An edge-coloring of a graph G is an assignment of colors to all the edges of G. A gc-coloring of a graph G is an edge-coloring of G such that each color appears at each vertex at least g(v) times. The maximum integer k such that G has a gc-coloring with k colors is called the gc-chromatic index of G and denoted by \(\chi\prime_{g_{c}}\)(G). In this paper, we extend a result on edge-covering coloring of Zhang and Liu in 2011, and give a new sufficient condition for a simple graph G to satisfy \(\chi\prime_{g_{c}}\)(G) = δg(G), where \(\delta_{g}\left(G\right) = min_{v\epsilon V (G)}\left\{\lfloor\frac{d\left(v\right)}{g\left(v\right)}\rfloor\right\}\). 相似文献
Let G be a 2-edge-connected simple graph on n vertices. For an edge e = uv ∈ E(G), define d(e) = d(u) + d(v). Let F denote the set of all simple 2-edge-connected graphs on n ≥ 4 vertices such that G ∈ F if and only if d(e) + d(e’) ≥ 2n for every pair of independent edges e, e’ of G. We prove in this paper that for each G ∈ F, G is not Z3-connected if and only if G is one of K2,n?2, K3,n?3, K2,n?2+, K3,n?3+ or one of the 16 specified graphs, which generalizes the results of X. Zhang et al. [Discrete Math., 2010, 310: 3390–3397] and G. Fan and X. Zhou [Discrete Math., 2008, 308: 6233–6240]. 相似文献
We study metabelian Alperin groups, i.e., metabelian groups in which every 2-generated subgroup has a cyclic commutator subgroup. It is known that, if the minimum number d(G) of generators of a finite Alperin p-group G is n ≥ 3, then d(G′) ≤ Cn2 for p≠ 3 and d(G′) ≤ Cn2 + Cn3 for p = 3. The first section of the paper deals with finite Alperin p-groups G with p≠ 3 and d(G) = n ≥ 3 that have a homocyclic commutator subgroup of rank Cn2. In addition, a corollary is deduced for infinite Alperin p-groups. In the second section, we prove that, if G is a finite Alperin 3-group with homocyclic commutator subgroup G- of rank Cn2 + Cn3, then G″ is an elementary abelian group. 相似文献
For any module V over the two-dimensional non-abelian Lie algebra b and scalar α ∈ C, we define a class of weight modules Fα(V) with zero central charge over the affine Lie algebra A1(1). These weight modules have infinitedimensional weight spaces if and only if V is infinite dimensional. In this paper, we will determine necessary and sufficient conditions for these modules Fα(V) to be irreducible. In this way, we obtain a lot of irreducible weight A1(1)-modules with infinite-dimensional weight spaces. 相似文献
We introduce the notion of property (RD) for a locally compact, Hausdorff and r-discrete groupoid G, and show that the set S2l(G) of rapidly decreasing functions on G with respect to a continuous length function l is a dense spectral invariant and Fréchet *-subalgebra of the reduced groupoid C*-algebra Cr*(G) of G when G has property (RD) with respect to l, so the K-theories of both algebras are isomorphic under inclusion. Each normalized cocycle c on G, together with an invariant probability measure on the unit space G0 of G, gives rise to a canonical map τc on the algebra Cc(G) of complex continuous functions with compact support on G. We show that the map τc can be extended continuously to S2l(G) and plays the same role as an n-trace on Cr*(G) when G has property (RD) and c is of polynomial growth with respect to l, so the Connes’ fundament paring between the K-theory and the cyclic cohomology gives us the K-theory invariants on Cr*(G). 相似文献
Let ?: E(G) → {1, 2, · · ·, k} be an edge coloring of a graph G. A proper edge-k-coloring of G is called neighbor sum distinguishing if \(\sum\limits_{e \mathrel\backepsilon u} {\phi \left( e \right)} \ne \sum\limits_{e \mathrel\backepsilon v} {\phi \left( e \right)} \) for each edge uv ∈ E(G). The smallest value k for which G has such a coloring is denoted by χ′Σ(G), which makes sense for graphs containing no isolated edge (we call such graphs normal). It was conjectured by Flandrin et al. that χ′Σ(G) ≤ Δ(G) + 2 for all normal graphs, except for C5. Let mad(G) = \(\max \left\{ {\frac{{2\left| {E\left( h \right)} \right|}}{{\left| {V\left( H \right)} \right|}}|H \subseteq G} \right\}\) be the maximum average degree of G. In this paper, we prove that if G is a normal graph with Δ(G) ≥ 5 and mad(G) < 3 ? \(\frac{2}{{\Delta \left( G \right)}}\), then χ′Σ(G) ≤ Δ(G) + 1. This improves the previous results and the bound Δ(G) + 1 is sharp. 相似文献
We obtain exact constants in Jackson-type inequalities for smoothness characteristics Λk(f), k ∈ N, defined by averaging the kth-order finite differences of functions f ∈ L2. On the basis of this, for differentiable functions in the classes L2r, r ∈ N, we refine the constants in Jackson-type inequalities containing the kth-order modulus of continuity ωk. For classes of functions defined by their smoothness characteristics Λk(f) and majorants Φ satisfying a number of conditions, we calculate the exact values of certain n-widths. 相似文献
Let G be a simple graph, let d(v) denote the degree of a vertex v and let g be a nonnegative integer function on V (G) with 0 ≤ g(v) ≤ d(v) for each vertex v ∈ V (G). A gc-coloring of G is an edge coloring such that for each vertex v ∈ V (G) and each color c, there are at least g(v) edges colored c incident with v. The gc-chromatic index of G, denoted by χ′gc (G), is the maximum number of colors such that a gc-coloring of G exists. Any simple graph G has the gc-chromatic index equal to δg(G) or δg(G) ? 1, where \({\delta _g}\left( G \right) = \mathop {\min }\limits_{v \in V\left( G \right)} \left\lfloor {d\left( v \right)/g\left( v \right)} \right\rfloor \). A graph G is nearly bipartite, if G is not bipartite, but there is a vertex u ∈ V (G) such that G ? u is a bipartite graph. We give some new sufficient conditions for a nearly bipartite graph G to have χ′gc (G) = δg(G). Our results generalize some previous results due to Wang et al. in 2006 and Li and Liu in 2011. 相似文献
Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p and p′ are joined by an edge if there is an element in G of order pp′. We denote by k(Γ(G)) the number of isomorphism classes of finite groups H satisfying Γ(G) = Γ(H). Given a natural number r, a finite group G is called r-recognizable by prime graph if k(Γ(G)) = r. In Shen et al. (Sib. Math. J. 51(2):244–254, 2010), it is proved that if p is an odd prime, then Bp(3) is recognizable by element orders. In this paper as the main result, we show that if G is a finite group such that Γ(G) = Γ(Bp(3)), where p > 3 is an odd prime, then \({G\cong B_p(3)}\) or Cp(3). Also if Γ(G) = Γ(B3(3)), then \({G\cong B_3(3), C_3(3), D_4(3)}\), or \({G/O_2(G)\cong {\rm Aut}(^2B_2(8))}\). As a corollary, the main result of the above paper is obtained. 相似文献
We shall first present an explicit realization of the simple N = 4 superconformal vertex algebra LcN?=?4 with central charge c = ?9. This vertex superalgebra is realized inside of the bcβγ system and contains a subalgebra isomorphic to the simple affine vertex algebra LA1\( \left(-\frac{3}{2}{\varLambda}_0\right) \). Then we construct a functor from the category of LcN?=?4-modules with c = ?9 to the category of modules for the admissible affine vertex algebra LA1\( \left(-\frac{3}{2}{\varLambda}_0\right) \). By using this construction we construct a family of weight and logarithmic modules for LcN?=?4 and LA1\( \left(-\frac{3}{2}{\varLambda}_0\right) \). We also show that a coset subalgebra LA1\( \left(-\frac{3}{2}{\varLambda}_0\right) \) is a logarithmic extension of the W(2; 3)-algebra with c = ?10. We discuss some generalizations of our construction based on the extension of affine vertex algebra LA1 (kΛ0) such that k + 2 = 1/p and p is a positive integer. 相似文献
In our previous papers, we introduced the notion of a generalized solution to the initial-boundary value problem for the wave equation with a boundary function µ(t) such that the integral ∫0T (T ? t)|µ(t)|pdt exists. Here we prove that this solution is a unique solution to the problem in Lp that satisfies the corresponding integral identity. 相似文献
An r-acyclic edge chromatic number of a graph G, denoted by ar′r(G), is the minimum number of colors used to produce an edge coloring of the graph such that adjacent edges receive different colors and every cycle C has at least min {|C|, r} colors. We prove that ar′r(G) ≤ (4r + 1)Δ(G), when the girth of the graph G equals to max{50, Δ(G)} and 4 ≤ r ≤ 7. If we relax the restriction of the girth to max {220, Δ(G)}, the upper bound of ar′r(G) is not larger than (2r + 5)Δ(G) with 4 ≤ r ≤ 10. 相似文献
Given an indexing set I and a finite field Kα for each α ∈ I, let ? = {L2(Kα) | α ∈ I} and \(\mathfrak{N} = \{ SL_2 (K_\alpha )|\alpha \in I\}\). We prove that each periodic group G saturated with groups in \(\Re (\mathfrak{N})\) is isomorphic to L2(P) (respectively SL2(P)) for a suitable locally finite field P. 相似文献
Let k be an algebraically closed field, and V be a vector space of dimension n over k. For a set ω = (\(\vec d\)(1), ..., \(\vec d\)(m)) of sequences of positive integers, denote by Lω the ample line bundle corresponding to the polarization on the product X = Πi=1m Flag(V, \(\vec n\)(i)) of flag varieties of type \(\vec n\)(i) determined by ω. We study the SL(V)-linearization of the diagonal action of SL(V) on X with respect to Lω. We give a sufficient and necessary condition on ω such that Xss(Lω) ≠ \(\not 0\) (resp., Xs(Lω) ≠ \(\not 0\)). As a consequence, we characterize the SL(V)-ample cone (for the diagonal action of SL(V) on X), which turns out to be a polyhedral convex cone. 相似文献
We study the inverse problem of the reconstruction of the coefficient ?(x, t) = ?0(x, t) + r(x) multiplying ut in a nonstationary parabolic equation. Here ?0(x, t) ≥ ?0 > 0 is a given function, and r(x) ≥ 0 is an unknown function of the class L∞(Ω). In addition to the initial and boundary conditions (the data of the direct problem), we pose the problem of nonlocal observation in the form ∫0Tu(x, t) dμ(t) = χ(x) with a known measure dμ(t) and a function χ(x). We separately consider the case dμ(t) = ω(t)dt of integral observation with a smooth function ω(t). We obtain sufficient conditions for the existence and uniqueness of the solution of the inverse problem, which have the form of ready-to-verify inequalities. We suggest an iterative procedure for finding the solution and prove its convergence. Examples of particular inverse problems for which the assumptions of our theorems hold are presented. 相似文献
Let Tt: X → X be a C0-semigroup with generator A. We prove that if the abscissa of uniform boundedness of the resolvent s0(A) is greater than zero then for each nondecreasing function h(s): ?+ → R+ there are x′ ∈ X′ and x ∈ X satisfying ∫0∞h(|〈x′, Txx〉|)dt = ∞. If i? ∩ Sp(A) ≠ Ø then such x may be taken in D(A∞). 相似文献
We present an equivalence theorem, which includes all known characterizations of the class Bp, i.e., the weight class of Ariño and Muckenhoupt, and also some new equivalent characterizations. We also give equivalent characterizations for the classes Bp*, B∞* and RBp, and prove and apply a “gluing lemma” of independent interest. 相似文献
Let G be a finite group. The main result of this paper is as follows: If G is a finite group, such that Γ(G) = Γ(2G2(q)), where q = 32n+1 for some n ≥ 1, then G has a (unique) nonabelian composition factor isomorphic to 2G2(q). We infer that if G is a finite group satisfying |G| = |2G2(q)| and Γ(G) = Γ (2G2(q)) then G ? = 2G2(q). This enables us to give new proofs for some theorems; e.g., a conjecture of W. Shi and J. Bi. Some applications of this result are also considered to the problem of recognition by element orders of finite groups. 相似文献
Let G be a finite group. Let X1(G) be the first column of the ordinary character table of G. We will show that if X1(G) = X1(PGU3(q2)), then G ? PGU3(q2). As a consequence, we show that the projective general unitary groups PGU3(q2) are uniquely determined by the structure of their complex group algebras. 相似文献
Results on the solvability of boundary integral equations on a plane contour with a peak obtained in collaboration with V.G. Maz’ya are developed. Earlier, it was proved that, on a contour Γ with an outward peak, the operator of the boundary equation of the Dirichlet boundary value problem maps the space ?p, β + 1 (Γ) continuously onto \(\mathcal{N}_{p,\beta } (\Gamma )\). The norm of a function in ?p, β (Γ) is defined as
, where q± are the intersection points of Γ with the circle {z: |z| = |q|} and δ > 0 is a fixed small number. On a contour with an inward peak, the operator of the boundary equation of the Dirichlet problem continuously maps ?p, β + 1 (Γ) onto ?p, β(Γ), where ?p, β(Γ) is the direct sum of \(\mathcal{N}_{p,\beta }^ + (\Gamma )\) (Γ) and the space
(Γ) of functions on Γ of the form p(z) = Σk = 0mt(k)Rezk with the parameter m = [μ ? β ? p?1]. The operator I ? 2W of the boundary integral equation of plane elasticity theory, where W is the elastic double-layer potential, is considered. The main result is that the operator I ? 2W continuously maps the space ?p, β + 1 × ?p, β + 1(Γ) to the space \(\mathcal{N}_{p,\beta }^ - \times \mathcal{N}_{p,\beta }^ - (\Gamma )\).
On a contour with an inward peak, the obtained representation of the operator I ? 2W and theorems on the boundedness of auxiliary integral operators imply that the images of vector-valued functions from ?p, β + 1 × ?p, β + 1(Γ) have components representable as sums of functions from the spaces \(\mathcal{N}_{p,\beta }^ - (\Gamma )\)(Γ) and ?p, β(Γ). 相似文献