Mode beating in spot-size converter lasers

An anomalous modulation in the wavelength spectrum has been observed in lasers with spot-size converters. This intensity modulation is shown to be caused by beating between the fundamental lasing mode and radiation modes in the taper. This results in a periodic modulation in the net gain spectrum, which causes wavelength jumps between adjacent net gain maxima, and a drive current dependent spectral width that is expected to affect system performance. The amplitude of this spectral modulation is reduced significantly by either using an angled rear-facet which reflects the beating radiation modes away from the laser axis, or by using a nonlinear, adiabatic taper.     

Packing Non-Zero A-Paths In Group-Labelled Graphs

Let G=(V,E) be an oriented graph whose edges are labelled by the elements of a group Γ and let A⊂V. An A-path is a path whose ends are both in A. The weight of a path P in G is the sum of the group values on forward oriented arcs minus the sum of the backward oriented arcs in P. (If Γ is not abelian, we sum the labels in their order along the path.) We are interested in the maximum number of vertex-disjoint A-paths each of non-zero weight. When A = V this problem is equivalent to the maximum matching problem. The general case also includes Mader's S-paths problem. We prove that for any positive integer k, either there are k vertex-disjoint A-paths each of non-zero weight, or there is a set of at most 2k −2 vertices that meets each of the non-zero A-paths. This result is obtained as a consequence of an exact min-max theorem. These results were obtained at a workshop on Structural Graph Theory at the PIMS Institute in Vancouver, Canada. This research was partially conducted during the period the first author served as a Clay Mathematics Institute Long-Term Prize Fellow.     

Study of the peak effect phenomenon in single crystals of 2H-NbSe2

The weakly pinned single crystals of the hexagonal 2H-NbSe₂ compound have emerged as prototypes for determining and characterizing the phase boundaries of the possible order-disorder transformations in the vortex matter. We present here a status report based on the ac and dc magnetization measurements of the peak effect phenomenon in three crystals of 2H-NbSe₂, in which the critical current densities vary over two orders of magnitude. We sketch the generic vortex phase diagram of a weakly pinned superconductor, which also utilizes theoretical proposals. We also establish the connection between the metastability effects and pinning.     

Metric spaces with discrete topological fundamental group

With a certain natural topology, the fundamental group of a locally path connected metric space X becomes discrete if and only if X is semilocally simply connected. We include some related material and offer a counterexample to a similar result.     

Evolution in complex systems

What features characterize complex system dynamics? Power laws and scale invariance of fluctuations are often taken as the hallmarks of complexity, drawing on analogies with equilibrium critical phenomena. Here we argue that slow, directed dynamics, during which the system's properties change significantly, is fundamental. The underlying dynamics is related to a slow, decelerating but spasmodic release of an intrinsic strain or tension. Time series of a number of appropriate observables can be analyzed to confirm this effect. The strain arises from local frustration. As the strain is released through “quakes,” some system variable undergoes record statistics with accompanying log‐Poisson statistics for the quake event times. We demonstrate these phenomena via two very different systems: a model of magnetic relaxation in type II superconductors and the Tangled Nature model of evolutionary ecology and show how quantitative indications of aging can be found. © 2004 Wiley Periodicals, Inc. Complexity 10: 49–56, 2004     

Stark's conjecture over complex cubic number fields

Systematic computation of Stark units over nontotally real base fields is carried out for the first time. Since the information provided by Stark's conjecture is significantly less in this situation than the information provided over totally real base fields, new techniques are required. Precomputing Stark units in relative quadratic extensions (where the conjecture is already known to hold) and coupling this information with the Fincke-Pohst algorithm applied to certain quadratic forms leads to a significant reduction in search time for finding Stark units in larger extensions (where the conjecture is still unproven). Stark's conjecture is verified in each case for these Stark units in larger extensions and explicit generating polynomials for abelian extensions over complex cubic base fields, including Hilbert class fields, are obtained from the minimal polynomials of these new Stark units.

     

Stereoselective migration of sterically hindered organoboranes in cyclic and acyclic systems. A stereoselective allylic C-H activation reaction

The thermal migration of cyclic and acyclic organoboranes were studied. In most cases, a stereoselective 1,2-dyotropic migration was observed, allowing the stereocontrol of three contiguous chiral centers. Scope and limitations of this thermal migration are presented.     

Rhythmic pattern formations in gels and Matalon—Packter law: A fresh perspective

The periodic precipitation pattern formation in gelatinous media is interpreted as a moving boundary problem. The time law, spacing law and width law are revisited on the basis of the new scenario. The explicit dependence of the geometric structure on the initial concentrations of the reactants is derived. Matalon—Packter law, which relates the spacing coefficient with the initial concentrations is reformulated removing many ambiguities and impractical parameters. Experimental results are discussed to establish the significance of moving boundary concept in the diffusion controlled pattern forming systems