Simultaneous chromatic dispersion, polarization-mode-dispersion and OSNR monitoring at 40Gbit/s

A novel method for independent and simultaneous monitoring of chromatic dispersion ( CD), first-order PMD and OSNR in 40Gbit/s systems is proposed and demonstrated. This is performed using in-band tone monitoring of 5GHz, optically down-converted to a low intermediate-frequency (IF) of 10kHz. The measurement provides a large monitoring range with good accuracies for CD (4742 +/- 100ps/nm), differential group delay (DGD) (200 +/- 4ps) and OSNR (23 +/- 1dB), independently of the bit-rate. In addition, the use of electro-absorption modulators (EAM) for the simultaneous down-conversion of all channels and the use of low-speed detectors makes it cost effective for multi-channel operation. (C) 2008 Optical Society of Americ

An Algorithmic Study of Manufacturing Paperclips and Other Folded Structures

We study algorithmic aspects of bending wires and sheet metal into a specified structure. Problems of this type are closely related to the question of deciding whether a simple non-self-intersecting wire structure (a carpenter's ruler) can be straightened, a problem that was open for several years and has only recently been solved in the affirmative. If we impose some of the constraints that are imposed by the manufacturing process, we obtain quite different results. In particular, we study the variant of the carpenter's ruler problem in which there is a restriction that only one joint can be modified at a time. For a linkage that does not self-intersect or self-touch, the recent results of Connelly et al. and Streinu imply that it can always be straightened, modifying one joint at a time. However, we show that for a linkage with even a single vertex degeneracy, it becomes NP-hard to decide if it can be straightened while altering only one joint at a time. If we add the restriction that each joint can be altered at most once, we show that the problem is NP-complete even without vertex degeneracies. In the special case, arising in wire forming manufacturing, that each joint can be altered at most once, and must be done sequentially from one or both ends of the linkage, we give an efficient algorithm to determine if a linkage can be straightened.Comment: 28 pages, 14 figures, Latex, to appear in Computational Geometry - Theory and Application

Archetypal analysis of galaxy spectra

Archetypal analysis represents each individual member of a set of data vectors as a mixture (a constrained linear combination) of the pure types or archetypes of the data set. The archetypes are themselves required to be mixtures of the data vectors. Archetypal analysis may be particularly useful in analysing data sets comprising galaxy spectra, since each spectrum is, presumably, a superposition of the emission from the various stellar populations, nebular emissions and nuclear activity making up that galaxy, and each of these emission sources corresponds to a potential archetype of the entire data set. We demonstrate archetypal analysis using sets of composite synthetic galaxy spectra, showing that the method promises to be an effective and efficient way to classify spectra. We show that archetypal analysis is robust in the presence of various types of noise.Comment: 6 pages, 5 figures, 1 style-file. Accepted for publication by MNRA

Geometry and Topology of Escape I: Epistrophes

We consider a dynamical system given by an area-preserving map on a two-dimensional phase plane and consider a one-dimensional line of initial conditions within this plane. We record the number of iterates it takes a trajectory to escape from a bounded region of the plane as a function along the line of initial conditions, forming an ``escape-time plot''. For a chaotic system, this plot is in general not a smooth function, but rather has many singularities at which the escape time is infinite; these singularities form a complicated fractal set. In this article we prove the existence of regular repeated sequences, called ``epistrophes'', which occur at all levels of resolution within the escape-time plot. (The word ``epistrophe'' comes from rhetoric and means ``a repeated ending following a variable beginning''.) The epistrophes give the escape-time plot a certain self-similarity, called ``epistrophic'' self-similarity, which need not imply either strict or asymptotic self-similarity.Comment: 15 pages, 9 figures, to appear in Chaos, first of two paper

Range Queries on Uncertain Data

Given a set $P$ of $n$ uncertain points on the real line, each represented by its one-dimensional probability density function, we consider the problem of building data structures on $P$ to answer range queries of the following three types for any query interval $I$: (1) top-$1$ query: find the point in $P$ that lies in $I$ with the highest probability, (2) top-$k$ query: given any integer $k\leq n$ as part of the query, return the $k$ points in $P$ that lie in $I$ with the highest probabilities, and (3) threshold query: given any threshold $\tau$ as part of the query, return all points of $P$ that lie in $I$ with probabilities at least $\tau$. We present data structures for these range queries with linear or nearly linear space and efficient query time.Comment: 26 pages. A preliminary version of this paper appeared in ISAAC 2014. In this full version, we also present solutions to the most general case of the problem (i.e., the histogram bounded case), which were left as open problems in the preliminary versio

Global Dimension of Polynomial Rings in Partially Commuting Variables

For any free partially commutative monoid $M(E,I)$, we compute the global dimension of the category of $M(E,I)$-objects in an Abelian category with exact coproducts. As a corollary, we generalize Hilbert's Syzygy Theorem to polynomial rings in partially commuting variables.Comment: 11 pages, 2 figure

Geometry and Topology of Escape II: Homotopic Lobe Dynamics

We continue our study of the fractal structure of escape-time plots for chaotic maps. In the preceding paper, we showed that the escape-time plot contains regular sequences of successive escape segments, called epistrophes, which converge geometrically upon each endpoint of every escape segment. In the present paper, we use topological techniques to: (1) show that there exists a minimal required set of escape segments within the escape-time plot; (2) develop an algorithm which computes this minimal set; (3) show that the minimal set eventually displays a recursive structure governed by an ``Epistrophe Start Rule'': a new epistrophe is spawned Delta = D+1 iterates after the segment to which it converges, where D is the minimum delay time of the complex.Comment: 13 pages, 8 figures, to appear in Chaos, second of two paper

The conduciveness of CA-rule graphs

Given two subsets A and B of nodes in a directed graph, the conduciveness of the graph from A to B is the ratio representing how many of the edges outgoing from nodes in A are incoming to nodes in B. When the graph's nodes stand for the possible solutions to certain problems of combinatorial optimization, choosing its edges appropriately has been shown to lead to conduciveness properties that provide useful insight into the performance of algorithms to solve those problems. Here we study the conduciveness of CA-rule graphs, that is, graphs whose node set is the set of all CA rules given a cell's number of possible states and neighborhood size. We consider several different edge sets interconnecting these nodes, both deterministic and random ones, and derive analytical expressions for the resulting graph's conduciveness toward rules having a fixed number of non-quiescent entries. We demonstrate that one of the random edge sets, characterized by allowing nodes to be sparsely interconnected across any Hamming distance between the corresponding rules, has the potential of providing reasonable conduciveness toward the desired rules. We conjecture that this may lie at the bottom of the best strategies known to date for discovering complex rules to solve specific problems, all of an evolutionary nature