heuristics for the container loading problem abstract the knapsack container loading problem is the problem of loading a subset of rectangular boxes into a rectangular container of fixed dimensions such that the volume of the packed boxes is maximized a new heuristic based on the wall building approach is proposed which decomposes the problem into a number of layers which again are split into a number of strips the packing of a strip may be formulated and solved optimally as a knapsack problem with capacity equal to the width or height of the container the depth of a layer as well as the thickness of each strip is decided through a branch and bound approach where at each node only a subset of branches is explored several ranking rules for the selection of the most promising layer depths and strip widths are presented and the performance of the corresponding algorithms is experimentally compared for homogeneous and heterogeneous instances the best ranking rule is then used in a comprehensive computational study involving large sized instances these computational results show that instances with a total box volume up to 90 easily may be solved to optimality and that average fillings of the container volume exceeding 95 may be obtained for large sized instances

applications of the complex interpolation method to a von neumann algebra non commutative lp spaces non commutative lp spaces 1 p associated with a von neumann algebra are considered the paper consists of two parts in part i by making use of the complex interpolation method non commutative lp spaces are defined as interpolation spaces between the von neumann algebra in question and its predual also all expected properties such as duality and uniform convexity are proved in the frame of interpolaton theory and relative modular theory in part ii these lp spaces are compared with haagerup s lp spaces based on this comparison a non commutative analogue of the classical stein weiss interpolation theorem is obtained

an n tet graph approach for non guillotine packings of n dimensional boxes into an n container abstract in this paper we propose a simple recursive uniform algorithm for the problem of packing n dimensional boxes into an n container we are particularly concerned about the special case n 3 where the boxes can be packed in a given subset of their six possible positionings our method studies symmetries in the packings by the use of an ordered set of three directed graphs with the same edges a 3 tet or triad and induced smaller structures of the same kind named minors with the method degeneracy and symmetry issues which curtail the implicit enumeration to practically acceptable running times become transparent in order to illustrate the performance of the algorithm computational results from solving randomly generated 3 d examples are presented and compared with the ones of a layers and knapsack approach the present study has real world applications for the problems of pallet and container loading

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

space defragmentation for packing problems abstract one of main difficulties of multi dimensional packing problems is the fragmentation of free space into several unusable small parts after a few items are packed this study proposes a defragmentation technique to combine the fragmented space into a continuous usable space which potentially allows the packing of additional items we illustrate the effectiveness of this technique using the two and three dimensional bin packing problem where the aim is to load all given items represented by rectangular boxes into the minimum number of identical bins experimental results based on well known 2d and 3d bin packing data sets show that our defragmentation technique alone is able to produce solutions approaching the quality of considerably more complex meta heuristic approaches for the problem in conjunction with a bin shuffling strategy for incremental improvement our resultant algorithm outperforms all leading meta heuristic approaches based on the commonly used benchmark data by a significant margin

a new iterative doubling greedy lookahead algorithm for the single container loading problem the aim of the single container loading problem sclp is to pack three dimensional boxes into a three dimensional container so as to maximize the volume utilization of the container we propose a new block building approach that constructs packings by placing one block of boxes at a time until no more boxes can be loaded the key to obtaining high quality solutions is to select the right block to place into the right free space cuboid or residual space in the container we propose a new heuristic for evaluating the fitness of residual spaces and use a tree search to decide the best residual space block pair at each step the resultant algorithm outperforms the best known algorithms based on the 1600 commonly used benchmark instances even when given fewer computational resources we also adapted our approach to address the full support constraint the computational results for the full support support variant on the 1600 instances similarly show a significant improvement over existing techniques even when given substantially fewer computational resources

modular hamiltonians on the null plane and the markov property of the vacuum state we compute the modular hamiltonians of regions having the future horizon lying on a null plane for a cft this is equivalent to regions with a boundary of arbitrary shape lying on the null cone these hamiltonians have a local expression on the horizon formed by integrals of the stress tensor we prove this result in two different ways and show that the modular hamiltonians of these regions form an infinite dimensional lie algebra the corresponding group of unitary transformations moves the fields on the null surface locally along the null generators with arbitrary null line dependent velocities but act non locally outside the null plane we regain this result in greater generality using more abstract tools on the algebraic quantum field theory finally we show that modular hamiltonians on the null surface satisfy a markov property that leads to the saturation of the strong sub additive inequality for the entropies and to the strong super additivity of the relative entropy

the multiple container loading cost minimization problem abstract in the shipping and transportation industry there are several types of standard containers with different dimensions and different associated costs in this paper we examine the multiple container loading cost minimization problem mclcmp where the objective is to load products of various types into containers of various sizes so as to minimize the total cost we transform the mclcmp into an extended set cover problem that is formulated using linear integer programming and solve it with a heuristic to generate columns experiments on standard bin packing instances show our approach is superior to prior approaches additionally since the optimal solutions for existing test data is unknown we propose a technique to generate test data with known optimal solutions for mclcmp

heuristic algorithms for the three dimensional bin packing problem abstract the three dimensional bin packing problem 3bp consists of allocating without overlapping a given set of three dimensional rectangular items to the minimum number of three dimensional identical finite bins the problem is np hard in the strong sense and finds many industrial applications we introduce a tabu search framework exploiting a new constructive heuristic for the evaluation of the neighborhood extensive computational results on standard benchmark instances show the effectiveness of the approach with respect to exact and heuristic algorithms from the literature

two dimensional packing problems a survey we consider problems requiring to allocate a set of rectangular items to larger rectangular standardized units by minimizing the waste in two dimensional bin packing problems these units are finite rectangles and the objective is to pack all the items into the minimum number of units while in two dimensional strip packing problems there is a single standardized unit of given width and the objective is to pack all the items within the minimum height we discuss mathematical models and survey lower bounds classical approximation algorithms recent heuristic and metaheuristic methods and exact enumerative approaches the relevant special cases where the items have to be packed into rows forming levels are also discussed in detail

sufficient subalgebras and the relative entropy of states of a von neumann algebra a subalgebram 0 of a von neumann algebram is called weakly sufficient with respect to a pair of states if the relative entropy of and coincides with the relative entropy of their restrictions tom 0 the main result says thatm 0 is weakly sufficient for if and only ifm 0 contains the radon nikodym cocycle d d t other conditions are formulated in terms of generalized conditional expectations and the relative hamiltonian

conditional expectations in von neumann algebras and a theorem of takesaki conditional expectations play an important role in classical probability theory in the general context of von neumann algebras they were impliciteiy used by von neumann 41 chap ii and by dixmier 14 nakamura and turumaru 27 and umegaki 36 391 introduced an axiomatic definition of the concept of conditional expectation in the framework of von neumann or c algebras and established many properties of these objects especially in the context of von neumann algebras with a finite trace their starting point was the characterization given by moy 26 of the classical conditional expectations as operators on spaces of measurable functions tomiyama showed 33 that conditional expectations in the sense of the above mentioned authors can be characterized as norm one projection in c algebras the importance of norm one projection in the classification problem of von neumann algebras was recognized by hakeda and tomiyama 23 and subsequent research on this argument confirmed the usefulness of these objects this line of thought culminated in the fundamental work of connes 121 in which approximately finite von 245 0022 1236 82 020245 29 02 00 o

solving container loading problems by block arrangement abstract in order to solve heterogeneous single and multiple container loading problems an algorithm is presented that builds homogeneous blocks of identically orientated items first a greedy heuristic is presented that generates the desired block arrangements second the solutions provided by the greedy heuristic are improved by a tree search additional aspects such as load stability and weight distribution within the container are also taken into account the test cases of bischoff and ratcliff are used for benchmarking purposes

a hybrid genetic algorithm for the container loading problem abstract this paper presents a hybrid genetic algorithm ga for the container loading problem with boxes of different sizes and a single container for loading generated stowage plans include several vertical layers each containing several boxes within the procedure stowage plans are represented by complex data structures closely related to the problem to generate offspring specific genetic operators are used that are based on an integrated greedy heuristic the process takes several practical constraints into account extensive test calculations including procedures from other authors vouch for the good performance of the ga above all for problems with strongly heterogeneous boxes

the bottomn left bin packing heuristic an efficient implementation we study implementations of the bottom left heuristic for two dimensional bin packing to pack n rectangles into an infinite vertical strip of fixed width the strategy considered here places each rectangle in turn as low as possible in the strip in a left justified position for reasons of simplicity and good performance the bottom left heuristic has long been a favorite in practical applications however the best implementations found so far require a number of steps o n3 in this paper we present an implementation of the bottom left heuristic which requires linear space and quadratic time the algorithm is fairly practical and we believe that even for relatively small values of n it gives the most efficient implementation of the heuristic to date it proceeds by first determining all the possible locations where the next rectangle can fit then selecting the lowest of them it is optimal among all the algorithms based on this exhaustive strategy and its generality makes it adaptable to different packing heuristics

a heuristic for the container loading problem a tertiary tree based dynamic space decomposition approach increasing fuel costs post 911 security concerns and economic globalization provide a strong incentive for container carriers to use available container space more efficiently thereby minimizing the number of container trips and reducing socio economic vulnerability a heuristic algorithm based on a tertiary tree model is proposed to handle the container loading problem clp with weakly heterogeneous boxes a dynamic space decomposition method based on the tertiary tree structure is developed to partition the remaining container space after a block of homogeneous rectangular boxes is loaded into a container this decomposition approach together with an optimal fitting sequencing and an inner right corner occupying placement rule permits a holistic loading strategy to pack a container comparative studies with existing algorithms and an illustrative example demonstrate the efficiency of this algorithm

an effective quasi human based heuristic for solving the rectangle packing problem abstract in this paper we introduce an effective deterministic heuristic less flexibility first for solving the classical np complete rectangle packing problem many effective heuristics implemented for this problem are cpu intensive and non deterministic in nature others including the polynomial approximation methodology j assoc comput mach 32 1 1985 130 are too laborious for practical problem sizes the technique we propose is inspired and developed by enhancing some rule of thumb guidelines resulting from the generation long work experience of human professionals in ancient days although the less flexibility first heuristic is a deterministic algorithm the results are very encouraging this algorithm can consistently produce packing densities of around 99 on most randomly generated large examples as compared with the recent results of a well known simulated annealing based rectangle packing rp algorithm ieee trans computer aided design integrated circuits systems 17 1 1998 60 the results are much better both in less dead space 2 4 vs 6 7 and much less cpu time 9 57 vs 331 78 seconds experimenting our heuristics on a public rectangle packing data set covering instances of 16 97 rectangles the average unpack ratio is quite satisfactory 0 92 for bounding boxes limited to be optimum and 2 68 for the completed packing while most cases spend only a few minutes in cpu time

quantum conditional mutual information and approximate markov chains a state on a tripartite quantum system a otimes b otimes c forms a markov chain if it can be reconstructed from its marginal on a otimes b by a quantum operation from b to b otimes c we show that the quantum conditional mutual information i a c b of an arbitrary state is an upper bound on its distance to the closest reconstructed state it thus quantifies how well the markov chain property is approximated

half sided modular inclusion and the construction of the poincare group in this note we will give a construction of the poincare group out of the modular groups of the wedge algebras provided the groups act on the algebra of every double cone like the associated lorentz boosts this construction will use the concept of half sided modular inclusions instead of the first and second cohomology of the poincare group as used by brunetti guido and longo by our method we obtain directly the poincare group and not its covering group

ts2pack a two level tabu search for the three dimensional bin packing problem three dimensional orthogonal bin packing is a problem np hard in the strong sense where a set of boxes must be orthogonally packed into the minimum number of three dimensional bins we present a two level tabu search for this problem the first level aims to reduce the number of bins the second optimizes the packing of the bins this latter procedure is based on the interval graph representation of the packing proposed by fekete and schepers which reduces the size of the search space we also introduce a general method to increase the size of the associated neighborhoods and thus the quality of the search without increasing the overall complexity of the algorithm extensive computational results on benchmark problem instances show the effectiveness of the proposed approach obtaining better results compared to the existing ones

a two dimensional heuristic decomposition approach to a three dimensional multiple container loading problem abstract efficient container loading has the potential to considerably reduce logistics and transportation costs the container loading problem is computationally complex and despite extensive academic effort there remains room for algorithm improvement real world problems are not always addressed satisfactorily primarily due to the large number of complex constraints and objectives the problem addressed by this work is an unsolved multiple container loading problem introduced by renault on the occasion of the 2014 2015 esicup challenge organized by the euro special interest group on cutting and packing esicup renault s problem requires a large number of different items to be packed into containers of different types and sizes items must be grouped into stacks and respect the this side up constraint the primary objective is to minimize the volume of shipped containers the smallest volume container may be left behind for the next shipment and is excluded from the main objective nevertheless only a limited percentage of each product may be packed into this container additionally a set of secondary objectives is considered this work presents a decomposition approach embedded in a multi phase heuristic for the problem feasible solutions are generated quickly and subsequently improved by local search and post processing procedures experiments revealed that the approach generates optimal solutions for two instances in addition to good quality solutions for those remaining from the renault set the algorithmic contribution is however not restricted to the renault instances experiments on less constrained container loading instances from the literature demonstrate the approach s general applicability and competitiveness

half sided modular inclusions of von neumann algebras letn m be von neumann algebras on a hilbert spaceh a common cyclic and separating vector denote m n resp jm jn the associated modular operators and conjugations assume m it n it n fort 0 we call such an inclusion half sided modular then we prove the existence of a oneparameter unitary groupu a onh a r with generator frac 1 2 pi ln delta cal n ln delta m mathbin lower 3ex hbox buildrel over smash scriptstyle vphantom x 0 and relations 1 itu a it it nit u a nit u e 2 ta for alla t r 2 jmjn u 2 3 nit u 1 it u 1 for allt r 4 n u 1 u 1

nuclear maps and modular structures i general properties abstract we establish a correspondence between the split property of inclusions a b of von neumann algebras and nuclearity properties of the natural embeddings p l a lp b p 1 2 given by the modular theory

the three dimensional bin packing problem the problem addressed in this paper is that of orthogonally packing a given set of rectangular shaped items into the minimum number of three dimensional rectangular bins the problem is strongly np hard and extremely difficult to solve in practice lower bounds are discussed and it is proved that the asymptotic worst case performance ratio of the continuous lower bound is an exact algorithm for filling a single bin is developed leading to the definition of an exact branch and bound algorithm for the three dimensional bin packing problem which also incorporates original approximation algorithms extensive computational results involving instances with up to 90 items are presented it is shown that many instances can be solved to optimality within a reasonable time limit

non classical trigonal curves we study trigonal gorenstein curves of arithmetic genus g 5 such that g has a base point through realizing them as canonical curves lying over a cone using an explicit description of such curves we compute the dimension of their moduli spaces we also investigate non classical trigonal gorenstein curves in fact we give a complete classification of such curves for arithmetic genus g 5 more generally we classify the non classical curves when the characteristic of the constant field is g 1 g 2 or 2g 3 in characteristic 2 we also solve the case g 2n 1

recovery map for fermionic gaussian channels a recovery map effectively cancels the action of a quantum operation to a partial or full extent we study the petz recovery map in the case where the quantum channel and input states are fermionic and gaussian gaussian states are convenient because they are totally determined by their covariance matrix and because they form a closed set under so called gaussian channels using a grassmann representation of fermionic gaussian maps we show that the petz recovery map is also gaussian and determine it explicitly in terms of the covariance matrix of the reference state and the data of the channel as a by product we obtain a formula for the fidelity between two fermionic gaussian states we also discuss subtleties arising from the singularities of the involved matrices a recovery map effectively cancels the action of a quantum operation to a partial or full extent we study the petz recovery map in the case where the quantum channel and input states are fermionic and gaussian gaussian states are convenient because they are totally determined by their covariance matrix and because they form a closed set under so called gaussian channels using a grassmann representation of fermionic gaussian maps we show that the petz recovery map is also gaussian and determine it explicitly in terms of the covariance matrix of the reference state and the data of the channel as a by product we obtain a formula for the fidelity between two fermionic gaussian states we also discuss subtleties arising from the singularities of the involved matrices