probabilistic generation of finite simple groups ii abstract in earlier work it was shown that each nonabelian finite simple group g has a conjugacy class c such that whenever 1 x g the probability is greater than 1 10 that g x y for a random y c much stronger asymptotic results were also proved here we show that allowing equality the bound 1 10 can be replaced by 13 42 and excluding an explicitly listed set of simple groups the bound 2 3 holds we use these results to show that any nonabelian finite simple group g has a conjugacy class c such that if x 1 x 2 are nontrivial elements of g then there exists y c such that g x 1 y x 2 y similarly aside from one infinite family and a small explicit finite set of simple groups g has a conjugacy class c such that if x 1 x 2 x 3 are nontrivial elements of g then there exists y c such that g x 1 y x 2 y x 3 y we also prove analogous but weaker results for almost simple groups

the steiner tree problem i formulations compositions and extension of facets in this paper we give some integer programming formulations for the steiner tree problem on undirected and directed graphs and study the associated polyhedra we give some families of facets for the undirected case along with some compositions and extensions we also give a projection that relates the steiner tree polyhedron on an undirected graph to the polyhedron for the corresponding directed graph this is used to show that the lp relaxation of the directed formulation is superior to the lp relaxation of the undirected one

on a conjecture of gluck let mathbf f g and b g respectively denote the fitting subgroup and the largest degree of an irreducible complex character of a finite group g a well known conjecture of d gluck claims that if g is solvable then g mathbf f g le b g 2 we confirm this conjecture in the case where mathbf f g is coprime to 6 we also extend the problem to arbitrary finite groups and prove several results showing that the largest irreducible character degree of a finite group strongly controls the group structure

a mass transportation approach to quantitative isoperimetric inequalities a sharp quantitative version of the anisotropic isoperimetric inequality is established corresponding to a stability estimate for the wulff shape of a given surface tension energy this is achieved by exploiting mass transportation theory especially gromov s proof of the isoperimetric inequality and the brenier mccann theorem a sharp quantitative version of the brunn minkowski inequality for convex sets is proved as a corollary

the tree of hubs location problem this paper presents the tree of hubs location problem it is a network hub location problem with single assignment where a fixed number of hubs have to be located with the particularity that it is required that the hubs are connected by means of a tree the problem combines several aspects of location network design and routing problems potential applications appear in telecommunications and transportation systems when set up costs for links between hubs are so high that full interconnection between hub nodes is prohibitive we propose an integer programming formulation for the problem furthermore we present some families of valid inequalities that reinforce the formulation and we give an exact separation procedure for them finally we present computational results using the well known ap and cab data sets

solving group steiner problems as steiner problems abstract the generalized spanning tree or group steiner problem gsp is a generalization of the steiner problem in graphs spg one requires a tree spanning at least one vertex of each subset given in a family of vertex subsets while minimizing the sum of the corresponding edge costs specialized solution procedures have been developed for this problem in this paper we investigate the performance of a known but so far neglected transformation to the undirected steiner problem in graphs when combined with a recent metaheuristic for the spg this straightforward approach compares favorably with specialized gsp heuristics thus we set a standard for future algorithms

a branch and cut algorithm for the symmetric generalized traveling salesman problem we consider a variant of the classical symmetric traveling salesman problem in which the nodes are partitioned into clusters and the salesman has to visit at least one node for each cluster this np hard problem is known in the literature as the symmetric generalized traveling salesman problem gtsp and finds practical applications in routing scheduling and location routing in a companion paper fischetti et al fischetti m j j salazar p toth 1995 the symmetric generalized traveling salesman polytope networks 26 113 123 we modeled gtsp as an integer linear program and studied the facial structure of two polytopes associated with the problem here we propose exact and heuristic separation procedures for some classes of facet defining inequalities which are used within a branch and cut algorithm for the exact solution of gtsp heuristic procedures are also described extensive computational results for instances taken from the literature and involving up to 442 nodes are reported

subgroups of maximal rank in finite exceptional groups of lie type the purpose of this paper is to investigate a large and natural class of maximal subgroups of the finite exceptional groups of lie type which we call subgroups of maximal rank these subgroups play a prominent role both in the classification of local maximal subgroups in 9 theorem 1 and in the reduction theorem for subgroups in 28 theorem 2 in 9 it is shown that any local maximal subgroup of a finite exceptional group of lie type is either a subgroup of maximal rank or one of a small list of exceptions and in 28 using this result it is proved that any maximal subgroup is either of maximal rank or almost simple or in a known list to describe these subgroups we require the following notation let g be a simple adjoint algebraic group over an algebraically closed field k of characteristic p 0 and let o be an endomorphism of g such that l ga is a finite exceptional simple group of lie type over fq where q p let a be a group such that f x l the group aut l is generated by ga together with field and graph automorphisms see 5 38 all of which extend to morphisms of the abstract group g commuting with o thus there is a subgroup x of caut c a such that x x o and so x acts on the set of a stable subsets of g for a a stable subset y we write nx y for the stabilizer in x of y if d is a a stable closed connected reductive subgroup of g containing a maximal torus t of g and m nx d we call m a subgroup of maximal rank in x in this paper we determine the structure and conjugacy classes of those subgroups of maximal rank which are maximal in x

a new relaxation method for the generalized minimum spanning tree problem we consider a generalization of the minimum spanning tree problem called the generalized minimum spanning tree problem denoted by gmst it is known that the gmst problem is hard we present several mixed integer programming formulations of the problem based on a new formulation of the problem we give a new solution procedure that finds the optimal solution of the gmst problem for graphs with nodes up to 240 we discuss the advantages of our approach in comparison with earlier methods

guided local search and its application to the traveling salesman problem abstract the traveling salesman problem tsp is one of the most famous problems in combinatorial optimization in this paper we are going to examine how the techniques of guided local search gls and fast local search fls can be applied to the problem gls sits on top of local search heuristics and has as a main aim to guide these procedures in exploring efficiently and effectively the vast search spaces of combinatorial optimization problems gls can be combined with the neighborhood reduction scheme of fls which significantly speeds up the operations of the algorithm the combination of gls and fls with tsp local search heuristics of different eiciency and effectiveness is studied in an effort to determine the dependence of gls on the underlying local search heuristic used comparisons are made with some of the best tsp heuristic algorithms and general optimization techniques which demonstrate the advantages of gls over alternative heuristic approaches suggested for the problem

linear groups with orders having certain large prime divisors in this paper we obtain a classification of those subgroups of the finite general linear group gld q with orders divisible by a primitive prime divisor of qe 1 for some in the course of the analysis we obtain new results on modular representations of finite almost simple groups in particular in the last section we obtain substantial extensions of the results of landazuri and seitz on small cross characteristic representations of some of the finite classical groups 1991 mathematics subject classification primary 20g40 secondary 20c20 20c33 20c34 20e99

on the orders of primitive groups abstract almost all primitive permutation groups of degree n have order at most n i 0 log 2 n 1 n 2 i 1 log 2 n or have socle isomorphic to a direct power of some alternating group the mathieu groups m11 m12 m23 and m24 are the four exceptions as a corollary the sharp version of a theorem of praeger and saxl is established where m12 turns out to be the largest primitive group for an application a bound on the orders of permutation groups without large alternating composition factors is given this sharpens a lemma of babai cameron palfy and generalizes a theorem of dixon

lin kernighan heuristic adaptations for the generalized traveling salesman problem abstract the lin kernighan heuristic is known to be one of the most successful heuristics for the traveling salesman problem tsp it has also proven its efficiency in application to some other problems in this paper we discuss possible adaptations of tsp heuristics for the generalized traveling salesman problem gtsp and focus on the case of the lin kernighan algorithm at first we provide an easy to understand description of the original lin kernighan heuristic then we propose several adaptations both trivial and complicated finally we conduct a fair competition between all the variations of the lin kernighan adaptation and some other gtsp heuristics it appears that our adaptation of the lin kernighan algorithm for the gtsp reproduces the success of the original heuristic different variations of our adaptation outperform all other heuristics in a wide range of trade offs between solution quality and running time making lin kernighan the state of the art gtsp local search

non abelian composition factors of finite groups with the cut property abstract a finite group g is said to have the cut property if the integral group ring z g has only trivial central units as shown in 3 g has the cut property if and only if for every g g any generator of g is conjugate to either g or g 1 such a group is therefore quadratic rational i e q q 2 for every irreducible character irr g the possible non abelian composition factors of quadratic rational groups have been classified in 4 in this paper we determine which of these finite simple groups occur as composition factors of some finite group with the cut property

finite simple groups with short galois orbits on conjugacy classes abstract all finite simple groups are determined with the property that every galois orbit on conjugacy classes has size at most 4 from this we list all finite simple groups g for which the normalized group of central units of the integral group ring z g is an infinite cyclic group

the non coprime k gv problem abstract let v be a finite faithful completely reducible fg module for a finite field f and a finite group g in various cases explicit linear bounds in v are given for the numbers of conjugacy classes k g v and k g of the semidirect product gv and of the group g respectively these results concern the so called non coprime k g v problem

upper and lower bounding strategies for the generalized minimum spanning tree problem abstract we address the generalized minimum spanning tree problem gmst which requires spanning at least one vertex out of every set of disjoint vertices in a graph we show that the geometric version of this problem is np hard and we propose two stochastic heuristics the first one is a very fast randomized greedy search algorithm and the second one being a genetic algorithm also we investigate some existing integer programming formulations and present an new one a new lagrangian based lower bound is proposed and implemented to assess the performance of the heuristics computational experiments performed on a large set of randomly generated instances with up to 1000 vertices and 10 000 edges provide evidence of the good performance of the proposed heuristics

fuchsian groups finite simple groups and representation varieties let be a fuchsian group of genus at least 2 at least 3 if is non oriented we study the spaces of homomorphisms from to finite simple groups g and derive a number of applications concerning random generation and representation varieties precise asymptotic estimates for hom g are given implying in particular that as the rank of g tends to infinity this is of the form g 1 o 1 where is the measure of we then prove that a randomly chosen homomorphism from to g is surjective with probability tending to 1 as g combining our results with lang weil estimates from algebraic geometry we obtain the dimensions of the representation varieties text hom gamma bar g where bar g is gl n k or a simple algebraic group over k an algebraically closed field of arbitrary characteristic a key ingredient of our approach is character theory involving the study of the zeta function g s 1 s where the sum is over all irreducible complex characters of g

a two level solution approach for solving the generalized minimum spanning tree problem abstract in this paper we are addressing the generalized minimum spanning tree problem denoted by gmstp which is a variant of the classical minimum spanning tree mst problem the main characteristic of this problem is the fact that the vertices of the graph are partitioned into a given number of clusters and we are looking for a minimum cost tree spanning a subset of vertices which includes exactly one vertex from each cluster we describe a two level solution approach for solving the gmstp obtained by decomposing the problem into two logical and natural smaller subproblems an upper level global subproblem and a lower level local subproblem and solving them separately the goal of the first subproblem is to determine global trees spanning the clusters using a genetic algorithm with a diploid representation of the individuals while the goal of the second subproblem is to determine the best tree w r t cost minimization for the above mentioned global trees spanning exactly one vertex from each cluster the second subproblem is solved optimally using dynamic programming extensive computational results are reported and discussed for an often used set of benchmark instances the obtained results show an improvement in the quality of the achieved solutions and demonstrate the efficiency of our approach compared to the existing methods from the literature

on generalized minimum spanning trees the generalized minimum spanning tree problem consists of designing a minimum cost tree spanning several clusters the purpose of this note is to pinpoint several inaccuracies contained in a previous publication and to propose a valid formulation for this problem

an improved hybrid ant local search algorithm for the partition graph coloring problem in this paper we propose a hybrid ant colony optimization aco algorithm for the partition graph coloring problem pgcp given an undirected graph g v e whose nodes are partitioned into a given number of the sets the goal of the pgcp is to find a subset v v that contains exactly one node from each cluster and a coloring for v so that in the graph induced by v two adjacent nodes have different colors and the total number of used colors is minimal our hybrid algorithm is obtained by executing a local search procedure after every aco iteration the performance of our algorithm is evaluated on a set of instances commonly used as benchmark and the computational results show that compared to state of the art algorithms our improved hybrid aco algorithm achieves solid results in very short run times

the base size of a primitive diagonal group abstract a base b for a finite permutation group g acting on a set is a subset of with the property that only the identity of g can fix every point of b we prove that a primitive diagonal group g has a base of size 2 unless the top group of g is the alternating or symmetric group acting naturally in which case the minimal base size of g is determined up to two possible values we also prove that the minimal base size of g satisfies a well known conjecture of pyber moreover we prove that if the top group of g does not contain the alternating group then the proportion of pairs of points that are bases for g tends to 1 as g tends to infinity a similar result for the case when the degree of the top group is fixed is given

a polynomial bound for the orders of primitive solvable groups the structure of primitive solvable permutation groups has bee r thoroughly investigated since galois relying on the known structure theorems we determine the largest order a n of a primitive solvable group of degree ii theorem 3 in the course of the proof we obtain the largest order a n of a transitive solvable group of degree n as well theorem 2 although the formula describing the exact value of i is quite complicated we are able to give an asymptotic evaluation it turns out thar the order of a primitive solvable group of degree n is bounded by 24 12 for a constant c 3 24399 theorem i a similar asymptotic estimate obtained recentiy by babai cameron and pilfy 111 shows that the orders of those primitive permutation groups of degree iz in which all nonabelian simple sections have orders less than a iixed integer k are bounded by n where c c k depends on k only related estimates under the assumption of the classification of finite simple groups are given by cameron 2 strong motivation for such asymptotic estimates comes from recent results of luks 151 on the complexity of graph isomorphism testing the results of the present paper have immediate application to graphs of vaience k p we mean that the field multiplication can be defined on the k dimensional vector space over gf p so that this equ ality holds 127 002 8693 82 g g i27 i so2 gg g

an answer to a question of isaacs on character degree graphs abstract let n be a normal subgroup of a finite group g we consider the graph g n whose vertices are the prime divisors of the degrees of the irreducible characters of g whose kernel does not contain n and two vertices are joined by an edge if the product of the two primes divides the degree of some of the characters of g whose kernel does not contain n we prove that if g n is disconnected then g n is solvable this proves a strong form of a conjecture of isaacs

fixed conjugacy classes of normal subgroups and the k gv problem abstract we establish several new bounds for the number of conjugacy classes of a finite group all of which involve the maximal number c of conjugacy classes of a normal subgroup fixed by some element of a suitable subset of the group to apply these formulas effectively the parameter c which in general is hard to control is studied in some important situations these results are then used to provide a new shorter proof of the most difficult case of the well known k g v problem which occurs for p 5 and v induced from the natural module of a 5 complement of gl 2 5 we also show how for large p the new results reduce the k g v problem to the primitive case thereby improving previous work on this furthermore we discuss how they can be used in tackling the imprimitive case of the as of yet unsolved noncoprime k g v problem

diameters of finite simple groups sharp bounds and applications let g be a finite simple group and let s be a normal subset of g we determine the diameter of the cayley graph r g s associated with g and s up to a multiplicative constant many applications follow for example we deduce that there is a constant c such that every element of g is a product of c involutions and we generalize this to elements of arbitrary order we also show that for any word w w xl xd there is a constant c c w such that for any simple group g on which w does not vanish every element of g is a product of c values of w from this we deduce that every verbal subgroup of a semisimple profinite group is closed other applications concern covering numbers expanders and random walks on finite simple groups

bases of primitive linear groups abstract we correct and improve a result in 3 giving the structure of finite primitive linear groups of unbounded base size this confirms a well known conjecture of pyber on base sizes of primitive permutation groups in the case of affine groups whose associated linear group is primitive

generalized spanning trees abstract in this paper we propose a definition for the generalized minimal spanning tree gmst of a graph the gmst requires spanning at least one node out of every set of disjoint nodes node partition in a graph the analysis of the gmst problem is motivated by real life agricultural settings related to construction of irrigation networks in desert environments we prove that the gmst problem is np hard and examine a number of heuristic solutions for this problem computational experiments comparing these heuristics are presented

word maps conjugacy classes and a noncommutative waring type theorem let w w x xd 1 be a nontrivial group word we show that if g is a sufficiently large finite simple group then every element g e g can be expressed as a product of three values of w in g this improves many known results for powers commutators as well as a theorem on general words obtained in 19 the proof relies on probabilistic ideas algebraic geometry and character theory our methods which apply the zeta function g s x eirrg xo s giye rise to various additional results of independent interest including applications to conjectures of ore and thompson

quantitative stability for the brunn minkowski inequality abstract we prove a quantitative stability result for the brunn minkowski inequality if a b 1 t 1 with 0 and t a 1 t b 1 n 1 for some small then up to a translation both a and b are quantitatively close in terms of to a convex set k

maximal overgroups of singer elements in classical groups abstract in this paper we use the results of a recent paper by guralnick penttila praeger and saxl proc london math soc 78 1999 167 214 to calculate an explicit list of the maximal overgroups of the singer elements in the finite special linear general unitary special unitary symplectic and minus type orthogonal groups

bounds on the number and sizes of conjugacy classes in finite chevalley groups with applications to derangements we present explicit upper bounds for the number and size of conjugacy classes in finite chevalley groups and their variations these results have been used by many authors to study zeta functions associated to representations of finite simple groups random walks on chevalley groups the final solution to the ore conjecture about commutators in finite simple groups and other similar problems in this paper we solve a strong version of the boston shalev conjecture on derangements in simple groups for most of the families of primitive permutation group representations of finite simple groups the remaining cases are settled in two other papers of the authors and applications are given in a third

enumerating finite groups of given order let f n denote the number of isomorphism classes of groups of order n let n hw p9ig be the prime factorization of n and let a a n ek and u ii n maxkl1 gi the function f n has been subject to considerable study g higman 15 and c c sims 26 showed that if n is a prime power say n p then f n n 2 27 o l a2 for a x oc p m neumann 22 proved that if the number of isomorphism classes of simple groups of order n is bounded by nca2 for some constant c then f n n 1 c a2 holds in general combining the methods of 22 with some consequences of the classification theorem of finite simple groups a mciver and neumann 21 obtained that in fact the estimates f n na2 2 and f n n1 2 1 2 hold recently d f holt 16 found a rather elegant proof for the bound

a tabu search heuristic for the generalized minimum spanning tree problem this paper describes an attribute based tabu search heuristic for the generalized minimum spanning tree problem gmstp known to be np hard given a graph whose vertex set is partitioned into clusters the gmstp consists of designing a minimum cost tree spanning all clusters an attribute based tabu search heuristic employing new neighborhoods is proposed an extended set of tsplib test instances for the gmstp is generated and the heuristic is compared with recently proposed genetic algorithms the proposed heuristic yields the best results for all instances moreover an adaptation of the tabu search algorithm is proposed for a variation of the gmstp in which each cluster must be spanned at least once

survivable networks linear programming relaxations and the parsimonious property we consider the survivable network design problem the problem of designing at minimum cost a network with edge connectivity requirements as special cases this problem encompasses the steiner tree problem the traveling salesman problem and thek edge connected network design problem we establish a property referred to as the parsimonious property of the linear programming lp relaxation of a classical formulation for the problem the parsimonious property has numerous consequences for example we derive various structural properties of these lp relaxations we present some algorithmic improvements and we perform tight worst case analyses of two heuristics for the survivable network design problem

augmented lagrangean based algorithms for centralized network design capacitated spanning tree problems appear frequently as fundamental problems in many communication network design problems an integer programming formulation and a new set of valid inequalities are presented for the linear characterization of the problem a combination of a subgradient optimization procedure and an augmented lagrangean based procedure is used to generate tight lower bounds the procedure begins with an explicit representation of a subset of the constraints and the corresponding lagrangean problem is solved the solution is examined in order to identify implicit constraints that are violated those are added to the lagrangean problem forming an expanded problem and an efficient dual ascent procedure is then applied when no further improvement is possible through this procedure a subgradient optimization procedure is invoked in order to further tighten the lower bound value an exchange heuristic is applied to the nonfeasible lagrangean solution in an attempt to generate good feasible solutions to the problem the procedure has been tested and has generated bounds that are significantly better than ones obtained through previously published procedures

conjugacy classes in sylow p subgroups of gl n q taking s to consist of lower unitriangular matrices higman notes that for i 3 ic g j

character degrees and random walks in finite groups of lie type for a finite group h let irr h denote the set of irreducible characters of h and define the zeta function zeta h t sum chi in irr h chi 1 t for real t 0 we study the asymptotic behaviour of zeta h t for finite simple groups h of lie type and also of a corresponding zeta function defined in terms of conjugacy classes applications are given to the study of random walks on simple groups and on base sizes of primitive permutation groups

generation of finite quasisimple groups with an application to groups acting on beauville surfaces we verify a conjecture of bauer catanese and grunewald by proving a stronger result concerning quasisimple groups more specifically we prove that every finite quasisimple group apart from a5 and its cover sl 2 5 is a beauville group

upper bounds for the number of conjugacy classes of a finite group abstract for a finite group g let k g denote the number of conjugacy classes of g we prove that a simple group of lie type of untwisted rank l over the field of q elements has at most 6 q l conjugacy classes using this estimate we show that for completely reducible subgroups g of gl n q we have k g q 10 n confirming a conjecture of kovacs and robinson for finite groups g with f g a p group we prove that k g cp a where p a is the order of a sylow p subgroup of g and c is a constant for groups with o p g 1 we obtain that k g g p this latter result confirms a conjecture of iranzo navarro and monasor we also improve various earlier results concerning conjugacy classes of permutation groups and linear groups as a by product we show that any finite group g has a soluble subgroup s and a nilpotent subgroup n such that k g s and k g n 3

generalized network design problems abstract network design problems ndp consist of identifying an optimal subgraph of a graph subject to side constraints in generalized ndp the vertex set is partitioned into clusters and the feasibility conditions are expressed in terms of the clusters several applications of generalized ndp arise in the fields of telecommunications transportation and biology the aim of this review article is to formally define generalized ndp to study their properties and to provide some applications

on a conjecture of g e wall abstract we prove that if g is a finite almost simple group having socle of lie type of rank r then the number of maximal subgroups of g is at most c r 2 3 g where c is an absolute constant this verifies a conjecture of wall for groups of sufficiently large rank using this we prove that any finite group g has at most 2 c g 3 2 maximal subgroups

a new rule for source connection problems abstract in this paper we study situations where a group of agents require a service that can only be provided from a source the so called source connection problems these problems contain the standard fixed tree the classical minimum spanning tree and some other related problems such as the k hop the degree constrained and the generalized minimum spanning tree problems among others our goal is to divide the cost of a network among the agents to this end we introduce a rule which will be referred to as a painting rule because it can be interpreted by means of a story about painting some meaningful properties in this context and a characterization of the rule are provided

a strong cutting plane algorithm for production scheduling with changeover costs changeover costs and times are central to numerous manufacturing operations these costs arise whenever work centers capable of processing only one product at a time switch from the manufacture of one product to another although many researchers have contributed to the solution of scheduling problems that include changeover costs due to the problem s combinatorial explosiveness optimization based methods have met with limited success in this paper we develop and apply polyhedral methods from integer programming for a dynamic version of the problem computational tests with problems containing one to five products and up to 225 integer variables show that polyhedral methods based upon a set of facet inequalities developed in this paper can effectively reduce the gap between the value of an integer programming formulation of the problem and its linear programming relaxation by a factor of 94 to 100 these results suggest the use of a combined cutting plane branch and bound procedure as a solution approach in a test with a five product problem this procedure when compared with a standard linear programming based branch and bound approach reduced computation time by a factor of seven

globally and locally minimal weight spanning tree networks abstract the competition between local and global driving forces is significant in a wide variety of naturally occurring branched networks we have investigated the impact of a global minimization criterion versus a local one on the structure of spanning trees to do so we consider two spanning tree structures the generalized minimal spanning tree gmst defined by dror et al eur j oper res 120 2000 583 and an analogous structure based on the invasion percolation network which we term the generalized invasive spanning tree gist in general these two structures represent extremes of global and local optimality respectively structural characteristics are compared between the gmst and gist for a fixed lattice in addition we demonstrate a method for creating a series of structures which enable one to span the range between these two extremes two structural characterizations the occupied edge density i e the fraction of edges in the graph that are included in the tree and the tortuosity of the arcs in the trees are shown to correlate well with the degree to which an intermediate structure resembles the gmst or gist both characterizations are straightforward to determine from an image and are potentially useful tools in the analysis of the formation of network structures

asymptotic results for primitive permutation groups abstract we prove that the number of conjugacy classes of primitive permutation groups of degreenis at mostnc n where n denotes the maximal exponent occurring in the prime factorization ofn this result is applied to investigating maximal subgroup growth of infinite groups we then proceed by showing that if the point stabilizerg of a primitive groupgof degreendoes not have the alternating group alt d as a section then the order ofgis bounded by a polynomial inn this result extends a well known theorem of babai cameron and palfy it is used to prove for example that ifhis a subgroup of indexnin a groupg andhis a product ofbcyclic groups then g hg ncwherecdepends onb

equality in borell brascamp lieb inequalities on curved spaces abstract by using optimal mass transportation and a quantitative holder inequality we provide estimates for the borell brascamp lieb deficit on complete riemannian manifolds accordingly equality cases in borell brascamp lieb inequalities including brunn minkowski and prekopa leindler inequalities are characterized in terms of the optimal transport map between suitable marginal probability measures these results provide several qualitative applications both in the flat and non flat frameworks in particular by using caffarelli s regularity result for the monge ampere equation we give a new proof of dubuc s characterization of the equality in borell brascamp lieb inequalities in the euclidean setting when the n dimensional riemannian manifold has ricci curvature ric m n 1 k for some k r it turns out that equality in the borell brascamp lieb inequality is expected only when a particular region of the manifold between the marginal supports has constant sectional curvature k a precise characterization is provided for the equality in the lott sturm villani type distorted brunn minkowski inequality on riemannian manifolds related results for not necessarily reversible finsler manifolds are also presented

normalizers of primitive permutation groups abstract let g be a transitive normal subgroup of a permutation group a of finite degree n the factor group a g can be considered as a certain galois group and one would like to bound its size one of the results of the paper is that a g n if g is primitive unless n 3 4 54 38 58 or 316 this bound is sharp when n is prime in fact when g is primitive out g n unless g is a member of a given infinite sequence of primitive groups and n is different from the previously listed integers many other results of this flavor are established not only for permutation groups but also for linear groups and galois groups

the full steiner tree problem motivated by the reconstruction of phylogenetic tree in biology we study the full steiner tree problem in this paper given a complete graph g v e with a length function on e and a proper subset r v the problem is to find a full steiner tree of minimum length in g which is a kind of steiner tree with all the vertices of r as its leaves in this paper we show that this problem is np complete and max snp hard even when the lengths of the edges are restricted to either 1 or 2 for the instances with lengths either 1 or 2 we give a 8 5 approximation algorithm to find an approximate solution for the problem

a dual ascent approach for steiner tree problems on a directed graph the steiner tree problem on a directed graph stdg is to find a directed subtree that connects a root node to every node in a designated node setv we give a dual ascent procedure for obtaining lower bounds to the optimal solution value the ascent information is also used in a heuristic procedure for obtaining feasible solutions to the stdg computational results indicate that the two procedures are very effective in solving a class of stdg s containing up to 60 nodes and 240 directed 120 undirected arcs