a probabilistic comparison of split and type 1 triangle cuts for two row mixed integer programs we provide a probabilistic comparison of split and type 1 triangle cuts for mixed integer programs with two rows and two integer variables in terms of cut coefficients and volume cutoff under a specific probabilistic model of the problem parameters we show that for the above measure the probability that a split cut is better than a type 1 triangle cut is higher than the probability that a type 1 triangle cut is better than a split cut

a new class of cutting planes for the symmetric travelling salesman problem a comprehensive class of cutting planes for the symmetric travelling salesman problem tsp is proposed which contains the known comb inequalities the path inequalities and the 3 star constraints as special cases its relation to the clique tree inequalities is discussed the cutting planes are shown to be valid for a relaxed version of the tsp the travelling salesman problem on a road network and under certain conditions to define facets of the polyhedron associated with this problem

the triangle closure is a polyhedron recently cutting planes derived from maximal lattice free convex sets have been studied intensively by the integer programming community an important question in this research area has been to decide whether the closures associated with certain families of lattice free sets are polyhedra for a long time the only result known was the celebrated theorem of cook kannan and schrijver who showed that the split closure is a polyhedron although some fairly general results were obtained by andersen et al math oper res 35 1 233 256 2010 and averkov discret optimiz 9 4 209 215 2012 some basic questions have remained unresolved for example maximal lattice free triangles are the natural family to study beyond the family of splits and it has been a standing open problem to decide whether the triangle closure is a polyhedron in this paper we show that when the number of integer variables m 2 the triangle closure is indeed a polyhedron and its number of facets can be bounded by a polynomial in the size of the input data the techniques of this proof are also used to give a refinement of necessary conditions for valid inequalities being facet defining due to cornuejols and margot math program 120 429 456 2009 and obtain polynomial complexity results about the mixed integer hull

measurements of isothermal complex moduli of viscoelastic materials over a large range of frequencies a method of determination of the young s and shear complex moduli of viscoelastic materials using a transmissibility function is presented the experimental set up shows that it is possible to evaluate both moduli on the same samples we will show that with appropriate inverse methods isothermal complex moduli are obtained over a large range of frequencies

the symmetric traveling salesman polytope and its graphical relaxation composition of valid inequalities the graphical relaxation of the traveling salesman problem is the relaxation obtained by requiring that the salesman visit each city at least once instead of exactly once this relaxation has already led to a better understanding of the traveling salesman polytope in cornuejols fonlupt and naddef 1985 we show here how one can compose facet inducing inequalities for the graphical traveling salesman polyhedron and obtain other facet inducing inequalities this leads to new valid inequalities for the symmetric traveling salesman polytope this paper is the first of a series of three papers on the symmetric traveling salesman polytope the next one studies the strong relationship between that polytope and its graphical relaxation and the last one applies all the theoretical developments of the two first papers to prove some new facet inducing results

the traveling salesman problem in graphs with some excluded minors given a graph and a length function defined on its edge set the traveling salesman problem can be described as the problem of finding a family of edges an edge may be chosen several times which forms a spanning eulerian subgraph of minimum length in this paper we characterize those graphs for which the convex hull of all solutions is given by the nonnegativity constraints and the classical cut constraints this characterization is given in terms of excluded minors a constructive characterization is also given which uses a small number of basic graphs

small travelling salesman polytopes we introduce a new class of valid inequalities for the polytope of the symmetric travelling salesman problem we also gave complete characterizations of the polytope for 6 and 7 cities for the latter case the new inequalities are needed these results are related to work of r z norman in the 1950 s

the binested inequalities for the symmetric traveling salesman polytope in this paper we define a family of valid inequalities for the symmetric travelling salesman polytope which are defined on two nested sets of vertices of the graph these inequalities generalize the comb inequalities of chvital grotschel and padberg the clique tree inequalities of grotschel and pulleyblank the path inequalities of cornuejols fonlupt and naddef and the hyperstar inequalities of fleischman this is the largest known family of valid inequalities known so far facet inducing inequalities for the symmetric travelling salesman polytope contained in this class and in no other one are given proving that this is a proper generalization of the previously mentioned families

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

orthogonal polynomials from hermitian matrices a unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions it can be considered as a matrix version of exactly solvable schrodinger equations the hermitian matrices factorizable hamiltonians are real symmetric tridiagonal jacobi matrices corresponding to second order difference equations by solving the eigenvalue problem in two different ways the duality relation of the eigenpolynomials and their dual polynomials is explicitly established through the techniques of exact heisenberg operator solution and shape invariance various quantities the two types of eigenvalues the eigenvalues and the sinusoidal coordinates the coefficients of the three term recurrence the normalization measures and the normalisation constants etc are determined explicitly

on the relative strength of families of intersection cuts arising from pairs of tableau constraints in mixed integer programs we compare the relative strength of valid inequalities for the integer hull of the feasible region of mixed integer linear programs with two equality constraints two unrestricted integer variables and any number of nonnegative continuous variables in particular we prove that the closure of type 2 triangle resp type 3 triangle quadrilateral inequalities are all within a factor of 1 5 1 5 of the integer hull and provide examples showing that the approximation factor is not less than 1 125 1 125 there is no fixed approximation ratio for split or type 1 triangle inequalities however

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 traveling salesman problem on a graph and some related integer polyhedra given a graphg n e and a length functionl e the graphical traveling salesman problem is that of finding a minimum length cycle goingat least once through each node ofg this formulation has advantages over the traditional formulation where each node must be visited exactly once we give some facet inducing inequalities of the convex hull of the solutions to that problem in particular the so called comb inequalities of grotschel and padberg are generalized some related integer polyhedra are also investigated finally an efficient algorithm is given wheng is a series parallel graph

quasi exactly solvable systems and orthogonal polynomials this paper shows that there is a correspondence between quasi exactly solvable models in quantum mechanics and sets of orthogonal polynomials pn the quantum mechanical wave function is the generating function for the pn e which are polynomials in the energy e the condition of quasi exact solvability is reflected in the vanishing of the norm of all polynomials whose index n exceeds a critical value j the zeros of the critical polynomial pj e are the quasi exact energy eigenvalues of the system

on uniqueness of heine stieltjes polynomials for second order finite difference equations a second order finite difference equation has two linearly independent solutions it is shown here that like in the continuous case at most one of the two can be a polynomial solution the uniqueness in the classical continuous heine stieltjes theory is shown to hold under broader hypotheses than usually presented a difference between regularity condition and uniqueness is emphasized consistency in our uniqueness results is also checked against one of the shapiro problems an intrinsic relation between the heine stieltjes problem and the discrete bethe ansatz equations allows one to immediately extend the uniqueness result from the former to the latter the results have implications for nondegeneracy of polynomial solutions of physical models

non parametric identification of viscoelastic materials from wave propagation experiments we investigate how the frequency dependent wave propagation coefficient and complex modulus for a linearly viscoelastic material can be estimated from wave propagation experiments the strains at different sections of a bar specimen are measured as functions of time the time series are transformed into the frequency domain where a non parametric identification is made a thorough analysis of the quality of the non parametric estimate is made in which approximate expressions for the covariance matrices of the wave propagation coefficient and the complex modulus are derived the validity of these expressions are confirmed by numerical studies and real experiments

integrable systems on the lattice and orthogonal polynomials of discrete variable some particular examples of classical and quantum systems on the lattice are solved with the help of orthogonal polynomials and its connection to continuous models are explored

an algorithm for the separation of two row cuts we consider the question of finding deep cuts from a model with two rows of the type p i x s in mathbb z 2 times mathbb r n x f rs to do that we show how to reduce the complexity of setting up the polar of mathop mathrm conv p i from a quadratic number of integer hull computations to a linear number of integer hull computations furthermore we present an algorithm that avoids computing all integer hulls a polynomial running time is not guaranteed but computational results show that the algorithm runs quickly in practice

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

special nonuniform lattice snul orthogonal polynomials on discrete dense sets of points difference calculus compatible with polynomials i e such that the divided difference operator of first order applied to any polynomial must yield a polynomial of lower degree can only be made on special lattices well known in contemporary q calculus orthogonal polynomials satisfying difference relations on such lattices are presented in particular lattices which are dense on intervals q 1 are considered

minimum weight two connected spanning networks we consider the problem of constructing a minimum weight two connected network spanning all the points in a setv we assume a symmetric nonnegative distance functiond defined onv v which satisfies the triangle inequality we obtain a structural characterization of optimal solutions specifically there exists an optimal two connected solution whose vertices all have degree 2 or 3 and such that the removal of any edge or pair of edges leaves a bridge in the resulting connected components these are the strongest possible conditions on the structure of an optimal solution since we also show thatany two connected graph satisfying these conditions is theunique optimal solution for a particular choice of canonical distances satisfying the triangle inequality we use these properties to show that the weight of an optimal traveling salesman cycle is at most 4 3 times the weight of an optimal two connected solution examples are provided which approach this bound arbitrarily closely in addition we obtain similar results for the variation of this problem where the network need only span a prespecified subset of the points

worst case comparison of valid inequalities for the tsp we consider most of the known classes of valid inequalities for the graphical travelling salesman polyhedron and compute the worst case improvement resulting from their addition to the subtour polyhedron for example we show that the comb inequalities cannot improve the subtour bound by a factor greater than 10 9 the corresponding factor for the class of clique tree inequalities is 8 7 while it is 4 3 for the path configuration inequalities

hill s determinant approach to single mode spin boson model an efficient and stable numerical scheme based on hill s determinant is proposed for computing the whole spectrum of the single mode spin boson model its shown that for the low lying levels a simple version of hill s determinant method works well while for high levels a normalization of the determinant is necessary and this normalization results in an entire function whose zeros correspond to the energy levels it is very convenient to use this entire function to compute all the energy levels numerically some high levels of the model are computed and compared with irish s general rotating wave approximation a scaling law in the large photon number limit is numerically tested and several statistical properties of the spectrum are investigated

solution of the schrodinger equation by a spectral method abstract a new computational method for determining the eigenvalues and eigenfunctions of the schrodinger equation is described conventional methods for solving this problem rely on diagonalization of a hamiltonian matrix or iterative numerical solutions of a time independent wave equation the new method in contrast is based on the spectral properties of solutions to the time dependent schrodinger equation the method requires the computation of a correlation function r 0 r t from a numerical solution r t fourier analysis of this correlation function reveals a set of resonant peaks that correspond to the stationary states of the system analysis of the location of these peaks reveals the eigenvalues with high accuracy additional fourier transforms of r t with respect to time generate the eigenfunctions the effectiveness of the method is demonstrated for a one dimensional asymmetric double well potential and for the two dimensional henon heiles potential

identification of complex moduli and poisson s ratio from measured strains on an impacted bar identification of complex moduli and poisson s ratio from measured strains on an impacted bar

on the facets of mixed integer programs with two integer variables and two constraints in this paper we consider an infinite relaxation of the mixed integer linear program with two integer variables nonnegative continuous variables and two equality constraints and we give a complete characterization of its facets we also derive an analogous characterization of the facets of the underlying finite integer program

minimal valid inequalities for integer constraints in this paper we consider a semi infinite relaxation of mixed integer linear programs we show that minimal valid inequalities for this relaxation correspond to maximal lattice free convex sets and that they arise from nonnegative piecewise linear positively homogeneous convex functions

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

chva tal closures for mixed integer programming problems chvatal introduced the idea of viewing cutting planes as a system for proving that every integral solution of a given set of linear inequalities satisfies another given linear inequality this viewpoint has proven to be very useful in many studies of combinatorial and integer programming problems the basic ingredient in these cutting plane proofs is that for a polyhedronp and integral vectorw if max wx x p wx integer t thenwx t is valid for all integral vectors inp we consider the variant of this step where the requirement thatwx be integer may be replaced by the requirement that bar wx be integer for some other integral vector bar w the cutting plane proofs thus obtained may be seen either as an abstraction of gomory s mixed integer cutting plane technique or as a proof version of a simple class of the disjunctive cutting planes studied by balas and jeroslow our main result is that for a given polyhedronp the set of vectors that satisfy every cutting plane forp with respect to a specified subset of integer variables is again a polyhedron this allows us to obtain a finite recursive procedure for generating the mixed integer hull of a polyhedron analogous to the process of repeatedly taking chvatal closures in the integer programming case these results are illustrated with a number of examples from combinatorial optimization our work can be seen as a continuation of that of nemhauser and wolsey on mixed integer cutting planes

a distributional study of discrete classical orthogonal polynomials abstract for the sequences of discrete classical orthogonal polynomials charlier meixner hahn we can find a functional u which satisfies the difference distributional equation u u where and are polynomials of degrees 2 and 1 respectively from this it follows that these polynomials are solutions of a second order difference equation also they can be represented by a rodrigues type formula the sequence of difference polynomials derived from them constitutes an orthogonal polynomial sequence their weight functions satisfy a pearson type difference equation a structure relation pn 1 anpn 2 bnpn 1 cnpn also holds

optimizing over the subtour polytope of the travelling salesman problem a commonly studied relaxation of the travelling salesman problem is obtained by adding subtour elimination constraints to the constraints of a 2 factor problem and removing the integrality requirement we investigate the problem of solving this relaxation for a special type of objective function we also discuss some ways in which this relates to the concept of rank introduced by chvatal

a simple method for the determination of the complex modulus of resilient materials using a longitudinally vibrating three layer specimen abstract this paper proposes a new hybrid analytical experimental methodology for the determination of the complex modulus of resilient materials as for e g composition cork like materials it is based on a three layer specimen test and on a simple equation from the analytical model of the specimen the analytical formulation solves the problem based on the elementary theory of longitudinal vibrations and without taking the transversal motions into account this methodology is developed to avoid geometric constraints and boundary condition difficulties found on other tests as well as to minimize the computational time and complexity of the coding the experimental setup and the deduction of the simple equation are described metallic layers are used at the extremities of the specimen for which the young modulus and mass density are known while its loss factor may be neglected for the intermediate resilient layer which is assumed to present a linear viscoelastic behavior the mass density is known thus the only unknowns are the storage modulus and loss factor of that intermediate layer the modal loss factor and the first resonant frequency of the specimen are identified from the experimental receptance frequency response function frf curve using the rational fraction polynomial method rfp by using the first resonance frequency obtained experimentally the analytical response becomes only a function of the storage modulus resulting in a simple scalar nonlinear equation then a numerical routine is used to determine the storage modulus by calculating the first zero of the nonlinear equation to validate the proposed methodology the frequency response assurance criterion frac and the frequency amplitude assurance criterion faac are used for the five composition cork materials tested the complex modulus shown low frequency dependence in the 40 hz to 1600 hz range correlation values between the analytical and the experimental curves are greater than 95 thus validating the obtained results and the proposed methodology

quantum models with spectrum generated by the flows of polynomial zeros a class of purely bosonic models is characterized having the following properties in a hilbert space of analytic functions i wave function is the generating function for orthogonal polynomials of a discrete energy variable ii any hamiltonian has nondegenerate purely point spectrum that corresponds to infinite discrete support of measure in the orthogonality relation of the polynomials iii the support is determined exclusively by the points of discontinuity of iv the spectrum of can be numerically determined as fixed points of monotonic flows of the zeros of orthogonal polynomials v one can compute practically an unlimited number of energy levels e g in double precision if a model of is exactly solvable its spectrum can only assume one of four qualitatively different types the results are applied to spin boson quantum models that are at least partially diagonalizable and have at least single one dimensional irreducible component in the spin subspace examples include the rabi model and its various generalizations

quasi exactly solvable problems and sl 2 algebra recently discovered quasi exactly solvable problems of quantum mechanics are shown to be related to the existence of the finite dimensional representations of the groupsl 2 q whereq r c it is proven that the bilinear formh a j j b j stand for the generators allows one to generate a set of quasi exactly solvable problems of different types including those that are already known we get in particular problems in which the spectral riemannian surface containing an infinite number of sheets is split off one or two finite sheet pieces in the general case the transitionh h d2 dx2 v x is realized with the aim of the elliptic functions all known exactly solvable quantum problems with known spectrum and factorized riemannian surface can be obtained in this approach

the effects of kinematic model approximations on natural frequencies and modal damping of laminated composite plates abstract the present work deals with the effects of kinematics on the natural frequencies and modal damping of laminated composite plates three theories are considered the classical laminated plate theory clpt the first order shear deformation theory fsdt and the third order shear deformation theory tsdt the displacement field corresponding to a simply supported square laminated composite plate is introduced in the energy equation the governing equations are then formulated for the laminated composite plates using the hamiltonian principle equations of motion are established and then the harmonic free vibrations are studied the complex frequencies are obtained from the characteristic equation which gives the natural frequencies and the modal damping for each m n mode the effects of stacking sequence rotary inertia and thickness to side ratio h a on the natural frequencies and modal damping ratios are analyzed the results lead to conclusions about the applicability of each of the plate theories clpt fsdt and hsdt

brief paper structure testing of wave propagation models used in identification of viscoelastic materials in this paper matrix perturbation theory is applied to test the structure of wave propagation models used to identify the complex modulus of a viscoelastic material the analysis is based on a data matrix containing the measured data from a number of independent experiments the key observation is that if the structure of the model is correct then the unperturbed noise free matrix is rank deficient of a known order this means that the noise corrupted matrix will have a known number of singular values that diverge from zero only due to the measurement noise a test quantity based on the distribution of these perturbed singular values is used assuming that the signal to noise ratio is large and that measurement noise is white and gaussian distributed if the magnitudes of the smallest singular values are too large to be explained by the measurement noise only the model is rejected data from two different types of experimental setups is explored longitudinal wave propagation in a slender bar and the non equilibrium shpb procedure it is shown that the model can be accepted in the first case but should be rejected in the second

on the polyhedrality of cross and quadrilateral closures split cuts form a well known class of valid inequalities for mixed integer programming problems cook et al math program 47 155 174 1990 showed that the split closure of a rational polyhedron p is again a polyhedron in this paper we extend this result from a single rational polyhedron to the union of a finite number of rational polyhedra we then use this result to prove that cross cuts yield closures that are rational polyhedra cross cuts are a generalization of split cuts introduced by dash et al math program 135 221 254 2012 finally we show that the quadrilateral closure of the two row continuous group relaxation is a polyhedron answering an open question in basu et al math program 126 281 314 2011

theoretical challenges towards cutting plane selection while many classes of cutting planes are at the disposal of integer programming solvers our scientific understanding is far from complete with regards to cutting plane selection i e the task of selecting a portfolio of cutting planes to be added to the lp relaxation at a given node of the branch and bound tree in this paper we review the different classes of cutting planes available known theoretical results about their relative strength important issues pertaining to cut selection and discuss some possible new directions to be pursued in order to accomplish cutting plane selection in a more principled manner finally we review some lines of work that we undertook to provide a preliminary theoretical underpinning for some of the issues related to cut selection

generalised fulton gouterman transformation for systems of abelian symmetry a two level system which is coupled to a multitude of oscillators such that the total system formally or geometrically is governed by a mirror symmetry can be exactly diagonalised with respect to the two level subsystem fg transformation this has been thoroughly exploited for the two site exciton localisation problem as well as for two site quantum transport a generalisation to the n level system is given provided the n levels establish a regular representation of an abelian group and if the latter symmetry also governs the multi oscillatory subsystem implications for the quantum transport problem are discussed