Almost Optimal Dispersers

A disperser is a bipartite graph with the property that every subset A of of cardinality at least K, has at least fraction of the vertices of as neighbors. Such graphs have many applications in derandomization. Saks, Srinivasan and Zhou presented an explicit construction of a disperser with an almost optimal degree , for every . We extend their result for every parameter . Received November 12, 1998/Revised June 20, 2000 RID="*" ID="*" This work was partially done while the author was at the Department of Computer Science, Weizmann Institute of Science, Rehovot, Israel. This work was supported by a Phil Zacharia postdoctoral fellowship.     

Asymptotic analysis of dielectric coaxial fibers

Using an asymptotic analysis, we analytically calculate the dispersion and the field distribution of guided modes in an all-dielectric coaxial fiber. We compare the analytical results with those obtained from numerical calculations and find excellent agreement between them. We demonstrate that both the Bragg reflection and the total internal reflection play important roles in providing confinement and determining the dispersion characteristics of the coaxial fiber modes.     

Gbit/s rate bipolar pulse modulation of semiconductor injection lasers

Using a bipolar drive current pulse shape and very low threshold (9 mA) lasers, zero-background pseudorandom optical pulses were generated and detected at a rate of up to 4 Gbit/s without a pattern effect. The drive current pulse consists of a forward pulse followed by a reverse pulse, each of 70 ps duration and about 400 mA amplitude. No DC bias was applied, which is a necessary condition for pattern-effect-free modulation.     

Azimuthally polarized spatial dark solitons: exact solutions of Maxwell's equations in a Kerr medium

Spatial Kerr solitons, typically associated with the standard paraxial nonlinear Schro dinger equation, are shown to exist to all nonparaxial orders as exact solutions of Maxwell's equations in the presence of the vectorial Kerr effect. More precisely, we prove the existence of azimuthally polarized, spatial, dark soliton solutions of Maxwell's equations, while exact linearly polarized (2 + 1)D solitons do not exist. Our ab initio approach predicts the existence of dark solitons up to an upper value of the maximum field amplitude, corresponding to a minimum soliton width of about one-fourth of the wavelength.     

Nonparaxial dark solitons in optical Kerr media

We show that the nonlinear equation that describes nonparaxial Kerr propagation, together with the already reported bright-soliton solutions, admits of (1 + 1)D dark-soliton solutions. Unlike their paraxial counterparts, dark solitons can be excited only if their asymptotic normalized intensity u2infinity is below 3/7; their width becomes constant when u2infinity approaches this value.     

Engineering transverse Bragg resonance waveguides for large modal volume lasers

We recently analyzed a new class of laser amplifier based on transverse Bragg reflection. We show that the unique properties of Bragg confinement make it possible through modal loss discrimination to achieve single-transverse-mode operation with transverse modal size that is an order of magnitude larger than in lasers that depend on total internal reflection for transverse confinement.     

Derived Picard Groups of Finite-Dimensional Hereditary Algebras

Let A be a finite-dimensional algebra over a field k. The derived Picard group DPic _k (A) is the group of triangle auto-equivalences of D> ^b( mod A) induced by two-sided tilting complexes. We study the group DPic _k (A) when A is hereditary and k is algebraically closed. We obtain general results on the structure of DPic _k , as well as explicit calculations for many cases, including all finite and tame representation types. Our method is to construct a representation of DPic _k (A) on a certain infinite quiver ^irr. This representation is faithful when the quiver of A is a tree, and then DPic _k (A) is discrete. Otherwise a connected linear algebraic group can occur as a factor of DPic _k (A). When A is hereditary, DPic _k (A) coincides with the full group of k-linear triangle auto-equivalences of D^b( mod A). Hence, we can calculate the group of such auto-equivalences for any triangulated category D equivalent to D^b( mod A. These include the derived categories of piecewise hereditary algebras, and of certain noncommutative spaces introduced by Kontsevich and Rosenberg.     

Completion by Derived Double Centralizer

Let A be a commutative ring, and let $\mathfrak{a}$ be a weakly proregular ideal in A. (If A is noetherian then any ideal in it is weakly proregular.) Suppose M is a compact generator of the category of cohomologically $\mathfrak{a}$ -torsion complexes. We prove that the derived double centralizer of M is isomorphic to the $\mathfrak{a}$ -adic completion of A. The proof relies on the MGM equivalence from Porta et al. (Algebr Represent Theor, 2013) and on derived Morita equivalence. Our result extends earlier work of Dwyer et al. (Adv Math 200:357–402, 2006) and Efimov (2010).