On divisors of Lucas and Lehmer numbers

Let u _n be the nth term of a Lucas sequence or a Lehmer sequence. In this article we shall establish an estimate from below for the greatest prime factor of u _n which is of the form n exp(log n/104 log log n). In doing so, we are able to resolve a question of Schinzel from 1962 and a conjecture of Erd?s from 1965. In addition we are able to give the first general improvement on results of Bang from 1886 and Carmichael from 1912.     

On bijections that preserve complementarity of subspaces

The set G of all m-dimensional subspaces of a 2m-dimensional vector space V is endowed with two relations, complementarity and adjacency. We consider bijections from G onto G^′, where G^′ arises from a 2m^′-dimensional vector space V^′. If such a bijection ? and its inverse leave one of the relations from above invariant, then also the other. In case m?2 this yields that ? is induced by a semilinear bijection from V or from the dual space of V onto V^′.As far as possible, we include also the infinite-dimensional case into our considerations.     

Nonlinear approximation of functions from the class L r with respect to the Vilenkin system

In this paper we prove that for any function f from the class L ^r on [0, 1) one can find a function g from the same class (which differs from f on a set of arbitrarily small measure) whose greedy algorithm with respect to the Vilenkin system converges to f.     

An application of the Helly property to the partially ordered sets

Quilliot (Discrete Math. 1982.) showed that when the bowls of a connected graph satisfy the Helly property it is possible to deduce for this graph some fixed point and homomorphism extension theorems. For a partially ordered set E a special family of subsets is defined which, when it satisfies the Helly property, permits the deductions that every homomorphism from E into E has a fixed point, that every antitone function from E has “almost” a fixed point, and that there exists a simple criterion letting us know when a function f from a subset A of a partially ordered set G can be extended into a homomorphism from G to E.     

The reconstruction conjecture and edge ideals

Given a simple graph G on n vertices, we prove that it is possible to reconstruct several algebraic properties of the edge ideal from the deck of G, that is, from the collection of subgraphs obtained by removing a vertex from G. These properties include the Krull dimension, the Hilbert function, and all the graded Betti numbers β_i,j where j<n. We also state many further questions that arise from our study.     

Relative hemisystems on the Hermitian surface

Let S be a generalized quadrangle of order (q ²,q) containing a subquadrangle S′ of order (q,q). Then any line of S either meets S′ in q+1 points or is disjoint from S′. After Penttila and Williford (J. Comb. Theory, Ser. A 118:502–509, 2011), we call a subset H of the lines disjoint from S′ a relative hemisystem of S with respect to S′, provided that for each point x of S?S′, exactly half of the lines through x disjoint from S′ lie in H. A new infinite family of relative hemisystems on the generalized quadrangle $\mathcal{H}(3,q^{2})$ admitting the linear group PSL(2,q) as an automorphism group is constructed. The association schemes arising from our construction are not equivalent to those arising from the Penttila–Williford relative hemisystems.     

Operator-valued Fourier Haar multipliers

Criteria are given to ensure the boundedness of Fourier Haar multiplier operators from Lp([0,1],X) to Lq([0,1],Y) where the Fourier Haar multiplier sequences come not from R, as in the classical setting, but rather from the space of bounded linear operators from a Banach space X into a Banach space Y.     

Bounds on a graph's security number

Let G=(V,E) be a graph. A set S⊆V is a defensive alliance if |N[x]∩S|?|N[x]-S| for every x∈S. Thus, each vertex of a defensive alliance can, with the aid of its neighbors in S, be defended from attack by its neighbors outside of S. An entire set S is secure if any subset X⊆S, not just singletons, can be defended from an attack from outside of S, under an appropriate definition of what such a defense implies. The security number s(G) of G is the cardinality of a smallest secure set. Bounds on s(G) are presented.     

Farthest points and strong convexity of sets

We consider the existence and uniqueness of the farthest point of a given set A in a Banach space E from a given point x in the space E. It is assumed that A is a convex, closed, and bounded set in a uniformly convex Banach space E with Fréchet differentiable norm. It is shown that, for any point x sufficiently far from the set A, the point of the set A which is farthest from x exists, is unique, and depends continuously on the point x if and only if the set A in the Minkowski sum with some other set yields a ball. Moreover, the farthest (from x) point of the set A also depends continuously on the set A in the sense of the Hausdorff metric. If the norm ball of the space E is a generating set, these conditions on the set A are equivalent to its strong convexity.     

On derivable mappings

A linear mapping δ from an algebra A into an A-bimodule M is called derivable at c∈A if δ(a)b+aδ(b)=δ(c) for all a,b∈A with ab=c. For a norm-closed unital subalgebra A of operators on a Banach space X, we show that if C∈A has a right inverse in B(X) and the linear span of the range of rank-one operators in A is dense in X then the only derivable mappings at C from A into B(X) are derivations; in particular the result holds for all completely distributive subspace lattice algebras, J-subspace lattice algebras, and norm-closed unital standard algebras of B(X). As an application, every Jordan derivation from such an algebra into B(X) is a derivation. For a large class of reflexive algebras A on a Banach space X, we show that inner derivations from A into B(X) can be characterized by boundedness and derivability at any fixed C∈A, provided C has a right inverse in B(X). We also show that if A is a canonical subalgebra of an AF C^∗-algebra B and M is a unital Banach A-bimodule, then every bounded local derivation from A into M is a derivation; moreover, every bounded linear mapping from A into B that is derivable at the unit I is a derivation.     

Quadratic forms over polynomial rings over global fields

It is proved that a quadratic space over the polynomial extension of a global field K is extended from K if it is extended from K_v for every completion K_v of K.     

EDGE SUBDIVISIONS OF A GIVEN GEODETIC BLOCK PRESERVING ITS PROPERTY TO BE GEODETIC

1IntroductionInthispaperweshallconsideronlyundirected2-connectedsimplegraphs,i.e,graphsthatareloopless,finite,undirectedandwithoutmultipleedges.AgraphGissaidtobegeodeticifanypairofpointsofGarejoinedbyauniquepathofshortestlength,i.e,aulliquedistancepath[1].A2-connectedgeodeticgraphiscalledageodeticblock.Agrapllisgeodeticiffeachofitsblocksisgeodetic(seeStempleandWatkins['l).Obviously,oddcycle,tree,completegrapharegeodeticgraph,wecallthemthetrivialgeodeticgraph.Nowweonlycollsidertilenontrivial…     

Computing the stationary distribution for infinite Markov chains

In a situation where the unique stationary distribution vector of an infinite irreducible positive-recurrent stochastic matrix P is not analytically determinable, numerical approximations are needed. This paper partially synthesizes and extends work on finite-vector approximative solutions obtained from nxn northwest corner truncations _(n)P of P, from the standpoints of (pointwise convergence) algorithms as n→∞, and the manner of their computer implementation with a view to numerical stability and conditioning. The problem for finite n is connected with that of finding the unique stationary distribution of the finite stochastic matrix _(n)P? obtained from _(n)P by augmenting a column.     

Security in graphs

Let G=(V,E) be a graph. A set S⊆V is a defensive alliance if |N[x]∩S|?|N[x]-S| for every x∈S. Thus, each vertex of a defensive alliance can, with the aid of its neighbors in S, be defended from attack by its neighbors outside of S. An entire set S is secure if any subset X⊆S can be defended from an attack from outside of S, under an appropriate definition of what such a defense implies. Necessary and sufficient conditions for a set to be secure are determined.     

Matrix semigroup homomorphisms into higher dimensions

We study non-degenerate irreducible homomorphisms from the multiplicative semigroup of all n-by-n matrices over an algebraically closed field of characteristic zero to the semigroup of m-by-m matrices over the same field. We prove that every non-degenerate homomorphism from the multiplicative semigroup of all n-by-n matrices to the semigroup of (n + 1)-by-(n + 1) matrices when n ? 3 is reducible and that every non-degenerate homomorphism from the multiplicative semigroup of all 3-by-3 matrices to the semigroup of 5-by-5 matrices is reducible.     

A two-echelon inventory model with lost sales

This paper considers a single-item, two-echelon, continuous-review inventory model. A number of retailers have their stock replenished from a central warehouse. The warehouse in turn replenishes stock from an external supplier. The demand processes on the retailers are independent Poisson. Demand not met at a retailer is lost. The order quantity from each retailer on the warehouse and from the warehouse on the supplier takes the same fixed value Q, an exogenous variable determined by packaging and handling constraints. Retailer i follows a (Q, R_i) control policy. The warehouse operates an (SQ, (S − 1)Q) policy, with non-negative integer S. If the warehouse is in stock then the lead time for retailer i is the fixed transportation time L_i from the warehouse to that retailer. Otherwise retailer orders are met, after a delay, on a first-come first-served basis. The lead time on a warehouse order is fixed. Two further assumptions are made: that each retailer may only have one order outstanding at any time and that the transportation time from the warehouse to a retailer is not less than the warehouse lead time. The performance measures of interest are the average total stock in the system and the fraction of demand met in the retailers. Procedures for determining these performance measures and optimising the behaviour of the system are developed.     

The degree of multivalued maps from manifolds to spheres

In this paper, for closed connected oriented manifolds M and N of the same dimension, we study the degree of a triple (??, p, q), where p is a Vietoris map from a compact space ?? to M and q is a continuous map from ?? to N. In particular, we have Borsuk?CUlam-type degree theorems on manifolds with involutions.     

Module derivation problem for inverse semigroups

The derivation problem for a locally compact group G asserts that each bounded derivation from L ¹(G) to L ¹(G) is implemented by an element of M(G). Recently a simple proof of this result was announced. We show that basically the same argument with some extra manipulations with idempotents solves the module derivation problem for inverse semigroups, asserting that for an inverse semigroup S with set of idempotents E and maximal group homomorphic image G _S , if E acts on S trivially from the left and by multiplication from the right, any bounded module derivation from ? ¹(S) to ? ¹(G _S ) is inner.     

One-point extensions of generalized hexagons and octagons

In this note, we prove the uniqueness of the one-point extension S of a generalized hexagon of order 2 and prove the non-existence of such an extension S of any other finite generalized hexagon of classical order, different from the one of order 2, and of the known finite generalized octagons provided the following property holds: for any three points x, y and z of S, the graph-theoretic distance from y to z in the derived generalized hexagon S_x is the same as the distance from x to z in S_y.