Quasirecognition by prime graph of finite simple groups L n (2) and U n (2)

Let G be a finite group. The prime graph ??(G) of G is defined as follows. The vertices of ??(G) are the primes dividing the order of G and two distinct vertices p, p?? are joined by an edge if G has an element of order pp??. Let L=L _n (2) or U _n (2), where n?R17. We prove that L is quasirecognizable by prime graph, i.e. if G is a finite group such that ??(G)=??(L), then G has a unique nonabelian composition factor isomorphic to L. As a consequence of our result we give a new proof for the recognition by element orders of L _n (2). Also we conclude that the simple group U _n (2) is quasirecognizable by element orders.     

Multiplicative mappings at some points on matrix algebras

Let M_n be the algebra of all n×n matrices, and let φ:M_n→M_n be a linear mapping. We say that φ is a multiplicative mapping at G if φ(ST)=φ(S)φ(T) for any S,T∈M_n with ST=G. Fix G∈M_n, we say that G is an all-multiplicative point if every multiplicative linear bijection φ at G with φ(I_n)=I_n is a multiplicative mapping in M_n, where I_n is the unit matrix in M_n. We mainly show in this paper the following two results: (1) If G∈M_n with detG=0, then G is an all-multiplicative point in M_n; (2) If φ is an multiplicative mapping at I_n, then there exists an invertible matrix P∈M_n such that either φ(S)=PSP^-1 for any S∈M_n or φ(T)=PT^trP^-1 for any T∈M_n.     

On the geometry of flat pseudo-Riemannian homogeneous spaces

Let M = ? _s ⁿ /Γ be a complete flat pseudo-Riemannian homogeneous manifold, Γ ? Iso(? _s ⁿ ) its fundamental group and G the Zariski closure of Γ in Iso(? _s ⁿ ). We show that the G-orbits in ? _s ⁿ are affine subspaces and affinely diffeomorphic to G endowed with the (0)-connection. If the restriction of the pseudo-scalar product on ? _s ⁿ to the G-orbits is nondegenerate, then M has abelian linear holonomy. If additionally G is not abelian, then G contains a certain subgroup of dimension 6. In particular, for non-abelian G, orbits with non-degenerate metric can appear only if dim G ≥ 6. Moreover, we show that ? _s ⁿ is a trivial algebraic principal bundle G → M → ? ^n?k . As a consquence, M is a trivial smooth bundle G/Γ → M → ? ^n?k with compact fiber G/Γ.     

Conjugacy of Sylow 2-Subloops of the Chein Loops M 2n (G, 2)

In general, Sylow's Theorems do not hold for finite Moufang loops. It can be seen that if p is an odd prime then the Sylow p-subloops of the Chein loop M _2n (G, 2) are conjugate. Here we prove that it is also true that all the Sylow 2-subloops of M _2n (G, 2) are conjugate.     

L n/2-Curvature Gaps of the Weyl Tensor

We show a lower bound of the L ^n/2-norm of the Weyl tensor in terms of the Yamabe invariant if M ⁿ has Betti number b _n/2>0. This is a counterpart to a result by Akutagawa, Botvinnik, Kobayashi, and Seshadri, who proved that the L ^n/2-norm of the Weyl tensor can be arbitrarily large for conformal classes whose Yamabe invariant is close to the sigma invariant.     

On Connected 2-K n -Residual Graphs

A graph G is said to be K _n -residual if for every point u in G, the graph obtained by removing the closed neighborhood of u from G is isomorphic to K _n . We inductively define a multiply-K _n -residual graph by saying that G is m-K _n -residual if the removal of the closed neighborhood of any vertex of G results in an (m – 1)-K _n -residual graphs. Erdös, Harary and Klawe [2] determined the minimum order of the m?K _n -residual graphs for all m and n, which are not necessarily connected, the minimum order of connected; K _n -residual graphs, all K _n -residual extremal graphs. They also stated some conjectures regarding the connected case. In this paper, we determine the minimum order of a connected 2-K _n -residual graph and specify the extremal graphs, expect for n = 3. In particular, we determining only one connected 2-K ₄-residual graph of minimal order, and show that there is a connected 2-K ₆-residual graph non isomorphic to K ₈ × K ₃ with minimum order. Finally we present and a revised version of the conjecture in [2].     

Peixoto graph of Morse-Smale diffeomorphisms on manifolds of dimension greater than three

Let M ⁿ be a closed orientable manifold of dimension greater than three and G ₁(M ⁿ ) be the class of orientation-preserving Morse-Smale diffeomorphisms on M ⁿ such that the set of unstable separatrices of every f ∈ G ₁(M ⁿ ) is one-dimensional and does not contain heteroclinic orbits. We show that the Peixoto graph is a complete invariant of topological conjugacy in G ₁(M ⁿ ).     

More results on Ramsey—Turán type problems

The paper deals with common generalizations of classical results of Ramsey and Turán. The following is one of the main results. Assumek≧2, ε>0,G _n is a sequence of graphs ofn-vertices and at least 1/2((3k?5) / (3k?2)+ε)n ² edges, and the size of the largest independent set inG _n iso(n). LetH be any graph of arboricity at mostk. Then there exists ann ₀ such that allG _n withn>n ₀ contain a copy ofH. This result is best possible in caseH=K _2k .     

The evolution of the min-min random graph process

We study the following min-min random graph process G=(G₀,G₁,…): the initial state G₀ is an empty graph on n vertices (n even). Further, G_M+1 is obtained from G_M by choosing a pair {v,w} of distinct vertices of minimum degree uniformly at random among all such pairs in G_M and adding the edge {v,w}. The process may produce multiple edges. We show that G_M is asymptotically almost surely disconnected if M≤n, and that for M=(1+t)n, constant, the probability that G_M is connected increases from 0 to 1. Furthermore, we investigate the number X of vertices outside the giant component of G_M for M=(1+t)n. For constant we derive the precise limiting distribution of X. In addition, for n⁻¹ln⁴n≤t=o(1) we show that tX converges to a gamma distribution.     

Euler Tours of Maximum Girth in K2 n +1 and K2 n ,2 n

Given an eulerian graph G and an Euler tour T of G, the girth of T, denoted by g(T), is the minimum integer k such that some segment of k+1 consecutive vertices of T is a cycle of length k in G. Let g^E(G)= maxg(T) where the maximum is taken over all Euler tours of G.We prove that g^E(K_2n,2n)=4n–4 and 2n–3g^E(K_2n+1)2n–1 for any n2. We also show that g^E(K₇)=4. We use these results to prove the following:1)The graph K_2n,2n can be decomposed into edge disjoint paths of length k if and only if k4n–1 and the number of edges in K_2n,2n is divisible by k.2)The graph K_2n+1 can be decomposed into edge disjoint paths of length k if and only if k2n and the number edges in K_2n+1 is divisible by k.