A proof of a conjecture on the Randić index of graphs with given girth

The Randić index R(G) of a graph G is defined by , where is the degree of a vertex u in G and the summation extends over all edges uv of G. Aouchiche, Hansen and Zheng proposed the following conjecture: For any connected graph on n≥3 vertices with Randić index R and girth g,

with equalities if and only if . This paper is devoted to giving a confirmative proof to this conjecture.     

Monotone Jacobi parameters and non-Szegő weights

We relate asymptotics of Jacobi parameters to asymptotics of the spectral weights near the edges. Typical of our results is that for a_n≡1, b_n=−Cn^−β (), one has on (−2,2), and near x=2, where

     

A note on extreme cases of Sobolev embeddings

We study the spaces of functions on for which the generalized partial derivatives exist and belong to different Lorentz spaces L^p_k,s_k. For this kind of functions we prove a sharp version of the extreme case of the Sobolev embedding theorem using L(∞,s) spaces.     

Relative asymptotic of multiple orthogonal polynomials for Nikishin systems

We prove the relative asymptotic behavior for the ratio of two sequences of multiple orthogonal polynomials with respect to the Nikishin systems of measures. The first Nikishin system is such that for each k, σ_k has a constant sign on its compact support consisting of an interval , on which almost everywhere, and a discrete set without accumulation points in . If denotes the smallest interval containing , we assume that Δ_k∩Δ_k+1=0/, k=1,…,m−1. The second Nikishin system is a perturbation of the first by means of rational functions r_k, k=1,…,m, whose zeros and poles lie in .     

Regularity results for the gradient of solutions of a class of linear elliptic systems with data

We prove that if a vector-function f belongs to the Morrey space , with , n≥3, N≥2, λ]0,n−2], and u is the solution of the system

then Du belongs to the space , for any , provided the matrix of bounded measurable coefficients (A_ij) has sufficiently small dispersion of the eigenvalues.     

Engel graph associated with a group

Let G be a non-Engel group and let L(G) be the set of all left Engel elements of G. Associate with G a graph as follows: Take G L(G) as vertices of and join two distinct vertices x and y whenever [x,_ky]≠1 and [y,_kx]≠1 for all positive integers k. We call , the Engel graph of G. In this paper we study the graph theoretical properties of .     

Perfect packings with complete graphs minus an edge

Let denote the graph obtained from K_r by deleting one edge. We show that for every integer r≥4 there exists an integer n₀=n₀(r) such that every graph G whose order n≥n₀ is divisible by r and whose minimum degree is at least contains a perfect -packing, i.e. a collection of disjoint copies of which covers all vertices of G. Here is the critical chromatic number of . The bound on the minimum degree is best possible and confirms a conjecture of Kawarabayashi for large n.     

Conditional limiting distribution of beta-independent random vectors

The paper deals with random vectors in , possessing the stochastic representation , where R is a positive random radius independent of the random vector and is a non-singular matrix. If is uniformly distributed on the unit sphere of , then for any integer m<d we have the stochastic representations and , with W≥0, such that W² is a beta distributed random variable with parameters m/2,(d−m)/2 and (U₁,…,U_m),(U_m+1,…,U_d) are independent uniformly distributed on the unit spheres of and , respectively. Assuming a more general stochastic representation for in this paper we introduce the class of beta-independent random vectors. For this new class we derive several conditional limiting results assuming that R has a distribution function in the max-domain of attraction of a univariate extreme value distribution function. We provide two applications concerning the Kotz approximation of the conditional distributions and the tail asymptotic behaviour of beta-independent bivariate random vectors.     

Pointwise universal trigonometric series

A series is called a pointwise universal trigonometric series if for any , there exists a strictly increasing sequence of positive integers such that converges to f(z) pointwise on . We find growth conditions on coefficients allowing and forbidding the existence of a pointwise universal trigonometric series. For instance, if as |n|→∞ for some ε>0, then the series S_a cannot be pointwise universal. On the other hand, there exists a pointwise universal trigonometric series S_a with as |n|→∞.     

Gottlieb groups of spheres

This paper takes up the systematic study of the Gottlieb groups of spheres for k≤13 by means of the classical homotopy theory methods. We fully determine the groups for k≤13 except for the 2-primary components in the cases: k=9,n=53;k=11,n=115. In particular, we show if n=2ⁱ−7 for i≥4.     

Sparse p-version BEM for first kind boundary integral equations with random loading

We consider the weakly singular boundary integral equation on a deterministic smooth closed curve with random loading g(ω). Given the kth order statistical moment of g, the aim is the efficient deterministic computation of the kth order statistical moment of u. We derive a deterministic formulation for the kth statistical moment. It is posed in the tensor product Sobolev space and involves the k-fold tensor product operator . The standard full tensor product Galerkin BEM requires unknowns for the kth moment problem, where N is the number of unknowns needed to discretize Γ. Extending ideas of [V.N. Temlyakov, Approximation of functions with bounded mixed derivative, Proc. Steklov Inst. Math. (1989) vi+121. A translation of Trudy Mat. Inst. Steklov 178 (1986)], we develop the p-Sparse Grid Galerkin BEM to reduce the number of unknowns from to .     

A note on the signed edge domination number in graphs

Let G=(V(G),E(G)) be a graph. A function f:E(G)→{+1,−1} is called the signed edge domination function (SEDF) of G if ∑_e^′N[e]f(e^′)≥1 for every eE(G). The signed edge domination number of G is defined as is a SEDF of G}. Xu [Baogen Xu, Two classes of edge domination in graphs, Discrete Applied Mathematics 154 (2006) 1541–1546] researched on the edge domination in graphs and proved that for any graph G of order n(n≥4). In the article, he conjectured that: For any 2-connected graph G of order n(n≥2), . In this note, we present some counterexamples to the above conjecture and prove that there exists a family of k-connected graphs G_m,k with .     

All-derivable points in continuous nest algebras

Let be an operator algebra on a Hilbert space. We say that an element is an all-derivable point of for the strong operator topology if every strong operator topology continuous derivable linear mapping φ at G (i.e. φ(ST)=φ(S)T+Sφ(T) for any with ST=G) is a derivation. Let be a continuous nest on a complex and separable Hilbert space H. We show in this paper that every orthogonal projection operator P(M) () is an all-derivable point of for the strong operator topology.     

Polynomial-time approximation schemes for piercing and covering with applications in wireless networks

Let be a set of disks of arbitrary radii in the plane, and let be a set of points. We study the following three problems: (i) Assuming contains the set of center points of disks in , find a minimum-cardinality subset of (if exists), such that each disk in is pierced by at least h points of , where h is a given constant. We call this problem minimum h-piercing. (ii) Assuming is such that for each there exists a point in whose distance from D's center is at most αr(D), where r(D) is D's radius and 0α<1 is a given constant, find a minimum-cardinality subset of , such that each disk in is pierced by at least one point of . We call this problem minimum discrete piercing with cores. (iii) Assuming is the set of center points of disks in , and that each covers at most l points of , where l is a constant, find a minimum-cardinality subset of , such that each point of is covered by at least one disk of . We call this problem minimum center covering. For each of these problems we present a constant-factor approximation algorithm (trivial for problem (iii)), followed by a polynomial-time approximation scheme. The polynomial-time approximation schemes are based on an adapted and extended version of Chan's [T.M. Chan, Polynomial-time approximation schemes for packing and piercing fat objects, J. Algorithms 46 (2003) 178–189] separator theorem. Our PTAS for problem (ii) enables one, in practical cases, to obtain a (1+ε)-approximation for minimum discrete piercing (i.e., for arbitrary ).     

Chromatic Turán problems and a new upper bound for the Turán density of

We consider a new type of extremal hypergraph problem: given an r-graph and an integer k≥2 determine the maximum number of edges in an -free, k-colourable r-graph on n vertices.Our motivation for studying such problems is that it allows us to give a new upper bound for an old Turán problem. We show that a 3-graph in which any four points span at most two edges has density less than , improving previous bounds of due to de Caen [D. de Caen, Extension of a theorem of Moon and Moser on complete subgraphs, Ars Combin. 16 (1983) 5–10], and due to Mubayi [D. Mubayi, On hypergraphs with every four points spanning at most two triples, Electron. J. Combin. 10 (10) (2003)].     

Inheritance of hyper-duality in imprimitive Bose–Mesner algebras

We prove the following result concerning the inheritance of hyper-duality by block and quotient Bose–Mesner algebras associated with a hyper-dual pair of imprimitive Bose–Mesner algebras. Let and denote Bose–Mesner algebras. Suppose there is a hyper-duality ψ from the subconstituent algebra of with respect to p to the subconstituent algebra of with respect to . Also suppose that is imprimitive with respect to a subset of Hadamard idempotents, so is dual imprimitive with respect to the subset of primitive idempotents, where is the formal duality associated with ψ. Let denote the block Bose–Mesner algebra of on the block containing p, and let denote the quotient Bose–Mesner algebra of with respect to . Then there is a hyper-duality from the subconstituent algebra of with respect to p to the subconstituent algebra of with respect to .     

Varieties of nilpotent elements for simple Lie algebras I: Good primes

Let G be a simple algebraic group over k=C, or where p is good. Set g=LieG. Given rN and a faithful (restricted) representation , one can define a variety of nilpotent elements . In this paper we determine this variety when ρ is an irreducible representation of minimal dimension or the adjoint representation.     

Involutions of reductive Lie algebras in positive characteristic

Let G be a reductive group over a field k of characteristic ≠2, let , let θ be an involutive automorphism of G and let be the associated symmetric space decomposition. For the case of a ground field of characteristic zero, the action of the isotropy group G^θ on is well understood, since the well-known paper of Kostant and Rallis [B. Kostant, S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971) 753–809]. Such a theory in positive characteristic has proved more difficult to develop. Here we use an approach based on some tools from geometric invariant theory to establish corresponding results in (good) positive characteristic.Among other results, we prove that the variety of nilpotent elements of has a dense open orbit, and that the same is true for every fibre of the quotient map . However, we show that the corresponding statement for G, conjectured by Richardson, is not true. We provide a new, (mostly) calculation-free proof of the number of irreducible components of , extending a result of Sekiguchi for . Finally, we apply a theorem of Skryabin to describe the infinitesimal invariants .     

Ore-type and Dirac-type theorems for matroids

Let M be a connected binary matroid having no -minor. Let be a collection of cocircuits of M. We prove there is a circuit intersecting all cocircuits of if either one of two things hold:

(i) For any two disjoint cocircuits and in it holds that .

(ii) For any two disjoint cocircuits and in it holds that .

Part (ii) implies Ore's Theorem, a well-known theorem giving sufficient conditions for the existence of a hamilton cycle in a graph. As an application of part (i), it is shown that if M is a k-connected regular matroid and has cocircumference c^*2k, then there is a circuit which intersects each cocircuit of size c^*−k+2 or greater.We also extend a theorem of Dirac for graphs by showing that for any k-connected binary matroid M having no -minor, it holds that for any k cocircuits of M there is a circuit which intersects them.     

Semifields of order with left nucleus and center

In [G. Marino, O. Polverino, R. Trombetti, On -linear sets of PG(3,q³) and semifields, J. Combin. Theory Ser. A 114 (5) (2007) 769–788] it has been proven that there exist six non-isotopic families (i=0,…,5) of semifields of order q⁶ with left nucleus and center , according to the different geometric configurations of the associated -linear sets. In this paper we first prove that any semifield of order q⁶ with left nucleus , right and middle nuclei and center is isotopic to a cyclic semifield. Then, we focus on the family by proving that it can be partitioned into three further non-isotopic families: , , and we show that any semifield of order q⁶ with left nucleus , right and middle nuclei and center belongs to the family .