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1.
《Discrete Mathematics》2004,274(1-3):125-135
The classical Ramsey number r(m,n) can be defined as the smallest integer p such that in every two-coloring (R,B) of the edges of Kp, β(B)⩾m or β(R)⩾n, where β(G) denotes the independence number of a graph G. We define the upper domination Ramsey number u(m,n) as the smallest integer p such that in every two-coloring (R,B) of the edges of Kp, Γ(B)⩾m or Γ(R)⩾n, where Γ(G) is the maximum cardinality of a minimal dominating set of a graph G. The mixed domination Ramsey number v(m,n) is defined to be the smallest integer p such that in every two-coloring (R,B) of the edges of Kp, Γ(B)⩾m or β(R)⩾n. Since β(G)⩽Γ(G) for every graph G, u(m,n)⩽v(m,n)⩽r(m,n). We develop techniques to obtain upper bounds for upper domination Ramsey numbers of the form u(3,n) and mixed domination Ramsey numbers of the form v(3,n). We show that u(3,3)=v(3,3)=6, u(3,4)=8, v(3,4)=9, u(3,5)=v(3,5)=12 and u(3,6)=v(3,6)=15.  相似文献   

2.
Let G be a finite group and cd(G) be the set of all complex irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd(G) = cd(H), then G???H × A, where A is an abelian group. In this paper, we verify the conjecture for the family of simple exceptional groups of Lie type 3 D 4(q), when q?≥?3.  相似文献   

3.
Let φ be a homomorphism from the partially ordered abelian group (S, v) to the partially ordered abelian group (G, u) with φ(v) = u, where v and u are order units of S and G respectively. Then φ induces an affine map φ* from the state space St(G, u) to the state space St(S, v). Firstly, in this paper, we give some suitable conditions under which φ* is injective, surjective or bijective. Let R be a semilocal ring with the Jacobson radical J(R) and let π: RR/J(R) be a canonical map. We discuss the affine map (K 0 π)*. Secondly, for a semiprime right Goldie ring R with the maximal right quotient ring Q, we consider the relations between St(R) and St(Q). Some results from [ALFARO, R.: State spaces, finite algebras, and skew group rings, J. Algebra 139 (1991), 134–154] and [GOODEARL, K. R.-WARFIELD, R. B., Jr.: State spaces of K 0 of noetherian rings, J. Algebra 71 (1981), 322–378] are extended.  相似文献   

4.
Let G be a graph and d(u) denote the degree of a vertex u in G. The zeroth-order general Randi? index 0Rα(G) of the graph G is defined as ∑uV(G)d(u)α, where the summation goes over all vertices of G and α is an arbitrary real number. In this paper we correct the proof of the main Theorem 3.5 of the paper by Hu et al. [Y. Hu, X. Li, Y. Shi, T. Xu, Connected (n,m)-graphs with minimum and maximum zeroth-order general Randi? index, Discrete Appl. Math. 155 (8) (2007) 1044-1054] and give a more general Theorem. We finally characterize 1 for α<0 the connected G(n,m)-graphs with maximum value 0Rα(G(n,m)), where G(n,m) is a simple connected graph with n vertices and m edges.  相似文献   

5.
Jang-Ho Chun 《代数通讯》2013,41(10):3095-3102
For positive integers ? and n, several authors studied ??-gradings of the full matrix ring M n (k) over a field k. In this article, we show that every (G × H)-grading of M n (k) can be constructed by a pair of compatible G-grading and H-grading of M n (k), where G and H are any finite groups. When G and H are finite cyclic groups, we characterize all (G × H)-gradings which are isomorphic to a good grading. Moreover, the results can be generalized for any finite abelian group grading of M n (k).  相似文献   

6.
It was shown in Bíró et al. (2001) [7] that every cyclic subgroup C of the circle group T admits a characterizing sequence (un) of integers in the sense that unx→0 for some xT iff xC. More generally, for a subgroup H of a topological (abelian) group G one can define:
(a)
g(H) to be the set of all elements x of G such that unx→0 in G for all sequences (un) of integers such that unh→0 in G for all hH;
(b)
H to be g-closed if H=g(H).
We show then that an infinite compact abelian group G has all its cyclic subgroups g-closed iff GT.  相似文献   

7.
For two graphs G and H, let the mixed anti-Ramsey numbers, maxR(n;G,H), (minR(n;G,H)) be the maximum (minimum) number of colors used in an edge-coloring of a complete graph with n vertices having no monochromatic subgraph isomorphic to G and no totally multicolored (rainbow) subgraph isomorphic to H. These two numbers generalize the classical anti-Ramsey and Ramsey numbers, respectively.We show that maxR(n;G,H), in most cases, can be expressed in terms of vertex arboricity of H and it does not depend on the graph G. In particular, we determine maxR(n;G,H) asymptotically for all graphs G and H, where G is not a star and H has vertex arboricity at least 3.In studying minR(n;G,H) we primarily concentrate on the case when G=H=K3. We find minR(n;K3,K3) exactly, as well as all extremal colorings. Among others, by investigating minR(n;Kt,K3), we show that if an edge-coloring of Kn in k colors has no monochromatic Kt and no rainbow triangle, then n?2kt2.  相似文献   

8.
Suppose G is a locally compact noncompact group. For abelian such G's, it is shown in this paper that L1(G), C(G), and L(G) always have discontinuous translation-invariant linear forms(TILF's) while C0(G) and Lp(G) for 1 < p < ∞ have such forms if and only if GH is a torsion group for some open σ-compact subgroup H of G. For σ-compact amenable G's, all the above spaces have discontinuous left TILF's.  相似文献   

9.
Let G be a finite group. Let n be a positive integer and p a prime coprime to n. In this paper we prove that if the set of conjugacy class sizes of primary and biprimary elements of group G is {1,p a , p a n}, then GG 0 × H, where H is abelian and G 0 contains a normal subgroup M × P 0 of index p. Moreover, M × P 0 is the set of all elements of G 0 of conjugacy class sizes p a or 1, where M is an abelian π(n)-subgroup of G 0 and P 0 is an abelian p-subgroup of G 0, neither being central in G. Finally, p a = p and P/P 0 acts fixed-point-freely on M and ?(P) ≤ Z(P). This is an extension of Alan Camina’s theorems on the structure of groups whose set of conjugacy class size is {1,p a , p a q b }, where p and q are two distinct primes.  相似文献   

10.
Let V ?H be real separable Hilbert spaces. The abstract wave equation u′' + A(t)u = g(u), where u(t) ?V, A(t) maps V to its dual V1, and g is a nonlinear map from the ball S(R0) = {u?V: ∥u∥ < R0} into H, is considered. It is assumed that g is locally Lipschitz in S(R0) and possibly singular at the boundary. Local existence and continuation theorems are established for the Cauchy problem u(0) = u0?S(R0), u′(0) = u1?H. Global existence is shown for g(u) = εφ(u), where φ has a potential and ε is small. Global nonexistence is shown for g(u) = εφ(u), where φ satisfies an abstract convexity property and ε is large.  相似文献   

11.
Let A(R, S) denote the class of all m×n matrices of 0's and 1's having row sum vector R and column sum vector S. The interchange graph G(R, S) is the graph where the vertices are the matrices in A(R, S) and where two matrices are joined by an edge provided they differ by an interchange. We characterize those A(R, S) for which the graph G(R, S) has diameter at most 2 and those A(R, S) for which G(R, S) is bipartite.  相似文献   

12.
Motivated from [31], call a precompact group topology τ on an abelian group G ss-precompact (abbreviated from single sequence precompact  ) if there is a sequence u=(un)u=(un) in G such that τ is the finest precompact group topology on G   making u=(un)u=(un) converge to zero. It is proved that a metrizable precompact abelian group (G,τ)(G,τ) is ss-precompact iff it is countable. For every metrizable precompact group topology τ on a countably infinite abelian group G there exists a group topology η such that η is strictly finer than τ   and the groups (G,τ)(G,τ) and (G,η)(G,η) have the same Pontryagin dual groups (in other words, (G,τ)(G,τ) is not a Mackey group in the class of maximally almost periodic groups).  相似文献   

13.
Let F n be the free group of rank n, and let Aut+(F n ) be its special automorphism group. For an epimorphism π : F n G of the free group F n onto a finite group G we call the standard congruence subgroup of Aut+(F n ) associated to G and π. In the case n = 2 we fully describe the abelianization of Γ+(G, π) for finite abelian groups G. Moreover, we show that if G is a finite non-perfect group, then Γ+(G, π) ≤ Aut+(F 2) has infinite abelianization.  相似文献   

14.
For a finite group G and nonnegative integer n ≥ 0, one may consider the associated tower \(G \wr S_{n} := S_{n} \ltimes G^{n}\) of wreath product groups. Zelevinsky associated to such a tower the structure of a positive self-adjoint Hopf algebra (PSH-algebra) R(G) on the direct sum over integers n ≥ 0 of the Grothendieck groups K 0(R e p?G?S n ). In this paper, we study the interaction via induction and restriction of the PSH-algebras R(G) and R(H) associated to finite groups H ? G. A class of Hopf modules over PSH-algebras with a compatibility between the comultiplication and multiplication involving the Hopf k t h -power map arise naturally and are studied independently. We also give an explicit formula for the natural PSH-algebra morphisms R(H) → R(G) and R(G) → R(H) arising from induction and restriction. In an appendix, we consider a family of subgroups of wreath product groups analogous to the subgroups G(m, p, n) of the wreath product cyclotomic complex reflection groups G(m, 1, n).  相似文献   

15.
Given a group G and positive integers r,s≤|G|, we denote by μG(r,s) the least possible size of a product set AB={abaA,bB}, where A,B run over all subsets of G of size r,s, respectively. While the function μG is completely known when G is abelian [S. Eliahou, M. Kervaire, Minimal sumsets in infinite abelian groups, Journal of Algebra 287 (2005) 449-457], it is largely unknown for G non-abelian, in part because efficient tools for proving lower bounds on μG are still lacking in that case. Our main result here is a lower bound on μG for finite solvable groups, obtained by building it up from the abelian case with suitable combinatorial arguments. The result may be summarized as follows: if G is finite solvable of order m, then μG(r,s)≥μG(r,s), where G is any abelian group of the same order m. Equivalently, with our knowledge of μG, our formula reads .One nice application is the full determination of the function μG for the dihedral group G=Dn and all n≥1. Up to now, only the case where n is a prime power was known. We prove that, for all n≥1, the group Dn has the same μ-function as an abelian group of order |Dn|=2n.  相似文献   

16.
The Randi? index of a graph G is defined as , where d(u) is the degree of vertex u and the summation goes over all pairs of adjacent vertices u, v. A conjecture on R(G) for connected graph G is as follows: R(G)≥r(G)−1, where r(G) denotes the radius of G. We proved that the conjecture is true for biregular graphs, connected graphs with order n≤10 and tricyclic graphs.  相似文献   

17.
Straightening and bounded cohomology of hyperbolic groups   总被引:2,自引:0,他引:2  
It was stated by M. Gromov [Gr2] that, for any hyperbolic group G, the map from bounded cohomology Hnb(G,\Bbb R) H^n_b(G,{\Bbb R}) to Hn(G,\Bbb R) H^n(G,{\Bbb R}) induced by inclusion is surjective for n 3 2 n \ge 2 . We introduce a homological analogue of straightening simplices, which works for any hyperbolic group. This implies that the map Hnb(G,V) ? Hn(G,V) H^n_b(G,V) \to H^n(G,V) is surjective for n 3 2 n \ge 2 when V is any bounded \Bbb QG {\Bbb Q}G -module and when V is any finitely generated abelian group.  相似文献   

18.
Let A be an R G-module over a commutative ring R, where G is a group of infinite section p-rank (0-rank), C G (A) = 1, A is not a Noetherian R-module, and the quotient A/C A (H) is a Noetherian R-module for every proper subgroup H of infinite section p-rank (0-rank). We describe the structure of solvable groups G of this type.  相似文献   

19.
Let S be a subset of a finite abelian group G. The Cayley sum graph Cay+(G, S) of G with respect to S is a graph whose vertex set is G and two vertices g and h are joined by an edge if and only if g + hS. We call a finite abelian group G a Cayley sum integral group if for every subset S of G, Cay+(G, S) is integral i.e., all eigenvalues of its adjacency matrix are integers. In this paper, we prove that all Cayley sum integral groups are represented by Z3 and Zn2 n, n ≥ 1, where Zk is the group of integers modulo k. Also, we classify simple connected cubic integral Cayley sum graphs.  相似文献   

20.
Two natural extensions of Jensen’s functional equation on the real line are the equations f(xy) + f(xy −1) =  2f(x) and f(xy) + f(y −1 x) =  2f(x), where f is a map from a multiplicative group G into an abelian additive group H. In a series of papers (see Ng in Aequationes Math 39:85–99, 1990; Ng in Aequationes Math 58:311–320, 1999; Ng in Aequationes Math 62:143–159, 2001), Ng solved these functional equations for the case where G is a free group and the linear group GLn(R), R=\mathbbZ,\mathbbR{{GL_n(R), R=\mathbb{Z},\mathbb{R}}} , is a quadratically closed field or a finite field. He also mentioned, without a detailed proof, in the above papers and in (see Ng in Aequationes Math 70:131–153, 2005) that when G is the symmetric group S n , the group of all solutions of these functional equations coincides with the group of all homomorphisms from (S n , ·) to (H, + ). The aim of this paper is to give an elementary and direct proof of this fact.  相似文献   

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