Bulletin of the American Physical Society
76th Annual Meeting of the Southeastern Section of APS
Volume 54, Number 16
Wednesday–Saturday, November 11–14, 2009; Atlanta, Georgia
Session CD: Mathematical Methods |
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Chair: Toan Nguyen, Georgia Institute of Technology Room: Brussels |
Thursday, November 12, 2009 10:45AM - 10:57AM |
CD.00001: ABSTRACT WITHDRAWN |
Thursday, November 12, 2009 10:57AM - 11:09AM |
CD.00002: Non-Linear Model for the Disturbance of Electronics in by High Energy Electron Plasmas in the Van Allen Radiation Belts William Atkinson A model is presented that models the disturbance of electrical components by high energy electrons trapped in the Van Allen radiation belts; the model components consists of module computing the electron fluence rate given the altitude, the time of the year, and the sunspot number, a module that transports the electrons through the materials of the electrical component, and a module that computes the charge and electrical fields of the insulating materials as a function of time. A non-linear relationship (the Adameic-Calderwood equation) for the variation of the electrical conductivity with the electrical field strength is used as the field intensities can be quite high due to the small size of the electrical components and the high fluence rate of the electrons. The results show that the electric fields can often be as high as 10 MV/m in materials commonly used in cables such as Teflon and that the field can stay at high levels as long as an hour after the irradiation ends. [Preview Abstract] |
Thursday, November 12, 2009 11:09AM - 11:21AM |
CD.00003: A Matched Filter for Chaos Jonathan Blakely, Mark Stahl, Ned Corron In conventional communication theory, a matched filter provides optimal reception of information through additive Gaussian white noise. Chaos communication schemes have always performed worse than conventional schemes in the presence of noise due to the lack of a matched filter. We present a novel chaotic oscillator for which a matched filter exists. This oscillator differs from other known chaotic systems in that its waveform can be written as a linear superposition of a single basis function with random polarity at integer time steps. The matched filter is a linear filter with the time reversed basis function as its impulse response. We show experimental circuit implementations of both the oscillator and the matched filter. The matched filter output is shown to directly reveal the information content of the chaotic waveform. [Preview Abstract] |
Thursday, November 12, 2009 11:21AM - 11:33AM |
CD.00004: Averaging Method For Nonlinear Cubic Oscillator Ronald E. Mickens, 'Kale Oyedeji The nonlinear cubic equation is the following second-order differential equation (*) $\ddot {x}+x^3=\in f(x,\dot {x}),$ where $\in $ is a small parameter and f is a polynomial function of its arguments. In Mickens and Oyedeji (J. Sound and Vibration, Vol. 102, 579-582 (1985), we constructed a first-order averaging method for calculating approximations to the oscillatory solutions of Eq. (*). The purpose of the current work is to provide a generalization of this technique. This generalization comes from a new interpretation as to how the averaged equations should be solved. We compare solutions calculated using both the old and new procedures, and show that the new methodology eliminates certain mathematical inconsistencies inherent in the original formulation. [Preview Abstract] |
Thursday, November 12, 2009 11:33AM - 11:45AM |
CD.00005: Calculation of Approximations to the Periodic Solutions for the Cubic- Root Oscillator Dorian Wilkerson, R.E. Mickens The cubic-root oscillator (CRO) is modeled by the following second-order, nonlinear differential equation (*) $\begin{array}{l} \ddot {x}+x^{\frac{1}{3}}=0,x(0)=A,\dot {x}(0)=0. \\ \\ \end{array}$ First, we show that all solutions to Eq. (*) are periodic. Second, we calculate the exact value of for the period T(A). Third, two techniques are used to calculate approximations for the periodic solutions; these techniques are the methods of harmonic balance and iteration. Generalization of this methodology to other ``truly nonlinear (TNL)'' oscillators will also be discussed. [Preview Abstract] |
Thursday, November 12, 2009 11:45AM - 11:57AM |
CD.00006: ABSTRACT WITHDRAWN |
Thursday, November 12, 2009 11:57AM - 12:09PM |
CD.00007: Macroscopic E\&M {\it versus} resistive MHD, or plasma physics for the 21st century Robert Johnson Comparison is made of the theory of macroscopic electromagnetism with that of resistive magnetohydrodynamics. Inconsistencies in the MHD formalism are resolved only by respecting the macroscopic field formulation. Implications for current and future technologies are discussed. [Preview Abstract] |
Thursday, November 12, 2009 12:09PM - 12:21PM |
CD.00008: Neural Network and Least Squares Predictions for 43 ``new'' $s$ and $p$ Electron Diatomics Ray Hefferlin We combine least-squares and neural-network forecasts for vibration frequencies of diatomic molecules with 10 to 12 atomic valence electrons. We start with 108 least-squares forecasts based on CRC 2009 data,* and 1001 neural network forecasts based on Huber and Herzberg 1979 data,** and then insist that the standard-deviation bounds of one forecast overlap the bounds of the other; this requirement leaves 68 molecules, for 43 of which we find no tabulated data. The 68 composite predictions average 2.2{\%} on either side of the composite standard deviations. We find 41 literature values, for 22 molecules (none of which were in the set of 43) useable as tests; of them, 33 (80{\%}) fall within the 4.4{\%} composite standard deviation of the predictions. The 43 molecules are AlTe, AsI, AsSb, AsSn, AsTe, BeAt, BeBr, BeSe, BeTe, BPo, CAs, CBi, CPb, CSn, CTe, CaTe, GaTe, GeSb, GeSn, InS, LiSe, LiTe, MgTe, NAt, NGe, NI, NPb, NPo, NSn, NTe, NaSe, PSn, SbBr, SbCl, SbI, SbS, SbSe, SbTe, SiSb, SiSn, SnSb, SrSe, and TlO. *Hefferlin, R., Davis W.B., J. Ileto, \textit{J. Chem. Inf. Comp. Sci}. 43, 622-628, 2003. **Ms. Amy Beard assisted in the least-squares analysis. [Preview Abstract] |
Thursday, November 12, 2009 12:21PM - 12:33PM |
CD.00009: Generating Geometrical Elements for Any Space-Time Dennis Marks To distinguish time from space, use real Clifford algebras $\bf {R}$$_{n;s}$, where $n$ is the number of dimensions and $s$ is their signature ($s=-n, -n+2, {\ldots}$, or $n$). $\bf{R}$$_ {n;s}$ is isomorphic to algebras of real, complex, or quaternionic matrices ${\rm {\bf R}}(2^{\textstyle{n \over {\rm {\bf 2}}}})$, ${\rm {\bf C}}(2^{\textstyle{{n-1} \over 2}})$, or ${\rm {\bf H}}(2^{\textstyle{{n-2} \over 2}})$, or of block diagonal matrices ${ }^2{\rm {\bf R}}(2^{\textstyle{{n-1} \over 2}})\mbox{ }$ or ${ }^2{\rm {\bf H}}(2^{\textstyle{{n-3} \over 2}})\mbox{ }$, for $\vert (s~$+~3)$_{mod8}$~-~4$\vert $ = 1, 2, 3, 0, or 4, respectively. Each of the $n$ basis vectors $\bf{e} $$_{\nu}$ satisfies ${\rm {\bf e}}_\mu {\bf{\cdot}}~ {\rm {\bf e}}_\nu =\eta _{\mu \nu } {\rm {\bf I}}_{n;s} $, where the $\bf {e}$$_{\nu}$ are orthogonal $\eta_{\mu \nu }=0$ for $\mu \ne \nu $ and normalized $\eta_{\mu \nu }=+1$ for $p$ space-like dimensions and $\eta_{\mu \nu }=-1$ for $q$ time-like dimensions) and where $\bf{I}_{n;s}$ is the identity matrix whose rank is given by the isomorphisms above. The geometrical elements are the scalar $\bf{I}_{n;s}$, basis vectors $\bf{e}_ {\nu }$, and their products (bivectors, trivectors, etc.) up to the pseudo-scalar $n$-volume $\bf{J}_{n;s}~=~\bf{e}_{0}~\bf{e}_ {1}~\bf{\cdot ~\cdot ~\cdot ~e}_{n-1}$. Now $\left( {{\rm {\bf J}}_{n;s} } \right)^2=(-1)^{\textstyle{{s(s-1)} \over 2}}{\rm {\bf I}}_{n;s} =\sigma _s {\rm {\bf I}}_{n;s} $. The direct product of $\bf{R}$$_{n;s}$, with $n$ orthonormal basis vectors $\bf{e}$$_{\nu }$ with signature $s$, and $\bf{R}$$_{n';s'}$, with $n'$ orthonormal basis vectors $\bf{e}$$_{\nu '}$ with signature $s'$, is ${\rm {\bf R}}_{n+n';s+s'\sigma _s } $, with $n+n'$ orthonormal basis vectors { ${\bf{e}}_{\nu }\otimes \bf{I} _{n';s'}, \bf{J}_{n;s}\otimes \bf{e}$$_{\nu '}$ } with signature ${s+s'\sigma _{s}}$, for even positive $n$. Orthonormal basis vectors for any positive $n$ with any possible signature can be generated from the two orthonormal basis vectors of the Minkowskian plane. [Preview Abstract] |
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