a heuristic approach for the travelling purchaser problem abstract the travelling purchaser problem tpp is a known generalization of the travelling salesman problem and is defined as follows let us consider a set of products and a set of markets each market is provided with a limited amount of each product at a known price the tpp consists in selecting a subset of markets such that a given demand of each product can be purchased minimizing the routing cost and the purchasing cost this problem arises in several applications mainly in routing and scheduling contexts and it is np hard in the strong sense a new heuristic approach based on a local search scheme exploring a new neighbourhood structure is proposed the key idea is to perform a k exchange of markets instead of the classical 1 exchanges a neighbour of a given tpp solution is another tpp solution obtained by removing a path of consecutive markets and by inserting other markets so as to restore the feasibility this proposal is evaluated on a broad class of instances from literature where the routing costs are euclidean distances computational results show that our proposal is favourably compared to previous heuristic algorithms in literature

dimension of the orbit of marked subspaces given a nilpotent endomorphism we consider the manifold of invariant subspaces having a fixed segre characteristic in linear algebra appl 332 334 2001 569 the implicit form of a miniversal deformation of an invariant subspace with respect to the usual equivalence relation between subspaces is obtained here we obtain the explicit form of this deformation when the invariant subspace is marked and we use it to calculate the dimension of the orbit and in particular to characterize the stable marked subspaces those with open orbit moreover we study the rank of the endomorphisms in the quotient space by the subspaces in the miniversal deformation of the giving subspace

on the pointwise limits of bivariate lagrange projectors a linear algebra proof is given of the fact that the nullspace of a finite rank linear projector on polynomials in two complex variables is an ideal if and only if the projector is the bounded pointwise limit of lagrange projectors i e projectors whose nullspace is a radical ideal i e the set of all polynomials that vanish on a certain given finite set a characterization of such projectors is also given in the real case more generally a characterization is given of those finite rank linear projectors on polynomials in d complex variables with nullspace an ideal that are the bounded pointwise limit of lagrange projectors the characterization is in terms of a certain sequence of d commuting linear maps and so focuses attention on the algebra generated by such sequences

commuting pairs and triples of matrices and related varieties abstract in this note we show that the set of all commuting d tuples of commuting n n matrices that are contained in an n dimensional commutative algebra is a closed set and therefore gerstenhaber s theorem on commuting pairs of matrices is a consequence of the irreduciblity of the variety of commuting pairs we show that the variety of commuting triples of 4 4 matrices is irreducible we also study the variety of n dimensional commutative subalgebras of m n f and show that it is irreducible of dimension n 2 n for n 4 but reducible of dimension greater than n 2 n for n 7

a fast and effective heuristic for the orienteering problem abstract in the orienteering problem start and end points are specified along with other locations which have associated scores given a fixed amount of time the goal is to determine a path from the start point to the end point through a subset of locations in order to maximize the total path score in this paper a fast and extremely effective heuristic is presented and tested on 67 problems taken from the literature and 40 new test problems the computational results are presented in detail

matrix polynomials factorization via bisolvents abstract we reconsider the classification of all the factorizations of a matrix polynomial p as p q r with q a matrix polynomial and r t s a regular matrix pencil it is shown that the entire classification problem can be reduced to the simpler classification of factors r with commuting coefficients s t it is then shown that for these commuting factors s and t must satisfy a certain algebraic equation which we call the bisolvent equation this extends the generalized bezout theorem which associates monic factors i s with solutions s of a solvent equation in case p is regular the classification of commuting pairs s t of this type up to left equivalence is described in terms of enlarged standard pairs following a well known approach under a non derogatory generic condition on the roots of p the number of such pairs associated with degree minimal factorizations is finite and admits explicit description in terms of jordan chains

bond percolation on subsets of the square lattice and the threshold between one dimensional and two dimensional behaviour for each value of pi in 1 2 1 the author constructs a subgraph r pi of the square lattice z2 such that the bond percolation process on r pi has critical probability pi it is shown that the critical probability pi a of the region x y 0 or y or a ln x 1 0 or x infinity depends in a non trivial way upon the choice of the number a pi a is a continuous decreasing function of a and satisfies pi a to 1 2 as a to infinity and pi a to 1 as a down arrow 0 a family of one dimensional bond percolation processes is also constructed whose critical probabilities range over the interval 0 1 all proofs are rigorous

eigenvalue based algorithms for testing positive realness of siso systems a computational test for scalar positive real and strictly positive real functions is proposed in this paper the input for this algorithm is state space representation not necessarily minimal it is derived for the discrete time problem based on numerical stable eigenvalue computations the test can be used in the continuous counterpart which is in fact an improvement of an algorithm posed by bai and freund differences between the discrete and continuous time problems are also pointed out

a branch and cut approach to the cardinality constrained circuit problem the cardinality constrained circuit problem cccp is the problem of finding a minimum cost circuit in a graph where the circuit is constrained to have at most k edges the cccp is np hard we present classes of facet inducing inequalities for the convex hull of feasible circuits and a branch and cut solution approach using these inequalities

a tssp 1 decomposition strategy for the vehicle routing problem abstract the basic capacity constrained vehicle routing problem vrp is to determine a set of capacity feasible vehicle routes with minimum total travel cost so that all customers are visited by exactly one vehicle in this paper we introduce a new decomposition strategy for the vrp which separates the decisions of a dispatcher from those of the individual drivers the dispatcher is responsible for assigning a reward value to each of the customer cities so that each customer will be visited by exactly one vehicle based on the assigned reward values together with the given travel costs each vehicle driver is responsible for choosing which customers to visit and determining an individual feasible route the driver s problem is modeled as a traveling salesman subset tour problem with one additional constraint tssp 1 the tssp 1 decomposition is based on aagrangian relaxation and is capable of producing valid lower bounds for the vrp the best bound attainable is shown to be at least as good as the bound obtained by solving the linear programming relaxation of the classic set partitioning formulation of the vrp to tet the practical value of the decomposition we present a decomposition based heuristic and examine its performance on a collection of problems from the literature

bounds for the spectrum of a two parameter matrix eigenvalue problem abstract we consider the two parameter eigenvalue problem t j v j 1 a j 1 v j 2 a j 2 v j 0 where j c t j a j k j k 1 2 are matrices bounds for the spectral radius of that problem are suggested our main tool is a norm estimate for the operator inverse to the operator a 11 a 22 a 12 a 21 where means the tensor product in addition by virtue of that norm estimate and the ostrowsky schneider theorem we establish a condition that provides the conservation of the number of the eigenvalues of the considered problem in a half plane

approximating commuting operators the problem of approximating m tuples of commuting n n complex matrices by commuting m tuples of generic matrices is studied we narrow the gap for commuting triples by showing that they can be perturbed if n 29

on approximately simultaneously diagonalizable matrices abstract a collection a1 a2 ak of n n matrices over the complex numbers c has the asd property if the matrices can be perturbed by an arbitrarily small amount so that they become simultaneously diagonalizable such a collection must perforce be commuting we show by a direct matrix proof that the asd property holds for three commuting matrices when one of them is 2 regular dimension of eigenspaces is at most 2 corollaries include results of gerstenhaber and neubauer sethuraman on bounds for the dimension of the algebra generated by a1 a2 ak even when the asd property fails our techniques can produce a good bound on the dimension of this subalgebra for example we establish dim c a 1 a k 5 n 4 for commuting matrices a1 ak when one of them is 2 regular this bound is sharp one offshoot of our work is the introduction of a new canonical form the h form for matrices over an algebraically closed field the h form of a matrix is a sparse jordan like upper triangular matrix which allows us to assume that any commuting matrices are also upper triangular the jordan form itself does not accommodate this

an efficient composite heuristic for the symmetric generalized traveling salesman problem the main purpose of this paper is to introduce a new composite heuristic for solving the generalized traveling salesman problem the proposed heuristic is composed of three phases the construction of an initial partial solution the insertion of a node from each non visited node subset and a solution improvement phase we show that the heuristic performs very well on 36 tsplib problems which have been solved to optimality by other researchers we also propose some simple heuristics that can be used as basic blocks to construct more efficient composite heuristics

some thoughts on gerstenhaber s theorem abstract we weigh the current evidence for and against an extension of gerstenhaber s 1961 theorem to three commuting matrices over a field

stability of controlled invariant subspaces abstract let a b c n n c n m and m be an a b invariant subspace let c a b be the controllability subspace of the pair a b if m c a b m im b under additional hypotheses necessary and sufficient conditions for the stability of the subspace m are given

algorithms to solve the orienteering problem a comparison abstract the multiobjective vending problem mvp keller 1985 is a generalization of the traveling salesman problem tsp where it is not necessary to visit all nodes in the problem definition a special case of the mvp is the orienteering problem op tsiligirodes 1984 et al the problems are np hard a heuristic was designed to solve the general mvp keller 1985 three other heuristics were designed to solve the op tsiligirides 1984 golden et al 1985 the algorithms underlying these heuristics are outlined their performances are compared against three op problems

polynomials in a matrix and its commutant abstract given any norm u on the space of linear transformations matrices over a finite dimensional complex vector space and given commuting transformations t s and any e 0 there is a transformation t e such that u t e t p e such that p e t e s this result has implications for the theory of the numerical radius and other operator radii its proof involves the following fact of interest in itself if the jordan canonical form of s has k blocks then the transformations with k distinct eigenvalues are dense in the commutant of s

geometric characterization and classification of marked subspaces abstract given an endomorphism of a finite dimensional c vector space we obtain geometric conditions in order to characterize and to classify the marked invariant subspaces w e that is to say the invariant subspaces having some jordan basis which can be extended to a jordan basis of e both conditions are expressed in terms of the double family of subspaces ker f h im f d the starting point is a new criterion for extensibility of jordan bases

the cayley hamilton theorem and inverse problems for multiparameter systems abstract we review some of the current research in multiparameter spectral theory we prove a version of the cayley hamilton theorem for multiparameter systems and list a few inverse problems for such systems some consequences of results on determinantal representations proved by dixon dickson and vinnikov for the inverse problems are discussed

two generated commutative subalgebras of mn f abstract for two commuting matrices a b m n k k a field it has been known that dim k k a b n we show here that dim k cent m n k a b n and that equality for one implies equality for the other we also give both a module theoretic and a geometric characterization of those algebras for which equality is obtained furthermore we present a new proof of a result of gerstenhaber

brief paper a class of marked invariant subspaces with an application to algebraic riccati equations invariant subspaces of a matrix a are considered which are obtained by truncation of a jordan basis of a generalized eigenspace of a we characterize those subspaces which are independent of the choice of the jordan basis an application to hamilton matrices and algebraic riccati equations is given

the biflag manifold and the fixed points of a unipotent transformation on the flag manifold abstract we investigate the topology of the submanifold of the partial flag manifold consisting of those flags s 1 s d in f n f r or c that are fixed by a given unipotent a sl n f and for which the restrictions a s i have prescribed jordan type it is shown that this submanifold has a compact nonsingular projective variety as a strong deformation retract this variety the biflag manifold consists of rectangular arrays of subspaces which are nested along each row and column it is proven that the biflag manifold has the homology type mod 2 for f r of a product of grassmannians a partition of the biflag manifold into a finite union of affine spaces is also given in the special case when the biflag manifold is a flag manifold this partition coincides with the bruhat cell decomposition

on varieties of commuting nilpotent matrices abstract let n d n be the variety of all d tuples of commuting nilpotent n n matrices it is well known that n d n is irreducible if d 2 if n 3 or if d 3 and n 4 on the other hand n 3 n is known to be reducible for n 13 we study in this paper the reducibility of n d n for various values of d and n in particular we prove that n d n is reducible for all d n 4 in the case d 3 we show that it is irreducible for n 6

kronecker bases for linear matrix equations with application to two parameter eigenvalue problems abstract the general solutions of the homogeneous matrix equation axct bxdt 0 and the system of the matrix equations ax by 0 xct ydt 0 are described in terms of kronecker canonical forms i e in terms of kronecker invariants and kronecker bases for pairs of matrices a b and c d a canonical form for a pair of commuting matrices e f such that e2 f2 ef 0 is discussed these results are applied to construct a canonical basis for the second root subspace of a two parameter eigenvalue problem the corresponding relations for canonical invariants are given

a variety of commuting triples abstract the problem of approximating triples of commuting n n matrices by generic matrices is equivalent to the problem of whether the variety c 3 n of commuting triples is irreducible the answer to the problem is known to be positive for dimensions no greater than 5 and negative for dimensions no smaller than 30 using simultaneous commutative perturbations of pairs of matrices in the centralizer of the third matrix we are able to show that the answer is positive in dimension 6 as well all the proofs are given over an arbitrary algebraically closed field of characteristic zero

on a b t invariant subspaces having extendible brunovsky bases we consider a by invariant subspaces having a brunovsky basis which can be extended to a brunovsky basis of the whole space we obtain a geometrical characterization of this class of a by invariant subspaces and a complete family of numerical invariants to classify them 0 elsevier science inc 1997

a branch and cut algorithm for the undirected traveling purchaser problem the purpose of this paper is to present a branch and cut algorithm for the undirectedtraveling purchaser problem which consists of determining a minimum cost route through a subset of markets where the cost is the sum of travel and purchase costs the problem is formulated as an integer linear program and several families of valid inequalities are derived to strengthen the linear relaxation the polyhedral structure of the formulation is analyzed and several classes of valid inequalities are proved to be facet defining a branch and cut procedure is developed and tested over four classes of randomly generated instances results show that the proposed algorithm outperforms all previous approaches and can optimally solve instances containing up to 200 markets

jordan forms for mutually annihilating nilpotent pairs abstract in this paper we completely characterize all possible pairs of jordan canonical forms for mutually annihilating nilpotent pairs i e pairs a b of nilpotent matrices such that ab ba 0

on tensors of border rank l in cm n l notice this is the author s version of a work that was accepted for publication in linear algebra and its applications changes resulting from the publishing process such as peer review editing corrections structural formatting and other quality control mechanisms may not be reflected in this document changes may have been made to this work since it was submitted for publication a definitive version was subsequently published in linear algebra and its applications june 23 2011 doi 10 1016 j laa 2011 05 013 the original publication is available at www elsevier com

root vectors for geometrically simple two parameter eigenvalues a class of multiparameter eigenvalue problems involving generally non self adjoint and unbounded operators is studied bases for lower order root subspaces at geometrically simple eigenvalues of fredholm type of arbitrary finite index are computed in terms of the underlying multiparameter system

a note on the approximation of two sided rayleigh quotient to eigenvalues abstract we consider the two sided rayleigh quotient introduced by ostrowski as convergence acceleration tool for eigenvalue solvers an interesting property of the two sided rayleigh quotient is established its approximation to eigenvalues depends on the eigenvalue condition number this fact is valid not only for matrix but also for matrix polynomial multiparameter system and periodic matrix pairs

classification of isometries of spaces of constant curvature and invariant subspaces abstract we study the varieties of invariant totally geodesic submanifolds of isometries of the spherical euclidean and hyperbolic spaces in each finite dimension we show that the dimensions of the connected components of these varieties determine the orbit type or the z class of the isometry for this purpose we introduce the segre symbol of an isometry a discrete invariant encoding the structure of its normal form which parametrizes z classes we then provide a description of the isomorphism type of the varieties of invariant subspaces in terms of the segre symbol

the topology of the set of conditioned invariant subspaces we consider the topology of the set of conditioned invariant subspaces of an observable pair c a of a fixed dimension by fixing the observability indices of the restricted system a stratification by finitely many smooth manifolds is obtained termed brunovsky strata it is shown that each brunovsky stratum is homotopy equivalent to a generalized flag manifold from this description an effective formula for the betti numbers of the brunovsky strata can be derived

on varieties of commuting triples abstract the problem of approximating triples of commuting n n matrices by triples of generic matrices is equivalent to that whether the variety c 3 n of triples of commuting matrices is irreducible it is known that the variety is irreducible for n 7 and reducible for n 30 using simultaneous commutative perturbations of pairs of matrices in the centralizer of the third matrix we prove that c 3 8 is irreducible

pairs of commuting nilpotent matrices and hilbert function abstract let k be an infinite field there has been recent study of the family h n k of pairs of commuting nilpotent n n matrices relating this family to the fibre h n of the punctual hilbert scheme a n hilb n a 2 over the point np of the symmetric product sym n a 2 where p is a point of the affine plane a 2 v baranovsky the variety of pairs of commuting nilpotent matrices is irreducible transform groups 6 1 2001 3 8 r basili on the irreducibility of commuting varieties of nilpotent matrices j algebra 268 1 2003 56 80 a premet nilpotent commuting varieties of reductive lie algebras invent math 154 3 2003 653 683 in this study a pair of commuting nilpotent matrices a b is related to an artinian algebra k a b there has also been substantial study of the stratification of the local punctual hilbert scheme h n by the hilbert function as j briancon description de hilb n c x y invent math 41 1 1977 45 89 and others however these studies have been hitherto separate we first determine the stable partitions i e those for which p itself is the partition q p of a generic nilpotent element of the centralizer of the jordan nilpotent matrix j p we then explore the relation between h n k and its stratification by the hilbert function of k a b suppose that dim k k a b n and that k is algebraically closed of characteristic 0 or large enough p we show that a generic element of the pencil a b k has jordan partition the maximum partition p h whose diagonal lengths are the hilbert function h of k a b

singular graph and extension of jordan chains of an m matrix abstract let s be a singular graph we study conditions on the vertices of s to extend jordan chains for the generalized nullspace of each minus m matrix a with singular graph s the conditions are the interlevel the strong interlevel and the strong predecessor properties

finite dimensional multiparameter spectral theory the nonderogatory case abstract a completeness theorem for nonderogatory eigenvalues of multiparameter systems is proved in a finite dimensional setting also a basis for the second root subspace of a simple eigenvalue is given weakly elliptic multiparameter systems are introduced it is shown that simple real eigenvalues of such systems are nonderogatory

two generated commutative matrix subalgebras abstract let f be a field and let mn f be the algebra of n n matrices over f let a b mn f with ab ba and let a be the algebra generated by a b over f a theorem of gerstenhaber ann math 73 324 348 1991 states that the dimension of a is at most n gerstenhaber s proof uses the methods of algebraic geometry in this paper we obtain a purely matrix theoretic proof of the result we also examine the case when equality occurs the case where f is algebraically closed and a is indecomposable under similarity holds the key to the general situation and in the indecomposable case we obtain a cayley hamilton like theorem expressing bk as a polynomial in i b bk 1 with coefficients in f a where k denotes the number of blocks in the jordan form of a if all jordan blocks of a have the same size we obtain a nonderogatory like condition on b which is equivalent to dim f a n we also show that in this case dim f a n is equivalent to the maximality of a as a commutative subalgebra of mn f

nilpotent matrices having a given jordan type as maximum commuting nilpotent orbit abstract the jordan type of a nilpotent matrix is the partition giving the sizes of its jordan blocks we study pairs of partitions p q where q q p is the jordan type of a generic nilpotent matrix a commuting with a nilpotent matrix b of jordan type p t kosir and p oblak have shown that q has parts that differ pairwise by at least two such partitions which are also known as super distinct or rogers ramanujan are exactly those that are stable or self large in the sense that q q q in 2012 p oblak formulated a conjecture concerning the cardinality of q 1 q when q has two parts and proved some special cases r zhao refined this to posit that the partitions in q 1 q for q u u r with u r 1 could be arranged in an r 1 u r table t q where the entry in the k th row and l th column has k l parts we prove this table theorem and then generalize the statement to propose a box conjecture for the set of partitions q 1 q for an arbitrary partition q whose parts differ pairwise by at least two

differentiable structure of the set of controllable a b t invariant subspaces abstract given a b t hom c n c n m observable we prove that the set of a b t invariant subspaces having a fixed brunovsky kronecker structure is a connected manifold and we compute its dimension also we include some applications of these results

a numerical algorithm for computing a basis for the root subspace at a nonderogatory eigenvalue of a multiparameter system abstract a numerical algorithm for the calculation of a basis for the root subspace at a nonderogatory eigenvalue of a finite dimensional multiparameter system is given it is based on the algorithm proposed by kosir 1 the perturbation theory for the algorithm is studied and a new stopping criterion is proposed at the end two numerical examples are given

on the structure of commutative matrices ii a finite set a of n n nilpotent commutative matrices that have one dimensional joint kernel is considered the theorem due to suprunenko and tyshkevich that the algebra a generated by a and the identity matrix has dimension equal to n is proved a canonical basis for a is given and related structure constants are discussed

strong linear programming relaxations for the orienteering problem abstract consider a set of nodes each with an associated profit and a set of arcs each with an associated length the objective of the orienteering problem is to find the path beginning at a specified origin and terminating at a specified destination that maximizes total profit subject to 1 a constraint on the length of the path and 2 the condition that no node is visited more than once the problem may be formulated as a 0 1 integer program and it has been shown to be np hard prior research has focused on heuristics that obtain lower bounds on the optimal objective function value some recent research has proposed a branch and bound algorithm that solves a linear programming lp relaxation of the 0 1 model at each node and obtains an optimal orienteering path our research is concerned with tightening the lp relaxation by adding constraints and valid inequalities we propose a procedure to obtain upper bounds by solving three successive linear programs we test our procedure on datasets in existing literature and demonstrate that the average deviation between our upper bounds and the best known lower bounds feasible solutions is less than five percent the quality of upper bound obtained is significantly enhanced over the bound obtained from solving the basic lp relaxation

on stability of invariant subspaces of commuting matrices we study the stability of joint invariant subspaces of a finite set of commuting matrices we generalize some of the results of gohberg lancaster and rodman for the single matrix case for sets of two or more commuting matrices we exhibit some phenomena different from the single matrix case we show that each root subspace is a stable invariant subspace that each invariant subspace of a root subspace of a nonderogatory eigenvalue is stable and that even in the derogatory case the eigenspace is stable if it is one dimensional we prove that a pair of commuting matrices has only finitely many stable invariant subspaces at the end we discuss the stability of invariant subspaces of an algebraic multiparameter eigenvalue problem

critical points and intermediate phases on wedges of zd the authors examine the phase structure of self avoiding walks the ising magnet and bond percolation defined on subsets of zd d or 2 which have the geometry of wedges they prove that if the cross sectional area of the wedge diverges with its width then the high temperature critical point of any of these models defined on the wedge coincides with that of the corresponding model on the full lattice for the ising magnet and bond percolation they show that there is a non trivial low temperature critical point if the cross sectional area of the wedge diverges logarithmically with its width moreover for bond percolation they show that the low temperature critical point of a logarithmic wedge may be made arbitrarily close to that of the full lattice by taking the coefficient of the logarithm large enough corollaries to these theorems include the existence of an intermediate phase for the ising magnet and bond percolation on logarithmic wedges and the existence of a first order transition for percolation models on a subclass of these wedges

commuting pairs in the centralizers of 2 regular matrices inmn k kan algebraically closed field we call a matrixl regular if each eigenspace is at mostl dimensional we prove that the variety of commuting pairs in the centralizer of a 2 regular matrix is the direct product of various affine spaces and various determinantal varieties zl mobtained from matrices over truncated polynomial rings we prove that these varieties zl mare irreducible and apply this to the case of thek algebra generated by three commuting matrices we show that if one of the three matrices is 2 regular then the algebra has dimension at mostn we also show that such an algebra is always contained in a commutative subalgebra ofmn k of dimension exactlyn

an effective pso inspired algorithm for the team orienteering problem abstract the team orienteering problem top is a particular vehicle routing problem in which the aim is to maximize the profit gained from visiting customers without exceeding a travel cost time limit this paper proposes a new and fast evaluation process for top based on an interval graph model and a particle swarm optimization inspired algorithm psoia to solve the problem experiments conducted on the standard benchmark of top clearly show that our algorithm outperforms the existing solving methods psoia reached a relative error of 0 0005 whereas the best known relative error in the literature is 0 0394 our algorithm detects all but one of the best known solutions moreover a strict improvement was found for one instance of the benchmark and a new set of larger instances was introduced

a numerical method for the solution of two parameter eigenvalue problems abstract we consider a two parameter weakly coupled eigenvalue problem for matrix equations under known assumptions 4 such a problem can be reduced to a system of one parameter problems in a tensor product space after which it can be treated by known numerical methods the method is illustrated by a numerical example of the simple two parameter problem and the example of binding and browne 5 it turns out that the method is especially suitable for problems of smaller orders

the number of monodromy representations of abelian varieties of low p rank abstract let a g be an abelian variety of dimension g and p rank 1 over an algebraically closed field of characteristic p 0 we compute the number of homomorphisms from 1 et a g a to g l n f q where q is any power of p we show that for fixed g and n the number of such representations is polynomial in q and give an explicit formula for this polynomial we show that the set of such homomorphisms forms a constructible set and use the geometry of this space to deduce information about the coefficients and degree of the polynomial in the last section we prove a divisibility theorem about the number of homomorphisms from certain semidirect products of profinite groups into finite groups as a corollary we deduce that when 0 hom 1 et a g a g l n f q g l n f q is a laurent polynomial in q

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

stratification and bundle structure of the set of conditioned invariant subspaces in the general case abstract we extend some known results on the smooth stratification of the set of conditioned invariant subspaces to a general pair c a c p n n without any assumption on the observability more precisely we prove that the set of c a conditioned invariant subspaces having a fixed brunovsky kronecker structure is a submanifold of the corresponding grassmann manifold with a fiber bundle structure relating the observable and nonobservable part and we then compute its dimension we also prove that the set of all c a conditioned invariant subspaces having a fixed dimension is connected provided that the nonobservable part of c a has at most one eigenvalue this condition is in general necessary

enumerating submodules invariant under an endomorphism we study zeta functions enumerating submodules invariant under a given endomorphism of a finitely generated module over the ring of s integers of a number field in particular we compute explicit formulae involving dedekind zeta functions and establish meromorphic continuation of these zeta functions to the complex plane as an application we show that ideal zeta functions associated with nilpotent lie algebras of maximal class have abscissa of convergence 2

extensions of jordan bases for invariant subspaces of a matrix a characterization is obtained for the matrices a with the property that every some jordan basis of every a invariant subspace can be extended to a jordan basis of a these results are based on a criterion for a jordan basis of an invariant subspace to be extendable to a jordan basis of the whole space the criterion involves two concepts the constancy property and the depth property

centralizers in endomorphism rings we prove that the centralizer cen endr m of a nilpotent endomorphism of a finitely generated semisimple left r module mr over an arbitrary ring r is the homomorphic image of the opposite of a certain z r subalgebra of the full m m matrix algebra mm r z where m is the dimension of ker if r is a local ring then we give a complete characterization of cen and of the containment cen cen where is a not necessarily nilpotent element of endr m for a k linear map a of a finite dimensional vector space over a field k we determine the pi degree of cen a

the team orienteering problem with time windows an lp based granular variable neighborhood search abstract the team orienteering problem top is a known np hard problem that typically arises in vehicle routing and production scheduling contexts in this paper we introduce a new solution method to solve the top with hard time window constraints toptw we propose a variable neighborhood search vns procedure based on the idea of exploring most of the time granular instead of complete neighborhoods in order to improve the algorithm s efficiency without loosing effectiveness the method provides a general way to deal with granularity for those routing problems based on profits and complicated by time constraints extensive computational results are reported on standard benchmark instances performance of the proposed algorithm is compared to optimal solution values when available or to best known solution values obtained by state of the art algorithms the method comes out to be on average quite effective allowing to improve the best know values for 25 test instances

the tropical commuting variety we study tropical commuting matrices from two viewpoints linear algebra and algebraic geometry in classical linear algebra there exist various criteria to test whether two square matrices commut

versal deformations of invariant subspaces we describe a miniversal deformation of invariant subspaces with regard to a fixed endomorphism by means of a technique which can be applied to obtain explicitly miniversal deformations in a general orbit space in addition we present an application to the problem of classifying invariant subspaces

simultaneous versal deformations of endomorphisms and invariant subspaces abstract we study the set m of pairs f v defined by an endomorphism f of f n and a d dimensional f invariant subspace v it is shown that this set is a smooth manifold that defines a vector bundle on the grassmann manifold we apply this study to derive conditions for the lipschitz stability of invariant subspaces and determine versal deformations of the elements of m with respect to a natural equivalence relation introduced on it

new classes of examples satisfying the three matrix analog of gerstenhaber s theorem abstract in 1961 gerstenhaber proved the following theorem if k is a field and x and y are commuting d d matrices with entries in k then the unital k algebra generated by these matrices has dimension at most d the analog of this statement for four or more commuting matrices is false the three matrix version remains open we use commutative algebraic techniques to prove that the three matrix analog of gerstenhaber s theorem is true for some new classes of examples in particular we translate this three commuting matrix statement into an equivalent statement about certain maps between modules and prove that this commutative algebraic reformulation is true in special cases we end with ideas for an inductive approach intended to handle the three matrix analog of gerstenhaber s theorem more generally

a tabu search heuristic for the undirected selective travelling salesman problem the undirected selective travelling salesman problem stsp is defined on a graph g v e with positive profits associated with vertices and distances associated with edges the stsp consists of determining a maximal profit hamiltonian cycle over a subset of v whose length does not exceed a preset limit l we describe a tabu search ts heuristic for this problem the algorithm iteratively inserts clusters of vertices in the current tour or removes a chain of vertices tests performed on randomly generated instances with up to 300 vertices show that the algorithm consistently yields near optimal solutions

a cellular decomposition of the manifold of observable conditioned invariant subspaces abstract given an observable system defined by a pair of matrices c a we obtain a cellular decomposition of the manifold of c a conditioned invariant subspaces with fixed observability indices of the restricted system and we apply this decomposition to compute their homology groups

on a conjecture of hammersley and whittington concerning bond percolation on subsets of the simple cubic lattice the author verifies the truth of a conjecture of hammersley and whittington 1985 concerning bond percolation on certain subsets of the simple cubic lattice z3 let f and g be non decreasing non negative functions on 0 infinity and let z3 f g denote the f g wedge of z3 being the set of points x y z such that 0 or 0 the author shows that the condition 1 f x 1 g x to infinity as x to infinity is sufficient for the critical probability of the bond percolation process on z3 f g to be less than or equal to 1 2

an algebraic approach to multiparameter spectral theory root subspaces for multiprameter eigenvalue problems are described using coalgebraic techniques an algorithm is given to construct bases for the root subspaces

failure of the power laws on some subgraphs of the z2 lattice the author considers standard bernoulli bond percolation on a subgraph g of zd denote by theta p g and chi p g the percolation probability and the mean cluster size and pc g sup p theta p g 0 it is widely believed that the power laws hold as follows theta p g theta pc g g approximately p pc g beta p pc g and chi p g approximately p c g p gamma p pc g for some positive constants beta and gamma which only depend on d however the author finds this conjecture is not true for some non periodic subgraphs of z2

backward error condition numbers and pseudospectra for the multiparameter eigenvalue problem we define and evaluate the normwise backward error and condition numbers for the multiparameter eigenvalue problem mep the pseudospectrum for the mep is defined and characterized we show that the distance from a right definite mep to the closest non right definite mep is related to the smallest unbounded pseudospectrum some numerical results are given

characteristic and hyperinvariant subspaces over the field gf 2 let f be an endomorphism of a vector space v over a field k an f invariant subspace x v is called hyperinvariant respectively characteristic if x is invariant under all endomorphisms respectively automorphisms that commute with f if k 2 then all characteristic subspaces are hyperinvariant if k 2 then there are endomorphisms f with invariant subspaces that are characteristic but not hyperinvariant in this paper we give a new proof of a theorem of shoda which provides a necessary and sufficient condition for the existence of characteristic non hyperinvariant subspaces

mean field critical behaviour for percolation in high dimensions the triangle condition for percolation states that sum limits x y tau 0 x tau 0 y cdot tau y 0 is finite at the critical point where x y is the probability that the sitesx andy are connected we use an expansion related to the lace expansion for a self avoiding walk to prove that the triangle condition is satisfied in two situations i for nearest neighbour independent bond percolation on thed dimensional hypercubic lattice ifd is sufficiently large and ii in more than six dimensions for a class of spread out models of independent bond percolation which are believed to be in the same universality class as the nearest neighbour model the class of models in ii includes the case where the bond occupation probability is constant for bonds of length less than some large number and is zero otherwise in the course of the proof an infrared bound is obtained the triangle condition is known to imply that various critical exponents take their mean field bethe lattice values gamma beta 1 delta delta t 2 t underset raise0 3em hbox smash scriptscriptstyle geqslant 2 and that the percolation density is continuous at the critical point we also prove thatv2 in i and ii wherev2 is the critical exponent for the correlation length

perturbed marked reduced forms of invariant subspaces abstract the classification of invariant subspaces is an open problem related to other important ones like the carlson problem here we obtain a reduced form of these invariant subspaces as a new tool to tackle these problems in particular it allows us to prove quite easily partial results already known the key point is assigning to each invariant subspace a marked one its marked type in order to partition the set of invariant subspaces in a finite number of subsets the marked classes each one containing only one marked subspace next we parametrize minimally each marked class by means of the so called pm reduced families so that representatives of an invariant subspace its pm reduced forms appear in just one of these families

a branch and bound algorithm for the traveling purchaser problem an important generalization of the traveling salesman problem called the traveling purchaser problem is considered a branch and bound algorithm which solves a related simple plant location problem for calculating the bounds is proposed for this problem computational experiments with this algorithm provide the evidence that moderate size problems of upto 25 cities and 100 commodities can be solved in reasonable computation time the problem is shown to have applications in the areas of scheduling warehousing and routing

two parameter eigenvalue problems for matrices abstract the linked pair of two parameter eigenvalue problems t m 1 v m 1 2 v m 2 x m 0 x m 0 x m h m is considered for hermitian matrices t m v m1 v m2 acting on on the finite dimensional hilbert spaces h m m 1 2 we dispense with customary definiteness conditions although for m 1 2 we assume the existence of a linear combination of v m1 v m2 which is definite an embedding technique is used to study eigenvalues r 2 of specified variational index the transition between real and nonreal eigenvalues and algebraic multiplicity