nonstandard quantum groups and superization we obtain the universal r matrix of the nonstandard quantum group associated to the alexander conway knot polynomial we show further that this nonstandard quantum group is related to the super quantum group uq gl 1 1 by a general process of superization which we describe we also study a twisted variant of this nonstandard quantum group and obtain as a result a twisted version of uq gl 1 1 as a q supersymmetry of the exterior differential calculus of any quantum plane of hecke type acting by mixing the bosonic xi coordinates and the forms dxi

some applications of the hermite matrix polynomials series expansions this paper deals with hermite matrix polynomials expansions of some relevant matrix functions appearing in the solution of differential systems properties of hermite matrix polynomials such as the three terms recurrence formula permit an efficient computation of matrix functions avoiding important computational drawbacks of other well known methods results are applied to compute accurate approximations of certain differential systems in terms of hermite matrix polynomials

minimum cost capacity installation for multicommodity network flows consider a directed graphg v a and a set of traffic demands to be shipped between pairs of nodes inv capacity has to be installed on the edges of this graph in integer multiples of a base unit so that traffic can be routed in this paper we consider the problem of minimum cost installation of capacity on the arcs to ensure that the required demands can be shipped simultaneously between node pairs we study two different approaches for solving problems of this type the first one is based on the idea of metric inequalities see onaga and kakusho on feasibility conditions of multicommodity flows in networks ieee transactions on circuit theory ct 18 4 1971 425 429 and uses a formulation with only a variables the second uses an aggregated multicommodity flow formulation and has v a variables we first describe two classes of strong valid inequalities and use them to obtain a complete polyhedral description of the associated polyhedron for the complete graph on three nodes next we explain our solution methods for both of the approaches in detail and present computational results our computational experience shows that the two formulations are comparable and yield effective algorithms for solving real life problems 1998 the mathematical programming society inc published by elsevier science b v

a branch and cut algorithm for the dial a ride problem in the dial a ride problem users formulate requests for transportation from a specific origin to a specific destination transportation is carried out by vehicles providing a shared service the problem consists of designing a set of minimum cost vehicle routes satisfying capacity duration time window pairing precedence and ride time constraints this paper introduces a mixed integer programming formulation of the problem and a branch and cut algorithm the algorithm uses new valid inequalities for the dial a ride problem as well as known valid inequalities for the traveling salesman the vehicle routing and the pick up and delivery problems computational experiments performed on randomly generated instances show that the proposed approach can be used to solve small to medium size instances

the pickup and delivery problem with transfers formulation and a branch and cut solution method in this paper a strict formulation of a generalization of the classical pickup and delivery problem is presented here we add the flexibility of providing the option for passengers to transfer from one vehicle to another at specific locations as part of the mathematical formulation we include transfer nodes where vehicles may interact interchanging passengers additional variables to keep track of customers along their route are considered the formulation has been proven to work correctly and by means of a simple example instance we conclude that there exist some configurations in which a scheme allowing transfers results in better quality optimal solutions finally a solution method based on benders decomposition is addressed we compare the computational effort of this application with a straight branch and bound strategy we also provide insights to develop more efficient set partitioning formulations and associated algorithms for solving real size problems

the multi commodity one to one pickup and delivery traveling salesman problem this paper concerns a generalization of the traveling salesman problem tsp called multi commodity one to one pickup and delivery traveling salesman problem m pdtsp in which cities correspond to customers providing or requiring known amounts of m different commodities and the vehicle has a given upper limit capacity each commodity has exactly one origin and one destination and the vehicle must visit each customer exactly once the problem can also be defined as the capacitated version of the classical tsp with precedence constraints this paper presents two mixed integer linear programming models and describes a decomposition technique for each model to find the optimal solution computational experiments on instances from the literature and randomly generated compare the techniques and show the effectiveness of our implementation

bosonic representations of yangian double with on the basis of the rtt ttr formalism we introduce the quantum double of the yangian for with a central extension the gauss decomposition of the t matrices gives us the so called drinfel d generators using these generators we present some examples of both finite and infinite dimensional representations that are quite natural deformations of their corresponding affine counterpart

the crown inequalities for the symmetric traveling salesman polytope we define a new family of valid inequalities for the symmetric traveling salesman polytope stspn that is the convex hull of the incidence vectors of all hamiltonian cycles of a complete graph with n nodes the smallest value of n for which the inequalities are defined is 8 we call these inequalities crown inequalities and we prove that they are facet inducing for stspn for n 8 we describe some extensions of the crown inequalities and we prove that they induce facets of stspn as well finally we discuss some issues on the possible applications of these inequalities in a polyhedral cutting plane algorithm

a new extension of local search applied to the dial a ride problem abstract in this paper we propose a cheap yet effective extension to the traditional local improvement algorithm we present this extension in the context of the dial a ride problem a variant of the traveling salesman problem this extension which we call sacrificing yields significant improvements over its plain local improvement counterpart without adversely affecting the algorithm s running time

facet identification for the symmetric traveling salesman polytope several procedures for the identification of facet inducing inequalities for the symmetric traveling salesman polytope are given an identification procedure accepts as input the support graph of a point which does not belong to the polytope and returns as output some of the facet inducing inequalities violated by the point a procedure which always accomplishes this task is calledexact otherwise it is calledheuristic we give exact procedures for the subtour elimination and the 2 matching constraints based on the gomory hu and padberg rao algorithms respectively efficient reduction procedures for the input graph are proposed which accelerate these two algorithms substantially exact and heuristic shrinking conditions for the input graph are also given that yield efficient procedures for the identification of simple and general comb inequalities and of some elementary clique tree inequalities these procedures constitute the core of a polytopal cutting plane algorithm that we have devised and programmed to solve a substantial number of large scale problem instances with sizes up to 2392 nodes to optimality

the pickup and delivery problem faces and branch and cut algorithm abstract this paper formulates the pickup and delivery problem also known as the dial a ride problem as an integer program its polyhedral structure is explored and four classes of valid inequalities developed the results of a branch and cut algorithm based on these constraints are presented

approximating capacitated routing and delivery problems we provide approximation algorithms for some capacitated vehicle routing and delivery problems these problems can all be viewed as instances of the following k delivery tsp given n source points and n sink points in a metric space with exactly one item at each source find a minimum length tour by a vehicle of finite capacity k to pick up and deliver exactly one item to each sink the only known approximation algorithm for this family of problems is the 2 5 approximation algorithm of anily and hassin networks 22 1992 pp 419 433 for the special case k 1 for this case we use matroid intersection to obtain a 2 approximation algorithm based on this algorithm and some additional lower bound arguments we devise a factor approximation for k delivery tsp with arbitrary finite k we also present a 2 approximation algorithm for the case k infty we then initiate the study of dynamic variants of k delivery tsp that model problems in industrial robotics and other applications specifically we consider the situation where a robot arm with finite or infinite capacity must collect n point objects moving in the euclidean plane and deliver them to the origin the point objects are moving in the plane with known identical velocities they might for instance be on a moving conveyor belt we derive several useful structural properties that lead to constant factor approximations for problems of this type that are relevant to the robotics application along the way we show that maximum latency tsp is implicit in the dynamic problems and that the natural farthest neighbor heuristic produces a good approximation for several notions of latency

stronger multi commodity flow formulations of the capacitated vehicle routing problem the capacitated vehicle routing problem is a much studied and strongly np hard combinatorial optimization problem for which many integer programming formulations have been proposed we present two new multi commodity flow mcf formulations and show that they dominate all of the existing ones in the sense that their continuous relaxations yield stronger lower bounds moreover we show that the relaxations can be strengthened in pseudo polynomial time in such a way that all of the so called knapsack large multistar klm inequalities are satisfied the only other relaxation known to satisfy the klm inequalities based on set partitioning is strongly np hard to solve computational results demonstrate that the new mcf relaxations are significantly stronger than the previously known ones

drinfel d realization of quantum affine superalgebra using the super reshetikhin semenov tian shansky method and gauss decomposition we obtain drinfel d s currents realization of the quantum affine superalgebra the hopf structure for drinfel d s currents is also given

on the general linear coupled system for diffusion in media with two diffusivities abstract for the general linear coupled system of partial differential equations arising in the theory of diffusion in media with double diffusivity simple uniqueness criteria and a method of solution of boundary value problems are established the equations studied retain the so called cross terms which have been neglected in all previous investigations moreover these equations arise as generalizations of a number of existing theories for example heat flow in heterogeneous multicomponent systems flow of water in fissured rocks and a model of an arms race the simple inequalities obtained on the various constants of the theory which guarantee uniqueness of solutions and existence of source solutions might serve as guidelines in an experimental determination of these constants the solution procedure involves solving two boundary value problems for the classical diffusion equation and the formulae given mean that closed form expressions can be deduced for a number of commonly occurring boundary value problems the paper emphasizes the general equations without special reference to particular physical applications or boundary value problems

numerical solution with a priori error bounds of coupled time dependent hyperbolic systems abstract this paper deals with initial value problems for coupled time dependent hyperbolic partial differential systems first an exact solution is obtained using fourier transform then numerical solution of the underlying differential equations and numerical integration techniques permit the construction of numerical approximations with guaranteed a priori error bounds

the precedence constrained asymmetric traveling salesman polytope many applications of the traveling salesman problem require the introduction of additional constraints one of the most frequently occurring classes of such constraints are those requiring that certain cities be visited before others precedence constraints in this paper we study the precedence constrained asymmetric traveling salesman pcats polytope i e the convex hull of incidence vectors of tours in a precedence constrained directed graph we derive several families of valid inequalities and give polynomial time separation algorithms for important subfamilies we then establish the dimension of the pcats polytope and show that under reasonable assumptions the two main classes of inequalities derived are facet inducing

gauss decomposition of the yangian y mathfrak gl _ m n we describe a gauss decomposition for the yangian y mathfrak gl m n of the general linear lie superalgebra this gives a connection between this yangian and the yangian of the classical lie superalgebra y a m 1 n 1 for m n defined and studied in papers by stukopin and suggests natural definitions for the yangians y mathfrak sl n n and y a n n we also show that the coefficients of the quantum berezinian generate the centre of the yangian y mathfrak gl m n this was conjectured by nazarov in 1991

solving second order matrix differential equations with regular singular points avoiding the increase of the problem dimension in this paper we propose a method for solving second order matrix differential systems of the form t 2x ta t x b t x 0 where a and b are analytic matrix functions in a neighborhood of t 0 series solutions of the problem are constructed without increasing the problem dimension by introducing the concept of a fundamental set of solutions of the above equation an analogous result to the frobenius theorem to solve the corresponding scalar equation is presented

the graphical relaxation a new framework for the symmetric traveling salesman polytope a present trend in the study of thesymmetric traveling salesman polytope stsp n is to use as a relaxation of the polytope thegraphical relaxation gtsp n rather than the traditionalmonotone relaxation which seems to have attained its limits in this paper we show the very close relationship between stsp n and gtsp n in particular we prove that every non trivial facet of stsp n is the intersection ofn 1 facets of gtsp n n of which are defined by the degree inequalities this fact permits us to define a standard form for the facet defining inequalities for stsp n that we calltight triangular and to devise a proof technique that can be used to show that many known facet defining inequalities for gtsp n define also facets of stsp n in addition we give conditions that permit to obtain facet defining inequalities by composition of facet defining inequalities for stsp n and general lifting theorems to derive facet defining inequalities for stsp n k from inequalities defining facets of stsp n

a branch and cut algorithm for the resolution of large scale symmetric traveling salesman problems an algorithm is described for solving large scale instances of the symmetric traveling salesman problem stsp to optimality the core of the algorithm is a polyhedral cutting plane procedure that exploits a subset of the system of linear inequalities defining the convex hull of the incidence vectors of the hamiltonian cycles of a complete graph the cuts are generated by several identification procedures that have been described in a companion paper whenever the cutting plane procedure does not terminate with an optimal solution the algorithm uses a tree search strategy that as opposed to branch and bound keeps on producing cuts after branching the algorithm has been implemented in fortran two different linear programming lp packages have been used as the lp solver the implementation of the algorithm and the interface with one of the lp solvers is described in sufficient detail to permit the replication of our experiments computational results are reported with up to 42 stsps with sizes rangin

representations of super yangian the classification of the finite dimensional irreducible representations of the super yangian associated with the lie superalgebra gl 1 1 is presented in detail

solving coupled systems of linear second order differential equations knowing a part of the spectrum of the companion matrix abstract in this paper an explicit closed form solution of initial value problems for coupled systems of time invariant second order differential equations is given without computing the exponential of the associated companion matrix

the travelling salesman problem with pick up and delivery demand points for pick up and delivery are located in the plane a single vehicle of a given capacity performs both pick ups and deliveries the objective is to find the shortest feasible tour a real life application is described we compare this problem to the classic travelling salesman problem based on their similarity we develop a simple heuristic consisting of pick up and delivery along the travelling salesman tour asymptotic behavior and worst case analysis are provided we then introduce an alternative solution method which is an extension of the well known cheapest insertion heuristic performance of both heuristics is tested on various size simulated problems

analytic solution with a priori error bounds for a class of mixed coupled partial differential equations in this paper the existence of a sequence of stable solutions xn of the parametric matrix equation z2 az n2b 0 so that xnn converges to a stable matrix is studied the result is used for solving mixed problems related to the coupled second order partial differential system utt auxx but 0

the swapping problem on a line we consider the problem of optimally swapping objects between n workstations which we refer to as nodes located on a line there are m types of objects and the set of object types is denoted by s 1 m object type 0 is a dummy type the null object each node v contains one unit of a certain object type a v in s cup 0 and requires one unit of object type b v in s cup 0 we assume that the total supply equals the total demand for each of the object types separately a vehicle of unit capacity ships the objects so that the requirements of all nodes are satisfied the set of object types is partitioned into two sets objects that may be temporarily dropped at intermediate nodes before reaching their destination and objects that have to be shipped directly from their origin to their destination the objective is to design a route that starts and ends at the depot and a feasible assignment of object types to the route s arcs so that the total distance is minimized we propose an o n 2 algorithm to compute the optimal solution for this problem

stronger multi commodity flow formulations of the capacitated sequential ordering problem the sequential ordering problem sop is the generalisation of the asymmetric travelling salesman problem in which there are precedence relations between pairs of nodes hernandez salazar introduced a multi commodity flow mcf formulation for a generalisation of the sop in which the vehicle has a limited capacity we strengthen this mcf formulation by fixing variables and adding valid equations we then use polyhedral projection together with some known results on flows cuts and metrics to derive new families of strong valid inequalities for both problems finally we give computational results which show that our findings yield good lower bounds in practice

graded solutions of the yang baxter relation and link polynomials from a family of graded solvable models the authors derive representations of the braid group associated with the lie superalgebra gl m mod n and give explicitly a general form of the markov traces on the representations the braid operators thus obtained satisfy the hecke algebra the authors construct composite solvable models and obtain link polynomials from the braid operators for the composite models

super yangian double dy gl 1 1 and its gauss decomposition we extend the yangian double to the super or graded case and give its drinfel d generators a realization by gauss decomposition

the traveling salesman problem with pickup and delivery polyhedral results and a branch and cut algorithm the traveling salesman problem with pickup and delivery tsppd is defined on a graph containing pickup and delivery vertices between which there exists a one to one relationship the problem consists of determining a minimum cost tour such that each pickup vertex is visited before its corresponding delivery vertex in this paper the tsppd is modeled as an integer linear program and its polyhedral structure is analyzed in particular the dimension of the tsppd polytope is determined and several valid inequalities some of which are facet defining are introduced separation procedures and a branch and cut algorithm are developed computational results show that the algorithm is capable of solving to optimality instances involving up to 35 pickup and delivery requests thus more than doubling the previous record of 15

boundary value problems for second order differential operator equations abstract it is proved that the resolution problem of an operator boundary value problem for a second order differential operator equation with constant coefficients is solved in terms of solutions of certain algebraic operator equations explicit expressions of solutions are given

isomorphism of two realizations of quantum affine algebra u_q widehat mathfrak g mathfrak l text n text we establish an explicit isomorphism between two realizations of the quantum affine algebra u q widehat mathfrak g mathfrak l text n text given previously by drinfeld and reshetikhin semenov tian shansky our result can be considered as an affine version of the isomorphism between the drinfield jimbo and the faddeev reshetikhin takhtajan constructions of the quantum algebra u q mathfrak g mathfrak l n

yang baxter equation in long range interacting systems we consider the su p spin chains with long range interactions and the spin generalization of the calogero sutherland models we show that their properties derive from a transfer matrix obeying the yang baxter equation we obtain the expression of the conserved quantities of the dynamical models and we diagonalise them in the spin chain case we establish the connection between the degeneracies of the spectrum and the representation theory of the yangians we use a correspondence with the dynamical models to diagonalise the hamiltonian finally we extend the previous results to the case of a trigonometric r matrix

network design using cut inequalities we study the network loading problem with and without bifurcations we use a relaxation based on the cut condition for multicommodity flows we use a solution of the bifurcated case to derive a solution to the nonbifurcated problem a standard procedure is to aggregate the problem into a backbone network we applied this method to backbone networks coming from practical instances we obtained feasible solutions and bounds for the gap from optimality

clique tree inequalities and the symmetric travelling salesman problem the linear programming cutting plane approach for solving the travelling salesman problem has recently proven to be highly successful cf crowder and padberg crowder h p m w padberg 1980 solving large scale symmetric travelling salesman problems to optimality management sci 26 495 509 grotschel grotschel m 1980a on the symmetric travelling salesman problem solution of a 120 city problem math programming stud 12 61 77 padberg and hong padberg m w s hong 1980 on the symmetric travelling salesman problem a computational study math programming stud 12 78 107 one of the reasons for this success is certainly the fact that instead of ordinary cutting planes gomory cuts etc problem specific cutting planes could be used which define facets of the underlying integer programming polytopes in this paper we shall define a new class of inequalities clique tree inequalities valid for the travelling salesman polytope which properly contains many of the known classes of inequalities like subtour elimination constraints 2 matching constraints comb inequalities and we show that all these new inequalities induce facets of the travelling salesman polytope since the general structure of these new inequalities is quite simple we hope that it will be possible to use the inequalities efficiently in cutting plane procedures for the travelling salesman problem

an initial boundary value problem of physiological significance for equations of nerve conduction a model of nerve conduction that has gained wide acceptance in the biophysical community is that for the giant axon of the squid loligo of hodgkin and huxley 5 this nonlinear parabolic partial differential equation pde when considered as part of a particular initial boundary value problem ibvp models the electrical activity of an axon under conditions found both in nature and in the laboratory this ibvp for the hodgkin huxley equations is proved to be well posed in the sense of hadamard 3 and a priori bounds on the solution are derived

explicit solutions for second order operator differential equations with two boundary value conditions ii abstract boundary value problems for second order operator differential equations with two boundary value conditions are studied explicit expressions of the solutions in terms of data problems are given by means of the application of algebraic techniques analogous expressions to the ones known for the scalar case are obtained

projections of the capacitated network loading problem abstract consider an undirected network and a set of commodities with specified demands between various pairs of nodes of the network given two types of capacitated facilities that can be installed loaded for arc dependent costs we have to determine the integer number of facilities to load on each arc in order to send the required flow of all commodities at minimum total cost we present a natural mixed integer programming formulation of the problem and then consider its single commodity and multicommodity versions we develop equivalent formulations in a lower dimensional space by projecting out the flow variables and study the polyhedral properties of the corresponding projection cones our results strengthen an existing result for multicommodity flow problems we also characterize several classes of facet defining inequalities for this lower dimensional polyhedron and conclude by identifying some open problems and future research directions