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.     

{(3,4),4}-富勒烯图的极大共振性（英文）

A {(3,4), 4}-fullerene graph S is a 4-regular map on the sphere whose faces are of length 3 or 4.It follows from Euler s formula that the number of triangular faces is eight.A set H of disjoint quadrangular faces of S is called resonant pattern if S has a perfect matching M such that every quadrangular face in H is M-alternating.Let k be a positive integer,S is k-resonant if any i≤k disjoint quadrangular faces of S form a resonant pattern.Moreover,if graph S is k-resonant for any integer k,then ...     

可分裂的左GC-lpp半群

A left GC-lpp semigroup S is called split if the natural homomorphism γb of S onto S/γ induced by γ is split.It is proved that a left GC-lpp semigroup is split if and only if it has a left adequate transversal.In particular,a construction theorem for split left GC-lpp semigroups is established.     

Split Left GC-Lpp Semigroups

A left GC-lpp semigroup S is called split if the natural homomorphism γb of S onto S/γ induced by γ is split.It is proved that a left GC-lpp semigroup is split if and only if it has a left adequate transversal.In particular,a construction theorem for split left GC-lpp semigroups is established.     

HAMILTONIAN DECOMPOSITION OF CAYLEY GRAPHS OF ORDERS p~2 AND pq

1. IntroductionLet G be a finite group and S a subset of G such that S--1 ~ S, and 1 f S. The Cayleygraph Cay (G, S) is defined as the simple graph with V ~ G, and E = {glgZ I g,'g, or g,'g,6 S, gi, gi E G}. Cay (G, S) is vertex-transitive, and it is connected if and only if (S) = G,i.e. S is a generating set of G[1]. If G = Zn, then Cay (Zn, S) is called a circulant graph. Ithas been proved that any connected Cayley graph on a finite abelian group is hamiltonianl2].Furthermore, …     

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.     

Pseudo-Anosov Mapping Classes and Their Representations by Products of Two Dehn Twists

Let S be a Riemann surface of analytically finite type （p, n） with 3p-3＋n 〉 0. Let a ∈ S and S = S - {a}. In this article, the author studies those pseudo-Anosov maps on S that are isotopic to the identity on S and can be represented by products of Dehn twists. It is also proved that for any pseudo-Anosov map f of S isotopic to the identity on S, there are infinitely many pseudo-Anosov maps F on S - {b} = S - {a, b}, where b is a point on S, such that F is isotopic to f on S as b is filled in.     

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

Let G = （V, E） be a graph without isolated vertices. A set S lohtain in 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 lohtain in 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 lohtain in 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.     

On Levi-fiat hypersurfaces with given boundary in C~n

Let S■C~n be a compact connected 2-codimensional submanifold.If n≥3,essentially local conditions and the assumption:every complex point of S is elliptic imply the existence of a projection in C~n of a Levi-flat(2n-1)-subvariety whose boundary is S(Dolbeault,Tomassini,Zaitsev, 2005).We extend the result when S is homeomorphic to a sphere and has one hyperbolic point. For n=2 many results are known since the 1980's and a new result with a very technical hypothesis is announced.     

RESEARCH ANNOUNCEMENTS On Permutable Band

A semigroup S is called band if every element of S is idempotent.A semigroup Sis called permutable if ρσ=σρ for every two congruences ρ and σ on S.In thispaper we prove that if S is a permutable band,then |S|≤13. Let S be a semigroup.The lattice of congruences on S is denoted by C(S).LetP∈C(S),AS,A is called p-saturated,if A=Uap.     

A Construction for P-Regular Semigroups

A regular semigroup S with a special involution *, i.e., a unary operation on S satisfyingis called a regular *-semigroup[1]. It has been shown by Yamada[2] that a regular semigroup S isa regular *-semigroup if and only if it has a P-system, that is to say, there is a subset P of E(S)such that(2) As a generalization of regular semigroup and orthodox semigroup,Yamada[3] defined P-regular semigroup. Let S be a regularsemigroup. A subset P of E(S) is called a C-set in S if(c.2) (1) In this…     

On Gorenstein Projective Dimensions of Unbounded Complexes

Let R → S be a ring homomorphism and X be a complex of R-modules. Then the complex of S-modules S?_R~L X in the derived category D(S) is constructed in the natural way. This paper is devoted to dealing with the relationships of the Gorenstein projective dimension of an R-complex X(possibly unbounded) with those of the S-complex S?_R~L X.It is shown that if R is a Noetherian ring of finite Krull dimension and φ : R → S is a faithfully flat ring homomorphism, then for any homologically degree-wise finite complex X, there is an equality Gpd_RX = GpdS(S?_R~L X). Similar result is obtained for Ding projective dimension of the S-complex S?_R~L X.     

QUASI-FIELD AND SYSTEMS THEORY OVER QUASI-FIELD

Definition 1.Suppose that S is a set,(S,+,0)in a additive commutativemonoid,(S\O,·,1,-1)is a multiplication commutative group、 a∈S,o ·a=a·o=0and a,b,c∈S,a(b+c)=ab+ac.We call S a quasi-field. Theorem 1. Quasi—fields have only three kinds:     

Triangular Matrix Representations of Rings of Generalized Power Series

Let R be a ring and S a cancellative and torsion-free monoid and 〈 a strict order on S. If either （S,≤） satisfies the condition that 0 ≤ s for all s ∈ S, or R is reduced, then the ring [[R^S,≤]] of the generalized power series with coefficients in R and exponents in S has the same triangulating dimension as R. Furthermore, if R is a PWP ring, then so is [[R^S,≤]].     

On Semilattice Decomposition of an Abel–Grassmann’s Groupoid

In this paper, we have decomposed an AG-groupoid. Let S be an AG-groupoid with left identity and a relation γ be defined on S as: aγb if and only if there exist positive integers m and n such that bm ∈ ( Sa)S and a n ∈ ( Sb)S for all a and b in S. We have proved that S/γ is a maximal separative semilattice homomorphic image of S. Every AG-groupoid S is uniquely expressible as a semilattice Y of archimedean AG-groupoids Sα (α∈ Y ). The semilattice Y is isomorphic to S/γ and the S α (α∈ Y ) are the equivalence classes of S mod γ.     

Residual Finiteness of Finitely Generated Quasi-commutative Semigroups

A semigroup S is called residually finite if for any pair of distinct elements a,b∈S, there exists a congruence P on S such that S/p is finite and (a,b) P. In1958, Malcev proved the following theorem: Any finitely generated abelian semigroupis residually finite . In this paper,we prove that a finitely generated quasi-commu-tative semigroup is residually finite. It generalizes the above theorem.     

STRUCTURE OF OPTIMAL POLICIES FOR DISCOUNTED SEMI-MARKOV DECISION PROGRAMMING WITH UNBOUNDED REWARDS

In this paper, we discuss the structure of optimal policies for discountedsemi--Markov decision programming with unbounded rewards: {S, (A(i), i∈S), q, t,r,V_α}, where state space S is a countable set; in state i∈S, available action setA(i) is any set, and (A(i),(i)) is a measurable space; q is a time homogeneousfamily of jumps of states; t is a distributiou family of state jump's time, andonly depends on current state and current action too; V_αis the αa-discounted totalexpected reward.     

A characterization of Burniat surfaces with K2=4 and of non nodal type

Let S be a minimal surface of general type with pg(S) = 0 and K_S~2= 4. Assume the bicanonical map ψ of S is a morphism of degree 4 such that the image of ψ is smooth. Then we prove that the surface S is a Burniat surface with K~2= 4 and of non nodal type.     

关于拟AP内射模的注记

Let R be a ring.A right R-module M with S=End(MR)is called aquasi AP-injective module,if,for any s ∈ S,there exists a left ideal X_s of S such thatl_s(ker s)=SsX_s.Let M be a quasi AP-injective module which is a self-generator.We show that for such a module,if S is semiprime,then every maximal kernel of S isa direct summand of M.Furthermore,if ker(a_1)ker(a_2a_1)ker(a_3a_2a_1)satisfy the ascending conditions for any sequence a_1,a_2,a_3,...∈ S,then S is rightperfect.In this paper,we give a series of results which extend and generalize resultson AP-injective rings.     

On generalized inverse transversals

Let S be a regular semigroup, S° an inverse subsemigroup of S.S° is called a generalized inverse transversal of S, if V（x）∩S°≠Ф. In this paper, some properties of this kind of semigroups are discussed. In particular, a construction theorem is obtained which contains some recent results in the literature as its special cases.