representation zeta functions of compact p adic analytic groups and arithmetic groups we introduce new methods from p adic integration into the study of representation zeta functions associated to compact p adic analytic groups and arithmetic groups they allow us to establish that the representation zeta functions of generic members of families of p adic analytic pro p groups obtained from a global perfect lie lattice satisfy functional equations in the case of semisimple compact p adic analytic groups we exhibit a link between the relevant p adic integrals and a natural filtration of the locus of irregular elements in the associated semisimple lie algebra defined by centraliser dimension based on this algebro geometric description we compute explicit formulae for the representation zeta functions of principal congruence subgroups of the groups sl 3 o where o is a compact discrete valuation ring of characteristic 0 and of the corresponding unitary groups these formulae combined with approximative clifford theory allow us to determine the abscissae of convergence of representation zeta functions associated to arithmetic subgroups of algebraic groups of type a 2 assuming a conjecture of serre on the congruence subgroup problem we thereby prove a conjecture of larsen and lubotzky on lattices in higher rank semisimple groups for algebraic groups of type a 2 defined over number fields

solving symmetric vehicle routing problems asymmetrically abstract the vehicle routing problem can be regarded as a traveling salesman problem with additional constraints algorithms based on assignment relaxations provide better solutions for the symmetric traveling salesman problem if they are used on an asymmetric transformed distance matrix as shown by jonker et al in a paper in operations research the generalization of such a transformation to the distance matrix of symmetric vehicle routing problems is described the approach is illustrated within a heuristic algorithm although it can also be the basis for an exact algorithm for a number of standard problems computational results are given that are competitive to results of other algorithms

on dual solutions of the linear assignment problem abstract for the linear assignment problem we describe how to obtain different dual solutions it turns out that a shortest path algorithm can be used to compute such solutions with several interesting properties that enable to do better post optimality analysis two examples illustrate how different dual solutions can be used in the context of the traveling salesman problem

on quantum gauge theories in two dimensions two dimensional quantum yang mills theory is studied from three points of view i by standard physical methods ii by relating it to the largek limit of three dimensional chern simons theory and two dimensional conformal field theory iii by relating its weak coupling limit to the theory of reidemeister ray singer torsion the results obtained from the three points of view agree and give formulas for the volumes of the moduli spaces of representations of fundamental groups of two dimensional surfaces

extended deligne lusztig varieties for general and special linear groups abstract we give a generalisation of deligne lusztig varieties for general and special linear groups over finite quotients of the ring of integers in a non archimedean local field previously a generalisation was given by lusztig by attaching certain varieties to unramified maximal tori inside borel subgroups in this paper we associate a family of so called extended deligne lusztig varieties to all tamely ramified maximal tori of the group moreover we analyse the structure of various generalised deligne lusztig varieties and show that the unramified varieties including a certain natural generalisation do not produce all the irreducible representations in general on the other hand we prove results which together with some computations of lusztig show that for sl 2 f q 2 with odd q the extended deligne lusztig varieties do indeed afford all the irreducible representations

vanishing theorems for torsion automorphic sheaves on general pel type shimura varieties abstract we consider the interior cohomology and the hodge graded pieces in the case of the de rham realization of general not necessarily compact pel type shimura varieties with coefficients in the local systems corresponding to sufficiently regular algebraic representations of the associated reductive group for primes p bigger than an effective bound we prove that the f p and z p cohomology groups are concentrated in the middle degree that the z p cohomology groups are free of p torsion and that every f p cohomology class lifts to a z p cohomology class

an addendum on sensitivity analysis of the optimal assignment we point out that sensitivity results for the linear assignment problem can be produced by a shortest path based approach in a straightforward manner and as efficient as finding an optimal solution

dynamic bundle methods lagrangian relaxation is a popular technique to solve difficult optimization problems however the applicability of this technique depends on having a relatively low number of hard constraints to dualize when there are many hard constraints it may be preferable to relax them dynamically according to some rule depending on which multipliers are active from the dual point of view this approach yields multipliers with varying dimensions and a dual objective function that changes along iterations we discuss how to apply a bundle methodology to solve this kind of dual problems our framework covers many separation procedures to generate inequalities that can be found in the literature including but not limited to the most violated inequality we analyze the resulting dynamic bundle method giving a positive answer for its primal dual convergence properties and under suitable conditions show finite termination for polyhedral problems

a restricted lagrangean approach to the traveling salesman problem we describe an algorithm for the asymmetric traveling salesman problem tsp using a new restricted lagrangean relaxation based on the assignment problem ap the lagrange multipliers are constrained so as to guarantee the continued optimality of the initial ap solution thus eliminating the need for repeatedly solving ap in the process of computing multipliers we give several polynomially bounded procedures for generating valid inequalities and taking them into the lagrangean function with a positive multiplier without violating the constraints so as to strengthen the current lower bound upper bounds are generated by a fast tour building heuristic when the bound strengthening techniques are exhausted without matching the upper with the lower bound we branch by using two different rules according to the situation the usual subtour breaking disjunction and a new disjunction based on conditional bounds we discuss computational experience on 120 randomly generated asymmetric tsp s with up to 325 cities the maximum time used for any single problem being 82 seconds this is a considerable improvement upon earlier methods though the algorithm discussed here is for the asymmetric tsp the approach can be adapted to the symmetric tsp by using the 2 matching problem instead of ap

representation growth of compact linear groups we study the representation growth of simple compact lie groups and of sln o where o is a compact discrete valuation ring as well as the twist representation growth of gln o this amounts to a study of the abscissae of convergence of the corresponding twist representation zeta functions we determine the abscissae for a class of mellin zeta functions which include the witten zeta functions as a special case we obtain a new proof of the theorem of larsen and lubotzky that the abscissa of witten zeta functions is r where r is the rank and the number of positive roots we then show that the twist zeta function of gln o exists and has the same abscissa of convergence as the zeta function of sln o provided n does not divide char o we compute the twist zeta function of gl2 o when the residue characteristic p of o is odd and approximate the zeta function when p 2 to deduce that the abscissa is 1 finally we construct a large part of the representations of sl2 fq t q even and deduce that its abscissa lies in the interval 1 5 2

an inexact algorithm for the sequential ordering problem abstract given the directed g n a and the penalty matrix c the sequential ordering problem hereafter sop consists of finding the permutation of the nodes from the set n such that it minimizes a c based function and does not violate the precedence relationships given by the set a strong sufficient conditions for the infeasibility of a sop s instance are embedded in a procedure for the sop s pre processing note that it is one of the key steps in any algorithm that attempts to solve sop by dropping the constraints related to the precedence relationships sop can be converted in the classical asymmetric traveling salesman problem hereafter astp the algorithm obtains hopefully satisfactory solutions by modifying the optimal solution to the related assignment problem hereafter ap if it is not a feasible sequential ordering hereafter fso the new solution patches the subtours if any giving preference to the patches with zero reduced cost in the linking arcs the ap based lower bound on the optimal solution to atsp is tightened by using some of the procedures given in 1 in any case a local search for improving the initial fso is performed it uses 3 and 4 changed based procedures computational results on a broad set of cases are reported

genetic algorithm for asymmetric traveling salesman problem with imprecise travel times this paper presents a variant of the asymmetric traveling salesman problem atsp in which the traveling time between each pair of cities is represented by an interval of values wherein the actual travel time is expected to lie instead of a fixed deterministic value as in the classical atsp here the atsp with interval objective is formulated using the usual interval arithmetic to solve the interval atsp i atsp a genetic algorithm with interval valued fitness function is proposed for this purpose the existing revised definition of order relations between interval numbers for the case of pessimistic decision making is used the proposed algorithm is based on a previously published work and includes some new features of the basic genetic operators to analyze the performance and effectiveness of the proposed algorithm and different genetic operators computational studies of the proposed algorithm on some randomly generated test problems are reported

a reduction approach to the repeated assignment problem abstract we consider the repeated assignment problem rap which is a k fold repetition of the n n linear assignment problem lap with the additional requirement that no assignment can be repeated more than once in actual applications k is typically much smaller than n first we derive upper and lower bounds respectively by a heuristic together with local search and an efficient method solving the continuous relaxation the latter also solves a lagrangian relaxation such that the related pegging test to fix variables at zero or one decomposes into k independent pegging tests to laps these can be solved exactly by transforming them into all pairs shortest path problems together with these procedures we also employ a virtual pegging test and reduce rap in size numerical experiments show that the reduced instances with k n can be solved exactly using standard mip solvers

a lagrangean based branch and bound algorithm for single machine sequencing with precedence constraints to minimize total weighted completion time the single machine sequencing problem is considered in which there are precedence constraints on the jobs the objective is to minimize the sum of weighted completion times a lower bound is obtained by successively performing a lagrangean relaxation of appropriate constraints each lagrange multiplier is chosen to provide the maximum increment to the lower bound subject to retaining the nonnegativity of the coefficients of the variables when no further suitable constraints can be introduced into the lagrangean function the variables having zero cost coefficient are used to obtain a feasible sequence which provides an upper bound the gap between the lower and upper bound is reduced by removing some constraints from the lagrangean function and replacing them with others this lower bounding procedure is used in a branch and bound algorithm computational results indicate that the algorithm can satisfactorily solve problems with up to 100 jobs

representation zeta functions of nilpotent groups and generating functions for weyl groups of type b we study representation zeta functions of finitely generated torsion free nilpotent groups which are groups of rational points of unipotent group schemes over rings of integers of number fields using the kirillov orbit method and frak p adic integration we prove rationality and functional equations for almost all local factors of the euler products of these zeta functions we further give explicit formulae in terms of dedekind zeta functions for the zeta functions of class 2 nilpotent groups obtained from three infinite families of group schemes generalizing the integral heisenberg group as an immediate corollary we obtain precise asymptotics for the representation growth of these groups and key analytic properties of their zeta functions such as meromorphic continuation we express the local factors of these zeta functions in terms of generating functions for finite weyl groups of type b this allows us to establish a formula for the joint distribution of three functions or statistics on such weyl groups finally we compare our explicit formulae to frak p adic integrals associated to relative invariants of three infinite families of prehomogeneous vector spaces

similarity classes of integral p adic matrices and representation zeta functions of groups of type a2 we compute explicitly dirichlet generating functions enumerating finite dimensional irreducible complex representations of various p adic analytic and adelic profinite groups of type a2 this has consequences for the representation zeta functions of arithmetic groups h k where k is a number field and h is a k form of sl3 assuming that possesses the strong congruence subgroup property we obtain precise uniform estimates for the representation growth of our results are based on explicit uniform formulae for the representation zeta functions of the p adic analytic groups sl3 o and su3 o where o is a compact discrete valuation ring of characteristic 0 these formulae build on our classification of similarity classes of integral p adic 3 3 matrices in gl3 o and gu3 o where o is a compact discrete valuation ring of arbitrary characteristic organising the similarity classes by invariants which we call their shadows allows us to combine the kirillov orbit method with clifford theory to obtain explicit formulae for representation zeta functions in a different direction we introduce and compute certain similarity class zeta functions our methods also yield formulae for representation zeta functions of various finite subquotients of groups of the form sl3 o su3 o gl3 o and gu3 o arising from the respective congruence filtrations these formulae are valid in case that the characteristic of o is either 0 or sufficiently large analysis of some of these formulae leads us to observe p adic analogues of ennola duality

triangle inequality and symmetry in connection with the assignment and the traveling salesman problem abstract triangle inequality and symmetry play an important role in the treatment of the traveling salesman problem tsp examplarily it is shown how to transform the distance matrix in order to use a given heuristic algorithm furthermore related performance bounds will be given and studied the corresponding transformation originates in the dual assignment problem finally results concerning the exact case are mentioned as reducing the dimension of the original tsp and as well as the restricted use of non optimal edges the paper represents a survey of the above mentioned topic

a note on dual solutions of the assignment problem in connection with the traveling salesman problem abstract the algorithm of christofides is considered for the determination of a lower bound for the traveling salesman problem there exist many investigations for improving this bound but these are only heuristic proposals it is shown that the subgradients of the bound function can be easily calculated and a subgradient procedure is developed for computing a sharp bound

a lagrangian relaxation approach to the edge weighted clique problem the b clique polytope cp n b is the convex hull of the node and edge incidence vectors of all subcliques of size at most b of a complete graph on n nodes including the boolean quadric polytope qp n as a special case and being closely related to the quadratic knapsack polytope it has received considerable attention in the literature in particular the max cut problem is equivalent with optimizing a linear function over qp n n the problem of optimizing linear functions over cp n b has so far been approached via heuristic combinatorial algorithms and cutting plane methods we study the structure of cp n b in further detail and present a new computational approach to the linear optimization problem based on lucena s suggestion of integrating cutting planes into a lagrangian relaxation of an integer programming problem in particular we show that the separation problem for tree inequalities becomes polynomial in our lagrangian framework finally computational results are presented

the nef cone of toroidal compactifications of cal a _4 the moduli space of principally polarised abelian 4 folds can be compactified in several different ways by toroidal methods here we consider in detail the igusa compactification and the second voronoi compactification we describe in both cases the cone of nef cartier divisors the proof depends on a detailed description of the voronoi compactification which makes it possible to proceed by induction using the known description of the nef cone for compactifications of mathcal a 3 the igusa compactification has a non mathbb q factorial singularity which is resolved by a single blow up this resolution is the voronoi compactification the exceptional divisor e is a toric fano variety of dimension 9 the other boundary divisor d corresponds to degenerations with corank 1 after imposing a level structure in order to avoid certain technical complications we show that the closure of d in the voronoi compactification maps to the voronoi compactification of mathcal a 3 the toric description of the exceptional divisor allows us to describe the map in sufficient detail to estimate the intersection numbers needed this inductive process is only valid for the voronoi compactification the result for the igusa compactification is deduced from the voronoi compactification

strong limit multiplicity for arithmetic hyperbolic surfaces and 3 manifolds we show that every sequence of torsion free arithmetic congruence lattices in mathrm pgl 2 mathbb r or mathrm pgl 2 mathbb c satisfies a strong quantitative version of the limit multiplicity property we deduce that for r 0 in certain range growing linearly in the degree of the invariant trace field the volume of the r thin part of any congruence arithmetic hyperbolic surface or congruence arithmetic hyperbolic 3 manifold m is of order at most mathrm vol m 11 12 as an application we prove gelander s conjecture on homotopy type of arithmetic hyperbolic 3 manifolds we show that there are constants a b such that every such manifold m is homotopy equivalent to a simplicial complex with at most a mathrm vol m vertices all of degrees bounded by b

an optimal solution method for large scale multiple traveling salesmen problems we develop an efficient branch and bound based method for solving the multiple traveling salesman problem and develop lower bounds through a lagrangean relaxation that requires computing a degree constrained minimal spanning tree a subgradient optimization procedure updates the lagrange multipliers we use fast sensitivity analysis techniques to increase the underlying graph sparsity and reduce the problem size the algorithm was tested on 416 problems with up to 500 cities and 10 salesmen we also present computational results on different data sets and parameters in order to identify the major factors that influence the performance of the developed code

minimum directed 1 subtree relaxation for score orienteering problem score orienteering is a variant of the orienteering game where competitors start at a specified control point visit as many control points as possible and thereby collect prizes and return to the starting point within a prescribed amount of time the competitor with the highest amount of prizes wins the game in this paper we propose the minimum directed 1 subtree problem as a new relaxation to this problem and develop two methods to improve its lower bound we describe a cut and dual simplex method and a lagrangian relaxation method and construct an algorithm that combines these two methods in an appropriate way computational experiments are carried out to analyse the behavior of the proposed relaxation with respect to the characteristics of the test problems especially for large scale instances we find that the proposed relaxation is superior to the ordinary assignment relaxation

cluster based branching for the asymmetric traveling salesman problem abstract this paper presents a new branching scheme for the asymmetric traveling salesman problem atsp based on clusters a cluster is defined as a node set with the characteristic that there exists an optimal solution in which the nodes in the node set are visited consecutively the paper considers identification of clusters implementation of a cluster based branching scheme and cluster based dominance tests the new approach is implemented in a branch and bound algorithm using a well known additive bounding procedure considerable savings in computing time are obtained compared to previously published assignment based branch and bound algorithms for the atsp

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

eigenvarieties for reductive groups we develop the theory of overconvergent cohomology introduced by g stevens and we use it to give a construction of eigenvarieties associated to any reductive group g over q such that g r has discrete series we prove that the so called eigenvarieties are equidimensional and generically at over the weight space

the symmetric traveling salesman problem and edge exchanges in minimal 1 trees abstract in this article the effect of exchanging edges inside a minimal 1 tree with edges outside is analysed in combination with an upper bound this analysis enables the elimination of variables in the symmetric traveling salesman problem after discussion on a number of improvements for this analysis the implementation is described in a traveling salesman algorithm based on the 1 tree relaxation computational results show the advantages of the edges exchanges for euclidean problems up to 120 cities as well as for random table problems up to 200 cities

technical note a note on parametric analysis in linear assignment a classic application of the linear assignment problem is the assignment of people to jobs or jobs to people in this context it is interesting to measure competition for jobs and to generate a suitable list of jobs from which a person can choose the length of the list is a parameter a known list generation procedure is based on an interior point method followed by a parametric analysis we describe a more efficient procedure exploiting linear assignment theory and shortest path computations further we propose an alternative list generation procedure based on a special type of dual values for the linear assignment problem

solving some lexicographic multi objective combinatorial problems abstract the lexicographic multi objective optimization problem r lmop with r sum or bottleneck objectives optimizes a first objective then as far as a choice remains a second one then a third one and so on a general solution scheme from literature is based on scaling for the case that only sum criteria are involved the complexity is o rlogn t n m with t n m the complexity of optimization on combinatorial sum problem instances with m arcs and n nodes for cases with at least one bottleneck criterion the complexity is at least o rnlog2n t n m we describe a solution scheme that is suited to solve the shortest path r lmop the minimum spanning tree r lmop and the linear assignment r lmop the complexity is in all cases o r t n m

an additive bounding procedure for the asymmetric travelling salesman problem in this paper new lower bounds for the asymmetric travelling salesman problem are presented based on spanning arborescences the new bounds are combined in an additive procedure whose theoretical performance is compared with that of the balas and christofides procedure 1981 both procedures have been imbedded in a simple branch and bound algorithm and experimentally evaluated on hard test problems

classicite en theorie de hida in this article we prove an effective classicity theorem in hida s theory we focus more precisely on the case of the symplectic group and siegel modular forms we also prove that certain mod p siegel modular forms lift to characteristic zero and study carefully the divisibility by p of certain hecke operators

representation growth and rational singularities of the moduli space of local systems let g be a semisimple algebraic group defined over mathbb q p and let gamma be a compact open subgroup of g mathbb q p we relate the asymptotic representation theory of gamma and the singularities of the moduli space of g local systems on a smooth projective curve proving new theorems about both 1 we prove that there is a constant c independent of g such that the number of n dimensional representations of gamma grows slower than n c confirming a conjecture of larsen and lubotzky in fact we can take c 3 cdot text dim e 8 1 745 we also prove the same bounds for groups over local fields of large enough characteristic 2 we prove that the coarse moduli space of g local systems on a smooth projective curve of genus at least lceil c 2 rceil 1 374 has rational singularities for the proof we study the analytic properties of push forwards of smooth measures under algebraic maps more precisely we show that such push forwards have continuous density if the algebraic map is flat and all of its fibers have rational singularities