On Series of Ordinals and Combinatorics

This paper deals mainly with generalizations of results in finitary combinatorics to infinite ordinals. It is well-known that for finite ordinals ∑_bT<αβ is the number of 2-element subsets of an α-element set. It is shown here that for any well-ordered set of arbitrary infinite order type α, ∑_bT<αβ is the ordinal of the set M of 2-element subsets, where M is ordered in some natural way. The result is then extended to evaluating the ordinal of the set of all n-element subsets for each natural number n ≥ 2. Moreover, series ∑_β<αf(β) are investigated and evaluated, where α is a limit ordinal and the function f belongs to a certain class of functions containing polynomials with natural number coefficients. The tools developed for this result can be extended to cover all infinite α, but the case of finite α appears to be quite problematic.     

An experiment to measure collisionless radial transport ofenergetic electrons confined by a dipole magnetic field

A new laboratory terrella has been constructed in order to study collisionless radial diffusion of particles trapped within a dipole magnetic field. Columbia's collisionless terrella experiment (CTX) aims to reproduce the process of wave-induced radial transport and does not try to simulate magnetospheric structure. The first experiment planned for CTX is the direct measurement of stochastic radial diffusion induced from wave-particle drift resonances. The motivation for the CTX experiment is described, and the procedures to be used to measure the intensity and spectrum of fluctuations generating chaos, the rate of radial transport, and the evolution of the density and pressure profiles are illustrated. Because of the success of similar experiments conducted earlier in a long thin magnetic mirror, these dipole experiments can be performed with a high degree of confidence. An example from these earlier experiments is presented     

Some Fixed-Point Results for the Dynamic Assignment Problem

In previous work the authors consider the dynamic assignment problem, which involves solving sequences of assignment problems over time in the presence of uncertain information about the future. The algorithm proposed by the authors provides generally high-quality but non-optimal solutions. In this work, though, the authors prove that if the optimal solution to a dynamic assignment problem in one of two problem classes is unique, then the optimal solution is a fixed point under the algorithm.     

Application of a Near-Optimal Reinforcement Learning Controller to a Robotics Problem in Manufacturing: A Hybrid Approach

Optimization theory provides a framework for determining the best decisions or actions with respect to some mathematical model of a process. This paper focuses on learning to act in a near-optimal manner through reinforcement learning for problems that either have no model or the model is too complex. One approach to solving this class of problems is via approximate dynamic programming. The application of these methods are established primarily for the case of discrete state and action spaces. In this paper we develop efficient methods of learning which act in complex systems with continuous state and action spaces. Monte-Carlo approaches are employed to estimate function values in an iterative, incremental procedure. Derivative-free line search methods are used to obtain a near-optimal action in the continuous action space for a discrete subset of the state space. This near-optimal control policy is then extended to the entire continuous state space via a fuzzy additive model. To compensate for approximation errors, a modified procedure for perturbing the generated control policy is developed. Convergence results under moderate assumptions and stopping criteria are established.     

Crosslinked polystyrene latexes prepared with 12-(o-styryloxy)dodecyltrimethylammonium bromide

Crosslinked polystyrene latexes with high surface charge densities have been prepared with 1, 3, and 5 mol% divinylbenzene and 0.5–5.0 mol% of 12-(o-styryloxy)dodecyltrimethylammonium bromide ( 1 ). Within experimental error all of surfactant 1 was incorporated into the particles. Analysis of the latex particles by transmission electron microscopy showed nonspherical shapes, aggregates, and number average diameters ranging from 22 to 95 nm. The particle size decreases and the polydispersity increases with increasing amounts of surfactant 1 and of divinylbenzene in the monomer mixture. A mechanism is proposed for particle formation by precipitation of crosslinked polymer from monomer droplets to form primary particles, coagulation of the primary particles to colloidally stable aggregates, and growth by further polymerization.     

The solution of a convection–diffusion–reaction equation arising from cathodic reduction in a pulsating crack

This paper derives the convection–diffusion-reaction equation governing the reaction between the dissolved oxygen in sea-water and the steel walls of a pulsating crack. By the neglect of the diffusion term it is shown that an exact solution of the convection-reaction equation can be obtained. A numerical method for the solution of the complete convection–diffusion-reaction equation is derived by the use of finite differences. The numerical computation of the initial transient and the final periodic steady-state values is also discussed.     

Random orderings of the integers and card shuffling

In this paper we study random orderings of the integers with a certain invariance property. We describe all such orders in a simple way. We define and represent random shuffles of a countable set of labels and then give an interpretation of these orders in terms of a class of generalized riffle shuffles.     

Liquid crystalline hexa-amides and side chain polysiloxanes of azacrown [18]-N6

The hexa-4-dodecyloxybenzoyl derivative 1 of azacrown [18]-N₆ was originally reported to have a 'tubular' mesophase on the basis of its large central ring and 6-fold symmetry. Starting in the mesophase, annealing of 1 under a cover slip results in formation of a new crystalline phase that melts directly to an isotropic liquid at the temperature previously observed for the mesophase to isotropic transition. Thus the phase behaviour of 1 is kinetically controlled. The analogous hexa-3,4-bisdodecyloxybenzoyl derivative 2 of azacrown [18]-N₆ has no kinetic limitations to its phase changes and has an enantiotropic columnar liquid crystalline phase. We have synthesized side chain copolysiloxanes with a (CH₂)₁₁ spacer and 75-84 per cent by weight of the same 4-dodecyloxybenzoyl-[18]-N₆ mesogen. The polysiloxanes also display a liquid crystalline phase.     

Production of slow muonic hydrogen isotopes in vacuum

Muonic hydrogen isotopes (μ⁻ p, μ⁻ d, and μ⁻t) are simple quantum mechanical systems ideally suited for studies of numerous fundamental phenomena in electroweak and strong interactions as well as in applied areas such as muon chemistry or muon catalyzed fusion. Emission of muonic hydrogen isotopes into vacuum helps to overcome the limitations which are normally imposed on conventional investigations with gaseous and liquid targets. A proof of principle experiment for this new technique was performed at TRIUMF last year. Negative muons with 30 MeV/c momentum were stopped in a thin film of solid hydrogen and produced very low energy μ⁻d in vacuum. The distribution center of the normal velocity components of emitted μ⁻d atoms was measured to be ∼1 cm/μs. The yield of μ⁻d in vacuum is an increasing function of H₂ film thickness δ up to a value of δ≥1 mm.