Automorphisms of Extensions of Q by a Direct Sum of Finitely Many Copies of Q

Let G be an extension of Q by a direct sum of r copies of Q.(1) If G is abelian, then G is a direct sum of r + 1 copies of Q and Aut G = GL(r + 1, Q);(2) If G is non-abelian, then G is a direct product of an extraspecial Q-group E and m copies of Q, where E/ζ E is a linear space over Q with dimension 2 n and m + 2 n = r. Furthermore, let Aut_G'G be the normal subgroup of Aut G consisting of all elements of Aut G which act trivially on the derived subgroup G of G, and Aut_(G/ζG),_(ζG)G be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on the center ζ G of G. Then(i) The extension 1→ Aut_(G')G→ Aut G→ Aut G'→ 1 is split;(ii)Aut_(G')G/Aut_(G/ζG),_(ζG)G = Sp(2 n, Q) ×(GL(m, Q) Q~(m));(iii) Aut_(G/ζG),ζGG/Inn G= Q~(2 nm).     

Constructive characterizations of （γp, γ）- and （γp, γpr）-trees

Let G =(V,E) be a graph without isolated vertices.A set S  V is a domination set of G if every vertex in V -S is adjacent to a vertex in S,that is N[S] = V .The domination number of G,denoted by γ(G),is the minimum cardinality of a domination set of G.A set S  V is a paired-domination set of G if S is a domination set of G and the induced subgraph G[S]has a perfect matching.The paired-domination number,denoted by γpr(G),is defined to be the minimum cardinality of a paired-domination set S in G.A subset S  V is a power domination set of G if all vertices of V can be observed recursively by the following rules:(i) all vertices in N[S] are observed initially,and(ii) if an observed vertex u has all neighbors observed except one neighbor v,then v is observed(by u).The power domination number,denoted by γp(G),is the minimum cardinality of a power domination set of G.In this paper,the constructive characterizations for trees with γp = γ and γpr = γp are provided respectively.     

The Jacobson Radical of （R[G]）^G

In this paper,the Jacobson radical of (R[G])G, the fixed point subring of a group ring (R[G]) is studied. Some related properties of (R[G])G are discussed.     

Ladder图和Mbius Ladder图的彩虹点连通数（英文）

A vertex-colored graph G is said to be rainbow vertex-connected if every two vertices of G are connected by a path whose internal vertices have distinct colors, such a path is called a rainbow path. The rainbow vertex-connection number of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex-connected. If for every pair u, v of distinct vertices, G contains a rainbow u-v geodesic, then G is strong rainbow vertex-connected. The minimum number k for which there exists a k-vertex-coloring of G that results in a strongly rainbow vertex-connected graph is called the strong rainbow vertex-connection number of G, denoted by srvc(G). Observe that rvc(G) ≤ srvc(G) for any nontrivial connected graph G. In this paper, for a Ladder L_n,we determine the exact value of srvc(L_n) for n even. For n odd, upper and lower bounds of srvc(L_n) are obtained. We also give upper and lower bounds of the(strong) rainbow vertex-connection number of Mbius Ladder.     

RESEARCH ANNOUNCEMENTS Galos Correspondence in Field Algebra of G—spin Model

A C*-system is a pair (B, G) consisting of a unital C*-algebra B and a continuous group homomorphism α: G → Aut(B) where G is a compact group and Aut(B) the group of automor-phisms of B. If K is a normal subgroup of G and BK = {B∈ B: k(B) = B, k ∈ K}, then BK is a G-invariant C*-subalgebra of B. On the other hand, if A is a G-invariant C*-algebra with BG A B, set G (A) = {g ∈ G: g(A) = A, A ∈ A}, G (A) is a normal subgroup of G. Clearly K G(BK) and we call K Galois closed ifK = G(BK). Similarly, A BG(A) and we call A Galois closed if A = BG(A).     

On the p—Local Rank of Finite Groups

In this paper,we shall mainly study the p-solvable finite group in terms of p-local rank,and a group theoretic characterization will be given of finite p-solvabel groups with p-local rank two.Theorem A Let G be a finite p-solvable group with p-local rank plr(G)=2 and Op(G)=1.If P is a Sylow p-subgrounp of G,then P has a normal subgroup Q such that P/Q is cyclic or a generalized quaternion 2-group and the p-rank of Q is at most two.Theorem B Let G be a finite p-solvable group with Op(G)=1.Then the p-length lp(G)≤plr(G);if in addition plr(G)=lp (G) and p≥5 is odd,then plr(G)=0 or 1.     

On （g, f）-Uniform Graphs

A graph G is called a (g, f)-uniform graph if for each edge of G, there is a (g, f)-factor containing it and another (g, f)-factor excluding it. In this paper a necessary and sufficient condition for a graph to be a (g, f)-uniform graph is given and some applications of this condition are discussed. In particular, some simple sufficient conditions for a graph to be an [a, b]-uniform graph are obtained for a≤b.     

Constructions of （q,K, λ, t,Q） almost difference families

The concept of a （q, k, λ, t） almost dltterence tamlly （ADF） nas oeen introduced and studied by C. Ding and J. Yin as a useful generalization of the concept of an almost difference set. In this paper, we consider, more generally, （q, K,λ, t, Q）-ADFs, where K = {k1, k2,.…, kr} is a set of positive integers and Q = （q1,q2,... ,qr） is a given block-size distribution sequence. A necessary condition for the existence of a （q, K, λ, t, Q）-ADF is given, and several infinite classes of （q, K, A, t, Q）-ADFs are constructed.     

本刊英文版Vol.33(2017),No.6论文摘要（英文）

正A Comparison between the Metric Dimension and Zero Forcing Number of Trees and Unicyclic Graphs Linda EROH Cong X.KANG Eunjeong YI Abstract The metric dimension dim(G)of a graph G is the minimum number of vertices such that every vertex of G is uniquely determined by its vector of distances to the chosen vertices.The zero forcing number Z(G)of a graph G is the minimum cardinality of a set S of     

完全二部图K_(8,n)的点可区别IE-全染色（英文）

Let G be a simple graph. An IE-total coloring f of G is a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color. For each vertex x of G, let C(x) be the set of colors of vertex x and edges incident to x under f. For an IE-total coloring f of G using k colors, if C(u) ≠ C(v) for any two different vertices u and v of G, then f is called a k-vertex-distinguishing IE-total-coloring of G or a k-VDIET coloring of G for short. The minimum number of colors required for a VDIET coloring of G is denoted by χ_(vt)~(ie) (G) and is called vertex-distinguishing IE-total chromatic number or the VDIET chromatic number of G for short. The VDIET colorings of complete bipartite graphs K_(8,n)are discussed in this paper. Particularly, the VDIET chromatic number of K_(8,n) are obtained.     

The growth of solutions of f″+ e-zf′+ Q(z)f - 0 where the order (Q)＝ I

This paper investigates the growth of solutions of the equation f" + e-zf′ + Q(z)f＝ 0 wherethe order (Q)＝ 1. When Q(z) ＝ h(z)ebz, h(z) is nonzero polynomial, b ≠ -1 is a complex constant, every solution of the above equation has infinite order and the hyper-order 1. We improve the results of M. Frei, M.Ozawa, G. Gundersen and J. K. Langley.     

Characterizations of P(Q(4,Q),L)

In [2] we dealt with a characterization of the generalized quadrangle Q(4,q), q odd, by introducing the concept of (0,2)-set. The aim of this paper is to give a characterization of P(Q(4,q),L), q odd and L an arbitrary regular line of Q(4,q), by constructing these (0,2)-sets and using the result of [2].     

半单复形IMv(M,Q)

设M ,V ,Q是李普希茨流形 ,M是V的局部LIP平坦的紧子流形 ,V是开流形且dimV =dimQ .设U是M在V中的某开邻域且Δn 是Rn 中n维标准单形 .如f:Δn×U→Δn×Q是一个LIP浸入且P1f =P1,称f是一个n维单形 .令 (IMV(m ,Q) ) n 是上面所定义的所有n维单形的集合且令IMV(m ,Q) ={ (IMV(M ,Q) ) n} n 0 .本文证明了IMV(M ,Q)在我们所定义的面运算 i和退化运算Si下是一个半单复形 .     

算子的拟相似与(Q)类算子

本文引进定义于L(H)上的集值函数β(S)和(Q)类算子(指本质谱含于其一切拟相似算子的本质谱的算子),用β(S)刻划 (Q)类算子的特征;证明(Q)类算子范围广泛,次可分解算子(包括次标量算子,M-亚正常算子,半亚正常算子等等)是其中的一部分;(Q)类算子在L(H)中稠密.     

A_2(F)在G_2(F)中的扩群及其泛正规性

域Ｆ上Ａ２型Ｃｈｅｖａｌｌｅｙ群Ａ２（Ｆ）可视为Ｆ上Ｇ２型Ｃｈｅｖａｌｌｅｙ群Ｇ２（Ｆ）的子群．当 Ｆ是特征不为 ２，３的域且 Ｆ＝Ｆ３时，本文给出了 Ａ２（Ｆ）在 Ｇ２（Ｆ）中的所有扩群及其泛正规性．     

On existence subspaces inC(Q)

The property of a space to be an existence subspace is studied for subspacesE ofC(Q) such that eitherE orC(Q)/E is a Lindenstrauss space. For a Chebyshev subspaceL⊂C(Q)₁ an analytic representation of the nearest element in terms of the annihilatorL ^┴ is obtained. Translated fromMatematicheskie Zametki, Vol. 65, No. 5, pp. 726–737, May, 1999.     

Hypotheses implying Cesari's property (Q)

Property (Q) was originally introduced by Cesari in order to prove closure and existence theorems for optimal control problems with unbounded control spaces. Since that time, both Cesari and others have refined and modified this property in order to deal with current research problems. The present paper provides an addition to the list of hypotheses which imply Property (Q).This research was partially supported by the National Science Foundation, Grant No. GP-34480.     

Finite-codimensional Chebyshev subspaces in the complex spaceC(Q)

We consider finite-condimensional Chebyshev subspaces in the complex spaceC(Q), whereQ is a compact Hausdorff space, and prove analogs of some theorems established earlier for the real case by Garkavi and Brown (in particular, we characterize such subspaces). It is shown that if the real spaceC(Q) contains finite-codimensional Chebyshev subspaces, then the same is true of the complex spaceC(Q) (with the sameQ). Translated fromMatermaticheskie Zametki, Vol. 62, No. 2, pp. 178–191, August, 1997. Translated by V. E. Nazaikinskii     

On Cesari's property (Q)

In the present paper, we generalize, within the family of convex-valued multifunctions, a selection theorem of Cole (Ref. 1). This then leads us to prove some new results concerning Cesari's property (Q) and to the introduction of the Cesari limit of sets, which is then compared to the Kuratowski-Mosco limit.This work was done while the author was a visiting professor at the University of Pavia, Pavia, Italy. Financial support was provided by CNR and by NSF Grant No. 84-03135.     

Norm-attaining functionals onC(Q, X)

Functionals (vector measures) defined on the spaceC(Q, X) of continuous abstract functions (whereQ is a compact Hausdorff space andX is a Banach space) and attaining their norm on the unit sphere are considered. A characterization of such functionals is given in terms of the Radon-Nikodym derivative of the vector measure with respect to the variation of the measure and in terms of analogs of the derivative. Applications to the characterization of finite-codimensional subspaces with the best approximation property are given. Similar results are obtained for the spaceB(Q, Σ, X) of uniform limits of simple functions. Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 45–56, January, 1997. Translated by V. E. Nazaikinskii