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

complex symplectomorphisms and pseudo kahler islands in the quantization of toric manifolds let p be a delzant polytope we show that the quantization of the corresponding toric manifold x p in toric kahler polarizations and in the toric real polarization are related by analytic continuation of hamiltonian flows evaluated at time t sqrt 1 s we relate the quantization of x p in two different toric kahler polarizations by taking the time sqrt 1 s hamiltonian flow of strongly convex functions on the moment polytope p by taking s to infinity we obtain the quantization of x p in the singular real toric polarization recall that x p has an open dense subset which is biholomorphic to mathbb c n the quantization of x p in a toric kahler polarization can also be described by applying the complexified hamiltonian flow of the abreu guillemin symplectic potential g at time t sqrt 1 to an appropriate finite dimensional subspace of quantum states in the quantization of t mathbb t n in the vertical polarization by taking other imaginary times t k sqrt 1 k in mathbb r we describe toric kahler metrics with cone singularities along the toric divisors in x p for convex hamiltonian functions and sufficiently negative imaginary part of the complex time we obtain degenerate kahler structures which are negative definite in some regions of x p we show that the pointwise and l 2 norms of quantum states are asymptotically vanishing on negative definite regions

equivariant cohomology of k contact manifolds we investigate the equivariant cohomology of the natural torus action on a k contact manifold and its relation to the topology of the reeb flow using the contact moment map we show that the equivariant cohomology of this action is cohen macaulay the natural substitute of equivariant formality for torus actions without fixed points as a consequence generic components of the contact moment map are perfect morse bott functions for the basic cohomology of the orbit foliation mathcal f of the reeb flow assuming that the closed reeb orbits are isolated we show that the basic cohomology of mathcal f vanishes in odd degrees and that its dimension equals the number of closed reeb orbits we characterize k contact manifolds with minimal number of closed reeb orbits as real cohomology spheres we also prove a gkm type theorem for k contact manifolds which allows to calculate the equivariant cohomology algebra under the nonisolated gkm condition

symplectic surgery and the spinc dirac operator abstract letgbe a compact connected lie group and m a compact hamiltoniang space with moment mapj m g under the assumption that these data are pre quantizable one can construct an associated spinc dirac operator formula whose equivariant index yields a virtual representation ofg we prove a conjecture of guillemin and sternberg that if 0 is a regular value ofj the multiplicityn 0 of the trivial representation in the index space formula is equal to the index of the spinc dirac operator for the symplectic quotientm0 j 1 0 g this generalizes previous results for the case thatg tis abelian i e a torus

an improved bl algorithm for genetic algorithm of the orthogonal packing of rectangles this paper proposes an improved bl algorithm for genetic algorithm of the orthogonal packing of rectangles some improvements on the fitness function of genetic algorithm for the orthogonal packing of rectangles are also suggested solutions of two numerical examples show the effectiveness of these improvements

toric stacks ii intrinsic characterization of toric stacks the purpose of this paper and its prequel is to introduce and develop a theory of toric stacks which encompasses and extends several notions of toric stacks defined in the literature as well as classical toric varieties while the focus of the prequel is on how to work with toric stacks the focus of this paper is how to show a stack is toric for toric varieties a classical result says that a finite type scheme with an action of a dense open torus arises from a fan if and only if it is normal and separated in the same spirit the main result of this paper is that any artin stack with an action of a dense open torus arises from a stacky fan under reasonable hypotheses

geometric quantization and multiplicities of group representations the heisenberg uncertainty principle says that it is impossible to determine simultaneously the position and momentum of a quantum mechanical particle this can be rephrased as follows the smallest subsets of classical phase space in which the presence of a quantum mechanical particle can be detected are its lagrangian submanifolds for this reason it makes sense to regard the lagrangian submanifolds of phase space as being its true points see weinstein 17 now let g be a compact lie group and g x x x a hamiltonian action of g on x see w for definitions it is well known that the fixed points of this action form a symplectic submanifold of x see for instance guillemin and sternberg 5 however what can one say about the fixed points of g we will show that they are also the points of a symplectic manifold xc this manifold is the marsden weinstein reduction of x with respect to the zero orbit in g and will be described in sect 2 it was introduced in a completely different context fl om ours by marsden and weinstein 12 problems in classical mechanics can often be reduced to the study of hamiltonian systems on symplectic manifolds and problems in quantum mechanics to the study of linear operators on hilbert space this fact has inspired a number of efforts to quantize symplectic geometry by devising schemes for associating hilbert space to symplectic manifolds the no go theorems of groenwald and van hove impose some embarrassing limitations on all such schemes however it seems to be a useful idea heuristically to think of every symplectic manifold x a ic 1 as being symbiotically associated with a hilbert space xq a t in such a way that the classical observables on the first space correspond to quantum observables on the second space the heuristics further suggests that if g is a group of symmetries of x la i j it should also be a group of symmetries of xq a tum in this heuristic spirit we will state the main conjecture of this paper

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

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

a two stage intelligent search algorithm for the two dimensional strip packing problem this paper presents a two stage intelligent search algorithm for a two dimensional strip packing problem without guillotine constraint in the first stage a heuristic algorithm is proposed which is based on a simple scoring rule that selects one rectangle from all rectangles to be packed for a given space in the second stage a local search and a simulated annealing algorithm are combined to improve solutions of the problem in particular a multi start strategy is designed to enhance the search capability of the simulated annealing algorithm extensive computational experiments on a wide range of benchmark problems from zero waste to non zero waste instances are implemented computational results obtained in less than 60 seconds of computation time show that the proposed algorithm outperforms the supposedly excellent algorithms reported recently on average it performs particularly better for large instances

semi classical weights and equivariant spectral theory abstract we prove inverse spectral results for differential operators on manifolds and orbifolds invariant under a torus action these inverse spectral results involve the asymptotic equivariant spectrum which is the spectrum itself together with very large weights of the torus action on eigenspaces more precisely we show that the asymptotic equivariant spectrum of the laplace operator of any toric metric on a generic toric orbifold determines the equivariant biholomorphism class of the orbifold we also show that the asymptotic equivariant spectrum of a t n invariant schrodinger operator on r n determines its potential in some suitably convex cases in addition we prove that the asymptotic equivariant spectrum of an s 1 invariant metric on s 2 determines the metric itself in many cases finally we obtain an asymptotic equivariant inverse spectral result for weighted projective spaces as a crucial ingredient in these inverse results we derive a surprisingly simple formula for the asymptotic equivariant trace of a family of semi classical differential operators invariant under a torus action

frankel s theorem in the symplectic category we prove that if an n 1 dimensional torus acts symplectically on a 2n dimensional symplectic manifold then the action has a fixed point if and only if the action is hamiltonian one may regard it as a symplectic version of frankel s theorem which says that a kahler circle action has a fixed point if and only if it is hamiltonian the case of n 2 is the well known theorem by mcduff

kahler metrics on toric orbifolds a theorem of e lerman and s tolman generalizing a result of t delzant states that compact symplectic toric orbifolds are classified by their moment polytopes together with a positive integer label attached to each of their facets in this paper we use this result and the existence of global actionangle coordinates to give an effective parametrization of all compatible toric complex structures on a compact symplectic toric orbifold by means of smooth functions on the corresponding moment polytope this is equivalent to parametrizing all toric kahler metrics and generalizes an analogous result for toric manifolds a simple explicit description of interesting families of extremal kahler metrics arising from recent work of r bryant is given as an application of the approach in this paper the fact that in dimension four these metrics are self dual and conformally einstein is also discussed this gives rise in particular to a one parameter family of self dual einstein metrics connecting the well known eguchi hanson and taub nut metrics

toric generalized kahler ricci solitons with hamiltonian 2 form we show that the generalized kahler ricci soliton equation on 4 dimensional toric kahler orbifolds reduces to odes assuming there is a hamiltonian 2 form this leads to an explicit resolution of this equation on labelled triangles and convex labelled quadrilaterals in particular we give the explicit expression of the kahler ricci solitons of weighted projective planes as well as new examples

convex polytopes and quasilattices from the symplectic viewpoint we construct for each convex polytope possibly nonrational and nonsimple a family of compact spaces that are stratified by quasifolds i e each of these spaces is a collection of quasifolds glued together in an suitable way a quasifold is a space locally modelled on mathbb r k modulo the action of a discrete possibly infinite group the way strata are glued to each other also involves the action of an infinite discrete group each stratified space is endowed with a symplectic structure and a moment mapping having the property that its image gives the original polytope back these spaces may be viewed as a natural generalization of symplectic toric varieties to the nonrational setting

clustering of clusters abstract an algorithm for the clustering of existing clusters is introduced in this paper the algorithm was adopted from fuzzy c mean and modifications made to take into account the extra information i e some data samples already form clusters partition coefficients together with some other criteria are used for testing cluster validity the method was applied on chinese character recognition and an encouraging result was obtained

convergence properties of the generalized fuzzy c means clustering algorithms fuzzy c means fcm clustering algorithms have been widely used to solve clustering problems yang and yu 1 extended these to optimization procedures with respect to any probability distribution they showed that the optimal cluster centers are the fixed points of these generalized fcm clustering algorithms the convergence properties of algorithms are the important theoretical issue in this paper we present convergence properties of the generalized fcm clustering algorithms these are global convergence local convergence and its rate of convergence

new polytope decompositions and euler maclaurin formulas for simple integral polytopes abstract we give new weighted decompositions for simple polytopes generalizing previous results of lawrence varchenko and brianchon gram we start with witten s non abelian localization principle in equivariant cohomology for the norm square of the moment map in the context of toric varieties to obtain a decomposition for delzant polytopes then by a purely combinatorial argument we show that this formula holds for any simple polytope as an application we study euler maclaurin formulas

hamiltonian group actions on symplectic deligne mumford stacks and toric orbifolds abstract we develop differential and symplectic geometry of differentiable deligne mumford stacks orbifolds including hamiltonian group actions and symplectic reduction as an application we construct new examples of symplectic toric dm stacks

toric symplectic stacks abstract we give an intrinsic definition of compact toric symplectic stacks and show that they are classified by simple convex polytopes equipped with some additional combinatorial data this generalizes delzant s classification of compact toric symplectic manifolds as an application we show that any compact toric symplectic stack can be deformed to an ineffective toric orbifold

a multiple hyper ellipsoidal subclass model for an evolutionary classifier abstract a pattern classification scheme in which the classifier is able to grow and evolve during the operation process is presented the evolutionary property of the classifier is made possible by modeling the pattern vectors in multiple hyper ellipsoidal subclass distributions learning of the classifier takes place at the subclass levels only this property allows the classifier to retain its previously learned patterns while accepting and learning new pattern classes the classifier is suitable to operate in dynamical environments where continuous updating of the pattern class distributions is needed

group valued implosion and parabolic structures the purpose of this paper is twofold first we extend the notion of symplectic implosion to the category of quasi hamiltonian k manifolds where k is a simply connected compact lie group the imploded cross section of the double k k turns out to be universal in a suitable sense it is a singular space but some of its strata have a nonsingular closure this observation leads to interesting new examples of quasi hamiltonian k manifolds such as the spinning 2 n sphere for k su n secondly we construct a universal master moduli space of parabolic bundles with structure group k over a marked riemann surface the master moduli space carries a natural action of a maximal torus of k and a torus invariant stratification into manifolds each of which has a symplectic structure an essential ingredient in the construction is the universal implosion paradoxically although the universal implosion has no complex structure it is the four sphere for k su 2 the master moduli space turns out to be a complex algebraic variety

on strong consistency of the fuzzy generalized nearest neighbor rule the k nearest neighbor rule k nnr is a well known nonparametric decision rule in pattern classification fuzzy set theory has been widely used to represent the uncertainty of class membership several researchers extended conventional k nnr to fuzzy k nnr such as bezdek et al fuzzy sets and systems 18 1986 237 256 keller et al ieee trans syst man and cybern 15 4 1985 580 585 b6reau and dubuisson fuzzy sets and systems 44 1991 17 32 in this paper we describe a fuzzy generalized k nn algorithm this algorithm is a unified approach to a variety of fuzzy k nnr s then we create the strong consistency of posterior risk of the fuzzy generalized nnr

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

toric stacks i the theory of stacky fans the purpose of this paper and its sequel is to introduce and develop a theory of toric stacks which encompasses and extends several notions of toric stacks defined in the literature as well as classical toric varieties in this paper we define a toric stack as the stack quotient of a toric variety by a subgroup of its torus we also define a generically stacky version any toric stack arises from a combinatorial gadget called a stacky fan we develop a dictionary between the combinatorics of stacky fans and the geometry of toric stacks stressing stacky phenomena such as canonical stacks and good moduli space morphisms we also show that smooth toric stacks carry a moduli interpretation extending the usual moduli interpretations of p n and a 1 g m indeed smooth toric stacks precisely solve moduli problems specified by generalized effective cartier divisors with given linear relations and given intersection relations smooth toric stacks therefore form a natural closure to the class of moduli problems introduced for smooth toric varieties and smooth toric dm stacks in papers by cox and perroni respectively we include a plethora of examples to illustrate the general theory we hope that this theory of toric stacks can serve as a companion to an introduction to stacks in much the same way that toric varieties can serve as a companion to an introduction to schemes

foliation groupoids and their cyclic homology the purpose of this paper is to prove two theorems which concern the position of etale groupoids among general smooth or lie groupoids our motivation comes from the noncommutative geometry and algebraic topology concerning leaf spaces of foliations here one is concerned with invariants of the holonomy groupoid of a foliation such as the cohomology of its classifying space the cyclic homology of its smooth convolution algebra or the ktheory of the cconvolution algebras many results here depend on the fact that such a holonomy groupoid can be reduced to what is called a complete transversal of the foliation giving rise to an equivalent etale groupoid for etale groupoids sometimes called rdiscrete groupoids in the literature the cyclic homology sheaf theory and classifying spaces are each well understood as is the relation between these

convergence rate of the fuzzy generalized nearest neighbor rule tizzy k nearest neighbor rule k nnr has been applied in a variety of substantive areas yang and chen l described a fuzzy generalized k nn algorithm which is a unified approach to a variety of fuzzy k nnr s they created the strong consistency of posterior risk of the fuzzy generalized nnr in this paper we give their convergence rate that is the convergence rate of posterior risk of the fuzzy generalized nnr is exponentially fast

on a class of fuzzy c numbers clustering procedures for fuzzy data this paper describes a class of fuzzy clustering procedures for fuzzy data most fuzzy clustering techniques are designed for handling crisp data with their class memberships using the idea of fuzzy set theory here we derive new types of fuzzy clustering procedures in dealing with fuzzy data these procedures are called fuzzy c numbers fcn clusterings specially we construct these fcns for u type triangular trapezoidal and normal fuzzy numbers k jwoy s cluster analysis fuzzy clustering fuzzy data analysis fuzzy numbers fuzzy intervals fuzzy c means clustering fuzzy c numbers clustering

convexity properties of the moment mapping ii be its associated momen t mapping see w for definitions the set x is the union of co adjoint orbits the main result of this paper is a description of the orbit structure of this set to describe this result we first note that the coadjoint orbits in q can be parametr ized as follows let t be a caf tan subalgebra of g and let b be a positive definite g invariant bilinear form on by means of b we get a g equivariant identification q let t be the subspace of q corresponding to t then every co adjoint orbit intersects i in a weyl group orbit so there is a one one correspondence between co adjoint orbits in q and weyl group orbits in t to get an even sharper parametrization let t be a weyl chamber in then every weyl group orbit intersects t in a single point so t is a modul i space for the co adjoint orbits two orbits in x are said to be of the same type if they are isomorphic as homogeneous g spaces since x is compact there are at most a finite number of different types of orbits see 1113 let h be a closed subgroup of t of codimension r and let cit be the set of all orbits in x of type g h

stratified symplectic spaces and reduction let m w be a hamiltonian g space with proper momentum map j m g it is well known that if zero is a regular value of j and g acts freely on the level set j 0 then the reduced space mo j 0 g is a symplectic manifold we show that if the regularity assumptions are dropped the space mo is a union of symplectic manifolds i e it is a stratified symplectic space arms et al 2 proved that mo possesses a natural poisson bracket using their result we study hamiltonian dynamics on the reduced space in particular we show that hamiltonian flows are strata preserving and give a recipe for lifting a reduced hamiltonian flow to the level set j 0 finally we give a detailed description of the stratification of mo and prove the existence of a connected

the convexity package for hamiltonian actions on conformal symplectic manifolds consider a hamiltonian action of a compact connected lie group on a conformal symplectic manifold we prove a convexity theorem for the moment map under the assumption that the action is of lee type which establishes an analog of kirwan s convexity theorem in conformal symplectic geometry

hamiltonian 2 forms in kahler geometry ii global classification we present a classification of compact kahler manifolds admitting a hamiltonian 2 form which were classified locally in part i of this work this involves two components of independent interest the first is the notion of a rigid hamiltonian torus action this natural condition for torus actions on a kahler manifold was introduced locally in part i but such actions turn out to be remarkably well behaved globally leading to a fairly explicit classification up to a blow up compact kahler manifolds with a rigid hamiltonian torus action are bundles of toric kahler manifolds the second idea is a special case of toric geometry which we call orthotoric we prove that orthotoric kahler manifolds are diffeomorphic to complex projective space but we extend our analysis to orthotoric orbifolds where the geometry is much richer we thus obtain new examples of kahler einstein 4 orbifolds combining these two themes we prove that compact kahler manifolds with hamiltonian 2 forms are covered by blow downs of projective bundles over kahler products and we describe explicitly how the kahler metrics with a hamiltonian 2 form are parameterized we explain how this provides a context for constructing new examples of extremal kahler metrics in particular a subclass of such metrics which we call weakly bochner flat we also provide a self contained treatment of the theory of compact toric kahler manifolds since we need it and find the existing literature incomplete

flow reconstruction and invariant tori in the spatial three body problem abstract we deal with the spatial three body problem in the various regimes where the hamiltonian is split as the sum of two keplerian systems plus a small perturbation by averaging over the mean anomalies truncating higher order terms and using singular reduction theory we get a one degree of freedom hamiltonian system departing from the analysis performed in 39 concerning the relative equilibria of this reduced system we carry out the reconstruction of the kam tori surrounding the motions associated to each elliptic equilibrium the existence of five dimensional kam tori for the spatial three body problem is established these tori surround various types of motions from circular to near rectilinear passing through coplanar or perpendicular