Browsing by Author "Bowden, Robert L."

Now showing 1 - 20 of 22
  • Thumbnail Image
    Analysis of the dispersion function for anisotropic longitudinal plasma waves
    Arthur, M. D.; Bowden, Robert L.; Zweifel, Paul F. (AIP Publishing, 1979-10)
    An analysis of the zeros of the dispersion function for longitudinal plasma waves is made. In particular, the plasma equilibriumdistribution function is assumed to have two relative maxima and is not necessarily an even function. The results of this analysis are used to obtain the Wiener–Hopf factorization of the dispersion function. A brief analysis of the coupled nonlinear integral equations for the Wiener–Hopf factors is also presented.
  • Thumbnail Image
    Application of classical non-linear Liouville dynamic approximations
    Harter, Terry Lee (Virginia Polytechnic Institute and State University, 1988)
    This dissertation examines the application of the Liouville operator to problems in classical mechanics. An approximation scheme or methodology is sought that would allow the calculation of the position and momentum of an object at a specified later time, given the initial values of the object's position and momentum at some specified earlier time. The approximation scheme utilizes matrix techniques to represent the Liouville operator. An approximation scheme using the Liouville operator is formulated and applied to several simple one-dimensional physical problems, whose solution is obtainable in terms of known analytic functions. The scheme is shown to be extendable relative to cross products and powers of the variables involved. The approximation scheme is applied to a more complicated one-dimensional problem, a quartic perturbed simple harmonic oscillator, whose solution is not capable of being expressed in terms of simple analytic functions. Data produced by the application of the approximation scheme to the perturbed quartic harmonic oscillator is analyzed statistically and graphically. The scheme is reapplied to the solution of the same problem with the incorporation of a drag term, and the results analyzed. The scheme is then applied to a simple physical pendulum having a functionalized potential in order to ascertain the limits of the approximation technique. The approximation scheme is next applied to a two-dimensional non-perturbed Kepler problem. The data produced is analyzed statistically and graphically. Conclusions are drawn and suggestions are made in order to continue the research in several of the areas presented.
  • Thumbnail Image
    Bhabha scattering in e⁺e⁻ collisions at TRISTAN
    Lai, Anzhi (Virginia Tech, 1992-05-05)
    Bhabha scattering, the process of e⁺e⁻ → e⁺e⁻, has been studied at center-of-mass energies from 50 to 58 GeV with the AMY detector at the KEK e⁺e⁻ storage ring TRISTAN. The study is based on a data sample of 79.7 pb⁻¹ integrated luminosity. The differential cross section of Bhabha scattering has been measured. The measured cross section is found to agree fairly well with the Standard Model of the electroweak theory. The measured cross section is also compared with various four-fermion contact interaction models, and confidence level lower limits on the composite scale, A, are determined. In addition, the limits on VV model are converted to SM-break-down scales, which indicate the validity of the SM down to the distance of order ~ 10⁻¹⁷ cm and the electron charge radius of ~ 10⁻¹⁶ cm. Attempts are made in searching for an additional boson Z'. No clear signal of the existence of a Z' boson is found up to energy of ~160 GeV/c². The effect of transverse beam polarization on Bhabha scattering is also studied. The ϕ dependence of Bhabha events are fitted to the QED prediction and found to agree with the theory. However, no quantitative conclusion on polarization effect can be drawn based on current data sample, which does not provide enough statistics. More data is being accumulated and further study should be carried out.
  • Thumbnail Image
    The Born-Oppenheimer approximation in scattering theory
    Kargol, Armin (Virginia Tech, 1994-05-05)
    We analyze the Schrödinger equation i𝜖 ¬2â /â tΨ = H(𝜖)Ψ, where H(â ¬) = - f24 Î x + h(X) is the hamiltonian of a molecular system consisting of nuclei with masses of order 𝜖¬-4 and electrons with masses of order 1. The Born-Oppenheimer approximation consists of the adiabatic approximation to the motion of electrons and the semiclassical approximation to the time evolution of nuclei. The quantum propagator associated with this Schrödinger Equation is exp(-itH(â ¬)/â ¬2). We use the Born-Oppenheimer method to find the leading order asymptotic expansion in â ¬ to exp(_it~(t:»Ψ, i.e., we find Ψ(t) such that: (1) We show that if H(𝜖) describes a diatomic Molecule with smooth short range potentials, then the estimate (1) is uniform in time; hence the leading order approximation to the wave operators can be constructed. We also comment on the generalization of our method to polyatomic molecules and to Coulomb systems.
  • Thumbnail Image
    Conservative neutron-transport theory
    Bowden, Robert L.; Cameron, W. L.; Zweifel, Paul F. (AIP Publishing, 1977-02)
    A functional analytic development of the Case full_range and half_range expansions for the neutron transport equation for a conservative medium is presented. A technique suggested by Larsen is used to overcome the difficulties presented by the noninvertibility of the transport operator K _1 on its range. The method applied has considerable advantages over other approaches and is applicable to a class of abstract integro-differential equations.
  • Thumbnail Image
    The design, construction, and calibration of a generating voltmeter for a two million volt electrostatic accelerator
    Bowden, Robert L. (Virginia Polytechnic Institute, 1958)
    A generating voltmeter capable of measuring one, two, or found million volts has been designed and constructed for use with the Virginia Polytechnical Institute electrostatic accelerator. The voltmeter is a grounded shutter type, the rectified output of which is measured by a vacuum tube voltmeter. The voltmeter was calibrated by known nuclear resonances of fluoride. The calibration showed the meter to be accurate to within five percent at half scale deflation on the one million volts range the less than plus or minus three percent on the two million volts range.
  • Thumbnail Image
    Discrete dynamical systems in solving H-equations
    Chen, Jun (Virginia Tech, 1995-08-17)
    Three discrete dynamical models are used to solve the Chandrasekhar H-equation with a positive or negative characteristic function. Two of them produce series of continuous functions which converge to the solution of the H-equation. An iteration model of the nth approximation for the H-equation is discussed. This is a nonlinear n-dimensional dynamical system. We study not only the solutions of the nth approximation for the H-equation but also the mathematical structure and behavior of the orbits with respect to the parameter function, i.e. characteristic function. The dynamical system is controlled by a manifold. For n=2, stability of the fixed points is studied. The stable and unstable manifolds passing through the hyperbolically fixed point are obtained. Globally, the bounded orbits region is given. For parameter c in some region a periodic orbit of one dimension will cause periodic orbits in the higher dimensional system. For changing parameter c, the bifurcation points are discussed. For c ∈ (-5.6049, 1] the system has a series of double bifurcation points. For c ∈ (-8, -5.6049] chaos appears. For c in a window contained the chaos region, a new bifurcation phenomenon is found. For c ≤ -7 any periodic orbits appear. For c in the chaos region the behavior of attractor is discussed. Chaos occurs in the n-dimensional dynamical system.
  • Thumbnail Image
    Functional analytic treatment of linear transport equations in kinetic theory and neutron transport theory
    Cameron, William Lyle (Virginia Tech, 1978-01-08)
    The temperature-density equation of Kinetic Theory and the conservative neutron transport equation are studied. In both cases a modified version of the Larsen-Habetler resolvent integration technique is applied to obtain full-range and half-range expansions. For the neutron transport equation the method applied is seen to have notational advantages over previous approaches. In the case of the temperature-density equation this development extends previous results by enlarging the class of expandable functions and has the added advantage of rigor and simplicity. As a natural extension of the Kinetic Theory results, an integral equation for the surface density is derived for half-space problems involving the boundary condition of arbitrary accommodation.
  • Thumbnail Image
    A holographic system that records front-surface detail of a scene moving at high velocity
    Kurtz, Robert L. (Virginia Tech, 1971-06-05)
    It is known that any motion of the scene during the exposure of a hologram results in a spatial modulation of the recorded fringe contrast. On reconstruction this produces a spatial amplitude modulation of the reconstructed wavefront that tends to blur out the image. This paper discusses a novel holographic technique that uses an elliptical orientation for the holographic arrangement. It is shown that the degree of image degradation is not only a function of exposure time but also of the system used. The form of the functional system dependence is given, as well as the results of several systems tested, which verify this dependence. It is further demonstrated that the velocity of the target or the exposure time alone is inconsequential by itself and the important parameter is the total motion of the target Î X = VT. Using the resolution of front-surface detail from a target with a velocity of 17,546 cm/sec, we are able to predict an upper limit on target velocity for resolution of front-surface detail for a given system.
  • Thumbnail Image
    Maps of intervals with indifferent fixed points: thermodynamic formalism and phase transitions
    Prellberg, Thomas (Virginia Tech, 1991)
    We develop the thermodynamic formalism for a large class of maps of the interval with indifferent fixed points. For such systems the formalism yields one-dimensional systems with many-body infinite range interactions for which the thermodynamics is well defined while the Gibbs states are not. (Piecewise linear systems of this kind yield the soluble, in a sense, Fisher models.) We prove that such systems exhibit phase transitions, the order of which depends on the behavior at the indifferent fixed points. We obtain the critical exponent describing the singularity of the pressure and analyse the decay of correlations of the equilibrium states at all temperatures. Our technique relies on establishing and exploiting a relationship between the transfer operators of the original map and its suitable (expanding) induced version. The technique allows one to also obtain a version of the Bowen-Ruelle formula for the Hausdorff dimension of repellers for maps with indifferent fixed points, and to generalize Fisher results to some non-soluble models.
  • Thumbnail Image
    Multigroup neutron transport. I. Full range
    Bowden, Robert L.; Sancaktar, Selim; Zweifel, Paul F. (AIP Publishing, 1976-01)
    A functional analytic approach to the N_group, isotropic scattering, particle transport problem is presented. A full_range eigenfunction expansion is found in a particularly compact way, and the stage is set for the determination of the half_range expansion, which is discussed in a companion paper. The method is an extension of the work of Larsen and Habetler for the one_group case.
  • Thumbnail Image
    Multigroup neutron transport. II. Half range
    Bowden, Robert L.; Sancaktar, Selim; Zweifel, Paul F. (AIP Publishing, 1976-01)
    This paper accompanies a preceding one in which a functional analytic method was used to obtain the full_range expansion in multigoup neutron transport. In the present paper the analysis is extended to obtain the half_range expansion. The method used is an extension of the work of Larsen and Habetler for the one_group case. The results are given in terms of certain matrices which are solutions of coupled integral equations and which factor the dispersion matrix.
  • Thumbnail Image
    Nonlinear evolution of Vlasov equilibria
    Demeio, Lucio (Virginia Polytechnic Institute and State University, 1989)
    In this work, we investigate numerically the evolution of perturbed Vlasov equilibria. according to the full nonlinear system with particular emphasis on analyzing the asymptotic states towards which the system evolves. The simulations are carried out with the numerical code that we have implemented on the Cray X-MP of the Pittsburgh Supercomputing Center and which is based on the splitting scheme algorithm. Maxwellian symmetric and one-sided bump-on-tail and two-stream type of equilibrium distributions are considered: the only distribution which seems to evolve towards a BGK equilibrium is the two-stream while the asymptotic states for the other distributions are better described by superpositions of possible BGK modes. Perturbations with wave-like dependence in space and both symmetric and non-symmetric dependence on velocity are considered. For weakly unstable modes, the problem of the discrepancy between different theoretical models about the scaling of the saturation amplitude with the growth rate is addressed for the first time with the splitting scheme algorithm. The results are in agreement with the ones obtained in the past with less accurate algorithms and do not exhibit spurious numerical effects present in those. Finally, collisions are included in the splitting scheme in the form of the Krook model and some simulations are performed whose results are in agreement with existing theoretical models.
  • Thumbnail Image
    A note on the iteration of the Chandrasekhar non-linear H-equation
    Bowden, Robert L. (AIP Publishing, 1979-04)
    An iteration scheme to solve the Chandrasekhar H equation in the form H (_) ={1___F_ 1 0[_ (s) H (s)]/(s +_) d s}_1 is shown to converge monotonically and uniformly.
  • Thumbnail Image
    On the solution of nonlinear matrix integral equations in transport theory
    Bowden, Robert L.; Zweifel, Paul F.; Menikoff, R. (AIP Publishing, 1976-09)
    The coupled nonlinear matrix integral equations for the matrices X (z) and Y (z) which factor the dispersion matrix Λ (z) of multigroup transport theory are studied in a Banach space X. By utilizing fixed‐point theorems we are able to show that iterative solutions converge uniquely to the ’’physical solution’’ in a certain sphere of X. Both isotropic and anisotropic scattering are considered.
  • Thumbnail Image
    On the solutions to a class of nonlinear integral equations arising in transport theory
    Spiga, G.; Bowden, Robert L.; Boffi, V. C. (AIP Publishing, 1984-12)
    Existence and uniqueness for the solutions to a class of nonlinear equations arising in transport theory are investigated in terms of a real parameter _ which can take on positive and negative values. On the basis of contraction mapping and positivity properties of the relevant nonlinear operator, iteration schemes are proposed, and their convergence, either pointwise or in norm, is studied.
  • Thumbnail Image
    On the zeros of the dispersion function in particle transport theory
    Bowden, Robert L. (AIP Publishing, 1986-06)
    The zeros of the dispersion function that arise in particle transport with anisotropicscattering are studied. An algebraic test for the number of zeros is presented.
  • Thumbnail Image
    Oscillations and waves in anisotropic plasmas
    Arthur, Michael D. (Virginia Tech, 1979-05-05)
    The linearized Vlasov-Maxwell equations describing anisotropic plasma oscillations and waves are studied using an operator theoretic approach. The model considered is one dimensional so that after velocity averages perpendicular to this direction. have been taken, the equations can be naturally grouped into one set of equations for longitudinal modes and another set of equations for transverse modes. The problems of longitudinal and transverse plasma oscillations are studied by Fourier transforming the equations in the space variable and analyzing the resulting operator equations using the theory of semigroups. Existence and uniqueness theorems are proved, and solutions are constructed by the resolvent integration technique. The solutions are put into the form of a generalized eigenfunction expansion with eigenmodes corresponding to zeros of the appropriate plasma dispersion function. The expansion coefficients for eigenmodes corresponding to simple and second order real zeros of the plasma dispersion function are also presented, and constitute some of the new results obtained by our analysis. Existence and uniqueness of the solution to the longitudinal plasma wave boundary value problem is proved by writing the longitudinal equations in operator form and again using the theory of semigroups. The solution to the plasma wave boundary value problem is arrived at by a Fourier time transformation of the Vlasov equation coupled to Ampere's Law rather than Gauss‘ Law, and analyzing a scalar operator as opposed to the more complicated matrix operator that has previously been studied. Special care is used in constructing the half range transport operator whose resolution of the identity yields the solution in the form of a half range generalized eigenfunction expansion where again, new results are presented for the expansion coefficients for eigenfunctions corresponding to simple and second order real zeros of the fixed frequency longitudinal plasma dispersion function. Since this study is concerned with anisotropic plasmas, a non-even plasma equilibrium distribution function is assumed with the direct result that more stable and unstable plasma modes corresponding to real and complex zeros of the plasma dispersion function are possible that has previously been considered. Also, for the longitudinal plasma wave problem, the Wiener-Hopf factorization of the fixed frequency longitudinal plasma dispersion function is presented and the coupled nonlinear integral equations for the Wiener-Hopf factors are studied. These Wiener-Hopf factors are required in the construction of the half range transport operator.
  • Thumbnail Image
    Poisson-lie structures on infinite-dimensional jet groups and their quantization
    Stoyanov, Ognyan S. (Virginia Tech, 1993)
    We study the problem of classifying all Poisson-Lie structures on the group Gy of local diffeomorphisms of the real line R¹ which leave the origin fixed, as well as the extended group of diffeomorphisms G₀∞ ⊃ G∞ whose action on R¹ does not necessarily fix the origin. A complete classification of all Poisson-Lie structures on the group G∞ is given. All Poisson-Lie structures of coboundary type on the group G₀∞ are classified. This includes a classification of all Lie-bialgebra structures on the Lie algebra G∞ of G∞, which we prove to be all of coboundary type, and a classification of all Lie-bialgebra structures of coboundary type on the Lie algebra Go∞ of Go∞ which is the Witt algebra. A large class of Poisson structures on the space Vλ of λ-densities on the real line is found such that Vλ becomes a homogeneous Poisson space under the action of the Poisson-Lie group G∞. We construct a series of finite-dimensional quantum groups whose quasiclassical limits are finite-dimensional Poisson-Lie factor groups of G∞ and G₀∞.
  • Thumbnail Image
    Resolvent integration techniques for generalized transport equations
    Bowden, Robert L.; Greenberg, William; Zweifel, Paul F. (AIP Publishing, 1979-06)
    A generalized class of ’’transport type’’ equations is studied, including most of the known exactly solvable models; in particular, the transport operator K is a scalar type spectral operator. A spectral resolution for K is obtained by contour integration techniques applied to bounded functions of K. Explicit formulas are developed for the solutions of full and half range problems. The theory is applied to anisotropicneutron transport, yielding results which are proved to be equivalent to those of Mika.