≤a≤x≤b/η(b?a), here η is a non-increasing function such that η (0) = 1 and $\mathop {{\text{lim}}}\limits_{t \to {\text{ + }}\infty } \eta \left( t \right) = 0$ η (t) = 0. This maximal function was introduced by H. Aimar and L. L. Forzani [AF]. Let Φ be an N - function such that Φ and its complementary N - function satisfy Δ₂. It gives an A′_Φ(g) type characterization for the pairs of weights (u,v) such that the weak type inequality $$u\left( {\left\{ {x \in {\text{R}}\left| {M_g f\left( x \right) >\lambda } \right.} \right\}} \right) \leqslant \frac{C}{{\Phi \left( \lambda \right)}}\int_{\text{R}} {\Phi \left( {\left| f \right|v} \right)} $$ holds for every f in the Orlicz space L_Φ(v). And, there are no (nontrivial) weights w for which (w,w) satisfies the condition A′_Φ(g).     

Oscillation of nonlinear vector differential equations

Summary Oscillation criteria are obtained for vector partial differential equations of the type Δv+b(x, v)v=0, x∈G, v∈E^m, where G is an exterior domain in E_n, and b is a continuous nonnegative valued function in G × E^m. A solution v: G→E^m is called h-oscillatory in G whenever the scalar product [v(x), h] (|h|=1) has zeros x in G with |x| arbitrarily large. It is shown that the spherical mean of [v(x), h] over a hypersphere of radius r in Eⁿ satisfies a nonlinear ordinary differential inequality. As a consequence, the main theorems give sufficient conditions on b(x, t), depending upon the dimension n, for all solutions v to be h-oscillatory in G. Entrata in Redazione il 26 giugno 1975.