Browsing by Author "Greenberg, William"

Now showing 1 - 20 of 31
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    Analytical solutions of model equations for two phase gas mixtures: Transverse velocity perturbations
    Cavalier, J. F.; Greenberg, William (AIP Publishing, 1984)
    Model equations for a dilute binary gas system are derived, using a linear BGK scheme. Complete analytical solutions for the stationary half_space problem are obtained for transverse velocity perturbations. The method of solution relies on the resolvent integration technique.
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    Boundary value and Wiener-Hopf problems for abstract kinetic equations with nonregular collision operators
    Ganchev, Alexander Hristov (Virginia Polytechnic Institute and State University, 1986)
    We study the linear abstract kinetic equation T𝜑(x)′=-A𝜑(x) in the half space {x≥0} with partial range boundary conditions. The function ψ takes values in a Hilbert space H, T is a self adjoint injective operator on H and A is an accretive operator. The first step in the analysis of this boundary value problem is to show that T⁻¹A generates a holomorphic bisemigroup. We prove two theorems about perturbation of bisemigroups that are interesting in their own right. The second step is to obtain a special decomposition of H which is equivalent to a Wiener-Hopf factorization. The accretivity of A is crucial in this step. When A is of the form "identity plus a compact operator", we work in the original Hilbert space. For unbounded A’s we consider weak solutions in a larger space HT, which has a natural Krein space structure. Using the Krein space geometry considerably simplifies the analysis of the question of unique solvability.
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    Case eigenfunction expansion for a conservative medium
    Greenberg, William; Zweifel, Paul F. (AIP Publishing, 1976-02)
    By using the resolvent integration technique introduced by Larsen and Habetler, the one‐speed, isotropic scattering,neutron transport equation is treated in the infinite and semi‐infinite media. It is seen that the results previously obtained by Case’s ’’singular eigenfunction’’ approach are in agreement with those obtained by resolvent integration.
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    The Cauchy problem for the Diffusive-Vlasov-Enskog equations
    Lei, Peng (Virginia Tech, 1993-04-05)
    In order to better describe dense gases, a smooth attractive tail arising from a Coulomb-type potential is added to the hard core repulsion of the Enskog equation, along with a velocity diffusion. By choosing the diffusing term of Fokker-Planck type with or without dynamical friction forces. The Cauchy problem for the Diffusive-Vlasov-Poisson-Enskog equation (DVE) and the Cauchy problem for the Fokker-Planck-Vlasov-Poisson-Enskog equation (FPVE) are addressed.
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    The conditional entropy in the microcanonical ensemble
    Dietz, D.; Greenberg, William (AIP Publishing, 1975-08)
    The existence of the configurational microcanonical conditional entropy in classical statistical mechanics is proved in the thermodynamic limit for a class of long_range multiparticle observables. This result generalizes a theorem of Lanford for finite range observables.
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    Covering properties and quasi-uniformities of topological spaces
    Junnila, Heikki J. K. (Virginia Tech, 1978-06-05)
    This thesis deals with the relationships between covering properties and properties of compatible quasi-uniformities of a topological space. The covering properties considered in this work are orthocompactness, metacompactness and paracompactness; some generalizations of orthocompactness are also defined and studied.
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    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.
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    Equilibrium states of ferromagnetic abelian lattice systems
    Miekisz, Jacek (Virginia Polytechnic Institute and State University, 1984)
    Ferromagnetic abelian lattice systems are the topic of this paper. Namely, at each site of ZV-invariant lattice is placed a finite abelian group. The interaction is given by any real, negative definite, and translation invariant function on the space of configurations.Algebraic structure of the system is investigated. This allows a complete · description of the family of equilibrium states for given. interaction at low temperatures. At the same time it is proven that the low temperature expansion for Gibbs free energy is analytic. It is also shown that it is not necessary to consider gauge models in the case of Zm on ZV lattice.
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    The fine topology and other topologies on C(X,Y)
    Eklund, Anthony D. (Virginia Tech, 1978-05-05)
    "The Fine Topology" C(X,Y) where (Y,d) is a metric space is referred to, in an exercise in [14], as the topology generated by basic open neighborhoods of the form B(f,E) = {g: d(f(x),g(x)) < E(x)} where E is a positive continuous real valued function. So in the fine topology, a function g is close to f if g(x) is continuously close to f(x); whereas in the uniform topology, g(x) must be uniformly close to f(x), that is, within a constant distance of f(x). So the fine topology is an obvious refinement of the uniform topology. This topology has not been extensively studied before, and it is the purpose of this paper to see how the fine topology fits in with the lattice of other well studied topologies on C(X,Y), and to study some properties of this topology in itself. Furthermore, other results on these well studied topologies will-be examined and compared with the fine topology.
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    Functional calculus for symmetric multigroup transport operator
    Greenberg, William (AIP Publishing, 1976-02)
    A rigorous treatment of the symmetric multigroup transport equation is given by developing the functional calculus for the transport operator. Von Neumann spectral theory is applied to nonorthogonal cyclic subspaces, and the isometries onto C (N) are explicitly evaluated.
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    Global existence in L1 for the square-well kinetic equation
    Liu, Rongsheng (Virginia Tech, 1993-04-04)
    An attractive square-well is incorporated into the Enskog equation, in order to model the kinetic theory of a moderately dense gas with intermolecular potential. The existence of solutions to the Cauchy problem in L¹. global in time and for arbitrary initial data. is proved. A simple derivation of the square-well kinetic equation is given. Lewis's method is used~ which starts from the Liouville equation of statistical mechanics. Then various symmetries of the collisional integrals are established. An H-theorem for entropy, mass, and momentum conservation is obtained, as well as an energy estimate, and key gain-loss estimates. Approximate equations for the square-well kinetic equation are constructed that preserve symmetries of the collisional integral. Existence of nonnegative solutions of the approximate equations and weak compactness are obtained. The velocity averaging lemma of Golse is then a principal tool in demonstrating the convergence of the approximate solutions to a solution of the renormalized square well kinetic equation. The existence of weak solution of the initial value problem for the square well kinetic equation is thus proved.
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    The interaction function and lattice duals
    Greenberg, William (AIP Publishing, 1977-10)
    An interaction function is defined for lattice models in statistical mechanics. A correlation function expansion is derived, giving a direct proof of the duality relations for correlation functions.
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    Kinetic theory and global existence in Lp1s for a dense square-well fluid
    Yao, Aixiang (Virginia Tech, 1995)
    In this paper, we consider the kinetic equation for a dense square-well fluid and the geometric factor Y _ 1, provide the related kinetic theory, and prove a global existence theorem in L1 for the kinetic equation under rather general initial value condition. An analogue of the classical H-theorem is verified.
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    Large eddy simulation of turbulent vortices and mixing layers
    Sreedhar, Madhu K. (Virginia Tech, 1994)
    In this dissertation large-eddy simulation(LES) is used to study the transitional and turbulent structures of vortices and free shear layers. The recently developed dynamic model and the basic Smagorinsky model are utilized to model the subgrid-scale(SGS) stress tensor. The dynamic model has many advantages over the existing SGS models. This model has the ability to vary in time and space depending on the local turbulence conditions. This eliminates the need to tune the model constants a priori to suit the flow field being simulated. Three different flow fields are considered. First, the evolution of large-scale turbulent structures in centrifugally unstable vortices is studied. It is found that these structures appear as counter rotating vortex rings encircling the vortex core. The interaction of these structures with the core results in the transfer of angular momentum between the core and the surroundings. The mean tangential velocity decays due to this exchange of angular momentum. Second, the generation and decay of turbulent structures in a vortex with an axial velocity deficit are studied. The presence of a destabilizing wake-like axial velocity field in an otherwise centrifugally stable vortex results in a very complex flow field. The inflectional instability mechanism of the axial velocity deficit amplifies the initial disturbances and results in the generation of large-scale turbulent structures. These structures appear as branches sprouting out of the vortex core. The breakdown of these structures leads to small-scale motions. But the stabilizing effects of the rotational flow field tend to quench the small-scale motions and the vortex returns to its initial laminar state. The mean axial velocity deficit is weakened, but the mean tangential velocity shows no significant decay. Third, a transitional mixing layer calculation is performed.The growth and breakdown to small scales of vortical structures are studied. Emphasis is given to the identification of late transition structures and their subsequent break down. Formation of streamwise vortices in place of the original Kelvin-Helmholtz vortices and the subsequent appearance of hair-pin vortices at the edges of the mixing layer mark the completion of transition. The basic Smagorinsky model is also used in the mixing layer simulations. The performance of the dynamic model is compared with the previous results obtained using the basic Smagorinsky model. As expected, the basic Smagorinsky model is found to be more dissipative.
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    Multivariable Interpolation Problems
    Fang, Quanlei (Virginia Tech, 2008-07-07)
    In this dissertation, we solve multivariable Nevanlinna-Pick type interpolation problems. Particularly, we consider the left tangential interpolation problems on the commutative or noncommutative unit ball. For the commutative setting, we discuss left-tangential operator-argument interpolation problems for Schur-class multipliers on the Drury-Arveson space and for the noncommutative setting, we discuss interpolation problems for Schur-class multipliers on Fock space. We apply the Krein-space geometry approach (also known as the Grassmannian Approach). To implement this approach J-versions of Beurling-Lax representers for shift-invariant subspaces are required. Here we obtain these J-Beurling-Lax theorems by the state-space method for both settings. We see that the Krein-space geometry method is particularly simple in solving the interpolation problems when the Beurling-Lax representer is bounded. The Potapov approach applies equally well whether the representer is bounded or not.
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    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.
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    On the Cauchy problem for the linearized GPKdV and gauge transformations for a quadratic pencil and AKNS system
    Yordanov, Russi Georgiev (Virginia Tech, 1992)
    The present work in the area of soliton theory studies two problems which arise when seeking analytic solutions to certain nonlinear partial differential equations. In the first one, Lax pairs associated with prolonged eigenfunctions and prolonged squared eigenfunctions (prolonged squares) are derived for a Schrödinger equation with a potential depending polynomially on the spectral parameter (of degree N) and its respective hierarchy of nonlinear evolution equations (here named generalized polynomial Korteweg-de Vries equations or GPKdV). It is shown that the prolonged squares satisfy the linearized GPKdV equations. On that basis, the Cauchy problem for the linearized GPKdV has been solved by finding a complete set of such prolonged squares and applying an expansion formula derived in another work by the author. The results are a generalization of the ones by Sachs (SIAM J. Math. Anal. 14, 1983, 674-683). Moreover, a condition on the so-called recursion operator A is derived which generates the whole hierarchy of Lax pairs associated with the prolonged squares. As for the second part of the work, it developed a scheme for deriving gauge transformations between different linear spectral problems. Then the scheme is applied to obtain all known Darboux transformations for a quadratic pencil (the spectral problem considered in the first part at N = 2), Schrödinger equation (N = 1), Ablowitz-Kaup-Newell-Segur (AKNS) system and also derive the Jaulent-Miodek transformation. Moreover, the scheme yields a large family of new transformations of the above types. It also gives some insight on the structure of the transformations and emphasizes the symmetry with respect to inversion that they possess.
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    On the Spectrum of Neutron Transport Equations with Reflecting Boundary Conditions
    Song, Degong (Virginia Tech, 2000-02-14)
    This dissertation is devoted to investigating the time dependent neutron transport equations with reflecting boundary conditions. Two typical geometries --- slab geometry and spherical geometry --- are considered in the setting of L^p including L^1. Some aspects of the spectral properties of the transport operator A and the strongly continuous semigroup T(t) generated by A are studied. It is shown under fairly general assumptions that the accumulation points of { m Pas}(A):=sigma (A) cap { lambda :{ m Re}lambda > -lambda^{ast} }, if they exist, could only appear on the line { m Re}lambda =-lambda^{ast}, where lambda^{ast} is the essential infimum of the total collision frequency. The spectrum of T(t) outside the disk {lambda : |lambda| leq exp (-lambda^{ast} t)} consists of isolated eigenvalues of T(t) with finite algebraic multiplicity, and the accumulation points of sigma (T(t)) igcap{ lambda : |lambda| > exp (-lambda^{ast} t)}, if they exist, could only appear on the circle {lambda :|lambda| =exp (-lambda^{ast} t)}. Consequently, the asymptotic behavior of the time dependent solution is obtained.
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    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.
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    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∞.