The Eta-Invariant and Pontryagin Duality in K-Theory

The topological significance of the spectral Atiyah--Patodi--Singer -invariant is investigated. We show that twice the fractional part of the invariant is computed by the linking pairing in K-theory with the orientation bundle of the manifold. Pontryagin duality implies the nondegeneracy of the linking form. An example of a nontrivial fractional part for an even-order operator is presented.     

On the continuity of biconjugate convex functions

We show that a Banach space is a Grothendieck space if and only if every continuous convex function on has a continuous biconjugate function on , thus also answering a question raised by S. Simons. Related characterizations and examples are given.

     

磷脂核苷缀合物甘油结构修饰新方法及应用

通过环甘油磷脂核苷缀合物的亲核开环,为甘油结构修饰提供了一条简便有效的新路线.三乙胺对环甘油磷脂核苷缀合物的亲核开环,在室温下就能进行,合成了一类结构新型的以卵磷脂类似物作载体的核苷前药.     

Monopoles without strings: a conflict between the one-photon condition and duality invariance

We prove a new no-go theorem in the Dirac-algebra formulation of generalized electromagnetic theory, which includes magnetic monopoles and uses two potentialsA andM : It is impossible to construct a Lagrangian which is duality invariant and satisfies the one-photon assumption, from which Maxwell's equations and the equations of motion can be derived. Such a Lagrangian can be found only if either duality invariance or the one-photon assumption is sacrificed. These constraints as well as others discussed here are based on recently published results on monopoles without strings in the Dirac algebra, but they do not arise from any artificial restrictions in the Dirac-algebra formulation.     

Local convergence analysis for the REQP algorithm using conjugate basis matrices

The REQP algorithm solves constrained minimization problems using a sequential quadratic programming technique based on the properties of penalty functions. The convergence of REQP has been studied elsewhere (Refs. 1, 2). This paper uses a novel approach to the analysis of the method near to the solution, based on the use of conjugate subspaces. The stepp taken by a constrained minimization algorithm can be thought of as having two components,h in the subspace tangential to the constraints andv in the subspace spanned by the constraint normals. It is usual forh andv to be orthogonal components. Recently, Dixon (Ref. 3) has suggested constructingp from components which are not orthogonal. That is, we writep=h + v, whereh is in the subspace tangential to the constraints and wherev andh are conjugate with respect to the Hessian of the Lagrangian function. By looking at the conjugate components of the REQP search directions, it is possible to simplify the analysis of the behavior near the solution and to obtain new results about the local rate of convergence of the method.This work was supported by a SERC Studentship (TTN).     

Theq-state Potts model in the standard Pirogov-Sinai theory: Surface tensions and Wilson loops

Theq-state Potts model (both scalar and gauge versions) is rewritten, with the help of the duality transformation, into a form of the Pirogov-Sinai theory with noninteracting contours that can be controlled by cluster expansions onceq is large enough. This is then used in a new proof of the existence of a unique transition (inverse) temperature _t , where the mean internal energy is discontinuous. Moreover, we prove for the scalar model (again forq large enough) that there are discontinuities at _t of the magnetization and of the mass gap, with the magnetization vanishing below _t and the mass gap vanishing above _t . We also show that the surface tensions between ordered stable phases are strictly positive up to _t , and the surface tension between an ordered phase and the disordered one is strictly positive at _t . For the three-dimensional gauge model, the Wilson parameter exhibits a direct transition from an area law decay (quark confinement) to a perimeter law decay (deconfinement).On leave from ENS Rabat, Morocco.     

Tight integral duality gap in the Chinese Postman problem

LetG = (V, E) be a graph and letw be a weight functionw:E Z ⁺. Let be an even subset of the vertices ofG. AT-cut is an edge-cutset of the graph which dividesT into two odd sets. AT-join is a minimal subset of edges that meets everyT-cut (a generalization of solutions to the Chinese Postman problem). The main theorem of this paper gives a tight upper bound on the difference between the minimum weightT-join and the maximum weight integral packing ofT-cuts. This difference is called the (T-join) integral duality gap. Let _w be the minimum weight of aT-join, and letv _w be the maximum weight of an integral packing ofT-cuts. IfF is a non-empty minimum weightT-join, andn _F is the number of components ofF, then we prove that _w —v _w n _F –1.This result unifies and generalizes Fulkerson's result for |T|=2 and Seymour's result for |T|= 4.For a certain integral multicommodity flow problem in the plane, which was recently proved to be NP-complete, the above result gives a solution such that for every commodity the flow is less than the demand by at most one unit.Research of the first author was partially supported by Technion V.P.R. Fund — C. and J. Bishop Research Fund, and Argentinian Research Fund.Part of this work was done as part of the author's D.Sc. Thesis in the Faculty of Industrial and Management Engineering, Technion — Israel Institute of Technology, Haifa. Research of this author was partially supported by Technion V.P.R. Fund — The Baltimore Fund for Industrial Engineering and Management Research.     

Asymptotic behaviour of the loss probability of theM/G/1/K andG/M/1/K queues

This paper investigates the asymptotic behaviour of the loss probability of theM / G/1/K and G/M/1/K queues as the buffer size increases. It is shown that the loss probability approaches its limiting value, which depends on the offered load, with an exponential decay in essentially all cases. The value of the decay rate can be easily computed from the main queue parameters. Moreover, the close relation existing between the loss behaviour of the two examined queueing systems is highlighted and a duality concept is introduced. Finally some numerical examples are given to illustrate on the usefulness of the asymptotic approximation.     

Strong stationary duality for continuous-time Markov chains. Part I: Theory

LetX(t), 0t<, be an ergodic continuous-time Markov chain with finite or countably infinite state space. We construct astrong stationary dual chainX ^* whose first hitting times yield bounds on the convergence to stationarity forX. The development follows closely the discrete-time theory of Diaconis and Fill.^(2,3) However, for applicability it is important that we formulate our results in terms of infinitesimal rates, and this raises new issues.     

A one-phase algorithm for semi-infinite linear programming

We present an algorithm for solving a large class of semi-infinite linear programming problems. This algorithm has several advantages: it handles feasibility and optimality together; it has very weak restrictions on the constraints; it allows cuts that are not near the most violated cut; and it solves the primal and the dual problems simultaneously. We prove the convergence of this algorithm in two steps. First, we show that the algorithm can find an-optimal solution after finitely many iterations. Then, we use this result to show that it can find an optimal solution in the limit. We also estimate how good an-optimal solution is compared to an optimal solution and give an upper bound on the total number of iterations needed for finding an-optimal solution under some assumptions. This algorithm is generalized to solve a class of nonlinear semi-infinite programming problems. Applications to convex programming are discussed.