2D FDTD simulation of low loss small angle bend and Y branch configurations in a photonic crystal waveguide layout with a Mach–Zehnder device design configuration

A technique to produce low loss small angle bends in photonic crystal waveguides is presented. The technique consists of bridging parallel input and output waveguide segments with an inclined waveguide region of the same basic design that has a lateral dielectric shift. Results are presented for waveguides produced by enlarging the silicon gap along the central line, separating air holes in a square array photonic crystal for the TE polarization and an operating wavelength of λ_o = 1.55 μm. This low loss waveguide bending technique is applied to the design of Y branch and Mach–Zehnder photonic crystal structures. Simulation of the performance of the dielectric structures is achieved using 2-D FDTD, similar results are anticipated when applied to 3-D waveguide configurations and for other photonic crystal layouts.     

On the instability of the Riemann Hypothesis for varieties over finite fields

There exist perturbations of a rational function which remove zeroes and poles from a prescribed region as well as perturbations which add zeroes and poles to a prescribed region. We employ this to show the instability of the Riemann Hypothesis for zeta-functions of smooth projective varieties over finite fields.     

Bloch constants for planar harmonic mappings

We give a lower estimate for the Bloch constant for planar harmonic mappings which are quasiregular and for those which are open. The latter includes the classical Bloch theorem for holomorphic functions as a special case. Also, for bounded planar harmonic mappings, we obtain results similar to a theorem of Landau on bounded holomorphic functions.

     

Tests of resonance recognition criteria in coupled channel problems with separable potentials

Exactly soluble two-body two-channel problems are set up with separable potentials. Resonance is defined theoretically then the empirical resonance recognition criteria used in two-body data analyses are tested. Results are summarized in the last section of the paper.     

Quantum mechanics and the local observer

A notion of local observer inspired by the work of Segal is introduced here in the Hilbert space theory of quantum mechanics. The local observer finds a mathematical place in the Hilbert space through local negation or complementation. A logicomathematical theory of local negation is presented and its implications for quantum logic and the problem of measurement are discussed. The setting is constructivist mathematics and the main result of the paper states that the introduction of a local observer implies the nonorthocomplementability of the whole Hilbert space even in the finite-dimensional case. Making a mathematical place for the observer (the “projector”) thus modifies the structure of the observables or the system of the projections, in accordance with a nonclassical theory of quantum-mechanical measurement.     

Experimental investigation of high-quality synchronization of coupled oscillators

We describe two experiments in which we investigate the synchronization of coupled periodic oscillators. Each experimental system consists of two identical coupled electronic periodic oscillators that display bursts of desynchronization events similar to those observed previously in coupled chaotic systems. We measure the degree of synchronization as a function of coupling strength. In the first experiment, high-quality synchronization is achieved for all coupling strengths above a critical value. In the second experiment, no high-quality synchronization is observed. We compare our results to the predictions of the several proposed criteria for synchronization. We find that none of the criteria accurately predict the range of coupling strengths over which high-quality synchronization is observed. (c) 2000 American Institute of Physics.     

Generic singularities of the 3D-contact sub-Riemannian conjugate locus

In this paper, we extend and complete the classification of the generic singularities of the 3D-contact sub-Riemmanian conjugate locus in a neighborhood of the origin.     

Synthesis of macrocycles via cobalt-mediated [2 + 2 + 2] cycloadditions

We investigated the formation of macrocycles from alpha,omega-diynes in cobalt-mediated co-cyclotrimerization reactions. Long-chain alpha,omega-diynes underwent metal-mediated [2 + 2 + 2] cycloadditions with nitriles, cyanamides, or isocyanates in the presence of CpCo(CO)2 (Cp = cyclopentadienide) to yield pyridine-containing macrocycles, i.e., meta- and para-pyridinophanes, such as 5m/5p, 35m/35p, and 41m/41p. The regioselectivity of these reactions was affected by the length and type of linker unit between the alkyne groups, as well as by certain stereoelectronic factors. An analogous alpha,omega-cyano-alkyne, 28, combined with an alkyne to yield two isomeric meta-pyridinophanes, such as 5m and 29m, and an ortho cycloadduct (benzannulation product), such as 29o. We developed a reaction protocol for these cobalt-based [2 + 2 + 2] cycloadditions that involves markedly improved conditions such that this process offers a convenient, flexible synthetic approach to macrocyclic pyridine-containing compounds. For example, diyne 6 reacted with p-tolunitrile in 1,4-dioxane to give 7p and 7m (7:1 ratio) in 87% yield at a moderate temperature of ca. 100 degrees C in 24 h without photoirradiation or syringe-pump addition. Isocyanates were also effective reactants, as exemplified by the formation of 44p almost exclusively (44p:44m > 50:1) in 64% yield from diyne 8 and 2-phenylethylisocyanate. By using this improved protocol we were able to co-cyclotrimerize long-chain alpha,omega-diynes with alkynes in certain cases to demonstrate a successful macrocyclic variant of the Vollhardt reaction. For instance, diyne 6 reacted with dipropylacetylene to give paracyclophane 57p and benzannulene 57o (2:1 ratio) in 29% yield.