A MIP approach for some practical packing problems: Balancing constraints and tetris-like items

This paper considers packing problems with balancing conditions and items consisting of clusters of parallelepipeds (mutually orthogonal, i.e. tetris-like items). This issue is quite frequent in space engineering and a real-world application deals with the Automated Transfer Vehicle project (funded by the European Space Agency), at present under development. A Mixed Integer Programming (MIP) approach is proposed. The three-dimensional single bin packing problem is considered. It consists of orthogonally placing, with possibility of rotation, the maximum number of parallelepipeds into a given parallelepiped. A MIP formulation of the problem is reported together with a MIP-based heuristic approach. Balancing conditions are furthermore examined, as well as the orthogonal placement (with rotation) of tetris-like items into a rectangular domain.Received: September 2003, Revised: February 2004, AMS classification: 90B99, 05B40, 90C90, 90C59Thanks are due to T. A. Ciriani for the important suggestions given for the whole paper and to S. Gliozzi (IBM, Business Consulting Services) for the significant support offered, in particular in discussing the topics presented in Sect. 2.1.     

Higher-order energy expansions and spike locations

We consider the following singularly perturbed semilinear elliptic problem: where is a bounded domain in R ^N with smooth boundary , is a small constant and f is some superlinear but subcritical nonlinearity. Associated with (I) is the energy functional defined by where . Ni and Takagi ([29, 30]) proved that for a single boundary spike solution , the following asymptotic expansion holds: where c ₁ > 0 is a generic constant, is the unique local maximum point of and is the boundary mean curvature function at . In this paper, we obtain a higher-order expansion of where c ₂, c ₃ are generic constants and is the scalar curvature at . In particular c ₃ > 0. Some applications of this expansion are given.Received: 14 January 2003, Accepted: 28 July 2003, Published online: 15 October 2003Mathematics Subject Classification (2000): Primary 35B40, 35B45; Secondary 35J25     

Rotations of Hypercyclic and Supercyclic Operators

Let T B(X) be a hypercyclic operator and a complex number of modulus 1. Then T is hypercyclic and has the same set of hypercyclic vectors as T. A version of this results gives for a wide class of supercyclic operators that x X is supercyclic for T if and only if the set {tTⁿ x : t > 0, n = 0, 1, ...} is dense in X. This gives answers to several questions studied in the literature.     

Maximal ergodic theorems and applications to Riemannian geometry

We prove new ergodic theorems in the context of infinite ergodic theory, and give some applications to Riemannian and Kähler manifolds without conjugate points. One of the consequences of these ideas is that a complete manifold without conjugate points has nonpositive integral of the infimum of Ricci curvatures, whenever this integral makes sense. We also show that a complete Kähler manifold with nonnegative holomorphic curvature is flat if it has no conjugate points.     

Optimal accumulation of Jacobian matrices by elimination methods on the dual computational graph

The accumulation of the Jacobian matrix F of a vector function can be regarded as a transformation of its linearized computational graph into a subgraph of the directed complete bipartite graph K_n,m. This transformation can be performed by applying different elimination techniques that may lead to varying costs for computing F. This paper introduces face elimination as the basic technique for accumulating Jacobian matrices by using a minimal number of arithmetic operations. Its superiority over both edge and vertex elimination methods is shown. The intention is to establish the conceptual basis for the ongoing development of algorithms for optimizing the computation of Jacobian matrices.     

The -conjecture and equivariant <Emphasis Type="Italic">e</Emphasis><Subscript><Emphasis Type="Italic">C</Emphasis></Subscript>-invariants

Let X be a smooth closed oriented non-spin 4-manifold with even intersection form kE₈nH (n1). The -conjecture states that n is greater than or equal to |k|. In this paper we give a proof of the -conjecture. The strategy of this paper is to use the finite dimensional approximation of the map induced from the Seiberg-Witten equations and equivariant e_C-invariants as in the paper of M. Furuta and Y. Kametani.Mathematics Subject Classification (1991): 57R55This work was supported by Korea Research Foundation Grant (KRF–2002–003–C00011).     

Generation of Analytic Semigroups by Elliptic Operators with Dirichlet Boundary Conditions in a Cylindrical Domain

We consider an elliptic operator of second order with Dirichlet boundary conditions in a cylindrical domain. We show that a suitable interpretation of this operator in a certain space of continuous functions vanishing on the boundary is the infinitesimal generator of an analytic semigroup in the space of continuous functions. We prove several inclusions of the domain of the infinitesimal generator and of the real interpolation spaces between this domain and the basic space of continuous functions.     

Tower techniques for cosimplicial resolutions

Let be a simplicial model category and J : a simplicial coaugmented functor. Given an object X, the assignment nJⁿ⁺¹X defines a cofacial resolution (an augmented cosimplicial space without its codegeneracy maps). Following Bousfield and Kan we define J^sX = tot^s([n] Jⁿ⁺¹X). An object X is called J-injective if it is a retract of JX in Ho() via the natural map. We show that certain homotopy limits of J-injective objects are J_s-injective. Our method is to use the notion of pro-weak equivalences which was first introduced in a different language and context by David Edwards and Harold Hastings. The key observation is that a cofacial resolution X (-1) X which admits a left contraction gives rise to a pro-weak equivalence of towers {X(-1)}_s0{tot^SX}_{s 0}.The first author was supported in part by National Science Foundation grant DMS-0296117     

Semilocal rings with <Emphasis Type="Italic">n</Emphasis>-Engel multiplicative group

An associative ring R with unit element is called semilocal if R modulo its Jacobson radical is an artinian ring. It is proved that the multiplicative group R* of a semilocal ring R generated by R* satisfies an n-Engel condition for some positive integer n if and only if R is m-Engel as a Lie ring for some positive integer m depending only on n.Received: 21 January 2003     

Semistrictly quasiconvex mappings and non-convex vector optimization

This paper introduces a new class of non-convex vector functions strictly larger than that of P-quasiconvexity, with P ^m being the underlying ordering cone, called semistrictly ( ^m\ –int P)-quasiconvex functions. This notion allows us to unify various results on existence of weakly efficient (weakly Pareto) optima. By imposing a coercivity condition we establish also the compactness of the set of weakly Pareto solutions. In addition, we provide various characterizations for the non-emptiness, convexity and compactness of the solution set for a subclass of quasiconvex vector optimization problems on the real-line. Finally, it is also introduced the notion of explicit ( ^m\ –int P)-quasiconvexity (equivalently explicit (int P)-quasiconvexity) which plays the role of explicit quasiconvexity (quasiconvexity and semistrict quasiconvexity) of real-valued functions.Acknowldegements.The author wishes to thank both referees for their careful reading of the paper, their comments, remarks, helped to improve the presentation of some results. One of the referee provided the references [5, 6] and indirectly [20].