an exact algorithm for higher dimensional orthogonal packing higher dimensional orthogonal packing problems have a wide range of practical applications including packing cutting and scheduling combining the use of our data structure for characterizing feasible packings with our new classes of lower bounds and other heuristics we develop a two level tree search algorithm for solving higher dimensional packing problems to optimality computational results are reported including optimal solutions for all two dimensional test problems from recent literature this is the third in a series of articles describing new approaches to higher dimensional packing

spectral theory for the periodic sine gordon equation a concrete viewpoint a summary of the spectral theory for quasiperiodic sine and sinh gordon equations is given analogies with whole line solitons and scattering theory motivates the discussion the relation between the ingredients in the inverse spectral solution of the periodic sine gordon equation and physical characteristics of sine gordon waves is emphasized the explicit topics covered are summarized in the table of contents in the introduction

homoclinic orbits in the near integrable double discrete sine gordon equation we establish the existence of homoclinic orbits for the near integrable double discrete sine gordon ddsg equation under periodic boundary conditions the hyperbolic structure and homoclinic orbits are constructed through the backlund transformation and lax pair a geometric perturbation method based on mel nikov analysis is used to establish necessary criteria for the persistent of temporally homoclinic orbits for the class of ddsg equations with dissipative perturbations

geometry of the modulational instability iii homoclinic orbits for the periodic sine gordon equation in this paper the homoclinic geometric structure of the integrable sine gordon equation under periodic boundary conditions is developed specifically focus is given to orbits homoclinic to n tori simple examples of such homoclinic orbits are constructed and a physical interpretation of these states is given a labeling is provided which identifies and catalogues all such orbits these orbits are related in a one to one manner to linearized instabilities explicit formulas for all homoclinic orbits are given in terms of backlund transformations

multiplicity one representations of divergence free lie algebras abstract divergence free lie algebras also known as the special lie algebras of cartan type are lie algebras of volume preserving transformation groups they are simple in generic case dokovic and zhao found a certain graded generalization of them in this paper we classify all the irreducible and indecomposable multiplicity one modules of the simple generalized divergence free lie algebras

complex ginzburg landau equations as perturbations of nonlinear schrodinger equations a melnikov approach abstract we study the persistence of quasiperiodic and homoclinic solutions of generalized nonlinear schrodinger equations under ginzburg landau perturbations in this paper the first of a series melnikov criteria for the persistence of quasiperiodic and homoclinic solutions are derived directly from the governing partial differential equations via an averaging technique for families of tori of quasiperiodic solutions such as rotating waves and traveling waves that arise within critical sets of linear combinations of conserved functionals we find that usually only isolated tori will satisfy these selection criteria moreover in some simple cases these criteria are sufficient to conclude that a torus persists we also demonstrate the nonpersistence of solutions that are homoclinic to rotating waves under a broad class of ginzburg landau perturbations which satisfy a convexity condition

lp bounds in an interval graph algorithm for orthogonal packing feasibility we consider the feasibility problem opp orthogonal packing problem in higher dimensional orthogonal packing given a set of d dimensional d 2 rectangular items decide whether all of them can be orthogonally packed in the given rectangular container without rotation the one dimensional 1d bar lp relaxation of opp reduces the latter to a 1d cutting stock problem where the packing of each stock bar represents a possible 1d stitch through an opp layout the dual multipliers of the lp provide us with another kind of powerful bounding information conservative scales we investigate how the set of possible 1d packings can be tightened using the overlapping information of item projections on the axes with the goal to tighten the relaxation we integrate the bar relaxation into an interval graph algorithm for opp which operates on such overlapping relations numerical results on 2d and 3d instances demonstrate the efficiency of tightening leading to a speedup and stabilization of the algorithm

homoclinic crossings and pattern selection the correlations between a homoclinic orbit hmo and coherent patterns in the nonlinear schr odinger model are discussed in a hamiltonian situation two independent patterns are revealed to exist one corresponds to a motion within an hmo and the other outside of an hmo the study further illustrates when the hamiltonian constraint is released the significance of an hmo on the patterns dynamics by presenting the irregular hmo crossings and the resulting chaotic selections between the patterns

an exact algorithm for the two dimensional strip packing problem this paper considers the two dimensional strip packing problem 2sp in which a set of rectangular items have to be orthogonally packed without overlapping into a strip of a given width and infinite height by minimizing the overall height of the packing the 2sp is np hard in the strong sense and finds many practical applications we propose reduction procedures lower and upper bounds and an exact algorithm for the 2sp the new lower bounds are both combinatorial bounds and bounds derived from different relaxations of mathematical formulations of the 2sp the new upper bounds are obtained by constructive heuristics based on different strategies to place the items into the strip the new exact method is based on a branch and bound approach computational results on different sets of test problems derived from the literature show the effectiveness of the new lower and upper bounds and of the new exact algorithm

numerical chaos symplectic integrators and exponentially small splitting distances two discrete versions of duffing s equation are derived as reductions of discretizations of the nonlinear schrodinger nls equation inheriting the properties of the nsl discretizations one is shown to be integrable and using a mel nikov type analysis the other discretization is shown to be nonintegrable arguments are given why it is possible for the nonintegrable scheme to restore the homoclinic orbit at an exponentially fast rate which is faster than the order of the approximation as h 0 this suggests why there can be fundamental differences between symplectic and nonsymplectic discretizations of certain continuous hamiltonian systems

spectral and modulational stability of periodic wavetrains for the nonlinear klein gordon equation abstract this paper is a detailed and self contained study of the stability properties of periodic traveling wave solutions of the nonlinear klein gordon equation u t t u x x v u 0 where u is a scalar valued function of x and t and the potential v u is of class c 2 and periodic stability is considered both from the point of view of spectral analysis of the linearized problem spectral stability analysis and from the point of view of wave modulation theory the strongly nonlinear theory due to whitham as well as the weakly nonlinear theory of wave packets the aim is to develop and present new spectral stability results for periodic traveling waves and to make a solid connection between these results and predictions of the formal modulation theory which has been developed by others but which we review for completeness

doubly z graded lie algebras containing a virasoro algebra abstract in this paper we study z z graded lie algebras a i j zai j over a characteristic 0 field f with dim ai j 1 for each i and j where z denotes the integers we shall assume that a satisfies the following three properties i vir the centerless virasoro algebra ii 0 and ad and ad act faithfully on the subspace iii dim dim 1 and is generated by and the main result of the paper is the classification of these algebras more precisely a is isomorphic to a simple block algebra l a which has basis ei j i j z z 1 a 0 2 a 0 and subject to the bracket formula for some suitable constant a f 0 1 2 or is isomorphic to a one or two dimensional central extension of l a with a f 0 1 2 or a is isomorphic a one dimensional derivation extension of l 1 and l 2 or a two dimensional extension of l 1 by a center element and a derivation

rogue waves dissipation and downshifting abstract we investigate the effects of dissipation on the development of rogue waves and downshifting by adding nonlinear and linear damping terms to the one dimensional dysthe equation significantly rogue waves do not develop after the downshifting becomes permanent thus in our experiments permanent downshifting serves as an indicator that damping is sufficient to prevent the further development of rogue waves using the inverse spectral theory of the nls equation simulations of the damped dysthe equation for sea states characterized by jonswap spectrum consistently show that rogue wave events are well predicted by proximity to homoclinic data as measured by the spectral splitting distance the cut off distance c u t o f f decreases as the strength of the damping increases indicating that for stronger damping the jonswap initial data must be closer to homoclinic data for rogue waves to occur

the virasoro algebra three theroems are proved with suitable hypotheses in each case characterizations are found for the virasoro algebra for some of its representations and for the ramond neveu schwarz superalgebra built around the virasoro algebra

easily computable facets of the knapsack polytope it is known that facets and valid inequalities for the knapsack polytope p can be obtained by lifting a simple inequality derived from each minimal cover we study the computational complexity of such lifting in particular we show that the task of computing a lifted facet can be accomplished in o ns where s n is the cardinality of the minimal cover also for a lifted inequality with integer coefficients we show that the dual tasks of recognizing whether the inequality is valid for p or is a facet of p can be done within the same time bound

orbits homoclinic to resonances with an application to chaos in a model of the forced and damped sine gordon equation abstract in this paper we develop new global perturbation techniques for detecting homoclinic and heteroclinic orbits in a class of four dimensional ordinary differential equations that are perturbations of completely integrable two degree of freedom hamiltonian systems our methods are fundamentally different than other global perturbation methods e g standard melnikov theory in that we are seeking orbits homoclinic and heteroclinic to fixed points that are created in a resonance resulting from the perturbation our methods combine the higher dimensional melnikov theory with geometrical singular perturbation theory and the theory of foliations of invariant manifolds we apply our methods to a modified model of the forced and damped sine gordon equation developed by bishop et al we give explicit conditions in terms of the system parameters for the model to possess a symmetric pair of homoclinic orbits to a fixed point of saddle focus type chaotic dynamics follow from a theorem of silnikov this provides a mechanism for chaotic dynamics geometrically similar to that observed by bishop et al namely a random jumping between two spatially dependent states with an intermediate passage through a spatially independent state however in order for this type of silnikov dynamics to exist we require a different and unphysical type of damping compared to that used by bishop et al

bidimensional packing by bilinear programming we consider geometric problems in which rectangles have to be packed into identical squares these problems turn out to be very hard in practice and ilp formulations in which variables specify the coordinates in the packing perform very poorly while most methods developed so far are based on simple geometric considerations a recent landmark result of fekete and schepers suggests to put these geometric aspects aside and use the most advanced tools for the one dimensional case in this paper we make additional progress in this direction especially on the basic question does a given set of rectangles fit into a square which turns out to be the bottleneck of all the approaches known given a set of rectangles and the associated convex hull of rectangle subsets that fit into a square we derive a wide class of valid inequalities for this convex hull from a complete description of the two knapsack polytopes associated with the widths and the heights of the rectangles respectively in addition we illustrate how to solve the associated separation problem as a bilinear program for which we develop a solution method that turns out to be fast in practice and show that the integer solutions that satisfy all these constraints generally correspond to vertices of the original convex hull for the benchmark instances in the literature the same tools are used to derive lower bounds for the two dimensional bin packing problem corresponding to the determination of an optimal pair of so called dual feasible functions these lower bounds in many cases equal those obtained by the customary set covering formulation for which column generation is very hard but are computable within a time that is smaller by some orders of magnitude this allows us to close a few of the benchmark instances in the literature

frequency and phase locking of spatially periodic perturbed sine gordon breather trains this paper considers the damped driven sine gordon equation with even periodic boundary conditions on a finite length nearly integrable perturbation methods are applied to infer approximate solution branches of space and time periodic breather trains of order one amplitude these approximate solutions consist of coherent spatial structures either homogeneous or with one breather per period which lock in frequency and phase to the small time periodic perturbation in the presence of uniform weak damping the predicted resonant breather trains are then verified by overman via direct simulation of the perturbed sine gordon pde and shown to be accurate within the order of the perturbation theory

multi dimensional homoclinic jumping and the discretized nls equation we consider a class of dynamical systems that arise frequently in multi mode truncations and discretizations of partial differential equations including the perturbed nls we develop a general method to detect the existence of multi pulse solutions that are doubly asymptotic to an invariant manifold with two different time scales we use our method together with some recent results of li and mclaughlin to show the existence of several families of multi pulse orbits for the ablowitz ladik discretization of the perturbed nls these orbits include n pulse heteroclinic orbits and n pulse silnikov type orbits for arbitrarily large n

benjamin feir type instability of sine gordon equation and spectrum of lame equation the benjamin feir type instability of the nonhomoclinic wavetrain solution of the sine gordon equation against the spatially periodic small perturbation is observed by relating the fourier coefficients of the first order approximate solution of fourier type to the band structure of the spectrum of the lame equation

numerical simulation of quasi periodic solutions of the sine gordon equation abstract analytic solutions of the sine gordon equation corresponding to periodic boundary conditions can be complicated especially if initial values are chosen in the vicinity of homoclinic orbits for such initial values it has been demonstrated that numerical solutions develop instabilities and may become chaotic because of the analytical complexities one cannot easily calculate the accuracy of the numerical solutions and therefore compare different numerical schemes in a straightforward manner in this note we evaluate numerical methods in terms of the nonlinear spectrum in particular it provides one with a means of comparing symplectic and nonsymplectic integrators for integrable infinite dimensional hamiltonian systems

harish chandra modules over the twisted heisenberg virasoro algebra in this paper we classify all indecomposable harish chandra modules of the intermediate series over the twisted heisenberg virasoro algebra meanwhile some bosonic modules are also studied

an exact algorithm for the two dimensional orthogonal packing problem with unloading constraints this paper describes an exact algorithm for solving a two dimensional orthogonal packing problem with unloading constraints which occurs as a subproblem of mixed vehicle routing and loading problems the packing considered in this work is basically a feasibility problem involving a single bin the problem is addressed through a decomposition approach wherein a branch and cut algorithm is designed for solving a one dimensional relaxation of the original problem when an integer solution is found in the branching tree a subsidiary problem is solved to identify a two dimensional packing that does not lead to any overlap and satisfies the unloading constraints cuts are added when the subsidiary problem proves to be infeasible several preprocessing techniques aimed at reducing the size of the solution space and uncovering infeasibility are also described a numerical comparison with the best known exact method is reported at the end based on benchmark instances

a characterization of the block lie algebra and itsq forms in characteristic 0 abstract the block algebra l referred to here is the lie algebra over a field f of characteristic 0 with basis e r s r s z z 0 0 and subject to the commutation relations e h k e r s rk sh e h r k s let 0 1 q f the q form l q of l is the lie algebra with basis e r s q r s z z 0 0 and subject to the commutation relations e h k q e r s q q rk q sh e h r k s q we study here a z z graded lie algebra formula with a 0 0 0 and dim a i j 1 for all i j z z 0 0 in this paper we show that a is necessarily isomorphic to l or l q l q l q for some q f if a satisfies a certain nondegeneracy condition

multi pulse jumping orbits and homoclinic trees in a modal truncation of the damped forced nonlinear schro dinger equation abstract in this paper we prove the existence of multi pulse orbits homoclinic to a slow manifold in a two mode truncation of the damped forced nonlinear schrodinger equation first suggested by bishop et al these orbits are jumping i e the corresponding solutions keep switching in time between neighborhoods of the two charateristic breathers of the integrable limit in the case of no damping we find multi pulse smale horseshoes in the two mode model while in the dissipative case we establish the existence of structurally stable multi pulse heterolinic connections between two unstable equilibria the orbits we construct are not amenable to melnikov type perturbation methods in both the hamiltonian and the dissipative case we find homoclinic trees which describe the repeated bifurcations of multi pulse solutions to illustrate the theoretical predictions we also present visualizations of these complicated structures

exact algorithms for the two dimensional strip packing problem with and without rotations we propose exact algorithms for the two dimensional strip packing problem 2sp with and without 90 rotations we first focus on the perfect packing problem pp which is a special case of 2sp wherein all given rectangles are required to be packed without wasted space and design branch and bound algorithms introducing several branching rules and bounding operations a combination of these rules yields an algorithm that is especially efficient for feasible instances of pp we then propose several methods of applying the pp algorithms to 2sp our algorithms succeed in efficiently solving benchmark instances of pp with up to 500 rectangles and those of 2sp with up to 200 rectangles they are often faster than existing exact algorithms specially tailored for problems without rotations

the geometry of real sine gordon wavetrains the characterization ofreal n phase quasiperiodic solutions of the sine gordon equation has been an open problem in this paper we achieve this result employing techniques of classical algebraic geometry which have not previously been exploited in the soliton literature a significant by product of this approach is a naturalalgebraic representation of the full complex isospectral manifolds and an understanding of how the real isospectral manifolds are embedded by placing the problem in this general context these methods apply directly to all soliton equations whose multiphase solutions are related to hyperelliptic functions

parametric instabilities in the discrete sine gordon equation abstract the one dimensional sine gordon equation is an example of an exactly solvable nonlinear partial differential equation when discretized on a periodic lattice in space and in time it corresponds to a lattice of pendula coupled by linear springs we show that the discretized system is unstable to a parametric mode when the frequency of time discretization is sufficiently small and we obtain the instability condition and growth rate by considering the effects of two finite amplitude modes we also obtain the nonlinear saturation of the instability we also examine how solutions of the exact sine gordon equation behave under this map both in and away from the parametrically unstable regime

a combinatorial characterization of higher dimensional orthogonal packing higher dimensional orthogonal packing problems have a wide range of practical applications including packing cutting and scheduling previous efforts for exact algorithms have been unable to avoid structural problems that appear for instance in two or higher dimensional space we present a new approach for modeling packings using a graph theoretical characterization of feasible packings our characterization allows it to deal with classes of packings that share a certain combinatorial structure instead of having to consider one packing at a time in addition we can make use of elegant algorithmic properties of certain classes of graphs this allows our characterization to be the basis for a successful branch and bound framework this is the first in a series of papers describing new approaches to higher dimensional packing

existence of homoclinic tubes in the sine gordon equation with hamiltonian perturbation we consider the sine gordon equation under hamiltonian perturbation with even periodic boundary conditions we give an analytic expression for homoclinic orbits and produce a useful representation of the gradient of an important integral of motion we establish the existence of homoclinic tubes based on mel nikov analysis and an implicit function theorem argument

on the numerical solution of the sine gordon equation in this the first of two papers on the numerical solution of the sine gordon equation we investigate the numerical behavior of a double discrete completely integrable discretization of the sine gordon equation for certain initial values in the vicinity of homoclinic manifolds this discretization admits an instability in the form of grid scale oscillations we clarify the nature of the instability through an analytical investigation supported by numerical experiments in particular a perturbation analysis of the associated linear spectral problem shows that the initial values used for the numerical experiments lie exponentially close to a homoclinic manifold this paves the way for the second paper where we use the non linear spectrum as a basis for comparing different numerical schemes

combinatorial benders cuts for mixed integer linear programming mixed integer programs mips involving logical implications modeled through big m coefficients are notoriously among the hardest to solve in this paper we propose and analyze computationally an automatic problem reformulation of quite general applicability aimed at removing the model dependency on the big m coefficients our solution scheme defines a master integer linear problem ilp with no continuous variables which contains combinatorial information on the feasible integer variable combinations that can be distilled from the original mip model the master solutions are sent to a slave linear program lp which validates them and possibly returns combinatorial inequalities to be added to the current master ilp the inequalities are associated to minimal or irreducible infeasible subsystems of a certain linear system and can be separated efficiently in case the master solution is integer the overall solution mechanism closely resembles the benders one but the cuts we produce are purely co

stability and bifurcation of spatially coherent solutions of the damped driven nls equation an analytical study is conducted of the structure stability and bifurcation of the spatially dependent time periodic solutions of the damped driven sine gordon equation in the nonlinear schrodinger approximation locked states are found for which the spatial structure consists of coherent excitations localized about x 0 or l 2 a bifurcation analysis reveals the relationship of these spatially localized solutions to the spatially independent ones and provides a cutoff wavenumber above which there are no spatially dependent solutions this establishes an upper bound on the number of local excitations comprising the spatial pattern a linear stability analysis shows that the spatially localized solutions undergo a hopf bifurcation to temporal quasi periodicity as the driver amplitude gamma is increased for sufficiently high driver frequencies the temporally periodic solution regains its stability via another hopf bifurcation in a gamma window of finite width before undergoing a third hop

a new exact method for the two dimensional orthogonal packing problem abstract the two dimensional orthogonal packing problem 2 opp consists in determining if a set of rectangles items can be packed into one rectangle of fixed size bin in this paper we propose two exact algorithms for solving this problem the first algorithm is an improvement on a classical branch bound method whereas the second algorithm is based on a new relaxation of the problem we also describe reduction procedures and lower bounds which can be used within enumerative methods we report computational experiments for randomly generated benchmarks which demonstrate the efficiency of both methods the second method is competitive compared to the best previous methods it can be seen that our new relaxation allows an efficient detection of non feasible instances

spectral stability analysis for periodic traveling wave solutions of nls and cgl perturbations abstract we consider cnoidal traveling wave solutions to the focusing nonlinear schrodinger equation nls that have been shown to persist when the nls is perturbed to the complex ginzburg landau equation cgl we show that while these periodic traveling waves are spectrally stable solutions of nls with respect to periodic perturbations they are unstable with respect to bounded perturbations furthermore we use an argument based on the fredholm alternative to find an instability criterion for the persisting solutions to cgl

combinatorial benders cuts for the strip packing problem we study the strip packing problem in which a set of two dimensional rectangular items has to be packed in a rectangular strip of fixed width and infinite height with the aim of minimizing the height used the problem is important because it models a large number of real world applications including cutting operations where stocks of materials such as paper or wood come in large rolls and have to be cut with minimum waste scheduling problems in which tasks require a contiguous subset of identical resources and container loading problems arising in the transportation of items that cannot be stacked one over the other the strip packing problem has been attacked in the literature with several heuristic and exact algorithms nevertheless benchmark instances of small size remain unsolved to proven optimality in this paper we propose a new exact method that solves a large number of the open benchmark instances within a limited computational effort our method is based on a benders decomposition in whi

a quasi periodic route to chaos in a near integrable pde abstract pattern formation and transitions to chaos are described for the damped ac driven one dimensional periodic sine gordon equation in a nonlinear schrodinger regime a generic quasi periodic route to intermittent chaos is exhibited in detail using a range of dynamical systems diagnostics in addition a nonlinear spectral transform is exploited i to identify and quantify coordinates of space time attractors in terms of a small number of soliton modes of the underlying integrable system ii to use these analytic coordinates to identify homoclinic orbits as possible sources of chaos and iii to demonstrate the significance of energy transfer between coherent and extended states in this chaotic system

numerical inverse spectral transform for the periodic sine gordon equation theta function solutions and their linearized stability abstract the inverse spectral transform for integrable nonlinear ordinary and partial differential equations such as the toda lattice korteweg de vries sine gordon and nonlinear schrodinger equations provides explicit algorithms to generate exact solutions under periodic or quasiperiodic boundary conditions these oscillatory wavetrains may be prescribed a priori to consist of a nonlinear superposition of n phases j x t j x j t 0 j j 1 n where the wave is 2 periodic independently in each phase this paper exhibits the numerical implementation of the inverse spectral solution of the sine gordon equation the general construction is outlined and then implemented for n 1 2 and 3 we compute 1 the exact theta function solutions 2 the floquet spectrum of x periodic solutions 3 the labelling of linearized instabilities of n phase solutions in terms of spectral data and 4 the linearized growth rate in each unstable mode the associated surfaces q n x t are displayed to illustrate a variety of spatial and dynamical phenomena in the oscillatory solution space of this integrable system