Common supports as fixed points

A family S of sets in R ^d is sundered if for each way of choosing a point from rd+1 members of S, the chosen points form the vertex-set of an (r–1)-simplex. Bisztriczky proved that for each sundered family S of d convex bodies in R ^d , and for each partition (S , S ), of S, there are exactly two hyperplanes each of which supports all the members of S and separates the members of S from the members of S . This note provides an alternate proof by obtaining each of the desired supports as (in effect) a fixed point of a continuous self-mapping of the cartesian product of the bodies.     

A “4π-type” study of the reaction p(d,pp)n at Ed = 26.5 MeV

Experimental data on the three-nucleon reaction p(d, pp)n at E_d = 26.5 MeV obtained with the multidetector system BOL, covering most of the phase space, are presented. The data are compared with Faddeev calculations using a local S-wave potential for the nucleon-nucleon interaction. The calculated cross sections, which are in general not sensitive to the precise choice of the nucleon-nucleon potential, agree well with the data over the major part of phase space. We have, though, discovered a region in phase space with a strong discrepancy. In the same region, around a neutron c.m. angle of 140°, the calculations are sensitive to the N-N potential. Theoretical and experimental angular distributions in the p-n recoil c.m. subsystem for proton- neutron energies below 0.5 MeV agree in absolute magnitude, both being isotropic. For larger relative energies, a discrepancy develops, part of which might be related to missing higher par- tial wave components in the N-N interaction, rather than to Coulomb or S-wave scattering effects. The total breakup cross section has been deduced from the data as 169 ± 36 mb.     

Comment on a note on duality gaps in linear programming over convex sets

In Ref. 1, Soyster has given a rather complicated proof of the absence of a duality gap, under a certain interiority condition, for a variant of a pair of optimization problems introduced by Ben-Israel, Charnes, and Kortanek (Ref. 2). A proof can be given directly (and under weaker conditions) by a simple application of a Lagrange multiplier theorem on convex programming in abstract spaces (Ref. 3).     

Nuclear Uptake of Gold Nanoparticles Deduced Using Dual‐Angle X‐Ray Fluorescence Mapping

Studies into the cell nucleus' incorporation of gold nanoparticles (AuNPs) are often limited by ambiguities arising from conventional imaging techniques. Indeed, it is suggested that to date there is no unambiguous imaging evidence for such uptake in whole cells, particularly at the single nanoparticle level. This shortcoming in understanding exists despite the nucleus being the most important subcellular compartment in eukaryotes and gold being the most commonly used metal nanoparticle in medical applications. Here, dual‐angle X‐ray flouresence is used to show individually resolved nanoparticles within the cell nucleus, finding them to be well separated and 79% of the intranuclear population to be monodispersed. These findings have important implications for nanomedicine, illustrated here through a specific exemplar of the predicted enhancement of radiation effects arising from the observed AuNPs, finding intranuclear dose enhancements spanning nearly five orders of magnitude.     

An extension of Prohorov's theorem for transition probabilities with applications to infinite-dimensional lower closure problems

A criterion for the relative weak compactness of sets of transition probabilitieson T×B(S) was given by Balder (1984 a),T being a space equipped with a fixed measure andB(S) denoting the σ-algebra of a standard Borel spaceS. This result, which generalizes Prohorov's classical tightness criterion, is extended here so as to cover the case where the spaceS is σ-standard Borel. As a consequence, the method applied in Balder (1984 a) now yields a new general infinite, dimensional lower closure result that can be used in the existence theory for optimal control.     

Comments on purification in continuum games

It is pointed out that Corollary 1 in a recent paper by Khan et al. (Int J Game Theory 34:91–104, 2006), presented there as an extension of the Dvoretzky–Wald–Wolfowitz theorem, is a special case of Lyapunov’s theorem for Young measures (Balder in Rend Instit Mat Univ Trieste 31 Suppl. 1:1–69) It is also pointed out that Theorems 1–4 in Khan et al. (Int J Game Theory 34:91–104, 2006) follow from a single strong purification per se result that is already contained, as an implementation of that Lyapunov theorem for Young measures, in the proof of Theorem 2.2.1 in Balder (J Econ Theory 102:437–470, 2002).     

A useful approximation scheme for Lagrangians

We present a new, useful approximation scheme for the integrand of an integral functional, revolving around a generalized bipolarity result. This scheme leads immediately to lower semicontinuity and lower closure results for the integral functional, as well as to other, more general seminormality properties.     

Simplicial decomposition in nonlinear programming algorithms

Simplicial decomposition is a special version of the Dantzig—Wolfe decomposition principle, based on Carathéodory's theorem. The associated class of algorithms has the following features and advantages: The master and the subprogram are constructed without dual variables; the methods remain therefore well-defined for non-concave objective functions, and pseudo-concavity suffices for convergence to global maxima. The subprogram produces affinely independent sets of feasible generator points defining simplices, which the master program keeps minimal by dropping redundant generator points and finding maximizers in the relative interiors of the resulting subsimplices. The use of parallel subspaces allows the direct application of any unrestricted optimization method in the master program; thus the best unconstrained procedure for any type of objective function can be used to find constrained maximizers for it.The paper presents the theory for this class of algorithms, the APL-code of a demonstration method and some computational experience with Colville's test problems.I am grateful to Philip Wolfe for encouraging me to write this paper, and I am indebted to him and a referee for helpful comments.Research was partially supported by a grant of the University of Alberta.     

An existence result for optimal economic growth problems

An existence result for optimal control problems of Lagrange type with unbounded time domain is derived very directly from a corresponding result for problems with bounded time domain. This subsumes the main existence result of R. F. Baum ¦J. Optim. Theory Appl.19 (1976), 89–116¦ and has the existence results for optimal economic growth problems of S.-I. Takekuma ¦J. Math. Econom.7 (1980), 193–208¦ and M. J. P. Magill ¦Econometrica49 (1981), 679–711; J. Math. Anal. Appl.82 (1981), 66–74¦ as simple corollaries. In addition, a new notion of uniform integrability is used, which coincides with the classical notion if the time domain is bounded.