Optimal Control for Switched Systems with Pre-defined Order and Switch-Dependent Dynamics

The optimal control problem for switched systems, with a predefined order of switches, is considered. The differential equations may depend explicitly on previous switching instants, and the latter may be state dependent. The solution is based on the Calculus-of-Variations, which leads to a single two-point boundary-value problem. A new condition for the Hamiltonian jump at the switching instants is obtained. Simple numerical examples demonstrate the results.     

On parabolic induction on inner forms of the general linear group over a non-archimedean local field

We give new criteria for the irreducibility of parabolic induction on the general linear group and its inner forms over a local non-archimedean field. In particular, we give a necessary and sufficient condition when the inducing data is of the form \(\pi \otimes \sigma \) where \(\pi \) is a ladder representation and \(\sigma \) is an arbitrary irreducible representation. As an application we simplify the proof of the classification of the unitary dual.     

Phases of the infinite U Hubbard model on square lattices

We apply the density matrix renormalization group to study the phase diagram of the infinite U Hubbard model on 2- to 6-leg ladders. Where the results are largely insensitive to the ladder width, we consider the results representative of the 2D square lattice. We find a fully polarized ferromagnetic Fermi liquid phase when n, the density of electrons per site, is in the range 1>n?0.800. For n=3/4 we find an unexpected insulating checkerboard phase with coexisting bond-density order with 4 sites per unit cell and block-spin antiferromagnetic order with 8 sites per unit cell. For 3/4>n, all ladders with width >2 have unpolarized ground states.     

Study of the membrane effect on turbulent mixing measurements in shock tubes

The effect of the thin membrane on the time evolution of the shock wave induced turbulent mixing between the two gases initially separated by it is investigated using two different sets of experiments. In the first set, in which a single-mode large-amplitude initial perturbation was studied, two gas combinations (air/SF and air/air) and two membrane thicknesses were used. The main conclusion of these experiments was that the tested membrane has a negligible effect on the evolution of the mixing zone, which evolves as predicted theoretically. In the second set, in which similar gas combinations and membrane thicknesses were used, small amplitude random-mode initial perturbation, caused by the membrane rupture, rather than the large amplitude single-mode initial perturbation used in the first set, was studied. The conclusions of these experiments were: (1) The membrane has a significant effect on the mixing zone during the initial stages of its growth. This has also been observed in the air/air experiment where theoretically no growth should exist. (2) The membrane effect on the late time evolution, where the mixing zone width has reached a relatively large-amplitude, was relatively small and in good agreement with full numerical simulations. The main conclusion from the present experiments is that the effect of the membrane is important only during the initial stages of the evolution (before the re-shock), when the perturbations have very small amplitudes, and is negligible when the perturbations reach relatively large amplitudes. Received 29 August 1998 / Accepted 25 December 1998     

From DNA to transistors

The rapid advance in molecular biology and nanotechnology opens up the possibility to explore the interface between biology and electronics at the single-molecule level. We focus on the organization of molecular electronic circuits. Interconnecting an immense number of molecular devices into a functional circuit and constructing a framework for integrated molecular electronics requires new concepts. A promising avenue relies on bottom-up assembly where the information for the circuit connectivity and functionality is embedded in the molecular building blocks. Biology can provide concepts and mechanisms for advancing this approach, but there is no straightforward way to apply them to electronics since biological molecules are essentially electrically insulating. Bridging the chasm between biology and electronics therefore presents great challenges. Circuit organization on the molecular scale is considered and contrasted with the levels of organization presented by the living world. The discussion then focuses on our proposal to harness DNA and molecular biology to construct the scaffold for integrated molecular electronics. DNA metallization is used to convert the DNA scaffold into a conductive one. We present the framework of sequence-specific molecular lithography based on the biological mechanism of homologous genetic recombination and carried out by the bacterial protein RecA. Molecular lithography enables us to use the information encoded in the scaffold DNA molecules for directing the construction of an electronic circuit. We show that it can lead all the way from DNA molecules to working transistors in a test-tube. Carbon nanotubes are incorporated as the active electronic components in the DNA-templated transistors. Our approach can, in principle, be applied to the fabrication of larger-scale electronic circuits. The realization of complex DNA-based circuits will, however, require new concepts and additional biological machinery allowing, for example, feedback from the electronic functionality to direct the assembly process and adaptation mechanisms.     

The bimolecular hydrogen-deuterium exchange behavior of protonated alkyl dipeptides in the gas phase

As part of an ongoing characterization of the intrinsic chemical properties of peptides, thermal hydrogen-deuterium exchange has been studied for a series of fast-atom-bombardment-generated protonated alkyldipeptides and related model compounds in the reaction with D₂O, CH₃OD, and ND₃ in a Fourier transform ion cyclotron resonance mass spectrometer. Despite the very large basicity difference between the dipeptides and the D₂O and CH₃OD exchange reagents, efficient exchange of all active hydrogen atoms occurs. From the kinetic data it appears that exchange of the amino, amide, and hydroxyl hydrogens proceeds with different efficiencies, which implies that the proton in thermal protonated dipeptides is immobile. The selectivity of the exchange at the different basic sites is governed by the nature of both the dipeptide and the exchange reagent. The results indicate that reversible proton transfer in the reaction complexes, which effectuates the deuterium incorporation, is assisted by formation of multiple hydrogen bonds between the reagents. Exchange is considered to proceed via the intermediacy of different competing intermediate complexes, each of which specifically leads to deuterium incorporation at different basic sites. The relative stabilization of the competing intermediate complexes can be related to the relative efficiencies of deuterium incorporation at different basic sites in the dipeptide. For all protonated dipeptides studied, the exchange in the reaction with ND₃ proceeds with unit efficiency, whereas all active hydrogen atoms are exchanged equally efficiently. Evidently specific multiple hydrogen bond formations are far less important in the reversible proton transfers with the relatively basic ammonia, which allows effective randomization of all active hydrogen atoms in the reaction complexes.     

“If You Can Turn a Rectangle into a Square, You Can Turn a Square into a Rectangle ...” Young Students Experience the Dragging Tool

This paper describes a study of the cognitive complexity of young students, in the pre-formal stage, experiencing the dragging tool. Our goal was to study how various conditions of geometric knowledge and various mental models of dragging interact and influence the learning of central concepts of quadrilaterals. We present three situations that reflect this interaction. Each situation is characterized by a specific interaction between the students’ knowledge of quadrilaterals and their understanding of the dragging tool. The analyses of these cases offer a prism for viewing the challenge involved in changing concept images of quadrilaterals while lacking understanding of the geometrical logic that underlies dragging. Understanding dragging as a manipulation that preserves the critical attributes of the shape is necessary for constructing the concept images of the shapes.     

Two-photon enhanced optogalvanic signal and population inversion

Two-photon enhancement of the optogalvanic signal is shown to be possible, both theoretically and experimentally, even in the case of population inversion of the first transition. This is shown, in particular, for the neon 1s₂ → 2p_i transitions.