INCOMPLETE SEMI-ITERATIVE METHODS FOR SOLVING SINGULAR LINEAR OPERATOR EQUATIONS IN BANACH SPACE WITH APPLICATIONS IN MARKOV CHAIN MODELING

1　 IntroductionLet Cbe the open complex plane,let X be a complex Banach space.The set of allbounded linear operators from X into X is denoted by B[X] which is also a Banach space.If X=Cn,the n-dimensional Euclidean space,then B[X] is the set of all n×n matrices,denoted by Cn,n. We denote the spectrum of an operator T∈ B[X] byσ( T) and its resol-vent operator R( λ,T) =( λI-T) - 1 ,where I is the identity operator andλ∈C.The spectral radius of T is denoted by r( T) .N( T) and…     

0-Semidistributive Inverse Semigroups

For an inverse semigroup S,the set L(S)of all inverse subsemigroups(including the empty set)of S forms a lattice with respect to intersection denoted as usual by ∩ and union,where the union is the inverse subsemingroup generated by inverse subsemigroups A,B of S.The set     

The number of all isormorphic classes of atomic subfields in the power set of an infinite set

Let X be an infinite set, the power set of X is denoted by P(X). = {G:G and F are isomorphic in the sense of Boolean algebra}, C=(:F is an atomicsubfield in P(X)}. Define C_1(C) is an incomparable class if (_1,_2∈C_1)(F_1≠F_2→F_1 and F_2 are incomparable), then |C| = 2~2~(|x|)= max{|C_1|}. In proof of this result, the AC and GCH are used. The sense of "isomorphism" in [2] is understood as"φ(φ is a bejection on     

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 C 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.     

AUTOMORPHISM GROUPS OF 4-VALENT CONNECTED CAYLEY GRAPHS OF p-GROUPS

61. IntroductionLet G be a trite grouP and S a subs6t of G such thst 1' S and S = S--1. The Cayleygraph X = Cay(G, S) Of G with respect to S is defined to have vertex set V(X) = G and edgeset E(X) = {(g, ag) I g E G, s E' S}. ~ the defection the following two faCts are obvious:(1) the automorphism group Ant(X) of X contains GR, the right regular representation ofG, as a subgroup, and (2) X is cormected if and only if S generates the group G.FOr a Cayley graph X = Cay(G, S) Of …     

Finite Groups in Which Each Irreducible Character has at Most Two Zeros

Let G be a finite group, Irr(G) denotes the set of irreducible complex characters of G and gG the conjugacy class of G containing element g. A well-known theorem of Burnside([1,Theorem 3.15]) states that every nonlinear X ∈ Irr(G) has a zero on G, that is, an element x (or a conjugacy class xG) of G with X(x) = 0. So, if the number of zeros of character table is very small, we may expect, the structure of group is heavily restricted. For example, [2, Proposition 2.7] claimes that G is a Frobenius group with a complement of order 2 if each row in charcter table has at most one zero (its proof uses the classification of simple groups). In this note, we characterize the finite group G satisfying the following hypothesis:     

Resolvability in circulant graphs

A set W of the vertices of a connected graph G is called a resolving set for G if for every two distinct vertices u, v ∈ V (G) there is a vertex w ∈ W such that d(u, w) ≠ d(v, w). A resolving set of minimum cardinality is called a metric basis for G and the number of vertices in a metric basis is called the metric dimension of G, denoted by dim(G). For a vertex u of G and a subset S of V (G), the distance between u and S is the number min s∈S d(u, s). A k-partition Π = {S 1 , S 2 , . . . , S k } of V (G) is called a resolving partition if for every two distinct vertices u, v ∈ V (G) there is a set S i in Π such that d(u, Si )≠ d(v, Si ). The minimum k for which there is a resolving k-partition of V (G) is called the partition dimension of G, denoted by pd(G). The circulant graph is a graph with vertex set Zn , an additive group of integers modulo n, and two vertices labeled i and j adjacent if and only if i-j (mod n) ∈ C , where CZn has the property that C =-C and 0 ■ C. The circulant graph is denoted by Xn, Δ where Δ = |C|. In this paper, we study the metric dimension of a family of circulant graphs Xn, 3 with connection set C = {1, n/2 , n-1} and prove that dim(Xn, 3 ) is independent of choice of n by showing that dim(Xn, 3 ) ={3 for all n ≡ 0 (mod 4), 4 for all n ≡ 2 (mod 4). We also study the partition dimension of a family of circulant graphs Xn,4 with connection set C = {±1, ±2} and prove that pd(Xn, 4 ) is independent of choice of n and show that pd(X5,4 ) = 5 and pd(Xn,4 ) ={3 for all odd n ≥ 9, 4 for all even n ≥ 6 and n = 7.     

RESEARCH ANNOUNOEMENTS

Let G be a finite group, Irr(G) denotes the set of irreducible complex characters of G and gGthe conjugacy class of G containing element g. A well-known theorem of Burnside([1,Theorem3. 15]) states that every nonlinear X E Irr(G) has a zero on G, that is, an element x (or a conjugacyclass xG) of G with x(x) = 0. So, if the number of zeros of character table is very small, we mayexpect, the structure of group is heavily restricted. For example, [2, Proposition 2.7] claimesthat G is a Fro…     

THE MAXIMAL NUMBER OF I_π-CHARACTERS OVER NORMAL SUBGROUPS

Let G be a π-separable group for a set of primes, let N be a normal subgroup of G, and let θ be an Iπ-character(i.e., irreducible π-partial character) of N. We obtain a necessary and suffcient condition for the number of Iπ-characters of G over θ to take the possible maximum |G : N |π. Some applications are given.     

O-Semidistributive Inverse Semigroups

For an inverse semigroup S, the set L(S) of all inverse subsemigroups (including the emptyset) of S forms a lattice with respect to intersection denoted as usual by ∩ and union, wherethe union is the inverse subsemigroup generated by inverse subsemigroups A, B of S. The setLF(S) of all full inverse subsemigroups of S forms a complete sublattice of L(S), with E_s as     

Longest Cycles in 3-Connected k-Regular Claw-Free Graphs

All graphs considered here are undirected aud finite without loop or multipleedges. A graph is called claw-free if it do not contain a K_(1,3) as an inducedsubgraph. Let δ(G) denote the minimum degree of a graph G, and let V(G) andE(G) be the vertex set and edge set of G, respectively. For a subset S of V(G)and a subgraph H of G,G[S] and G-H denote the subgraphs of G induced by the     

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.     

Power Semigroups of a Class of Cliford Semigroups

Let S =∪(Gα : α∈ E) be a semilattice of groups(i.e., a Cliford semigroup) and n a natural number. E is called an n-element chain of groups if it is an n-element chain. Denote by Cn the set of all n-element chains of groups. In this paper we shall show that for any natural number n, the class of semigroups Cn satisfies the strong isomorphism property.     

s-REGULAR DIHEDRAL COVERINGS OF THE COMPLETE GRAPH OF ORDER 4

§1. Introduction For a ?nite, simple, and undirected graph X, every edge of X gives rise to a pair ofopposite arcs, and we denote by V (X), E(X), A(X) and Aut(X) the vertex set, the edgeset, the arc set and the automorphism group of X, respectively. …     

Approximative compactness and continuity of metric projector in Banach spaces and applications

First we prove that the approximative compactness of a nonempty set C in a normed linear space can be reformulated equivalently in another way.It is known that if C is a semi-Chebyshev closed and approximately compact set in a Banach space X,then the metric projectorπC from X onto C is continuous.Under the assumption that X is midpoint locally uniformly rotund,we prove that the approximative compactness of C is also necessary for the continuity of the projectorπC by the method of geometry of Banach spaces.Using this general result we find some necessary and sufficient conditions for T to have a continuous Moore-Penrose metric generalized inverse T~ ,where T is a bounded linear operator from an approximative compact and a rotund Banach space X into a midpoint locally uniformly rotund Banach space Y.     

REVERSE ORDER LAW FOR GROUP INVERSE OF PRODUCT OF MATRICES ON SKEW FIELDS

Let K~(n×n) be the set of all n×n matrices and K_r~(n×n) the set {A∈K~(n×n)|rankA=r} on askew field K. Zhuang [1] denotes by A~# the group inverse of A∈K~(n×n) which is the solu-tion of the euqations:AXA=A, XAX=X, AX=AX.     

On inclusion of permutation modules

Let H1,H2 be subgroups of a finite group G. Assume that G=∪i=1mH2yiH1=∪j=1nH1gjH1 and that y1=1,g1=1.Let Di be the set consisting of right cosets of H2 contained in H2yiH1 and let dj(j=1, . . . ,n) be the set consisting of right cosets contained in H1gjH1.We define the n×m matrix Mz(z=1, . . . ,m) whose columns and rows are indexed by Di and dj respectively and the (dk,Dl) entry is |Dzgk∩Dl|. Let M=(M1, . . . ,Mm). Assume that 1H1G and 1H2G are semisimple permutation modules of a finite group G. In this paper, by using the matrix M , we give some sufficient and necessary conditions such that 1H1G is isomorphic to a submodule of 1H2G.As an application, we prove Foulkes' conjecture in special cases.     

On solubility and supersolubility of some classes of finite groups

Let G be a finite group and H a subgroup of G. Then H is said to be S-permutable in G if HP = PH for all Sylow subgroups P of G. Let HsG be the subgroup of H generated by all those subgroups of H which are S-permutable in G. Then we say that H is S-embedded in G if G has a normal subgroup T and an S-permutable subgroup C such that T ∩ H HsG and HT = C. Our main result is the following Theorem A. A group G is supersoluble if and only if for every non-cyclic Sylow subgroup P of the generalized Fitting subgrou...     

Completely Settling of the Multiplier Conjecture for the Case of n=3p~r

A k-subset D of a group G of order v (1 < k < v -1 ) is called a (v, k, )-difference set ifevery nonidentity element of G appears in the list of d1d2-1(d1, d2 D) exactly times. Setn = k - , n is called the order of difference set D.An taomorphism of a finite group G is called a (right) multiplier of a difference set D inG if sends D onto Da for some a G. If G is abelian and if is given by g gt, where t isan integer prime to the order of G, then and also t itself are called a numerical mul…