real forms of complex surfaces of constant mean curvature it is known that complex constant mean curvature cmc for short immersions in c 3 are natural complexifications of cmc immersions in r 3 in this paper conversely we consider real form surfaces of a complex cmc immersion which are defined from real forms of the twisted sl 2 c loop algebra asl 2 and classify all such surfaces according to the classification of real forms of asl 2 there are seven classes of surfaces which are called integrable surfaces and all integrable surfaces will be characterized by the lorentz harmonicities of their gauss maps into the symmetric spaces s 2 h 2 s 1 1 or the 4 symmetric space sl 2 u 1 we also give a unification to all integrable surfaces via the generalized weierstrass type representation

thickenings and conformal gravity a twistor correspondence is given for complex conformal space times with vanishing bach and eastwood dighton tensors when the weyl curvature is algebraically general these equations are precisely the conformal version of einstein s vacuum equations with cosmological constant this gives a fully curved version of the linearized correspondence of baston and mason b m

a multi objective programming approach to 1 5 dimensional assortment problem in this paper we study a 1 5 dimensional cutting stock and assortment problem which includes determination of the number of different widths of roll stocks to be maintained as inventory and determination of how these roll stocks should be cut by choosing the optimal cutting pattern combinations we propose a new multi objective mixed integer linear programming milp model in the form of simultaneously minimization two contradicting objectives related to the trim loss cost and the combined inventory cost in order to fulfill a given set of cutting orders an equivalent nonlinear version and a particular case related to the situation when a producer is interested in choosing only a few number of types among all possible roll sizes have also been considered a new method called the conic scalarization is proposed for scalarizing non convex multi objective problems and several experimental tests are reported in order to demonstrate the validity of the developed modeling and solving approaches

superconformal geometry in three dimensions a category of superconformal 3 2 dimensional supermanifolds is introduced and studied it is shown that each superconformal structure has associated with it a distinguished vector bundle or local supertwistor bundle equipped with a superconnection that has to be regarded as a supersymmetric extension of three dimensional conformal cartan s connection the curvature of the superconnection provides a superconformal invariant that vanishes if and only if the 3 2 supermanifold is locally isomorphic to isotropic super grassmanian gi 0 2 c1 4 b with b 0sp 1 4 a choice of scale of superconformal structure on a 3 2 supermanifold m determines canonically a volume form on m that induces on the underlying manifold mrd the action of n 1 d 3 supergravity with cosmological constant it is proved that a superconformal structure on m can be encoded into the holomorphic structure of the associated with m 3 1 dimensional complex superspace of null supergeodesics n a twistor correspondence between free holomor

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

representation properties racah sum rule and biedenharn elliott identity for uq osp 1 2 it is shown that the universal r matrix in the tensor product of two irreducible representation spaces of the quantum superalgebra uq osp 1 2 can be expressed by clebsch gordan coefficients the racah sum rule satisfied by uq osp 1 2 racah coefficients and 6 j symbols is derived from the properties of the universal r matrix in the tensor product of three representation spaces considering the tensor product of four irreducible representations it is shown that biedenharn elliott identity holds for uq osp 1 2 racah coefficients and 6 j symbols a recursion relation for uq osp 1 2 6 j symbols is derived from the biedenharn elliott identity it is shown that the universal r matrix in the tensor product of two irreducible representation spaces of the quantum superalgebra uq osp 1 2 can be expressed by clebsch gordan coefficients the racah sum rule satisfied by uq osp 1 2 racah coefficients and 6 j symbols is derived from the properties of the universal r matrix in the tensor product of three representation spaces considering the tensor product of four irreducible representations it is shown that biedenharn elliott identity holds for uq osp 1 2 racah coefficients and 6 j symbols a recursion relation for uq osp 1 2 6 j symbols is derived from the biedenharn elliott identity

physical space time and nonrealizable text cr structures space time views leading up to einstein s general relativity are described in relation to some of poincare s early ideas on the subject the basic geometry of twistor theory is introduced as it arises both from minkowski space time and the more general curved einstein models it is shown how this provides a cr structure this being in essence another of poincare s poioneering concepts in a natural way nonrealizable cr structure can arise and an example is presented due to c d hill g a j sparling and the author of a complex manifold with boundary which cannot be extended as a complex manifold beyond its c sup infinity boundary

global rigidity of holomorphic riemannian metrics on compact complex 3 manifolds we study compact complex 3 manifolds m admitting a locally homogeneous holomorphic riemannian metric g we prove the following i if the killing lie algebra of g has a non trivial semi simple part then it preserves some holomorphic riemannian metric on m with constant sectional curvature ii if the killing lie algebra of g is solvable then up to a finite unramified cover m is a quotient g where is a lattice in g and g is either the complex heisenberg group or the complex sol group

null geodesics in complex conformal manifolds and the lebrun correspondence in the complex riemannian framework we show that a con formal manifold containing a compact simply connected null geodesic is conformally flat in dimension 3 we use the lebrun correspondence that views a conformal 3 manifold as the conformal infinity of a selfdual four manifolds we also find a relation between the conformal invariants of the conformal infinity and its ambient

toric self dual einstein metrics as quotients we use the quaternion kahler reduction technique to study old and new self dual einstein metrics of negative scalar curvature with at least a two dimensional isometry group and relate the quotient construction to the hyperbolic eigenfunction ansatz we focus in particular on the semi quaternion kahler quotients of semi quaternion kahler hyperboloids analysing the completeness and topology and relating them to the self dual einstein hermitian metrics of apostolov gauduchon and bryant

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

analysis on analytic spaces and non self dual yang mills fields on donne une description cohomologique de la generalisation de witten isenberg yasskin green au cas non autodual de la construction torseur de ward pour des champs de yang mills autoduaux

a reduction approach for solving the rectangle packing area minimization problem abstract in the rectangle packing area minimization problem rpamp we are given a set of rectangles with known dimensions we have to determine an arrangement of all rectangles without overlapping inside an enveloping rectangle of minimum area the paper presents a generic approach for solving the rpamp that is based on two algorithms one for the 2d knapsack problem kp and the other for the 2d strip packing problem spp in this way solving an instance of the rpamp is reduced to solving multiple spp and kp instances a fast constructive heuristic is used as spp algorithm while the kp algorithm is instantiated by a tree search method and a genetic algorithm alternatively all these spp and kp methods have been published previously finally the best variants of the resulting rpamp heuristics are combined within one procedure the guillotine cutting condition is always observed as an additional constraint the approach was tested on 15 well known rpamp instances above all mcnc and gsrc instances and new best solutions were obtained for 10 instances the computational effort remains acceptable moreover 24 new benchmark instances are introduced and promising results are reported

the super rotation racah wigner calculus revisited it is shown that since the finite dimensional representations of the super rotation algebra are characterized by the superspin j and the parity of the representation space all features of the racah wigner calculus clebsch gordan coefficients recoupling coefficients as well as the wigner and racah symbols depend on both j and however it is noticed that the dependence on the parities of the wigner and racah symbols can be factorized out into phases so that one can define parity independent super s3 j and s6 j symbols the properties of these symbols are analyzed in particular it is shown that the s6 j symbols possess a symmetry similar to the regge symmetry satisfied by the rotation 6 j symbols analytical and numerical tables of the symbols are given for the lowest values of their arguments

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

twistor cr manifolds and three dimensional conformal geometry a cr i e partially complex 5 manifold is contructed as a sphere bundle over an arbitrary 3 manifold with conformal metric this so called twistor cr manifold is show to capture completely the original geometry and necessary and sufficient conditions are given for an abstract cr manifold to arise via the construction the above correspondence is then used to prove that a twistor cr manifold is locally imbeddable as a real hypersurface in c3 only if it is real analytic with respect to a suitable atlas introduction there is a very beautiful interplay between the conformal geometry of riemannian manifolds and complex analysis examples of which are found both in the classical theorem asserting the existence of isothermal coordinates on a surface and in the theory of self dual riemannian 4 manifolds penrose 1976 atiyah et al 1978 intermediating between these two theories is a relationship between 3dimensional conformal geometry and cr partially complex manifolds of dimension 5 in this paper we will explain this relationship focusing on the imbedding problem when can these partially complex manifolds be realized as real hypersurfaces in complex 3 manifolds the idea of associating a cr manifold with a conformal riemannian 3 manifold originated with penrose 1975 1983 who starting with a real analytic space like hypersurface in a real analytic lorentzian 4 manifold constructed a real hypersurface in a complex 3 manifold if the space like hypersurface is totally geodesic or merely all umbilic penroses s cr manifold coincides in our terminology with the twistor cr manifold of the conformal structure induced on the space like hypersurface after establishing notation and conventions in 0 principal results on cr manifolds are reviewed and the notions of cr vector bundle and cr contact form are introduced in 1 a general machine for constructing involutive distributions is produced and in 2 this construction is used to associate to every conformal riemannian 3 manifold a cr 5 manifold the cr manifolds arising in this way are characterized abstractly in 4 in terms of properties presented in 3 then in 5 it is shown that these cr manifolds are isomorphic to real hypersurfaces of complex received by the editors march 15 1983 and in revised form july 8 1983 1980 mathematics subject classificcation primary 32f25 secondary 53a30 83c99 work supported in part by nsf grant mcs 812 0790 001984 american mathematical society 0002 9947 84 1 00 25 per page

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

a local existence result for poincare einstein metrics abstract given a closed riemannian manifold m g m of dimension n 3 we prove the existence of a conformally compact einstein metric g defined on a collar neighborhood m 0 1 whose conformal infinity is g m

a genetic algorithm for the set covering problem abstract in this paper we present a genetic algorithm based heuristic for non unicost set covering problems we propose several modifications to the basic genetic procedures including a new fitness based crossover operator fusion a variable mutation rate and a heuristic feasibility operator tailored specifically for the set covering problem the performance of our algorithm was evaluated on a large set of randomly generated problems computational results showed that the genetic algorithm based heuristic is capable of producing high quality solutions

a bidirectional building approach for the 2d constrained guillotine knapsack packing problem abstract this paper investigates the 2d guillotine knapsack packing problem in which the objective is to select and cut a set of rectangles from a sheet with fixed size and maximize the total profit of the selected rectangles the orientation of the rectangles is fixed and the guillotine cut in which the cut must be parallel to the sides of the sheet to divide it into two completely separated sheets is required two well known methods namely the top down and bottom up approaches are combined into a coherent algorithm to address this problem computational experiments on benchmark test sets show that the approach finds the optimal solution for almost all moderately sized instances and outperforms all existing approaches for larger instances

a population heuristic for constrained two dimensional non guillotine cutting abstract in this paper we present a heuristic algorithm for the constrained two dimensional non guillotine cutting problem this is the problem of cutting a number of rectangular pieces from a single large rectangle so as to maximise the value of the pieces cut in addition the number of pieces of each type that are cut must lie within prescribed limits our heuristic algorithm is a population heuristic where a population of solutions to the problem are progressively evolved this heuristic is based on a new non linear formulation of the problem computational results are presented for a number of standard test problems taken from the literature and for a number of large randomly generated problems

the group with grassmann structure uosp 1 2 the finite dimensional representations of the lie superalgebraosp 1 2 and the group with grassmann structureosp 1 2 have been studied the explicit expression of the projection operator of the superalgebraosp 1 2 has been found the operator permits an arbitrary finite dimensional representation to be expanded in the components multiple to the irreducible ones the clebsch gordan coefficients for the tensor product of two arbitrary irreducible representations have been obtained the matrix elements of the irreducible representations of the groupuosp 1 2 the analoque of the compact form of the grouposp 1 2 are studied the explicit form of these matrix elements the differential equations satisfied by them and the integral of their product have been found

mathscr h space with a cosmological constant it is demonstrated that a real analytic 3 manifold with riemannian conformal metric is naturally the conformal infinity of a germ unique real analytic 4 manifold with real analytic riemannian metric satisfying the self dual einstein equations with cosmological constant 1 moreover this result holds if riemannian is replaced in the first case by lorentzian i e signature and in the second case by pseudo riemannian with signature or if real analytic is replaced by complex analytic and 9riemannian is replaced by holomorphic this provides a cosmological constant analogue of newman s space construction newman 1976 1977

congruences of null strings in complex space times and some cauchy kovalevski like problems it is shown that a problem of construction of a local congruence of null strings is equivalent to a natural cauchy kovalevski like problem related to an equation for a spinor field ka defining the congruence initial data are specified on two dimensional submanifolds in left conformally flat spaces the solution of that problem exists for arbitrary initial data

spaces of complex null geodesics in complex riemannian geometry the notion of a conlplex rienlanmianl 11 mcanifold meaning a complex 1 manifold with a nondegenerate complex quadratic form on each tangent space which varies holomorphically from point to point is briefly developed it is shown that provided ii 4 every such manifold locally arises canonically as the moduli space of all quadrics of a fixed normal bundle type in an associated space of comple x liull geodesics this relationship between local geometry and global complex analysis is stable under deformations introduction the present work provides a general framework analogous to but distinct from penrose s twistor correspondence penrose 15 atiyah et al 2 for the study of real analytic pseudo riemannian geometry on manifolds whose dimension exceeds three in which points of the original manifold correspond to compact complex submanifolds quadrics in an associated complex manifold the natural intermediate step in the construction is to consider complexification of the geometry leading to the notion of a complex riemannian manifold implicitly present for instance in the work of penrose in fact it is really the conformal structure that is of fundamental importance and so having briefly introduced holomorphic metrics in chapter i we proceed to establish the fundamental facts of holomorphic conformal geometry in chapter ii finally in chapter iii we develop our generalized twistor correspondence a brief historical note is in order a special case of this correspondence where the complex riemannian manifold is complex euclidean 4 space proved to be useful for the study of yang mills fields issenberg et at 7 witten 18 several years ago in the wake of this discovery the present author first sketched his ideas on the subject in 1979 in the informal twistor newsletter of the oxford mathematical institute a more fully developed version became the author s ph d thesis lebrun 10 which forms the basis of much of the present more refined paper several results are completely new however such as the theorems in 111 4 5 in dimension four some recent results have shown these notions to be of genuine physical interest within the last year a beguiling theory of the dirac manin 13 and wave lebrun 11 operators fitting neatly within this general framework has emerged for the present however we will examine only the big picture relegating this arguably more beautiful restricted case to treatment elsewhere received by the editors march 10 1982 1980 mathematics subject classification primary 32g10 53a30 secondary 32d15 53c15 53c22 83c99 d1983 american mathematical society 0002 9947 82 00001440 06 75

on the weyl tensor of a self dual complex 4 manifold we study complex 4 manifolds with holomorphic self dual conformal structures and we obtain an interpretation of the weyl tensor of such a manifold as the projective curvature of a field of cones on the ambitwistor space in particular its vanishing is implied by the existence of some compact simply connected null geodesics we also show that the projective structure of the surfaces of a self dual manifold is flat all these results are illustrated in detail in the case of the complexification of cp 2

a hybrid heuristic algorithm for the 2d variable sized bin packing problem in this paper we consider the two dimensional variable sized bin packing problem 2dvsbpp with guillotine constraint 2dvsbpp is a well known np hard optimization problem which has several real applications a mixed bin packing algorithm mixpacking which combines a heuristic packing algorithm with the best fit algorithm is proposed to solve the single bin problem and then a backtracking algorithm which embeds mixpacking is developed to solve the 2dvsbpp a hybrid heuristic algorithm based on iterative simulated annealing and binary search named hha is then developed to further improve the results of our backtracking algorithm computational experiments on the benchmark instances for 2dvsbpp show that hha has achieved good results and outperforms existing algorithms

racah wigner calculus for the super rotation algebra i the symmetry properties and the pseudo orthogonality relations of the super rotation clebsh gordan coefficients for the tensor product of two irreducible representations of the super rotation algebra are derived the symmetric super rotation 3 j symbol and the symmetric and invariant super rotation 6 j symbol are defined their basic properties are described and their relations to the usual 3 j and 6 j symbols are given

relations for clebsch gordan and racah coefficients in suq 2 and yang baxter equations relations are exploited among clebsch gordan cg and racah coefficients in the algebra suq 2 known as a deformation of su 2 these are used to show that the yang baxter yb relation for the irf interaction round a face model results from one of the symmetry relations for the 9 j symbol specific to suq 2 and that in an asymptotic limit this yb relation becomes the yb relation for the two dimensional vertex model the racah coefficient which has a particularly simple dependence on q is efficiently used such that an asymptotic limit of the racah coefficient is the cg coefficient and another limit gives the factorized s matrix of the vertex model

de rham wu decomposition of holomorphic riemannian manifolds it is shown that a de rham wu decomposition for holomorphic riemannian manifolds is obtained easily from the underlying real geometry of holomorphic riemannian manifolds by appealing to h wu s extension to pseudo riemannian manifolds of de rham s decomposition theorem

clebsch gordan coefficients for the quantum superalgebra uq osp 1 2 the structure of the finite dimensional representations of the quantum superalgebra u q osp 1 2 is considered and the projection operator for this quantum superalgebra is derived application of the projection operator permits one to obtain an explicit analytical formula for the q analog of the clebsch gordan coefficients pseudo orthogonality relations and some other properties of clebsch gordan coefficients are given

hyperkahler and quaternionic kahler geometry a quaternion hermitian manifold of dimension at least 12 with closed fundamental 4 form is shown to be quaternionic kahler a similar result is proved for 8 manifolds hyperkahler metrics are constructed on the fundamental quaternionic line bundle with the zero section removed of a quaternionic kahler manifold indefinite if the scalar curvature is negative this construction is compatible with the quaternionic kahler and hyperkahier quotient constructions and allows quaternionic kahler geometry to be subsumed into the theory of hyperkahler manifolds it is shown that the hyperkahler metrics that arise admit a certain type of su 2 action possess functions which are kahler potentials for each of the complex structures simultaneously and determine quaternionic kahler structures via a variant of the moment map construction quaternionic kahler metrics are also constructed on the fundamental quaternionic line bundle and a twistor space analogy leads to a construction of hyperkahler metrics with circle actions on complex line bundles over kahler einstein complex contact manifolds nilpotent orbits in a complex semi simple lie algebra with the hyperkahler metrics defined by kronheimer are shown to give rise to quaternionic kahler metrics and various examples of these metrics are identified it is shown that any quaternionic kahler manifold with positive scalar curvature and sufficiently large isometry group may be embedded in one of these manifolds the twistor space structure of the projectivised nilpotent orbits is studied

the space of harmonic maps of s 2 into s 4 toute surface superminimale ramifiee d aire 4 d dans s 4 provient d une paire de fonctions mesomorphes f 1 f 2 de bidegre d d telles que f 1 et f 2 ont le meme diviseur de ramification on deduit des conditions sous lesquelles des surfaces superminimales ramifiees peuvent etre engendrees a partir de telles paires de fonctions

tensor operators and wigner eckart theorem for the quantum superalgebra uq osp 1 2 tensor operators in graded representations of z2 graded hopf algebras are defined and their elementary properties are derived the wigner eckart theorem for irreducible tensor operators for uq osp 1 2 is proven examples of tensor operators in the irreducible representation space of hopf algebra uq osp 1 2 are considered the reduced matrix elements for the irreducible tensor operators are calculated a construction of some elements of the centre of uq osp 1 2 is given

an infinite number of hidden variables in hyper kahler metrics three types of hidden variables both independent and dependent are shown to underlie the hyper kahler geometry in a complexified setting the variables satisfy an infinite set of differential equations hierarchy just as in the case of most nonlinear integrable systems the notion of the plebanski key functions j math phys 16 2395 1975 is extended to this hierarchy to give an analog of the notion of the function two examples of special solutions which are reminiscent of several solution techniques in the theory of nonlinear integrable systems are presented for illustration

einstein metrics spinning top motions and monopoles on considere la construction de solutions pour l equation d einstein autoduale on applique la construction de lebrun a la sphere de berger la 3 sphere de metrique 1 2 2 2 i 3 3 2 ou 1 2 3 est une base de 1 formes invariante a gauche satisfaisant d i j k e ijk j k et i 3 est une constante

cutting stock problems and solution procedures this paper discusses some of the basic formulation issues and solution procedures for solving one and two dimensional cutting stock problems linear programming sequential heuristic and hybrid solution procedures are described for two dimensional cutting stock problems with rectangular shapes we also propose an approach for solving large problems with limits on the number of times an ordered size may appear in a pattern

racah coefficients and 6 j symbols for the quantum superalgebra uq osp 1 2 the tensor product of three irreducible representations of the quantum superalgebra uq osp 1 2 is studied and the super q analogs of racah coefficients and 6 j symbols for the quantum superalgebra uq osp 1 2 are defined racah coefficients and 6 j symbols depend on the superspin l and the parity which characterize the irreducible representations of uq osp 1 2 but the dependence on parities can be factored out so that one can define parity independent 6 j symbols it is shown that the 6 j symbols for the quantum superalgebra uq osp 1 2 satisfy symmetry properties and orthogonality relations similar to those of the 6 j symbols for quantum algebra uq su 2

hidden symmetries and integrable hierarchy of the mathcal n 4 supersymmetric yang mills equations we describe an infinite dimensional algebra of hidden symmetries of mathcal n 4 supersymmetric yang mills sym theory our derivation is based on a generalization of the supertwistor correspondence using the latter we construct an infinite sequence of flows on the solution space of the mathcal n 4 sym equations the dependence of the sym fields on the parameters along the flows can be recovered by solving the equations of the hierarchy we embed the mathcal n 4 sym equations in the infinite system of the hierarchy equations and show that this sym hierarchy is associated with an infinite set of graded symmetries recursively generated from supertranslations presumably the existence of such nonlocal symmetries underlies the observed integrable structures in quantum mathcal n 4 sym theory

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

characterization and modelling of guillotine constraints this paper focuses on guillotine cuts which often arise in real life cutting stock problems in order to construct a solution verifying guillotine constraints the first step is to know how to determine whether a given cutting pattern is a guillotine pattern for this purpose we first characterize guillotine patterns by proving a necessary and sufficient condition then we propose a polynomial algorithm to check this condition based on this mathematical characterization of guillotine patterns we then show that guillotine constraints can be formulated into linear inequalities the performance of the algorithm to check guillotine cutting patterns is evaluated by means of computational results

conformally invariant differential operators on minkowski space and their curved analogues this article describes the construction of a natural family of conformally invariant differential operators on a four dimensional pseudo riemannian manifold included in this family are the usual massless field equations for arbitrary helicity but there are many more besides the article begins by classifying the invariant operators on flat space this is a fairly straightforward task in representation theory best solved through the theory of verma modules the method generates conformally invariant operators in the curved case by means of penrose s local twistor transport

twistor theory at fifty from contour integrals to twistor strings we review aspects of twistor theory its aims and achievements spanning the last five decades in the twistor approach space time is secondary with events being derived objects that correspond to compact holomorphic curves in a complex threefold the twistor space after giving an elementary construction of this space we demonstrate how solutions to linear and nonlinear equations of mathematical physics anti self duality equations on yang mills or conformal curvature can be encoded into twistor cohomology these twistor correspondences yield explicit examples of yang mills and gravitational instantons which we review they also underlie the twistor approach to integrability the solitonic systems arise as symmetry reductions of anti self dual asd yang mills equations and einstein weyl dispersionless systems are reductions of asd conformal equations we then review the holomorphic string theories in twistor and ambitwistor spaces and explain how these theories give rise to remarkable new formulae for the computation of quantum scattering amplitudes finally we discuss the newtonian limit of twistor theory and its possible role in penrose s proposal for a role of gravity in quantum collapse of a wave function

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

almost complex and almost product einstein manifolds from a variational principle it is shown that the first order palatini variational principle for a generic nonlinear metric affine lagrangian depending on the symmetrized ricci square invariant leads to an almost product einstein structure or to an almost complex anti hermitian einstein structure on a manifold it is proved that a real anti hermitian metric on a complex manifold satisfies the kahler condition on the same manifold treated as a real manifold if and only if the metric is the real part of a holomorphic metric a characterization of anti kahler einstein manifolds and almost product einstein manifolds is obtained examples of such manifolds are considered

a genetic algorithm for the two dimensional strip packing problem with rectangular pieces given a set of rectangular pieces and a container of fixed width and variable length the two dimensional strip packing problem 2d spp consists of orthogonally placing all the pieces within the container without overlapping such that the overall length of the layout is minimised until now mainly heuristics for example genetic algorithms ga were proposed for the 2d spp which use encoded solutions that are manipulated by standard operators in this paper a ga for the 2d spp is suggested that works without any encoding of solutions rather fully defined layouts are manipulated as such by means of specific genetic operators two additional constraints namely the orientation constraint and the guillotine constraint can be taken into account the ga is subjected to a comprehensive test using benchmark instances with up to 5000 pieces compared to eleven competing methods from the literature the ga performs best

einstein metrics and complex singularities this paper is concerned with the construction of special metrics on non compact 4 manifolds which arise as resolutions of complex orbifold singularities our study is close in spirit to the construction of the hyperkahler gravitational instantons but we focus on a different class of singularities we show that any resolution x of an isolated cyclic quotient singularity admits a complete scalar flat kahler metric which is hyperkahler if and only if k x is trivial and that if k x is strictly nef then x also admits a complete non kahler self dual einstein metric of negative scalar curvature in particular complete self dual einstein metrics are constructed on simply connected non compact 4 manifolds with arbitrary second betti number

analytical formulas for racah coefficients and 6 j symbols of the quantum superalgebra uq osp 1 2 using the method of projection operators analytical formulas for racah coefficients and 6 j symbols of the quantum superalgebra uq osp 1 2 are derived the formulas obtained by this method are transformed by means of algebraic identities into symmetrical analytical formulas the form of which are very similar to the classical formulas obtained by racah and regge for su 2 racah coefficients and 6 j symbols symmetry properties of uq osp 1 2 racah coefficients and 6 j symbols following from these analytical formulas are studied in particular it is shown that similarly to the su 2 classical case in addition to the usual tetrahedral symmetry 6 j symbols of the quantum superalgebra uq osp 1 2 satisfy a regge type symmetry

a tabu search algorithm for a two dimensional non guillotine cutting problem in this paper we study a two dimensional non guillotine cutting problem the problem of cutting rectangular pieces from a large stock rectangle so as to maximize the total value of the pieces cut the problem has many industrial applications whenever small pieces have to be cut from or packed into a large stock sheet we propose a tabu search algorithm several moves based on reducing and inserting blocks of pieces have been defined intensification and diversification procedures based on long term memory have been included the computational results on large sets of test instances show that the algorithm is very efficient for a wide range of packing and cutting problems