new q matrices and their functional equations for the eight vertex model at elliptic roots of unity the q matrix invented by baxter in 1972 to solve the eight vertex model at roots of unity exists for all values of n the number of sites in the chain but only for a subset of roots of unity we show in this paper that a new q matrix which has recently been introduced and is non zero only for n even exists for all roots of unity in addition we consider the relations between all of the known q matrices of the eight vertex model and conjecture functional equations for them

long time evolution of wavefronts in random media this work studies the evolution of the front of an impulsive plane wave in a layered random medium a general second order hyperbolic equation with an inhomogeneous plane wave solution in a frame moving with random propagation speed is considered in this frame it takes the form of a coupled system of ordinary differential equations along characteristics for the propagation modes the author shows that in the asymptotic limit in which the amplitude of random inhomogeneities decreases but the distance at which the pulse is observed increases the leading part of the pulse the wavefront stabilizes to a deterministic waveform this waveform depends on the statistical properties of the medium and the initial pulse only and it is explicitly calculable the theory is applied to the elastic wave equations in a two dimensional medium

sov approach for integrable quantum models associated with general representations on spin 1 2 chains of the 8 vertex reflection algebra the analysis of the transfer matrices associated with the most general representations of the 8 vertex reflection algebra on spin 1 2 chains is here implemented by introducing a quantum separation of variables sov method which generalizes to these integrable quantum models the method first introduced by sklyanin for representations reproducing in their homogeneous limits the open xyz spin 1 2 quantum chains with the most general integrable boundary conditions we explicitly construct representations of the 8 vertex reflection algebras for which the transfer matrix spectral problem is separated then in these sov representations we get the complete characterization of the transfer matrix spectrum eigenvalues and eigenstates and its non degeneracy moreover we present the first fundamental step toward the characterization of the dynamics of these models by deriving determinant formulae for the matrix elements of the identity on separated states which particularly apply to transfer matrix eigenstates a comparison of our analysis of the 8 vertex reflection algebra with that of niccoli g 2012 j stat mech p10025 faldella s et al 2014 j stat mech p01011 for the 6 vertex leads to an interesting remark in that there is a profound similarity in both the characterization of the spectral problems and the scalar products which exists for these two different realizations of the reflection algebra once they are described by the sov method as will be shown in a future publication this remarkable similarity will be the basis of a simultaneous determination of the form factors of local operators of integrable quantum models associated with general reflection algebra representations of both 8 vertex and 6 vertex type

a survey of exact algorithms for the simple assembly line balancing problem in this survey paper we discuss the development of the simple assembly line balancing problem salbp modifications and generalizations over time present alternate 0 1 programming formulations and a general integer programming formulation of the problem discuss other well known problems related to salbp describe and comment on a number of exact i e optimum seeking methods and present a summary of the reported computational experiences all models discussed here are deterministic i e all input parameters are assumed to be known with certainty and all the algorithms discussed are exact the problem is termed simple in the sense that no mixed models subassembly lines zoning restrictions etc are considered due to the richness of the literature we exclude from discussion here a the inexact i e heuristic approximate algorithms for salpb and b the algorithms for the general assembly line balancing problem including the stochastic models

chiral potts model as a descendant of the six vertex model we observe that then state integrable chiral potts model can be considered as a part of some new algebraic structure related to the six vertex model as a result we obtain a functional equation which is supposed to determine all the eigenvalues of the chiral potts model transfer matrix

optimizing glass coating lines mip model and valid inequalities glass coating is a specific transformation aiming at improving glass performance the work presented in this paper deals with the determination of the optimal configuration of the production lines used to perform this operation we propose a first mip formulation of the problem and then discuss several types of valid inequalities for improving it the main idea is to exploit explicit or implicit binary exclusion constraints to derive stronger valid inequalities the maximal clique constraints efficient polynomial time separation algorithms exploiting special structure of the problem are described giving rise to a cutting plane generation procedure for strengthening the initial formulation the computational study carried out shows that with the enhanced formulation good solutions can be obtained within reasonable computation times using currently available integer programming software

the tq equation of the eight vertex model for complex elliptic roots of unity we extend our studies of the tq equation introduced by baxter in his 1972 solution of the eight vertex model with parameter given by 2l 2m1k im2k from m2 0 to the more general case of complex we find that there are several different cases depending on the parity of m1 and m2

traces on the sklyanin algebra and correlation functions of the eight vertex model we propose a conjectural formula for correlation functions of the z invariant inhomogeneous eight vertex model we refer to this conjecture as ansatz it states that correlation functions are linear combinations of products of three transcendental functions with theta functions and derivatives as coefficients the transcendental functions are essentially logarithmic derivatives of the partition function per site the coefficients are given in terms of a linear functional tr on the sklyanin algebra which interpolates the usual trace on finite dimensional representations we establish the existence of tr and discuss the connection to the geometry of the classical limit we also conjecture that the ansatz satisfies the reduced qkz equation as a non trivial example of the ansatz we present a new formula for the next nearest neighbour correlation functions

time reversal invariance for a one dimensional model of contact acoustic nonlinearity the interaction of a one dimensional 1d wave packet with a contact interface characterized by a unilateral contact law is investigated analytically and through a finite difference model it is shown that this interaction leads to the generation of higher harmonic sub harmonic and zero frequency components in the reflected wave resulting in a pulse distortion that is attributable to contact acoustic nonlinearity however the results also show that the re emission of a time reversed version of this distorted first reflection results in a healing of the distortions and a perfect recovery of the original pulse shape thereby demonstrating time reversal invariance for this type of contact acoustic nonlinearity a step by step analysis of the contact interaction provides insights into both the distortion arising from the first interaction and the subsequent healing during the second interaction these findings suggest that time reversal invariance should also apply more generally for scatterers exhibiting non dissipative contact acoustic nonlinearity

the importance of being odd in this letter i mainly consider a finite xxz spin chain with periodic boundary conditions and an odd number of sites this system is described by the hamiltonian hxxz j 1n jx j 1x jy j 1y jz j 1z as it turns out the ground state energy is proportional to the number of sites e 3n 2 for a special value of the asymmetry parameter 1 2 the trigonometric polynomial q u the zeros of which are parameters of the ground state bethe eigenvector is explicitly constructed this polynomial of degree n n 1 2 satisfies the baxter t q equation using the second independent solution of this equation that corresponds to the same eigenvalue of the transfer matrix it is possible to find a derivative of the ground state energy w r t the asymmetry parameter this derivative is closely connected with the correlation function jz j 1z 1 2 3 2n2 this correlation function is related to the average number of spin strings for the ground state nstring n 1 n i would like to stress that all the above simple formulae are not applicable to the case of an even number of sites which is usually considered

shock structure due to stochastic forcing and the time reversal of nonlinear waves this paper is concerned with the study of the deformation of a nonlinear pulse traveling in a random medium we consider shallow water waves with a spatially random depth we demonstrate that in the presence of properly scaled stochastic forcing the solution to the nonlinear conservation law is regularized leading to a viscous shock profile this enables us to perform time reversal experiments beyond the critical time for shock formation

the xxz spin chain at δ 1 2 bethe roots symmetric functions and determinants a number of conjectures have been given recently concerning the connection between the antiferromagnetic xxz spin chain at 1 2 and various symmetry classes of alternating sign matrices here we use the integrability of the xxz chain to gain further insight into these developments in doing so we obtain a number of new results using baxter s q function for the xxz chain for periodic twisted and open boundary conditions these include expressions for the elementary symmetric functions evaluated at the ground state solution of the bethe roots in this approach schur functions play a central role and enable us to derive determinant expressions which appear in certain natural double products over the bethe roots when evaluated these give rise to the numbers counting different symmetry classes of alternating sign matrices

valid inequalities and facets of the capacitated plant location problem recently several successful applications of strong cutting plane methods to combinatorial optimization problems have renewed interest in cutting plane methods and polyhedral characterizations of integer programming problems in this paper we investigate the polyhedral structure of the capacitated plant location problem our purpose is to identify facets and valid inequalities for a wide range of capacitated fixed charge problems that contain this prototype problem as a substructure

spin chains and combinatorics in this paper we continue the investigation of finite xxz spin chains with periodic boundary conditions and odd number of sites initiated in our previous article stroganov yu g 2001 j phys a math gen 34 l179 85 as it turned out for a special value of the asymmetry parameter 1 2 the hamiltonian of the system has an eigenvalue which is exactly proportional to the number of sites e 3n 2 using mathematica we have found explicitly the corresponding eigenvectors for n 17 the obtained results support the conjecture our paper that this special eigenvalue corresponds to the ground state vector we make a lot of conjectures concerning the correlations of the model many remarkable relations between the wavefunction components are noted it turns out for example that the ratio of the largest component to the least one is equal to the number of the alternating sign matrices

frequency content of randomly scattered signals the statistical properties of acoustic signals reflected by a randomly layered medium are analyzed when a pulsed spherical wave issuing from a point source is incident upon it the asymptotic analy

regular xxz bethe states at roots of unity as highest weight vectors of the sl2 loop algebra we show that every regular bethe ansatz eigenvector of the xxz spin chain at roots of unity is a highest weight vector of the sl2 loop algebra for some restricted sectors with respect to eigenvalues of the total spin operator sz and evaluate explicitly the highest weight in terms of the bethe roots we also discuss whether a given regular bethe state in the sectors generates an irreducible representation or not in fact we present such a regular bethe state in the inhomogeneous case that generates a reducible weyl module here we call a solution of the bethe ansatz equations which is given by a set of distinct and finite rapidities regular bethe roots we call a nonzero bethe ansatz eigenvector with regular bethe roots a regular bethe state

new developments in the eight vertex model ii chains of odd length we study the transfer matrix of the 8 vertex model with an odd number of lattice sites n for systems at the root of unity points mk l with m odd the transfer matrix is known to satisfy the famous tq equation where q is a specifically known matrix we demonstrate that the location of the zeroes of this q matrix is qualitatively different from the case of evenn and in particular they satisfy a previously unknown equation which is more general than what is often called bethe s equation for the case of even m where no q matrix is known we demonstrate that there are many states which are not obtained from the formalism of the sos model but which do satisfy the tq equation the ground state for the particular case of 2k 3 and n odd is investigated in detail

breaking of time reversal invariance in nonlinear acoustics time reversal invariance of nonlinear acoustic wave propagation is experimentally investigated reversibility is studied for propagation shorter or longer than shock formation distance in the first case time reversal invariance holds and a sinusoid distorted by nonlinearities during forward propagation progressively recovers its initial shape after the time reversal operation in the second case reversibility is broken locally at the shock front as a time reversal operation transforms a stable compression shock into an unstable expansion shock achieving experimentally the time reversal process with a time reversal mirror made of reversible piezoelectric transducers for very broadband signals would require transducers with huge bandwidths to date such transducers remain unavailable in order to overcome this technical limitation we restricted ourselves in this study to one dimensional 1d propagation for which an experimental ersatz of a time reversal mirror can be used indeed in a 1d case the time reversal operation applied on a plane wave can be mimicked for an antisymmetric wave form by a reflection of the plane wave onto a pressure release interface

bethe s equation is incomplete for the xxz model at roots of unity we demonstrate for the six vertex and xxz model parameterized by q q 1 2 1 that when q2n 1 for integer n 2 the bethe s ansatz equations determine only the eigenvectors which are the highest weights of the infinite dimensional sl2 loop algebra symmetry group of the model therefore in this case the bethe s ansatz equations are incomplete and further conditions need to be imposed in order to completely specify the wave function we discuss how the evaluation parameters of the finite dimensional representations of the sl2 loop algebra can be used to complete this specification

higher harmonics in the far field due to dynamic crack face contacting the dynamic problem of contacting crack faces due to incident wave motion is numerically solved by using the boundary element method the crack is a penny shaped crack in a stressed elastic solid of infinite extent the crack is subjected to normal incidence of a longitudinal wave the contact boundary conditions with no overlap and no friction on the crack faces give rise to a boundary type nonlinear problem a time domain boundary integral equation is numerically solved by a step by step time marching scheme crack face solutions are calculated as well as scattered far fields fourier amplitudes of higher harmonics of the scattered far fields are obtained as functions of the state of pre stress

valid linear inequalities for fixed charge problems many problems in the operations research management science literature can be formulated with both zero one and continuous variables however the exact optimization of such mixed zero one models remains a computational challenge in this paper we propose to study mixed problems from a mathematical point of view that is similar in spirit to recent research on purely combinatorial problems that has investigated systems of defining linear inequalities or facets of the underlying polytope at least two numerical studies have validated this line of research computationally and the advances in the problem solving capabilities are considerable we expect that similar gains are possible for the mixed zero one problem more precisely we consider the mixed integer programs whose feasible region x is composed of i a simple additive constraint in the continuous variables xj for j 1 2 n and ii constraints 0 xj mjyj defined by binary variables yj for j 1 2 n this type of feasible region arises in a variety of mixed integer problems and particularly in network problems with fixed charges on the arcs we derive two classes of facet defining linear inequalities of the convex hull of x and show that the second of these classes gives a complete description of the convex hull when mj m for all j we also develop methods to detect violated inequalities from these classes so that these facets can be used as cutting planes to strengthen the formulations of certain mixed integer problems

the q operator and functional relations of the eight vertex model at root of unity for odd n following baxter s method of producing q 72 operator we construct the q operator of the root of unity eight vertex model for the crossing parameter 2mk n with odd n where q 72 does not exist we use this new q operator to study the functional relations in the fabricius mccoy comparison between the root of unity eight vertex model and the superintegrable n state chiral potts model by the compatibility of the constructed q operator with the structure of baxter s eight vertex solid on solid sos model we verify the set of functional relations of the root of unity eight vertex model using the explicit form of the q operator and fusion weights of the sos model

bethe equations on the wrong side of the equator we analyse baxter s famous t q equations for the xxx xxz spin chain and show that apart from its usual polynomial trigonometric solution which provides the solution of bethe ansatz equations there also exists a second solution which should correspond to the bethe ansatz beyond this second solution of baxter s equation plays an essential role and together with the first one gives rise to all fusion relations

completing bethe s equations at roots of unity in a previous paper we demonstrated that bethe s equations are not sufficient to specify the eigenvectors of the xxz model at roots of unity for states where the hamiltonian has degenerate eigenvalues we here find the equations which will complete the specification of the eigenvectors in these degenerate cases and present evidence that the sl 2 loop algebra symmetry is sufficiently powerful to determine that the highest weight of each irreducible representation is given by bethe s ansatz

properties of the string operator in the eight vertex model the construction of creation operators of exact strings in eigenvectors of the eight vertex model at elliptic roots of unity of the crossing parameter which allow the generation of the complete set of degenerate eigenstates is based on the conjecture that the naive string operator vanishes in this paper we present a proof of this conjecture furthermore we show that for chains of odd length the string operator is either proportional to the symmetry operator s or vanishes depending on the precise form of the crossing parameter

integrable structure of conformal field theory ii q operator and ddv equation this paper is a direct continuation of 1 where we began the study of the integrable structures in conformal field theory we show here how to construct the operators bf q pm lambda which act in the highest weight virasoro module and commute for different values of the parameter these operators appear to be the cft analogs of the q matrix of baxter 2 in particular they satisfy baxter s famous t q equation we also show that under natural assumptions about analytic properties of the operators as the functions of the baxter s relation allows one to derive the nonlinear integral equations of destri de vega ddv 3 for the eigenvalues of the q operators we then use the ddv equation to obtain the asymptotic expansions of the q operators at large it is remarkable that unlike the expansions of the t operators of 1 the asymptotic series for q contains the dual nonlocal integrals of motion along with the local ones we also discuss an intriguing relation between the vacuum eigenvalues of the q operators and the stationary transport properties in the boundary sine gordon model on this basis we propose a number of new exact results about finite voltage charge transport through the point contact in the quantum hall system

an antiperiodic dynamical six vertex model i complete spectrum by sov matrix elements of the identity on separate states and connections to the periodic eight vertex model the spin 1 2 highest weight representations of the dynamical six vertex and the standard eight vertex yang baxter algebra on a finite chain are considered in this paper in particular the integrable quantum models associated with the corresponding transfer matrices under antiperiodic boundary conditions for the dynamical six vertex case and periodic boundary conditions for the eightvertexcaseareanalyzedhere fortheantiperiodicdynamicalsix vertextransfer matrix defined on chains with an odd number of sites we adapt sklyanin s quantum separation of variable sov method and explicitly construct the sov representations from the original space of the representations in this way we provide the complete characterization of the eigenvalues and the eigenstates proving also the simplicity of its spectrum moreover we characterize the matrix elements of the identity on separated states of this model by determinant formulae the matrices entering these determinants have elements given by sums over the sov spectrum of the product of the coefficients of the separate states this sov analysis is done without any need to be reduced to the case of the so called elliptic roots of unit and the results derived here define the required setup to extend to the dynamical six vertex model the approach recently developed by the author and collaborators to compute the form factors of the local operators in the sov framework for the periodic eight vertex transfer matrix we prove that its eigenvalues have to satisfy a fixed system of equations in the case of a chain with an odd number of sites this system of equations is the same entering in the sov characterization of the antiperiodic dynamical six vertex transfer matrix spectrum this implies that the set of the periodic eight vertex eigenvalues is contained in the set of the antiperiodic dynamical six vertex eigenvalues a criterion is introduced to find simultaneous eigenvalues of these two transfer matrices and associate with any of such eigenvalues one nonzero eigenstate of the periodic eight vertex transfer matrix by using the sov results moreover a preliminary discussion

assembly system design a branch and cut approach this paper addresses the single product assembly system design problem asdp which seeks to minimize total cost by optimally integrating design selecting the machine type to locate at each activated station and operating issues assigning tasks to observe precedence relationships and cycle time restrictions we propose an effective branch and cut approach for solving single product asdps adapting inequalities known to be valid for embedded line balancing structures to form inequalities that are valid for the asdp the implementation alsoinvolves a specialized preprocessor a heuristic separation procedures and an enumeration scheme computational tests establish benchmark results for this first implementation of cutting planes for solving the asdp

new developments in the eight vertex model we demonstrate that the q matrix introduced in baxter s 1972 solution of the eight vertex model has some eigenvectors which are not eigenvectors of the spin reflection operator and conjecture a new functional equation for q v which both contains the bethe equation that gives the eigenvalues of the transfer matrix and computes the degeneracies of these eigenvalues

completeness of the bethe ansatz for the six and eight vertex models we discuss some of the difficulties that have been mentioned in the literature in connection with the bethe ansatz for the six vertex model and xxz chain and for the eight vertex model in particular we discuss the beyond the equator infinite momenta and exact complete string problems we show how they can be overcome and conclude that the coordinate bethe ansatz does indeed give a complete set of states as expected

a cutting plane algorithm for the linear ordering problem the linear ordering problem is an np hard combinatorial optimization problem with a large number of applications including triangulation of input output matrices archaeological senation minimizing total weighted completion time in one machine scheduling and aggregation of individual preferences in a former paper we have investigated the facet structure of the 0 1 polytope associated with the linear ordering problem here we report on a new algorithm that is based on these theoretical results the main part of the algorithm is a cutting plane procedure using facet defining inequalities this procedure is combined with various heuristics and branch and bound techniques our computational results compare favorably with the results of existing codes in particular we could triangulate all input output matrices of size up to 60 60 available to us within acceptable time bounds