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

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

Entangled photons, nonlocality and Bell inequalities in the undergraduate laboratory

We use polarization-entangled photon pairs to demonstrate quantum nonlocality in an experiment suitable for advanced undergraduates. The photons are produced by spontaneous parametric downconversion using a violet diode laser and two nonlinear crystals. The polarization state of the photons is tunable. Using an entangled state analogous to that described in the Einstein-Podolsky-Rosen ``paradox,'' we demonstrate strong polarization correlations of the entanged photons. Bell's idea of a hidden variable theory is presented by way of an example and compared to the quantum prediction. A test of the Clauser, Horne, Shimony and Holt version of the Bell inequality finds $S = 2.307 \pm 0.035$, in clear contradiciton of hidden variable theories. The experiments described can be performed in an afternoon.Comment: 10 pages, 6 figure

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

Configuration study for a 30 GHz monolithic receive array, volume 2

The formalism of the sidelobe suppression algorithm and the method used to calculate the system noise figure for a 30 GHz monolithic receive array are presented. Results of array element weight determination and performance studies of a Gregorian aperture image system are also given

Configuration study for a 30 GHz monolithic receive array, volume 1

Gregorian, Cassegrain, and single reflector systems were analyzed in configuration studies for communications satellite receive antennas. Parametric design and performance curves were generated. A preliminary design of each reflector/feed system was derived including radiating elements, beam-former network, beamsteering system, and MMIC module architecture. Performance estimates and component requirements were developed for each design. A recommended design was selected for both the scanning beam and the fixed beam case. Detailed design and performance analysis results are presented for the selected Cassegrain configurations. The final design point is characterized in detail and performance measures evaluated in terms of gain, sidelobe level, noise figure, carrier-to-interference ratio, prime power, and beamsteering. The effects of mutual coupling and excitation errors (including phase and amplitude quantization errors) are evaluated. Mechanical assembly drawings are given for the final design point. Thermal design requirements are addressed in the mechanical design

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

ALLY: An operator's associate for satellite ground control systems

The key characteristics of an intelligent advisory system is explored. A central feature is that human-machine cooperation should be based on a metaphor of human-to-human cooperation. ALLY, a computer-based operator's associate which is based on a preliminary theory of human-to-human cooperation, is discussed. ALLY assists the operator in carrying out the supervisory control functions for a simulated NASA ground control system. Experimental evaluation of ALLY indicates that operators using ALLY performed at least as well as they did when using a human associate and in some cases even better