Existence and multiplicity of solutions for asymptotically linear, noncoercive elliptic equations

First we prove the existence of a nontrivial smooth solution for a p-Laplacian equation with a (p − 1)-linear nonlinearity and a noncoercive Euler functional, under hypotheses including resonant problems with respect to the principal eigenvalue of (-D_p, W^1,p₀(Z)){(-{\it \Delta}_p,\,W^{1,p}_0(Z))} . Then, for the semilinear problem (i.e., p = 2), assuming nonuniform nonresonance at infinity and zero, we prove a multiplicity theorem which provides the existence of at least three nontrivial solutions, two being of opposite constant sign. Our approach combines minimax techniques with Morse theory and truncation arguments.     

Towards experiments with highly charged ions at HESR

The atomic physics collaboration SPARC is a part of the APPA pillar at the future Facility for Antiproton and Ion Research. It aims at atomic-physics research across virtually the full range of atomic matter. An emphasis of this contribution are the atomic physics experiments addressing the collision dynamics in strong electro-magnetic fields as well as the fundamental interactions between electrons and heavy nuclei at the HESR. Here we give a short overview about the central instruments for SPARC experiments at this storage ring.     

A Mechanistically and Operationally Simple Route to Metal–N-Heterocyclic Carbene (NHC) Complexes

We have been puzzled by the involvement of weak organic and inorganic bases in the synthesis of metal–N-heterocyclic carbene (NHC) complexes. Such bases are insufficiently strong to permit the presumed required deprotonation of the azolium salt (the carbene precursor) prior to metal binding. Experimental and computational studies provide support for a base-assisted concerted process that does not require free NHC formation. The synthetic protocol was found applicable to a number of transition-metal- and main-group-centered NHC compounds and could become the synthetic route of choice to form M–NHC bonds.     

Semidefinite relaxations for quadratically constrained quadratic programming: A review and comparisons

At the intersection of nonlinear and combinatorial optimization, quadratic programming has attracted significant interest over the past several decades. A variety of relaxations for quadratically constrained quadratic programming (QCQP) can be formulated as semidefinite programs (SDPs). The primary purpose of this paper is to present a systematic comparison of SDP relaxations for QCQP. Using theoretical analysis, it is shown that the recently developed doubly nonnegative relaxation is equivalent to the Shor relaxation, when the latter is enhanced with a partial first-order relaxation-linearization technique. These two relaxations are shown to theoretically dominate six other SDP relaxations. A computational comparison reveals that the two dominant relaxations require three orders of magnitude more computational time than the weaker relaxations, while providing relaxation gaps averaging 3% as opposed to gaps of up to 19% for weaker relaxations, on 700 randomly generated problems with up to 60 variables. An SDP relaxation derived from Lagrangian relaxation, after the addition of redundant nonlinear constraints to the primal, achieves gaps averaging 13% in a few CPU seconds.     

Convergence theory of extrapolated iterative methods for a certain class of non-symmetric linear systems

Summary A variety of iterative methods considered in [3] are applied to linear algebraic systems of the formAu=b, where the matrixA is consistently ordered [12] and the iteration matrix of the Jacobi method is skew-symmetric. The related theory of convergence is developed and the optimum values of the involved parameters for each considered scheme are determined. It reveals that under the aforementioned assumptions the Extrapolated Successive Underrelaxation method attains a rate of convergence which is clearly superior over the Successive Underrelaxation method [5] when the Jacobi iteration matrix is non-singular.     

Nonlinear elliptic differential equations with multivalued nonlinearities

In this paper we study nonlinear elliptic boundary value problems with monotone and nonmonotone multivalued nonlinearities. First we consider the case of monotone nonlinearities. In the first result we assume that the multivalued nonlinearity is defined on all ℝ. Assuming the existence of an upper and of a lower solution, we prove the existence of a solution between them. Also for a special version of the problem, we prove the existence of extremal solutions in the order interval formed by the upper and lower solutions. Then we drop the requirement that the monotone nonlinearity is defined on all of ℝ. This case is important because it covers variational inequalities. Using the theory of operators of monotone type we show that the problem has a solution. Finally in the last part we consider an eigenvalue problem with a nonmonotone multivalued nonlinearity. Using the critical point theory for nonsmooth locally Lipschitz functionals we prove the existence of at least two nontrivial solutions (multiplicity theorem).     

Global Optimization of 0-1 Hyperbolic Programs

We develop eight different mixed-integer convex programming reformulations of 0-1 hyperbolic programs. We obtain analytical results on the relative tightness of these formulations and propose a branch and bound algorithm for 0-1 hyperbolic programs. The main feature of the algorithm is that it reformulates the problem at every node of the search tree. We demonstrate that this algorithm has a superior convergence behavior than directly solving the relaxation derived at the root node. The algorithm is used to solve a discrete p-choice facility location problem for locating ten restaurants in the city of Edmonton.The research was supported in part by NSF awards DMII 95-02722 and BES 98-73586 to NVS.     

Optimal control and admissible relaxation for subdifferential evolution inclusions

In this paper we study the relaxation of optimal control problems monitored by subdifferential evolution inclusions. First under appropriate convexity conditions, we establish an existence result. Then we introduce the relaxed problem and show that it always has a solution under fairly general hypotheses on the data. Subsequently we examine when the relaxation is admissible. So we show that every relaxed trajectory can be approximated by extremal original ones (i.e. original trajectories generated by bang-bang controls) and that the values of the original and relaxed problems are equal. Some examples are also presented.