Strongly meager sets of size continuum

We will construct several models where there are no strongly meager sets of size 2⁰. First author partially supported by NSF grant DMS 0200671.Second author partially supported by Israel Science Foundation and NSF grant DMS 0072560. Publication 807. Mathematics Subject Classification (2000):03E15, 03E20     

Schwarz reflection geometry I: Continuous iteration of reflection

Differential equations are derived for a continous limit of iterated Schwarzian reflection of analytic curves, and solutions are interpreted as geodesics in an infinite-dimensional symmetric space geometry. Mathematics Subject Classification (1991):53C35, 53A30, 30D05.     

Coding lemmata in L

Under the assumption that there exists an elementary embedding(henceforth abbreviated as and in particular under we prove a Coding Lemma for and find certain versions of it which are equivalent to strong regularity of cardinals below . We also prove that a stronger version of the Coding Lemma holds for a stationary set of ordinals below .Mathematics Subject Classification (2000):03E55     

A Conservative Space-time Mesh Refinement Method for the 1-D Wave Equation. Part II: Analysis

Summary. We introduced in [2] a new method for space-time refinement for the 1-D wave equation. This method is based on the conservation of a discrete energy through two different discretization grids which guarantees the stability of the scheme. In this second part, we analyse the accuracy of this scheme in a detailed way by means of a plane wave analysis and numerical experiments that permit us to point out spurious numerical phenomena and explain how to control them. Mathematics Subject Classification (2000):65N12     

Spectral theory for tensor products of Hermitian holomorphic line bundles

Let L be a quasipositive line bundle over a compact complex Hermitian manifold. It has been conjectured that the smallest positive eigenvalue of the Laplacian, acting on the kth tensor power of L, is uniformly bounded independent of k. Various special cases of the conjecture are proved, but a counterexample is given to the general conjecture. This research was partially supported by NSF DMS 0203070.Mathematics Subject Classification (2000):58G25     

Convex approximations for complete integer recourse models

We consider convex approximations of the expected value function of a two-stage integer recourse problem. The convex approximations are obtained by perturbing the distribution of the random right-hand side vector. It is shown that the approximation is optimal for the class of problems with totally unimodular recourse matrices. For problems not in this class, the result is a convex lower bound that is strictly better than the one obtained from the LP relaxation.This research has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences.Key words.integer recourse – convex approximationMathematics Subject Classification (1991):90C15, 90C11     

Strong test modules and multiplier ideals

We introduce the notion of strong test module and show that a large number of such modules appear in the tight closure theory of complete domains: the test ideal (this has already been known), the parameter test module, and the module of relative test elements. They also appear as certain multiplier ideals, a concept of interest in algebraic geometry. Mathematics Subject Classification (2000):13A35     

Lift-and-project cuts and perfect graphs

We analyze the application of lift-and-project to the clique relaxation of the stable set polytope. We characterize all the inequalities that can be generated through the application of the lift-and-project procedure, introduce the concept of 1-perfection and prove its equivalence to minimal imperfection. This characterization of inequalities and minimal imperfection leads to a generalization of the Perfect Graph Theorem of Lovász, as proved by Aguilera, Escalante and Nasini [1].Mathematics Subject Classification:05C17, 90C57     

Minimal concave cost rebalance of a portfolio to the efficient frontier

One usually constructs a portfolio on the efficient frontier, but it may not be efficient after, say three months since the efficient frontier will shift as the elapse of time. We then have to rebalance the portfolio if the deviation is no longer acceptable. The method to be proposed in this paper is to find a portfolio on the new efficient frontier such that the total transaction cost required for this rebalancing is minimal. This problem results in a nonconvex minimization problem, if we use mean-variance model. In this paper we will formulate this problem by using absolute deviation as the measure of risk and solve the resulting linearly constrained concave minimization problem by a branch and bound algorithm successfully applied to portfolio optimization problem under concave transaction costs. It will be demonstrated that this method is efficient and that it leads to a significant reduction of transaction costs. Key words.portfolio optimization – rebalance – mean-absolute deviation model – concave cost minimization – optimization over the efficient set – global optimizationMathematics Subject Classification (1991):20E28, 20G40, 20C20     

Tight, not semi–fillable contact circle bundles

Extending our earlier results, we prove that certain tight contact structures on circle bundles over surfaces are not symplectically semi–fillable, thus confirming a conjecture of Ko Honda.Mathematics Subject Classification (2000)57R57, 57R17Partially supported by MURST and member of EDGE, Research Training Network HPRN-CT-2000-00101, supported by The European Human Potential ProgrammePartially supported by OTKA T034885     

Filtration and Finite-Dimensional Characterization of Logarithmically Convex Measures

We study the classes C(, ) and C _H(, ) of logarithmically convex measures that are a natural generalization of the notion of Boltzmann measure to an infinite-dimensional case. We prove a theorem on the characterization of these classes in terms of finite-dimensional projections of measures and describe some applications to the theory of random series.     

Integration of locally exponential Lie algebras of vector fields

A locally convex Lie algebra is said to be locally exponential if it belongs to some local Lie group in canonical coordinates. In this note we give criteria for locally exponential Lie algebras of vector fields on an infinite-dimensional manifold to integrate to global Lie group actions. Moreover, we show that all necessary conditions are satisfied if the manifold is finite-dimensional connected and σ-compact, which leads to a generalization of Palais’ Integrability Theorem.      

A subproblem-centric model and approach to the nurse scheduling problem

The number of hospitals in Japan exceeds 10,000, and every month nurses are scheduled to shifts in about 30,000 units in total. There is serious demand for automating this scheduling task. In this paper, we introduce a mathematical programming formulation of the nurse scheduling problem in Japan, and develop a meta-heuristic approach to solve the problem. This scheduling problem is a hard combinatorial problem due to tight constraints involving such factors as the skill level of a team, the need to balance workload among nurses, and the consideration of nurses' preferences, even though the number of the nurses to be scheduled is not large, at between 20 and 40. The performance of our approach is demonstrated by the successful solution of data taken from actual scheduling problems. The proposed model and approach can be adapted for the majority of hospitals in Japan, as well as for some hospitals in other countries, and is likely applicable to many other scheduling problems in the fields of business and logistics. Key words.nurse scheduling – block-angular problem – subproblem – integer programming – relaxation – tabu search – branch-and-boundMathematics Subject Classification (1991):20E28, 20G40, 20C20     

On the Uniqueness ofL 2-solutions in half-space of certain differential equations

Square integrable solutions to the equation{– ²/y² + P(D_x)+b(y)–}u(x, y) = f(x, y) are considered in the half-spacey>0, x ⁿ , whereP(D _x) is a constant coefficient operator. Under suitable conditions on lim_y0u(x, y), b(y), f(x, y) and , it is shown that suppu = suppf. This generalizes a result due to Walter Littman.Research partially supported by USNSF Grant 79-02538-A02.     

Möbius geometry of hypersurfaces with constant mean curvature and scalar curvature

Let be a hypersurface in the (m+1)-dimensional unit sphere S^m+1 without umbilics. Four basic invariants of x under the Möbius transformation group in S^m+1 are a Riemannian metric g called Möbius metric, a 1-form called Möbius form, a symmetric (0,2) tensor A called Blaschke tensor and symmetric (0,2) tensor B called Möbius second fundamental form. In this paper, we prove the following classification theorem: let be a hypersurface, which satisfies (i) 0, (ii) A+g+B0 for some functions and , then and must be constant, and x is Möbius equivalent to either (i) a hypersurface with constant mean curvature and scalar curvature in S^m+1; or (ii) the pre-image of a stereographic projection of a hypersurface with constant mean curvature and scalar curvature in the Euclidean space R^m+1; or (iii) the image of the standard conformal map of a hypersurface with constant mean curvature and scalar curvature in the (m+1)-dimensional hyperbolic space H^m+1. This result shows that one can use Möbius differential geometry to unify the three different classes of hypersurface with constant mean curvature and scalar curvature in S^m+1, R^m+1 and H^m+1.Partially supported the Alexander Humboldt Stiftung and Zhongdian grant of NSFC.Partially supported by RFDP, Qiushi Award, 973 Project and Jiechu grant of NSFC.Mathematics Subject Classification (2000):Primary 53A30; Secondary 53B25     

The Eta Invariant and the Real Connective K-Theory of the Classifying Space for Quaternion Groups

We express the real connective K-theory groups o_4k–1(B Q ) ofthe quaternion group Q of order = 2 ^j 8 in terms of therepresentation theory of Q by showing o_4k–1(B Q ) = Sp(S ^4k+3/Q )where is any fixed point free representation of Q in U(2k + 2).     

A generalization of Le Potier’s vanishing theorem

The main result is a general vanishing theorem for the cohomology of the ample vector bundles obtained as Schur functors of some vectors bundle which is not assumed to be ample itself. This is a generalization of Le Potiers vanishing theorem. It is also proven that for two partitions I and J such that IJ, the ampleness of S_IE implies that of S_JE.Mathematics Subject Classification (2000):14F17     

A finite element approximation of the Griffith’s model in fracture mechanics

Summary. The Griffith model for the mechanics of fractures in brittle materials is consider in the weak formulation of SBD spaces. We suggest an approximation, in the sense of –convergence, by a sequence of discrete functionals defined on finite elements spaces over structured and adaptive triangulations. The quasi-static evolution for boundary value problems is also taken into account and some numerical results are shown. Mathematics Subject Classification (2000):65N30     

Local search algorithms for the rectangle packing problem with general spatial costs

We propose local search algorithms for the rectangle packing problem to minimize a general spatial cost associated with the locations of rectangles. The problem is to pack given rectangles without overlap in the plane so that the maximum cost of the rectangles is minimized. Each rectangle has a set of modes, where each mode specifies the width and height of the rectangle and its spatial cost function. The spatial costs are general piecewise linear functions which can be non-convex and discontinuous. Various types of packing problems and scheduling problems can be formulated in this form. To represent a solution of this problem, a pair of permutations of n rectangles is used to determine their horizontal and vertical partial orders, respectively. We show that, under the constraint specified by such a pair of permutations, optimal locations of the rectangles can be efficiently determined by using dynamic programming. The search for finding good pairs of permutations is conducted by local search and metaheuristic algorithms. We report computational results on various implementations using different neighborhoods, and compare their performance. We also compare our algorithms with other existing heuristic algorithms for the rectangle packing problem and scheduling problem. These computational results exhibit good prospects of the proposed algorithms. Key words.rectangle packing – sequence pair – general spatial cost – dynamic programming – metaheuristicsMathematics Subject Classification (1991):20E28, 20G40, 20C20     

Construction of a Class of Finite H ∞ Controllers for Infinite-Dimensional Systems

We consider the suppression of forced oscillations in distributed systems of a hyperbolic type by finite-dimensional controllers using an H objective. The system is split into a finite-dimensional and an infinite-dimensional subsystems. The controller receives a signal from the output of both systems. The class of controllers is described in the form of a system of ordinary differential equations.