Principal bundles whose restrictions to a curve are isomorphic

Let X be a normal projective variety defined over an algebraically closed field k. Let |O _X (1)| be a very ample invertible sheaf on X. Let G be an affine algebraic group defined over k. Let E _G and F _G be two principal G-bundles on X. Then there exists an integer n > > 0 (depending on E _G and F _G ) such that if the restrictions of E _G and F _G to a curve C ∈ |O _X (n)| are isomorphic, then they are isomorphic on all of X.     

A weaker version of Lovász' path removal conjecture

We prove there exists a function f(k) such that for every f(k)-connected graph G and for every edge eE(G), there exists an induced cycle C containing e such that G−E(C) is k-connected. This proves a weakening of a conjecture of Lovász due to Kriesell.     

Two Paths Joining Given Vertices in 2k-Edge-Connected Graphs

Let k≥2 be an integer and G = (V(G), E(G)) be a k-edge-connected graph. For X⊆V(G), e(X) denotes the number of edges between X and V(G) − X. Let {s_i, t_i}⊆X_i⊆V(G) (i=1,2) and X₁∩X₂=∅. We here prove that if k is even and e(X_i)≤2k−1 (i=1,2), then there exist paths P₁ and P₂ such that P_i joins s_i and t_i, V(P_i)⊆X_i (i=1,2) and G − E(P₁∪P₂) is (k−2)-edge-connected (for odd k, if e(X₁)≤2k−2 and e(X₂)≤2k−1, then the same result holds [10]), and we give a generalization of this result and some other results about paths not containing given edges.     

Some Notes on Signed Edge Domination in Graphs

The closed neighborhood N_G[e] of an edge e in a graph G is the set consisting of e and of all edges having an end-vertex in common with e. Let f be a function on E(G), the edge set of G, into the set {−1, 1}. If for each e ∈ E(G), then f is called a signed edge dominating function of G. The signed edge domination number γ_s^′(G) of G is defined as . Recently, Xu proved that γ_s^′(G) ≥ |V(G)| − |E(G)| for all graphs G without isolated vertices. In this paper we first characterize all simple connected graphs G for which γ_s^′(G) = |V(G)| − |E(G)|. This answers Problem 4.2 of [4]. Then we classify all simple connected graphs G with precisely k cycles and γ_s^′(G) = 1 − k, 2 − k. A. Khodkar: Research supported by a Faculty Research Grant, University of West Georgia. Send offprint requests to: Abdollah Khodkar.     

Affinely equivalent complete flat manifolds

Let E _Aff(Γ,G, m) be the set of affine equivalence classes of m-dimensional complete flat manifolds with a fixed fundamental group Γ and a fixed holonomy group G. Let n be the dimension of a closed flat manifold whose fundamental group is isomorphic to Γ. We describe E _Aff(Γ,G, m) in terms of equivalence classes of pairs (ε, ρ), consisting of epimorphisms of Γ onto G and representations of G in ℝ ^m-n . As an application we give some estimates of card E _Aff(Γ,G, m).     

Distinguishing labeling of the actions of almost simple groups

Suppose Γ is a group acting on a set X, written as (Γ,X). An r-labeling f: X→{1,2, ..., r} of X is called distinguishing for (Γ,X) if for all σ∈Γ,σ≠1, there exists an element x∈X such that f(x)≠f(x ^σ ). The distinguishing number d(Γ,X) of (Γ,X) is the minimum r for which there is a distinguishing r-labeling for (Γ,X). If Γ is the automorphism group of a graph G, then d(Γ,V (G)) is denoted by d(G), and is called the distinguishing number of the graph G. The distinguishing set of Γ-actions is defined to be D*(Γ)={d(Γ,X): Γ acts on X}, and the distinguishing set of Γ-graphs is defined to be D(Γ)={d(G): Aut(G)≅Γ}. This paper determines the distinguishing set of Γ-actions and the distinguishing set of Γ-graphs for almost simple groups Γ.     

Deformations of complex structures on Γ/SL 2(C)

LetG be a connected complex semisimple Lie group. Let Γ be a cocompact lattice inG. In this paper, we show that whenG isSL ₂(C), nontrivial deformations of the canonical complex structure onX exist if and only if the first Betti number of the lattice Γ is non-zero. It may be remarked that for a wide class of arithmetic groups Γ, one can find a subgroup Γ′ of finite index in Γ, such that Γ′/[Γ′,Γ′] is finite (it is a conjecture of Thurston that this is true for all cocompact lattices inSL(2, C)). We also show thatG acts trivially on the coherent cohomology groupsH ⁱ(Γ/G, O) for anyi≥0.     

On r-recognition by prime graph of Bp(3) where p is an odd prime

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 B _p (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(B _p (3)), where p > 3 is an odd prime, then G @ B_p(3){G\cong B_p(3)} or C _p (3). Also if Γ(G) = Γ(B ₃(3)), then G @ B₃(3), C₃(3), D₄(3){G\cong B_3(3), C_3(3), D_4(3)}, or G/O₂(G) @ Aut(²B₂(8)){G/O_2(G)\cong {\rm Aut}(^2B_2(8))}. As a corollary, the main result of the above paper is obtained.     

Group algebras

Given a group G and a commutative ring k with identity, one can define an k-algebra k[G] called the group algebra of G over k. An element α ∈ k[G] is said to be algebraic if f(α) = 0 for some non-zero polynomial f(X) ∈ k[X]. We will discuss some of the developments in the study of algebraic elements in group algebras.     

An upper bound for domination number of 5-regular graphs

Let G = (V, E) be a simple graph. A subset S ⊆ V is a dominating set of G, if for any vertex u ∈ V-S, there exists a vertex v ∈ S such that uv ∈ E. The domination number, denoted by γ(G), is the minimum cardinality of a dominating set. In this paper we will prove that if G is a 5-regular graph, then γ(G) ⩽ 5/14n.     

Principal bundles whose restriction to curves are trivial

Let k be an algebraically closed field and X a smooth projective variety defined over k. Let E_G be a principal G–bundle over X, where G is an algebraic group defined over k, with the property that for every smooth curve C in X the restriction of E_G to C is the trivial G–bundle. We prove that the principal G–bundle E_G over X is trivial. We also give examples of nontrivial principal bundle over a quasi-projective variety Y whose restriction to every smooth curve in Y is trivial.     

Geometrical implications of the existence of very smooth bump functions in Banach spaces

We show that ifX is a Banach space and if there is a non-zero real-valuedC ^∞-smooth function onX with bounded support, then eitherX contains an isomorphic copy ofc ₀(N), or there is an integerk greater than or equal to 1 such thatX is of exact cotype 2k and, in this case,X contains an isomorphic copy ofl ^2k(N). We also show that ifX is a Banach space such that there is onX a non-zero real-valuedC ⁴-smooth function with bounded support and ifX is of cotypeq forq<4, thenX is isomorphic to a Hilbert space.     

Indices of hyperelliptic curves over p-adic fields

Let k be a p-adic field of odd residue characteristic and let C be a hyperelliptic (or elliptic) curve defined by the affine equation Y ²=h(X). We discuss the index of C if h(X)=ɛf(X), where ɛ is either a non-square unit or a uniformising element in O _k and f(X) a monic, irreducible polynomial with integral coefficients. If a root θ of f generates an extension k(θ) with ramification index a power of 2, we completely determine the index of C in terms of data associated to θ. Theorem 3.11 summarizes our results and provides an algorithm to calculate the index for such curves C. Received: 14 July 1997 / Revised version: 16 February 1998     

On the prime graph of <Emphasis Type="Italic">PSL</Emphasis>(2, <Emphasis Type="Italic">p</Emphasis>) where <Emphasis Type="Italic">p</Emphasis> > 3 is a prime number

Let G be a finite group. We define the prime graph Γ(G) as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, q are joined by an edge if there is an element in G of order pq. Recently M. Hagie [5] determined finite groups G satisfying Γ(G) = Γ(S), where S is a sporadic simple group. Let p > 3 be a prime number. In this paper we determine finite groups G such that Γ(G) = Γ(PSL(2, p)). As a consequence of our results we prove that if p > 11 is a prime number and p ≢ 1 (mod 12), then PSL(2, p) is uniquely determined by its prime graph and so these groups are characterizable by their prime graph. The third author was supported in part by a grant from IPM (No. 84200024).     

On the length of commutators in Chevalley groups

Let G = G(Φ,R) be the simply connected Chevalley group with root system Φ over a ring R. Denote by E(Φ,R) its elementary subgroup. The main result of the article asserts that the set of commutators C = {[a, b]|a ∈ G(Φ, R), b ∈ E(Φ, R)} has bounded width with respect to elementary generators. More precisely, there exists a constant L depending on Φ and dimension of maximal spectrum of R such that any element from C is a product of at most L elementary root unipotent elements. A similar result for Φ = A _l , with a better bound, was earlier obtained by Sivatski and Stepanov.     

AF-Embedding of Crossed Products of Certain Graph C＊-algebras by Quasi-free Actions （II）

Let E be a row-finite directed graph, let G be a locally compact abelian group with dual group Ĝ = Γ, let ω be a labeling map from E* to Γ, and let (C*(E), G, α ^ω ) be the C*-dynamical system defined by ω. Some mappings concerning the AF-embedding construction of C*(E) ×_a^w GC*(E) \times _{\alpha ^\omega } G are studied in more detail. Several necessary conditions of AF-embedding and some properties of almost proper labeling map are obtained. Moreover it is proved that if E is constructed by attaching some 1-loops to a directed graph T consisting of some rooted directed trees and G is compact, then ω is almost proper, that is a sufficient condition for AF-embedding, if and only if Σ _j=1 ^k w_gj 1 1_G\omega _{\gamma _j } \ne 1_\Gamma for any loop γ _i , γ ₂, ..., γ _k attached to one path in T.     

Tiling Transitive Tournaments and Their Blow-ups

Let TT _k denote the transitive tournament on k vertices. Let TT(h,k) denote the graph obtained from TT _k by replacing each vertex with an independent set of size h≥1. The following result is proved: Let c ₂=1/2, c ₃=5/6 and c _k =1−2^{−k−log k} for k≥4. For every ∈>0 there exists N=N(∈,h,k) such that for every undirected graph G with n>N vertices and with δ(G)≥c _k n, every orientation of G contains vertex disjoint copies of TT(h,k) that cover all but at most ∈n vertices. In the cases k=2 and k=3 the result is asymptotically tight. For k≥4, c _k cannot be improved to less than 1−2^{−0.5k(1+o(1))}. This revised version was published online in June 2006 with corrections to the Cover Date.     

Lower bounds on signed edge total domination numbers in graphs

The open neighborhood N _G (e) of an edge e in a graph G is the set consisting of all edges having a common end-vertex with e. Let f be a function on E(G), the edge set of G, into the set {−1, 1}. If for each e ∈ E(G), then f is called a signed edge total dominating function of G. The minimum of the values , taken over all signed edge total dominating function f of G, is called the signed edge total domination number of G and is denoted by γ _st ′(G). Obviously, γ _st ′(G) is defined only for graphs G which have no connected components isomorphic to K ₂. In this paper we present some lower bounds for γ _st ′(G). In particular, we prove that γ _st ′(T) ⩾ 2 − m/3 for every tree T of size m ⩾ 2. We also classify all trees T with γ _st ′(T). Research supported by a Faculty Research Grant, University of West Georgia.     

A Property About Minimum Edge- and Minimum Clique-Cover of a Graph

 Let G be a graph with n vertices, and denote as γ(G) (as θ(G)) the cardinality of a minimum edge cover (of a minimum clique cover) of G. Let E (let C) be the edge-vertex (the clique-vertex) incidence matrix of G; write then P(E)={x∈ℜ ⁿ :Ex≤1,x≥0}, P(C)={x∈ℜ ⁿ :Cx≤1,x≥0}, α _E (G)=max{1 ^T x subject to x∈P(E)}, and α _C (G)= max{1 ^T x subject to x∈P(C)}. In this paper we prove that if α _E (G)=α _C (G), then γ(G)=θ(G). Received: May 20, 1998?Final version received: April 12, 1999     

On the adjacent-vertex-strongly-distinguishing total coloring of graphs

For any vertex u∈V(G), let T_N(U)={u}∪{uv|uv∈E(G), v∈v(G)}∪{v∈v(G)|uv∈E(G)}and let f be a total k-coloring of G. The total-color neighbor of a vertex u of G is the color set C_f(u)={f(x)|x∈TN(U)}. For any two adjacent vertices x and y of V(G)such that C_f(x)≠C_f(y), we refer to f as a k-avsdt-coloring of G("avsdt"is the abbreviation of"adjacent-vertex-strongly- distinguishing total"). The avsdt-coloring number of G, denoted by X_(ast)(G), is the minimal number of colors required for a avsdt-coloring of G. In this paper, the avsdt-coloring numbers on some familiar graphs are studied, such as paths, cycles, complete graphs, complete bipartite graphs and so on. We proveΔ(G) 1≤X_(ast)(G)≤Δ(G) 2 for any tree or unique cycle graph G.