generalized drazin invertibility of combinations of idempotents abstract the paper serves as a correction to j math anal appl 359 2009 731 738 dealing with the drazin invertibility of combinations of idempotents p q in a banach algebra as the arguments employed to do this work equally well for both the usual and generalized drazin inverse the latter is included in the discussion which covers the equivalence of the drazin invertibility of 1 pq p q p q and in general of p q with 0 as well as the equivalence of the drazin invertibility of the commutator pq qp and anticommutator pq qp of p q

optimal solution of vehicle routing problems using minimum k trees we consider the problem of optimally scheduling a fleet of k vehicles to make deliveries to n customers subject to vehicle capacity constraints given a graph with n 1 nodes a k tree is defined to be a set of n k edges that span the graph we show that the vehicle routing problem can be modeled as the problem of finding a minimum cost k tree with two k edges incident on the depot and subject to some side constraints that impose vehicle capacity and the requirement that each customer be visited exactly once the side constraints are dualized to obtain a lagrangian problem that provides lower bounds in a branch and bound algorithm this algorithm has produced proven optimal solutions for a number of difficult problems including a well known problem with 100 customers and several real problems with 25 71 customers

some results on the drazin invertibility and idempotents in this paper several equivalent conditions for the drazin invertibility of the sum difference and the product of idempotents in banach algebra are established

characterizations of the commutators and the anticommutator involving idempotents in this paper we characterize the idempotency the invertibility or generalized invertibility the positivity and the range relations of the commutator c pq qp and the anticommutator d pq qp involving idempotents particular attention is paid to the idempotent operators p and q and some further characteristics when p and q are positive are derived as well

the perturbation theory for the drazin inverse and its applications abstract let a and e be n n matrices and b a e denote the drazin inverse of a by ad we give an upper bound for the relative error b d a d a d under certain circumstances an error bound for the solution for the singular equations ax b b r a is also considered

semi infinite cohomology and string theory abstract we develop the theory of semi infinite cohomology of graded lie algebras first introduced by feigin we show that the relative semi infinite cohomology has a structure analogous to that of the de rham cohomology in kahler geometry we prove a vanishing theorem for a special class of modules and we apply our results to the case of the virasoro algebra and the fock module in this case the zero cohomology is identified as the physical subspace of the fock module and the no ghost theorem follows we reveal the profound relation of semi infinite cohomology theory to the gauge invariant free string theory constructed by banks and peskin we then indicate the connection between gauge invariant interacting string theories and the geometric realizations of the infinite dimensional lie algebras

determinants torsion and strings we apply the results of bf1 bf2 on determinants of dirac operators to string theory for the bosonic string we recover the holomorphic factorization of belavin and knizhik witten s global anomaly formula is used to give sufficient conditions for anomaly cancellation in the heterotic string for arbitrary background spacetimes to prove the latter result we develop certain torsion invariants related to characteristic classes of vector bundles and to index theory

string structures and the index of the dirac ramond operator on orbifolds we discuss the relationship between string structures and the topological class h4 m z on non simply connected manifolds we also investigate to what extent the index formula for the dirac ramond operator detects the class

a gentle guide to the basics of two projections theory abstract this paper is a survey of the basics of the theory of two projections it contains in particular the theorem by halmos on two orthogonal projections and roch silbermann gohberg and krupnik s theorem on two idempotents in banach algebras these two theorems which deliver the desired results usually very quickly and comfortably are missing or wrongly cited in many recent publications on the topic the paper is intended as a gentle guide to the field the basic theorems are precisely stated some of them are accompanied by full proofs others not but precise references are given in each case and many examples illustrate how to work with the theorems

new additive results for the g drazin inverse the paper studies additive properties of the generalized drazin inverse g drazin inverse in a banach algebra and finds an explicit expression for the g drazin inverse of the sum a b in terms of a and b and their g drazin inverses under fairly mild conditions on a and b

the representation and approximation of the drazin inverse of a linear operator in hilbert space we present a unified representation theorem for the drazin inverse of linear operators in hilbert space and a general error bound five specific expressions computational procedures and their error bounds for the drazin inverse are uniformly derived from the unified representation theorem

on the drazin inverse of the linear combinations of two idempotents in a complex banach algebra abstract let a be a complex banach algebra with unit and let p q be two idempotents in a for c 0 the authors obtain an explicit representation formula for the drazin inverse of p q and give the upper bound of the corresponding drazin index by using the presented method as a consequence the main previously published results on drazin inverse of p q are corollaries of our theorem finally the authors apply the theorem to an example which cannot be solved by previously known results

some additive results on drazin inverse some additive perturbation results for drazin inverses are given in particular a formula is given for the drazin inverse of a sum of two matrices when one of the products of these matrices vanishes some special applications of this are also considered

a multiperiod planning model for the capacitated minimal spanning tree problem the multiperiod capacitated minimal spanning tree mcmst problem consists of scheduling the installation of links in a network so as to connect a set of terminal nodes s 2 3 n to a central node node 1 with minimal present value of expenditures where link capacities limit the number of terminal nodes sharing a link some of the terminal nodes are active at the beginning of the planning horizon while others are activated over time we formulate this problem as an integer programming problem a branch exchange heuristic procedure for solving the problem is presented we also present a lagrangian relaxation method to find a lower bound for the optimal objective function value this lower bound may be used to estimate the quality of the solution given by the branch exchange heuristic experimental results over a wide range of problem structures show that the branch exchange heuristic method yields verifiably good solutions to this problem

the drazin invertibility of the difference and the sum of two idempotent operators in this note some equivalents are established of the drazin invertibility of differences and sums of idempotent operators on a hilbert space

the drazin and generalized drazin invertibility of linear combinations of idempotents abstract let a be a unital complex banach algebra and p q be two idempotents in a in this paper we prove that the drazin resp generalized drazin invertibility of p q p q is independent of the choice of scalars c 0 and c we also give a simple representation formula for the drazin inverse of p q as an application we study the left resp right drazin invertibility and the property of being browder of linear combinations of two idempotent operators on a banach space

the dirac ramond operator on loops in flat space in this paper a rigorous construction of the s1 equivariant dirac operator i e dirac ramond operator on the space of mean zero loops in rd is given and its equivariant l2 index computed essential use is made of infinite tensor product representations of the canonical anticommutation relations algebra

minimal spanning trees with a constraint on the number of leaves abstract in this paper we discuss minimal spanning trees with a constraint on the number of leaves tree topologies appear when designing centralized terminal networks the constraint on the number of leaves arises because the software and hardware associated to each terminal differs accordingly with its position in the tree usually the software and hardware associated to a degree 1 terminal is cheaper than the software and hardware used in the remaining terminals because for any intermediate terminal j one needs to check if the arrival message is destined to that node or to any other node located after node j as a consequence that particular terminal needs software and hardware for message routing on the other hand such equipment is not needed in degree 1 terminals assuming that the hardware and software for message routing in the nodes is already available the above discussion motivates a constraint stating that a tree solution has to contain exactly a certain number of degree 1 terminals we present two different formulations for this problem and some lower bounding schemes derived from them we discuss a simple local exchange heuristic and present computational results taken from a set of complete graphs with up to 40 nodes integer linear programming formulations for related problems are also discussed at the end

a classification of formulations for the time dependent traveling salesman problem abstract the time dependent traveling salesman problem tdtsp is a generalization of the classical traveling salesman problem where the cost of any given arc is dependent of its position in the tour the tdtsp can model several real world applications e g one machine sequencing in this paper we present a classification of formulations for the tdtsp this framework includes both new and old formulations the new formulation presented in this paper is derived from a quadratic assignment model for the tdtsp in a first step lawler s transformation procedure is used to derive an equivalent linearized version of the quadratic model in a second step a stronger formulation is obtained by tightening some constraints of the previous formulation it is shown that in terms of linear relaxations the latter formulation is either equivalent or better than other formulations already known from the literature finally we compare these formulations with other well known formulations for the classical traveling salesman problem

hodge elliptic genera and how they govern k3 theories the complex hodge elliptic genus and its conformal field theoretic counterpart were recently introduced by kachru and tripathy refining the traditional complex elliptic genus we construct a different so called chiral hodge elliptic genus which is expected to agree with the generic conformal field theoretic hodge elliptic genus in contrast to the complex hodge elliptic genus as originally defined for k3 surfaces x the chiral hodge elliptic genus is shown to be independent of all moduli moreover employing kapustin s results on infinite volume limits it is shown that it agrees with the generic conformal field theoretic hodge elliptic genus of k3 theories while the complex hodge elliptic genus does not this new invariant governs part of the field content of k3 theories supporting the idea that all their spaces of states have a common subspace which underlies the generic conformal field theoretic hodge elliptic genus and thereby the complex elliptic genus mathematically this space is modelled by the sheaf cohomology of the chiral de rham complex of x it decomposes into irreducible representations of the n 4 superconformal algebra such that the multiplicity spaces of all massive representations have precisely the dimensions required in order to furnish the representation of the mathieu group m 24 that is predicted by mathieu moonshine this is interpreted as evidence in favour of the ideas of symmetry surfing which have been proposed by taormina and the author along with the claim that the sheaf cohomology of the chiral de rham complex is a natural home for mathieu moonshine these investigations also imply that the generic chiral algebra of k3 theories is precisely the n 4 superconformal algebra at central charge c 6 if the usual predictions on infinite volume limits from string theory hold true

characterizations and representations of the group inverse involving idempotents this paper is to present some results on the group invertibility of products and differences of idempotents in addition some formulae for the group inverse of sums differences and products of idempotents are established by using some given idempotents

the drazin inverses of sum and difference of idempotents in this note the drazin inverses of sum and difference of idempotents on a hilbert space are established under some conditions

a 2n constraint formulation for the capacitated minimal spanning tree problem in this paper we present a new formulation for the capacitated minimal spanning tree cmst problem one advantage of the new formulation is that it is more compact in the number of constraints than a well known formulation additionally we show that the linear programming relaxation of both formulations produces optimal solutions with the same cost we present a brief discussion concerning valid inequalities for the cmst which are directly derived from the new formulation we show that some of the new inequalities are not dominated by some sets of facet inducing inequalities for the cmst we derive some lagrangian relaxation based methods from the new formulation and present computational evidence showing that reasonable improvements on the original linear programming bounds can be obtained if these methods are strengthened by the use of cutting planes the reported computational results indicate that one of the methods presented in this paper dominates in most of the cases the previous best methods reported in the literature the most significant improvements are obtained in the instances with the root in the corner

analytic index for a family of dirac ramond operators we derive a cohomological formula for the analytic index of a family of dirac ramond operators and exhibit its modular properties

robust branch cut and price for the capacitated minimum spanning tree problem over a large extended formulation this paper presents a robust branch cut and price algorithm for the capacitated minimum spanning tree problem cmst the variables are associated to q arbs a structure that arises from a relaxation of the capacitated prize collecting arborescence problem in order to make it solvable in pseudo polynomial time traditional inequalities over the arc formulation like capacity cuts are also used moreover a novel feature is introduced in such kind of algorithms powerful new cuts expressed over a very large set of variables are added without increasing the complexity of the pricing subproblem or the size of the lps that are actually solved computational results on benchmark instances from the or library show very significant improvements over previous algorithms several open instances could be solved to optimality

elliptic genera and quantum field theory it is shown that in elliptic cohomology as recently formulated in the mathematical literature the supercharge of the supersymmetric nonlinear signa model plays a role similar to the role of the dirac operator ink theory this leads to several insights concerning both elliptic cohomology and string theory some of the relevant calculations have been done previously by schellekens and warner in a different context

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

proof of character valued index theorems a character valued index is a generalization of the ordinary dirac index to manifolds with nontrivial automorphism groups a simple proof of the corresponding fixed point theorem is presented which uses the techniques of supersymmetric quantum mechanics this theorem relates the character valued index to a topological integral of curvature forms on the fixed point space of the automorphism in question

the drazin inverse of the linear combinations of two idempotents in the banach algebra abstract in this paper some drazin inverse representations of the linear combinations of two idempotents in a banach algebra are obtained moreover we present counter examples to and establish the corrected versions of two theorems by cvetkovic ilic and deng

on the perturbation of the group inverse and oblique projection for given matrices a and e which are n matrices and matrices and e we discuss upper bounds for b bb b a a and bb aa aa also a lower bound for b is derived

string theory and loop space index theorems we study index theorems for the dirac ramond operator on a compact riemannian manifold the existence of a group action on the loop space makes possible the definition of a character valued index which we calculate by using a two dimensional sigma model withn 1 2 supersymmetry we compute the euler characteristic the hirzebruch signature and the dirac ramond genus of loop space we compare our results to the calculations made by using the atiyah singer character valued index theorem

extended formulations and branch and cut algorithms for the black and white traveling salesman problem in this paper we study integer linear programming models and develop branch and cut algorithms to solve the black and white traveling salesman problem bwtsp bourgeois laporte semet 2003 which is a variant of the well known traveling salesman problem tsp two strategies to model the bwtsp have been used in the literature the problem is either modeled on the original graph as tsp using a single set of binary edge variables and with additional non trivial hop and distance constraints between every pair of black nodes see ghiani laporte semat 2006 or as a sequence of constrained paths composed of white nodes connecting pairs of black nodes see muter 2015 in this paper we study and develop an intermediate approach based on the observation that it is sufficient to guarantee the required distance and hop limit of the path from a given black node to the next black node without explicitly stating which one it is thus instead of stating the two constraints after adding appropriately defined variables for each pair of black nodes they are stated for each black node only that represents the source of each path based on this idea we develop several variants of position and distance dependent reformulations together with corresponding layered graph representations branch and cut algorithms are developed for all proposed formulations and empirically compared by an extensive computational study the obtained results allow us to provide insights into individual advantages and disadvantages of the different models

multiple center capacitated arc routing problems a tabu search algorithm using capacitated trees abstract in the capacitated arc routing problem with multiple centers m carp the objective is to find routes starting from the given depots or centers such that each required arc is served capacity and usually additional constraints are satisfied and total travel cost is minimized in this paper we consider a heuristic transformation of the m carp into a multiple center capacitated minimum spanning tree problem with arc constraints that we call arc constrained cmst an algorithm for determining initial feasible solutions as well as an improvement procedure for this problem are described and the re translation of cmst solutions into the carp context is explained it is shown that the objective function value of the obtained carp solution is easily derived from the respective value of the corresponding heuristic cmst solution furthermore the possibility of including side constraints and the consideration of additional objective functions is discussed computations on real world benchmark problems compare the results of the tabu search and simulated annealing metastrategies embedded in the improvement procedure

drazin inversion in the von neumann algebra generated by two orthogonal projections abstract criteria for drazin and moore penrose invertibility of operators in the von neumann algebra generated by two orthogonal projections are established and explicit representations for the corresponding inverses are given the results are illustrated by several examples that have recently been considered in the literature

the drazin inverses of products and differences of orthogonal projections in this note the drazin inverses of products and differences of orthogonal projections on a hilbert space are established

discretized formulations for capacitated location problems with modular distribution costs abstract in this paper we study a discretization reformulation technique in the context of a facility location problem with modular link costs we present a so called traditional model and a straightforward discretized model with a general objective function whose variable coefficients are computed by solving a simple knapsack problem we show that the linear programming relaxation of the discretized model dominates the linear programming relaxation of the original model the discretized model suggests quite intuitive valid inequalities that considerably improve the linear programming relaxation of the original model computational results based on randomly generated data show that in the context of problems with modular costs the proposed discretized models perform significantly better than the traditional models an important outcome of our research is the result whose proof is also presented in this paper that a restricted version of the discretized model gives an extended description of the convex hull of the integer solutions of a subproblem that usually arises in network design problems with modular costs

valid inequalities for non unit demand capacitated spanning tree problems with flow costs abstract we discuss valid inequalities for a variation of the capacitated minimum spanning tree problem which considers variable flow costs this variation with flow costs arises in the design of low voltage electricity distribution networks see bousba c wolsey l 1991 annals of operations research 33 285 303 or in the design of telephone networks see balakrishnan a magnanti t wong r 1995 operations research 43 58 76 we start by pointing out that most of the ideas discussed in the past for the unit demand versions of the problem are hardly applied to versions with general demands on the nodes then in the context of a flow based model we propose some valid inequalities for the problem which are based on the optimal solution of an adequate subset sum problem and which make sense only when the node demands are different we also consider a multicommodity reformulation of the original model and adapt for this model the inequalities proposed for the single commodity flow models we present an exponentially sized set of lower bounding constraints which are valid for the original flow based model and which are implied by the compact multicommodity reformulation finally computational results are presented

proof of the family index theorem via a path integral the integral formula of the family index theorem and its equivariant version are derived by use of a path integral analogous to such a proof of the atiyah singer index theorem but without supersymmetry some formulas explicitly presented here should be useful in applications of the family index theorem to physics

reverse order laws for the drazin inverses abstract this paper is to present some equivalent conditions concerning the reverse order law p q d q d p d for the drazin invertible operators p and q with partial commutative properties p p q 0 or q p q 0 or p p q q d 0 or q p p d q 0

a reduced cost iterated local search heuristic for the fixed charge transportation problem the fixed charge transportation problem fctp is a generalization of the transportation problem where an additional fixed cost is paid for sending a flow from an origin to a destination we propose an iterated local search heuristic based on the utilization of reduced costs for guiding the restart phase the reduced costs are obtained by applying a lower bounding procedure that computes a sequence of nondecreasing lower bounds by solving a three index mathematical formulation of the problem strengthened with valid inequalities the proposed method was tested on two sets of benchmark instances from the literature the first set was used to evaluate the state of the art heuristics for the problem the proposed heuristic was able to provide new best knownupper bounds on all 124 open instances on the second set of instances which was recently introduced for testing the currently best exact method for the problem the new heuristic was able to provide provably good upper bounds within short computing times