Enumerating reflexible 2-cell embeddings of connected graphs

Two 2-cell embeddings:X → S and j:X → S of a connected graph X into a closed orientable surface S are congruent if there are an orientation-preserving surface homeomorphism h on S and a graph automorphism γ of X such that h = γj.A 2-cell embedding:X → S of a graph X into a closed orientable surface S is described combinatorially by a pair(X;ρ) called a map,where ρ is a product of disjoint cycle permutations each of which is the permutation of the darts of X initiated at the same vertex following the orientation of S.The mirror image of a map(X;ρ) is the map(X;ρ 1),and one of the corresponding embeddings is called the mirror image of the other.A 2-cell embedding of X is reflexible if it is congruent to its mirror image.Mull et al.[Proc Amer Math Soc,1988,103:321-330] developed an approach for enumerating the congruence classes of 2-cell embeddings of graphs into closed orientable surfaces.In this paper we introduce a method for enumerating the congruence classes of reflexible 2-cell embeddings of graphs into closed orientable surfaces,and apply it to the complete graphs,the bouquets of circles,the dipoles and the wheel graphs to count their congruence classes of reflexible or nonreflexible(called chiral) embeddings.     

A Special Kind of Principal Congruences on MS-Algebras

In this article, for any a in the MS-algebra L, the author considers the con- gruence of the form defined by （x,y） ∈θa→←x∧a°° = y ∧a° and x ∨ a°° = y ∨ a°°. A characterization of subdirectly irreducible MS-algebras is given and it is also proved that the union of two principal congruences having this form is a principal congruence if and only if L ∈ K2 ∨ K3.     

Coexistence of unbounded solutions and periodic solutions of Liénard equations with asymmetric nonlinearities at resonance

In this paper, we deal with the existence of unbounded orbits of the mapping {θ1 = θ 2nπ 1/ρμ(θ) o(ρ-1),ρ1=ρ c-μ′(θ) o(1), ρ→∞,where n is a positive integer, c is a constant and μ(θ) is a 2π-periodic function. We prove that if c ＞ 0 and μ(θ) ≠ 0, θ∈ [0, 2π], then every orbit of the given mapping goes to infinity in the future for ρ large enough; if c ＜ 0 and μ(θ) ≠ 0, θ∈ [0, 2π], then every orbit of the given mapping goes to infinity in the past for ρ large enough. By using this result, we prove that the equation x″ f(x)x′ ax -bx- φ(x) =p(t) has unbounded solutions provided that a, b satisfy 1/√a 1/√b = 2/n and F(x)(= ∫x0 f(s)ds),and φ(x) satisfies some limit conditions. At the same time, we obtain the existence of 2π-periodic solutions of this equation.     

Approximate equivalence in von Neumann algebras

One formulation of D. Voiculescu's theorem on approximate unitary equivalence is that two unital representations π and ρ of a separable C*-algebra are approximately unitarily equivalent if and only if rank οπ = rank ορ. We study the analog when the ranges of π and ρ are contained in a von Neumann algebra R, the unitaries inducing the approximate equivalence must come from R, and "rank" is replaced with "R -rank" (defined as the Murray-von Neumann equivalence of the range projection).     

On E-solid Regular Semigroups With Inverse Transversals

Let S be a regular semigroup. An inverse subsemigroup S of S is called an inverse transversal if So contains an unique inverse x" for each x E S. In this case, I ~ {e E E(S)lee = e}and A = {g 6 E(S)gg = g}, are left and right regular subbands of S, respectively, where E(S)denotes the set of idempotehts in S. Denote E' = E(S). A regular semigroup S is calledE-solid if R|E(S) o LIE(S) = LIE(S) o R|E(s). A regular semigroup S is E-solid if and only ifthere is an inverse congruence…     

$D^2(\Omega,\rho)$的再生核与再生核诱导的度量

An important property of the reproducing kernel of D^2（Ω, ρ） is obtained and the reproducing kernels for D^2（Ω, ρ） are calculated when Ω = Bn× Bn and ρ are some special functions. A reproducing kernel is used to construct a semi-positive definite matrix and a distance function defined on Ω×Ω. An inequality is obtained about the distance function and the pseudodistance induced by the matrix.     

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.     

Groups with the same order and degree pattern

The degree pattern of a finite group has been introduced in [18].A group M is called k-fold OD- characterizable if there exist exactly k non-isomorphic finite groups having the same order and degree pattern as M .In particular,a 1-fold OD-characterizable group is simply called OD-characterizable.It is shown that the alternating groups A m and A m+1 ,for m = 27,35,51,57,65,77,87,93 and 95,are OD-characterizable,while their automorphism groups are 3-fold OD-characterizable.It is also shown that the symmetric groups S m+2 ,for m = 7,13,19,23,31,37,43,47,53,61,67,73,79,83,89 and 97,are 3-fold OD-characterizable.From this,the following theorem is derived.Let m be a natural number such that m 100.Then one of the following holds: (a) if m = 10,then the alternating groups A m are OD-characterizable,while the symmetric groups S m are OD- characterizable or 3-fold OD-characterizable;(b) the alternating group A 10 is 2-fold OD-characterizable;(c) the symmetric group S 10 is 8-fold OD-characterizable.This theorem completes the study of OD-characterizability of the alternating and symmetric groups A m and S m of degree m 100.     

纯正半群上的同余

Let S be an orthodox semigroup and γ the least inverse congruence on S. C(S) denotes the set of all congruences on S. In this paper we introduce the concept of admissible triples for S, where admissible triples are constructed by the congruences on S/γ. the equivalences on E(S)/L and E(S)/R. The notation Ca(S) denotes the set of all admissible triple for S. We prove that every congruence p on S can be uniquely determined by the admissible triple induced by p, and there exists a lattice isomomorphism between C(S) and Ca(S).     

THE NUMERICAL STABILITY OF THE BLOCK θ-METHODS FOR DELAY DIFFERENTIAL EQUATIONS

This paper focuses on the numerical stability of the block θ-methods adapted to differential equations with a dday argument. For the block θ-methods, an interpolation procedure is introduced which leads to the mumerical processes that satisfy an important asymptotic stability condition related to the class of test problems y＇ (t)=ay(t)+by(t-r) with a,b∈C, Re(a)＜- ｜b｜ and τ>0. We prove that the block θ-method is GP-stable if and only if the method is A-stable for ordinary differential equations. Furthermore, it is proved that the P-and GP-stability are equivalent for the block θ-method.     

Products of Involutions in Steinberg Group over Skew Fields

Consider the stable Steinberg group St(K) over a skew field K. An element x is called an involution if x2 = 1. In this paper, an involution is allowed to be the identity. The authors prove that an element A of GLn(K) up to conjugation can be represented as BC, where B is lower triangular and C is simultaneously upper triangular. Furthermore, B and C can be chosen so that the elements in the main diagonal of B areβ1,β2, ...,βn, and of C areγ1,γ2,... ,γnCn, where cn∈[K*, K*] and = det A. It is also proved that every element 6 in St(K) is a product of 10 involutions.     

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.     

Regular maps of graphs of order 4<Emphasis Type="Italic">p</Emphasis>

A 2-cell embedding f : X → S of a graph X into a closed orientable surface S can be described combinatorially by a pair M = (X;ρ ) called a map, where ρ is a product of disjoint cycle permutations each of which is the permutation of the arc set of X initiated at the same vertex following the orientation of S . It is well known that the automorphism group of M acts semi-regularly on the arc set of X and if the action is regular, then the map M and the embedding f are called regular. Let p and q be primes. Du et al. [J. Algebraic Combin., 19, 123-141 (2004)] classified the regular maps of graphs of order pq . In this paper all pairwise non-isomorphic regular maps of graphs of order 4 p are constructed explicitly and the genera of such regular maps are computed. As a result, there are twelve sporadic and six infinite families of regular maps of graphs of order 4 p ; two of the infinite families are regular maps with the complete bipartite graphs K2p,2p as underlying graphs and the other four infinite families are regular balanced Cayley maps on the groups Z4p , Z22 × Zp and D4p .     

A Class of Constrained Inverse Eigenproblem and Associated Approximation Problem for Symmetric Reflexive Matrices

Let S∈Rn×n be a symmetric and nontrival involution matrix. We say that A∈E R n×n is a symmetric reflexive matrix if AT = A and SAS = A. Let S R r n×n(S)={A|A= AT,A = SAS, A∈Rn×n}. This paper discusses the following two problems. The first one is as follows. Given Z∈Rn×m (m < n),∧= diag(λ1,...,λm)∈Rm×m, andα,β∈R withα<β. Find a subset (?)(Z,∧,α,β) of SRrn×n(S) such that AZ = Z∧holds for any A∈(?)(Z,∧,α,β) and the remaining eigenvaluesλm 1 ,...,λn of A are located in the interval [α,β], Moreover, for a given B∈Rn×n, the second problem is to find AB∈(?)(Z,∧,α,β) such that where ||.|| is the Frobenius norm. Using the properties of symmetric reflexive matrices, the two problems are essentially decomposed into the same kind of subproblems for two real symmetric matrices with smaller dimensions, and then the expressions of the general solution for the two problems are derived.     

LENTICULAR NONCOMMUTATIVE TORI

All C*-algebras of sections of locally trivial C* -algebra bundles over ∏i=1sLki(ni) with fibres Aw Mc(C) are constructed, under the assumption that every completely irrational noncommutative torus Aw is realized as an inductive limit of circle algebras, where Lki (ni) are lens spaces. Let Lcd be a cd-homogeneous C*-algebra over whose cd-homogeneous C*-subalgebra restricted to the subspace Tr × T2 is realized as C(Tr) A1/d Mc(C), and of which no non-trivial matrix algebra can be factored out.The lenticular noncommutative torus Lpcd is defined by twisting in by a totally skew multiplier p on Tr+2 × Zm-2. It is shown that is isomorphic to if and only if the set of prime factors of cd is a subset of the set of prime factors of p, and that Lpcd is not stablyisomorphic to if the cd-homogeneous C*-subalgebra of Lpcd restricted to some subspace LkiLki (ni) is realized as the crossed product by the obvious non-trivial action of Zki on a cd/ki-homogeneous C*-algebra over S2ni+1 for ki an integer greater than     

Congruence Pairs of Decomposable MS-Algebras*

In this paper, the authors first introduce the concept of congruence pairs on the class of decomposable MS-algebras generalizing that for principal MS-algebras (see [13]). They show that every congruence relation θ on a decomposable MS-algebra L can be uniquely determined by a congruence pair (θ₁, θ₂), where θ₁ is a congruence on the de Morgan subalgebra L^?? of L and θ₂ is a lattice congruence on the sublattice D(L) of L. They obtain certain congruence pairs of a decomposable MS-algebra L via central elements of L. Moreover, they characterize the permutability of congruences and the strong extensions of decomposable MS-algebras in terms of congruence pairs.     

A necessary and sufficient condition for a surface sum of two handlebodies to be a handlebody

The main results of the paper are that we give a necessary and sufficient condition for a surface sum of two handlebodies along a connected surface to be a handlebody as follows:(1) The annulus sum H = H_1∪_AH_2 of two handlebodies H_1 and H_2 is a handlebody if and only if the core curve of A is a longitude for either H_1 or H_2;(2) Let H = H_1 ∪S_(g,b) H_2 be a surface sum of two handlebodies H_1 and H_2 along a connected surface S = S_(g,b), b 1, n_i = g(H_i) 2, i = 1, 2. Suppose that S is incompressible in both H_1 and H_2. Then H is a handlebody if and only if there exists a basis J = {J_1,..., J_m} with a partition(J_1, J_2) of J such that J_1 is primitive in H_1 and J_2 is primitive in H_2.     

关于拟AP内射模的一些研究

Let R be a ring. R is called right AP-injective if, for any a ∈ R, there exists a left ideal of R such that lr(a) = Ra (?) Xa. We extend this notion to modules. A right .R-module M with 5 = End(MR) is called quasi AP-injective if, for any s ∈ S, there exists a left ideal Xs of S such that ls(Ker(s)) = Ss (?) Xs. In this paper, we give some characterizations and properties of quasi AP-injective modules which generalize results of Page and Zhou.     

Nordhaus-Gaddum results for the sum of the induced path number of a graph and its complement

The induced path number ρ(G) of a graph G is defined as the minimum number of subsets into which the vertex set of G can be partitioned so that each subset induces a path.Broere et al.proved that if G is a graph of order n,then n~(1/2) ≤ρ(G) + ρ(■) ≤ [3n/2].In this paper,we characterize the graphs G for which ρ(G) + ρ(■) = [3n/2],improve the lower bound on ρ(G) + ρ(■) by one when n is the square of an odd integer,and determine a best possible upper bound for ρ(G) + ρ(■) when neither G nor ■ has isolated vertices.     

有向循环图的同构

Let S belong to Zn-{0}.The circulant digraph DCn(S) is a directed graph with vertex set Zn and are set {(i,i s):i∈Zn,s∈S},A.Adam conjectured that DCn(S)≌DCn(T) if and only if T=uS for some unit u mod n.In this paper we prove that the conjecture is true if S is a minimal generating set of Zn and thus determine the full automorphism groups of such digraphs.The methods we employ are new and easy to be understood.