an exact algorithm for the pallet loading problem abstract the two dimensional packing problem of finding optimal layouts for identical rectangular boxes on a rectangular pallet has interested or practitioners for many years the problem is np complete and solution methods to date tend to be heuristic this paper discusses the development of an exact tree search algorithm based on a graph theoretic model of the problem

smooth and proper noncommutative schemes and gluing of dg categories abstract in this paper we discuss different properties of noncommutative schemes over a field we define a noncommutative scheme as a differential graded category of a special type we study regularity smoothness and properness for noncommutative schemes admissible subcategories of categories of perfect complexes on smooth projective schemes provide natural examples of smooth and proper noncommutative schemes that are called geometric noncommutative schemes in this paper we show that the world of all geometric noncommutative schemes is closed under an operation of a gluing of differential graded categories via bimodules as a consequence of the main theorem we obtain that for any finite dimensional algebra with separable semisimple part the category of perfect complexes over it is equivalent to a full subcategory of the category of perfect complexes on a smooth projective scheme moreover if the algebra has finite global dimension then the full subcategory is admissible we also provide a construction of a smooth projective scheme that admits a full exceptional collection and contains as a subcollection an exceptional collection given in advance as another application of the main theorem we obtain in characteristic 0 an existence of a full embedding for the category of perfect complexes on any proper scheme to the category of perfect complexes on a smooth projective scheme

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

polynomial parallel volume convexity and contact distributions of random sets we characterize convexity of a random compact set x in d via polynomial expected parallel volume of x the parallel volume of a compact set a is a function of r 0 and is defined here in two steps first we form the parallel set at distance r with respect to a one or two dimensional gauge body b then we integrate the volume of this relative parallel set with respect to all rotations of b we apply our results to characterize convexity of the typical grain of a boolean model via first contact distributions

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

the grothendieck duality theorem via bousfield s techniques and brown representability grothendieck proved that if f x y is a proper morphism of nice schemes then rf has a right adjoint which is given as tensor product with the relative canonical bundle the original proof was by patching local data deligne proved the existence of the adjoint by a global argument and verdier showed that this global adjoint may be computed locally in this article we show that the existence of the adjoint is an immediate consequence of brown s representability theorem lit follows almost as imme diately by smashing arguments that the adjoint is given by tensor product with a dualising complex verdier s base change theorem is an easy conse quence department of mathematics university of virginia charlottesville virginia 22903 e mail address an3rqvirginia edu this content downloaded from 157 55 39 27 on wed 07 sep 2016 06 26 19 utc all use subject to http about jstor org terms

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

a hybrid genetic algorithm for the two dimensional single large object placement problem in the two dimensional single large object placement problem we are given a rectangular master surface which has to be cut into a set of smaller rectangular items with the aim of maximizing the total value of the pieces cut we consider the special case in which the items cannot be rotated and must be cut with their edges always parallel to the edges of the surface we present new greedy algorithms and a hybrid genetic approach with elitist theory immigration rate heuristics on line and tailored crossover operators extensive computational results for a large number of small and large benchmark test problems are presented the results show that our approach outperforms existing heuristic algorithms

curvature measures and random sets ii in choosing models of stochastic geometry three general problems play a role which are closely connected with each other in the present paper second order local geometric properties of random subsets of r d are of interest these properties are described by signed curvature measures in a measure geometric context the theory of point processes on general spaces here on the space of subsets with positive reach provides an appropriate framework for solving construction and measurability problems mean value relations for random curvature measures associated with such set processes are derived by means of invariance properties ergodic interpretations of the curvature densities are also given the appendix provides auxiliary results for random signed radon measures in locally compact separable hausdorff spaces

on the weak computability of a four dimensional orthogonal packing and time scheduling problem this paper proposes a four dimensional orthogonal packing and time scheduling problem the problem differs from the classical packing problems in that the position and orientation of each item in the container can be changed over time in this way the four dimensional space time problem better uses the container time also we consider a general case that all parameters are real numbers which makes the problems more difficult to solve this paper proposes an algorithm and proves that the algorithm could solve the problem optimally by a finite number of operations we say this problem is weak computational meaning that if there exists a universal machine that could represent real numbers and could do unit arithmetic or logical operation on real numbers in finite time then the algorithm could find optimal solutions in finite time this paper also presents a proof of the weak computability over a general case of the three dimensional orthogonal packing problem where all parameters are positive real numbers

an exact two dimensional non guillotine cutting tree search procedure we consider the two dimensional cutting problem of cutting a number of rectangular pieces from a single large rectangle so as to maximize the value of the pieces cut we develop a lagrangean relaxation of a zero one integer programming formulation of the problem and use it as a bound in a tree search procedure subgradient optimization is used to optimize the bound derived from the lagrangean relaxation problem reduction tests derived from both the original problem and the lagrangean relaxation are given incorporating the bound and the reduction tests into a tree search procedure enables moderately sized problems to be solved

quantitative description and numerical simulation of random microstructures of composites and their effective elastic moduli abstract a digital image processing technique is used for measurement of centroid coordinates of fibers with forthcoming estimation of statistical parameters and functions describing the stochastic structure of a real fiber composite comparative statistical analysis of the real and numerically simulated structure are performed accompanying of known methods of the generation of random configurations by the random shaking procedure allows creating of the most homogenized and mixed structures that do not depend on the initial protocol of particle generation we consider a linearly elastic composite medium which consists of a homogeneous matrix containing a statistically homogeneous set of ellipsoidal inclusions the multiparticle effective field method see for references buryachenko appl mech rev 2001a 54 1 47 based on the theory of functions of random variables and green s functions is used for demonstration of the dependence of effective elastic moduli of fiber composites on the radial distribution functions estimated from the real experimental data as well as from the ensembles generated by the method proposed

an incremental linear time algorithm for recognizing interval graphs the fastest known algorithm for recognizing interval graphs s booth and s lucker j comput system sci 13 1976 pp 335 379 iteratively manipulates the system of all maximal cliques of the given graph in a rather complicated way in order to construct a consecutive arrangement more precisely a tree representation of all possible consecutive arrangements this paper presents a much simpler algorithm using a related but much more informative tree representation of interval graphs this tree is constructed in an incremental fashion by adding vertices to the graph in a predefined order such that adding a vertex u takes o operatorname adj u 1 amortized time

residual analysis for spatial point processes with discussion summary we define residuals for point process models fitted to spatial point pattern data and we propose diagnostic plots based on them the residuals apply to any point process model that has a conditional intensity the model may exhibit spatial heterogeneity interpoint interaction and dependence on spatial covariates some existing ad hoc methods for model checking quadrat counts scan statistic kernel smoothed intensity and berman s diagnostic are recovered as special cases diagnostic tools are developed systematically by using an analogy between our spatial residuals and the usual residuals for non spatial generalized linear models the conditional intensity plays the role of the mean response this makes it possible to adapt existing knowledge about model validation for generalized linear models to the spatial point process context giving recommendations for diagnostic plots a plot of smoothed residuals against spatial location or against a spatial covariate is effective in diagnosing spatial trend or co variate effects q q plots of the residuals are effective in diagnosing interpoint interaction

finite dimensional differential graded algebras and their geometric realizations abstract we prove that for any finite dimensional differential graded algebra with separable semisimple part the category of perfect modules is equivalent to a full subcategory of the category of perfect complexes on a smooth projective scheme with a full separable semi exceptional collection moreover we also show that it gives a characterization of such categories assuming that a subcategory is idempotent complete and has a classical generator

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

deriving fundamental measure theory from the virial series consistency with the zero dimensional limit fundamental measure theory fmt for hard particles has great potential for predicting the phase behavior of colloidal and nanometric shapes the modern versions of fmt are usually derived from the zero dimensional limit a system of at most one particle confined in a collection of cavities in the limit that all cavities shrink to the size of the particle in phys rev e 85 041150 2012 a derivation from an approximated and resummed virial expansion was presented whose result was not fully consistent with the fmt from the zero dimensional limit here we improve on this derivation and obtain exactly the same fmt functional as was obtained earlier from the zero dimensional limit as a result further improvements of fmt based on the virial expansion can now be formulated some of which we suggest in the outlook

the three dimensional knapsack problem with balancing constraints abstract in this paper we introduce a new packing problem the three dimensional knapsack problem with balancing constraints 3bkp the extension of the three dimensional knapsack problem 3kp where additional constraints related to the packing center of mass are given the 3bkp consists in orthogonally packing a subset of three dimensional weighted items into a knapsack in order to maximize the total profit of the loaded items the items must not overlap and the packing center of mass must lie into a predefined boxed domain inside the knapsack we assume that items can be rotated we give a mip model for the problem upper bounds and an efficient heuristic to solve large size instances the computational results show that the mip model cannot find optimal solutions except for small size instances but it can be used to calculate upper and lower bounds it is shown that our heuristic outperforms the solution quality both of the mip model and the heuristics available in the literature explicitly designed to solve the 3kp

on thermoelastostatics of composites with nonlocal properties of constituents ii estimation of effective material and field parameters abstract one considers a linear thermoelastic composite medium which consists of a homogeneous matrix containing a statistically homogeneous random set of ellipsoidal uncoated or coated heterogeneities it is assumed that the stress strain constitutive relations of constituents are described by the nonlocal integral operators whereas the equilibrium and compatibility equations remain unaltered as in classical local elasticity the general integral equations connecting the stress and strain fields in the point being considered and the surrounding points are obtained the method is based on a centering procedure of subtraction from both sides of a known initial integral equation their statistical averages obtained without any auxiliary assumptions such as e g effective field hypothesis implicitly exploited in the known centering methods in a simplified case of using of the effective field hypothesis for analyzing composites with one sort of heterogeneities one proves that the effective moduli explicitly depend on both the strain and stress concentrator factor for one heterogeneity inside the infinite matrix and does not directly depend on the elastic properties local or nonlocal of heterogeneities in such a case the levin s 1967 formula in micromechanics of composites with locally elastic constituents is generalized to their nonlocal counterpart a solution of a volume integral equation for one heterogeneity subjected to inhomogeneous remote loading inside an infinite matrix is proposed by the iteration method the operator representation of this solution is incorporated into the new general integral equation of micromechanics without exploiting of basic hypotheses of classical micromechanics such as both the effective field hypothesis and ellipsoidal symmetry assumption quantitative estimations of results obtained by the abandonment of the effective field hypothesis are presented

on the empty space function of some germ grain models abstract we derive and discuss formulas for the density and the hazard rate of the empty space function of a germ grain model in r d generated by a stationary point process and i i d convex primary grains n n n that are independent of our formulas are based on the palm probability of the germ process and the mean generalized curvature measure of the grain particular attention is paid to cluster models where the grains form a poisson cluster process our discussion of specific gauss poisson models with spherical grains provides some motivation for the use of the failure rate of f to detect clustering effects in the general case we propose a family of functions comparing the behaviour in the neighbourhood of a typical germ with the neighbourhood of an arbitrary point in space these characteristics can be used to measure effects of clustering and spatial interactions between the locations of the individual grains

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

entropic descriptor of a complex behaviour abstract we propose a new type of entropic descriptor that is able to quantify the statistical complexity a measure of complex behaviour by taking simultaneously into account the average departures of a system s entropy s from both its maximum possible value s max and its minimum possible value s min when these two departures are similar to each other the statistical complexity is maximal we apply the new concept to the variability over a range of length scales of spatial or grey level pattern arrangements in simple models the pertinent results confirm the fact that a highly non trivial length scale dependence of the entropic descriptor makes it an adequate complexity measure able to distinguish between structurally distinct configurational macrostates with the same degree of disorder a feature that makes it a good tool for discerning structures in complex patterns

exact solution of the two dimensional finite bon packing problem given a set of rectangular pieces to be cut from an unlimited number of standardized stock pieces bins the two dimensional finite bin packing problem is to determine the minimum number of stock pieces that provide all the pieces the problem is np hard in the strong sense and finds many practical applications in the cutting and packing area we analyze a well known lower bound and determine its worst case performance we propose not lower bounds which are used within a branch and bound algorithm for the exact solution of the problem extensive computational testing on problem instances from the literature involving up to 120 pieces shows the effectiveness of the proposed approach

the homotopy theory of dg categories and derived morita theory the main purpose of this work is to study the homotopy theory of dg categories up to quasi equivalences our main result is a description of the mapping spaces between two dg categories c and d in terms of the nerve of a certain category of c d bimodules we also prove that the homotopy category ho dg cat possesses internal hom s relative to the derived tensor product of dg categories we use these two results in order to prove a derived version of morita theory describing the morphisms between dg categories of modules over two dg categories c and d as the dg category of c d bi modules finally we give three applications of our results the first one expresses hochschild cohomology as endomorphisms of the identity functor as well as higher homotopy groups of the classifying space of dg categories i e the nerve of the category of dg categories and quasi equivalences between them the second application is the existence of a good theory of localization for dg categories defined in terms of a natural universal property our last application states that the dg category of continuous morphisms between the dg categories of quasi coherent resp perfect complexes on two schemes resp smooth and proper schemes is quasi equivalent to the dg category of quasi coherent resp perfect complexes on their product

a local steiner type formula for general closed sets and applications we introduce support curvature measures of an arbitrary closed set a in d and establish a local steiner type formula for the localized parallel volume of a we derive some of the basic properties of these support measures and explore how they are related to the curvature measures available in the literature then we use the support measures in analysing contact distributions of stationary random closed sets with a particular emphasis on the boolean model with general compact particles

random cell complexes and generalised sets on introduit les concepts de complexes cellulaires stochastiques d ensembles generalises et de leurs ensembles moyens

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

microstructure functions for a model of statistically inhomogeneous random media we propose a model for statistically inhomogeneous two phase random media including functionally graded materials consisting of inhomogeneous fully penetrable poisson distributed spheres this model can be constructed for any specified variation of volume fraction and permits one to represent and evaluate certain n point correlation functions that statistically characterize the microstructure for this model unlike the case of statistically homogeneous media the microstructure functions depend upon the absolute positions of their arguments however as with homogeneous random media this microstructural information will be useful in obtaining rigorous estimates of the effective properties of such systems copyright ital 1998 ital the american physical society

using tree search bounds to enhance a genetic algorithm approach to two rectangle packing problems a popular approach when using a genetic algorithm in the solution of constrained problems is to exploit problem specific information by representing individuals as ordered lists a construction heuristic is then often used as a decoder to produce a solution from each ordering in such a situation further information is often available in the form of bounds on the partial solutions this paper uses two two dimensional packing problems to illustrate how this information can be incorporated into the genetic operators to improve the overall performance of the search our objective is to use the packing problems as a vehicle for investigating a series of modifications of the genetic algorithm approach based on information from bounds on partial solutions

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

on support measures in minkowski spaces and contact distributions in stochastic geometry this paper is concerned with contact distribution functions of a random closed set n 1 n in r d where the n are assumed to be random nonempty convex bodies these distribution functions are defined here in terms of a distance function which is associated with a strictly convex gauge body structuring element that contains the origin in its interior support measures with respect to such distances will be introduced and extended to sets in the local convex ring these measures will then be used in a systematic way to derive and describe some of the basic properties of contact distribution functions most of the results are obtained in a general nonstationary setting only the final section deals with the stationary case

a random field model for generating synthetic microstructures of functionally graded materials this article presents a new level cut inhomogeneous filtered poisson random field model for representing two phase microstructures of statistically inhomogeneous functionally graded materials with fully penetrable embedded particles the model involves an inhomogeneous filtered poisson random field comprising a sum of deterministic kernel functions that are scaled by random variables and a cut of the filtered poisson field above a specified level the resulting level cut field depends on the poisson intensity level kernel functions random scaling variables and random rotation matrices a reconstruction algorithm including model calibration and monte carlo simulation is presented for generating samples of two phase microstructures of statistically inhomogeneous media numerical examples demonstrate that the model developed is capable of producing a wide variety of two and three dimensional microstructures of functionally graded composites containing particles of various sizes shapes densities gradations and orientations an example involving finite element analyses of random microstructures leading to statistics of effective properties of functionally graded composites illustrates the usefulness of the proposed model copyright 2008 john wiley sons ltd

blaschke and minkowski endomorphisms of convex bodies we consider maps of the family of convex bodies in euclidean d dimensional space into itself that are compatible with certain structures on this family a minkowski endomorphism is a continuous minkowski additive map that commutes with rotations for d 3 a representation theorem for such maps is given showing that they are mixtures of certain prototypes these prototypes are obtained by applying the generalized spherical radon transform to support functions we give a complete characterization of weakly monotonic minkowski endomorphisms a corresponding theory is developed for blaschke endomorphisms where additivity is now understood with respect to blaschke addition using a special mixed volume an adjoining operator can be introduced this operator allows one to identify the class of blaschke endomorphisms with the class of weakly monotonic non degenerate and translation covariant minkowski endomorphisms the following application is also shown if a weakly monotonic and non trivial endomorphism maps a convex body to a homothet of itself then this body must be a ball

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

categorical resolution of singularities abstract building on the concept of a smooth dg algebra we define the notion of a smooth derived category we then propose the definition of a categorical resolution of singularities our main example is the derived category d x of quasi coherent sheaves on a scheme x we prove that d x has a canonical categorical resolution if the base field is perfect and x is a separated scheme of finite type with a dualizing complex

uniqueness of enhancement for triangulated categories the paper contains general results on the uniqueness of a dg enhancement for trian gulated categories as a consequence we obtain such uniqueness for the unbounded categories of quasi coherent sheaves for the triangulated categories of perfect complexes and for the bounded de rived categories of coherent sheaves on quasi projective schemes if a scheme is projective then we also prove a strong uniqueness for the triangulated category of perfect complexes and for the bounded de rived categories of coherent sheaves these results directly imply that fully faithful functors from the bounded derived categories of coherent sheaves and the triangulated categories of perfect complexes on projective schemes can be represented by objects on the product

an exact algorithm for general orthogonal two dimensional knapsack problems abstract we present a new exact tree search procedure for solving two dimensional knapsack problems in which a number of small rectangular pieces each of a given size and value are required to be cut from a large rectangular stock plate the objective is to maximise the value of pieces cut or minimise the wastage we consider the case where there is a maximum number of times that a piece may be used in a cutting pattern the algorithm limits the size of the tree search by using a bound derived from a langrangean relaxation of a 0 1 integer programming formulaton of the problem subgradient optimisation is used to optimise this bound reduction tests derived from both the original problem and the lagrangean relaxation produce substantial computational gains the computational performance of the algorithm indicates that it is an effective procedure capable of solving optimally practical two dimensional cutting problems of medium size

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

packing d dimensional bins in d stages we consider the d dimensional bin packing problem the most relevant generalization of classical bin packing and show a general result about the asymptotic worst case ratio of a wide class of approximation algorithms that construct solutions in d stages containing many heuristics previously considered in the literature moreover we give the exact value of the asymptotic worst case ratio between the optimal solution and the best solution obtainable by packing the items in d stages showing how to achieve such a ratio efficiently the key property behind our result is the asymptotic optimality of the fractional relaxation of one dimensional bin packing an important by product of the approximation schemes for the problem from the 1980s which to the best of our knowledge is used here for the first time only for the sake of the analysis for the two dimensional case we push the approximablity threshold below 2 after more than 20 years reaching 1 691 but also improve and widely simplify the analysis of the previous best method from the 1980s moreover for d 3 we improve the previous approximability threshold by a factor of 1 691 a notable improvement when d is small

integral geometry in minkowski spaces abstract we investigate the possibility of extending classical integral geometric results involving lower dimensional areas from euclidean space to minkowski spaces finite dimensional banach spaces of the two natural notions of area in a minkowski space due respectively to busemann and to holmes and thompson the latter turns out to be the more tractable one for the holmes thompson area we derive a translative intersection formula and in the class of hypermetric minkowski spaces full analogues of the crofton formulae for rectifiable sets and for convex bodies for the busemann k area we give a short proof of the fact that it coincides for k rectifiable sets with the k dimensional hausdorff measure induced by the minkowski metric

an and or graph approach to the solution of two dimensional non guillotine cutting problems this work is concerned with unconstrained two dimensional non guillotine cutting problems where a rectangular plate is cut into a number of rectangular pieces in such a way as to optimize an objective function we reduce the state space of the problem by considering non guillotine cutting patterns which are combinations of guillotine and simple non guillotine cuts these cutting patterns are represented as complete paths in an and or graph and a branch and bound method is described we provide rules and heuristics that reduce the graph search and present computational results from randomly generated examples as well as from the literature

efficient measurement of the percolation threshold for fully penetrable discs we study the percolation threshold for fully penetrable discs by measuring the average location of the frontier for a statistically inhomogeneous distribution of fully penetrable discs we use two different algorithms to efficiently simulate the frontier including the continuum analogue of an algorithm previously used for gradient percolation on a square lattice we find that c 0 676 339 0 000 004 thus providing an extra significant digit of accuracy to this constant some figures in this article appear in colour in the electronic version see www iop org

on the second derivative of the spherical contact distribution function of smooth grain models we consider a stationary grain model in d with convex compact and smoothly bounded grains we study the spherical contact distribution function f of and derive under suitable assumptions an explicit formula for its second derivative f the value f 0 is of a simple form and admits a clear geometric interpretation for the boolean model we obtain an interesting new formula for the d 1 st quermass density

palm pairs and the general mass transport principle we consider a lcsc group g acting on a borel space s and on an underlying finite measure space our first main result is a transport formula connecting the palm pairs of jointly stationary random measures on s a key and new technical result is a measurable disintegration of the haar measure on g along the orbits the second main result is an intrinsic characterization of the palm pairs of a g invariant random measure we then proceed with deriving a general version of the mass transport principle for possibly non transitive and non unimodular group operations first in a deterministic and then in its full probabilistic form