Spectro-spatial analyses of a nonlinear metamaterial with multiple nonlinear local resonators

Nonlinear Dynamics - Recent focus has been given to spectro-spatial analysis of nonlinear metamaterials since they can predict interesting nonlinear phenomena not accessible by spectral analysis...     

The Essential Spectrum of Schrödinger, Jacobi, and CMV Operators

We provide a very general result which identifies the essential spectrum of broad classes of operators as exactly equal to the closure of the union of the spectra of suitable limits at infinity. Included is a new result on the essential spectra when potentials are asymptotic to isospectral tori. We also recover within a unified framework the HVZ Theorem and Krein's results on orthogonal polynomials with finite essential spectra. Supported in part by The Israel Science Foundation (grant No. 188/02). Supported in part by NSF grant DMS-01 40592. Research supported in part by grant No. 2002068 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel.     

Spray flame dynamics with oscillating flow and droplet grouping

The co-flow laminar spray diffusion flame in an oscillating flow field is investigated. Mild slip is permitted between the droplets and their host surroundings and droplet grouping resulting from the host flow oscillations is accounted for. The spray is modelled using the sectional approach and a perturbation analysis using a small sectional Stokes number is utilised for solving the liquid phase governing equations. The effect of droplet grouping is described through a specially constructed model for the vaporisation Damkohler number. The large chemical Damkohler number assumption is adopted and a formal analytical solution is developed for Schwab-Zeldovitch parameters through which the dynamics of the spray flame front shapes and thermal fields are deduced. Computed results based on the solutions demonstrate how the phenomenon of droplet grouping can lead to the existence of multiple flame sheets as a result of the dynamic change in the type of the main homogeneous flame from under- to over-ventilated as the flow field oscillates. Concomitant fluctuating thermal fields are also shown to be present indicating a potential impact on undesirable pollutants production.     

Children’s use of realistic considerations in problem solving: some English evidence

This is the second of two papers exploring children’s responses to an extended version of a division-with-remainder problem intended to elicit general rather than particular realistic considerations during mathematical problem solving. Responses to two problems are analyzed. The first is a ‘realistically’ contextualised item drawn from national tests in England whose ambiguities have been previously discussed (Cooper, 1992); the second is a version of this problem revised to encourage a wider range of realistic responses. In Cooper and Harries (2002), the responses of children at the end of their first year of secondary schooling were analyzed. Here the responses of children at the end of their primary schooling are analyzed and compared with the previous results. It is shown that many children, given suitable encouragement, are willing and able to enter into an extended form of realistic thinking during problem solving, although the original test item renders this invisible.     

Non-Uniformity and Generalised Sacks Splitting

We show that there do not exist computable functions f ₁(e, i), f ₂(e, i), g ₁(e, i), g ₂(e, i) such that for all e, i ∈ ω, (1) $ {\left( {W_{{f_{1} {\left( {e,i} \right)}}} - W_{{f_{2} {\left( {e,i} \right)}}} } \right)} \leqslant _{{\rm T}} {\left( {W_{e} - W_{i} } \right)}; $ (2) $ {\left( {W_{{g_{1} {\left( {e,i} \right)}}} - W_{{g_{2} {\left( {e,i} \right)}}} } \right)} \leqslant _{{\rm T}} {\left( {W_{e} - W_{i} } \right)}; $ (3) $ {\left( {W_{e} - W_{i} } \right)} \not\leqslant _{{\rm T}} {\left( {W_{{f_{1} {\left( {e,i} \right)}}} - W_{{f_{2} {\left( {e,i} \right)}}} } \right)} \oplus {\left( {W_{{g_{1} {\left( {e,i} \right)}}} - W_{{g_{2} {\left( {e,i} \right)}}} } \right)}; $ (4) $ {\left( {W_{e} - W_{i} } \right)} \not\leqslant _{{\rm T}} {\left( {W_{{f_{1} {\left( {e,i} \right)}}} - W_{{f_{2} {\left( {e,i} \right)}}} } \right)}{\text{unless}}{\left( {W_{e} - W_{i} } \right)} \leqslant _{{\rm T}} {\emptyset};{\text{and}} $ (5) $ {\left( {W_{e} - W_{i} } \right)} \leqslant _{{\rm T}} {\left( {W_{{g_{1} {\left( {e,i} \right)}}} - W_{{g_{2} {\left( {e,i} \right)}}} } \right)}{\text{unless}}{\left( {W_{e} - W_{i} } \right)} \leqslant _{{\rm T}} {\emptyset}. $ It follows that the splitting theorems of Sacks and Cooper cannot be combined uniformly.     

On a Conjecture of Kleene and Post

A proof is given that 0 ′ (the argest Turing degree containing a computably enumerable set) is definable in the structure of the degrees of unsolvability. This answers a long‐standing question of Kleene and Post, and has a number of corollaries including the definability of the jump operator.     

In situ device for salinity measurements (chloride detection) of ocean surface

Microbend sodium chloride sensors based on polymer swelling are the latest approach to fiberoptic sodium chloride sensors. We have designed, constructed, and tested a compact microbend fiberoptic sodium chloride sensor that utilizes microbend and polymer swelling effects through the use of modified polymer beads. The design of this sensor is an improvement over previously reported microbend sensors. This sodium chloride sensor has shown feasibility as a practical, stable, inexpensive device with excellent performance and most of the desirable sensor properties including reversibility. Further optimization of the sensor is possible for the various applications we envision.     

Spectral-domain optical coherence phase microscopy for quantitative phase-contrast imaging

We describe a novel microscopy technique for quantitative phase-contrast imaging of a transparent specimen. The technique is based on depth-resolved phase information provided by common path spectral-domain optical coherence tomography and can measure minute phase variations caused by changes in refractive index and thickness inside the specimen. We demonstrate subnanometer level path-length sensitivity and present images obtained on reflection from a known phase object and human epithelial cheek cells.     

Palladium-catalyzed asymmetric allylic alkylation of meso- and dl-1,2-divinylethylene carbonate

The palladium-catalyzed asymmetric allylic alkylation of a 1:1 mixture of dl- and meso-1,2-divinylethylene carbonate is reported. For the first time, both the ionization and nucleophilic addition steps of the catalytic cycle act as enantiodiscriminating steps to give a single product in high enantiomeric excess. The reactions proceed in >98% ee to efficiently generate useful chiral building blocks from acrolein. The absolute and relative configurations of iso-cladospolide B and 11-epi-iso-cladospolide B were verified by total synthesis, solving an apparent discrepancy in the literature.     

Pd and Mo Catalyzed Asymmetric Allylic Alkylation

The ability to control the alkylation of organic substrates becomes ever more powerful by using metal catalysts. Among the major benefits of metal catalysis is the possibility to perform such processes asymmetrically using only catalytic amounts of the chiral inducing agent which is a ligand to the metal of the catalyst. A unique aspect of asymmetric metal catalyzed processes is the fact that many mechanisms exist for stereoinduction. Furthermore, using the same catalyst system, many types of bonds including but not limited to C-C, C-N, C-O, C-S, C-P, and C-H can be formed asymmetrically. An overview of this process using palladium and molybdenum based metals being developed in my laboratories and how they influence strategy in synthesizing bioactive molecular targets is presented.