Browsing by Subject "mathematics, applied"

Now showing 1 - 20 of 22

Acoustic Radiation Force On A Sphere In Tissue
(2012-05) Ilinskii, Y. A.; Zabolotskaya, Evgenia A.; Hamilton, M. F.; Ilinskii, Yurii A.; Zabolotskaya, Evgenia A.; Hamilton, Mark F.
A theory is presented for the acoustic radiation force on a sphere embedded in a soft elastic medium that possesses a shear modulus several orders of magnitude smaller than its bulk modulus. Scattering of both compressional and shear waves is taken into account. There is no restriction on the size of the sphere or, apart from symmetry about an axis passing through the center of the sphere, the form of the incident compressional wave. If the medium is a liquid and the sphere is small in comparison with a wavelength the result reduces to the classical theory developed by Gor'kov [1] and others. The effect of shear elasticity on the radiation force is discussed in the context of parameters relevant to a gas bubble in tissue.
Bayesian Analysis Of RR Lyrae Luminosities And Kinematics
(2007-11) Jefferys, T. R.; Barnes, T. G.; Dambis, A.; Jefferys, W. H.; Jefferys, Thomas R.
We are using a hierarchical Bayes model to analyze the distances, luminosities, and kinematics of RR Lyrae stars. Our model relates these characteristics to the raw data of proper motions, radial velocities, apparent luminosities and metallicities of each star. A combination of Gibbs and Metropolis-Hastings sampling, using latent variables for the actual velocity and luminosity of each star, is used to draw a sample from the full posterior distribution of these variables, with consideration to identifiability and the properness of the hierarchical model, and draw inferences on the quantities of interest in the usual way. We have applied our model to the large HIPPARCOS database, and we have attempted to include metallicity and period in our model, which has not been done previously.
Bayesians Can Learn From Old Data
(2007-11) Jefferys, W. H.; Jefferys, William H.
In a widely-cited paper, Glymour (Theory and Evidence, Princeton, N. J.: Princeton University Press, 1980, pp. 63-93) claims to show that Bayesians cannot team from old data. His argument contains an elementary error. I explain exactly where Glymour went wrong, and how the problem should be handled correctly. When the problem is fixed, it is seen that Bayesians, just like logicians, can indeed learn from old data.
Boundaries of Siegel Disks: Numerical Studies of their Dynamics and Regularity
(2008-09) de la Llave, Rafael; Petrov, Nikola P.; de la Llave, Rafael
Siegel disks are domains around fixed points of holomorphic maps in which the maps are locally linearizable (i.e., become a rotation under an appropriate change of coordinates which is analytic in a neighborhood of the origin). The dynamical behavior of the iterates of the map on the boundary of the Siegel disk exhibits strong scaling properties which have been intensively studied in the physical and mathematical literature. In the cases we study, the boundary of the Siegel disk is a Jordan curve containing a critical point of the map (we consider critical maps of different orders), and there exists a natural parametrization which transforms the dynamics on the boundary into a rotation. We compute numerically this parameterization and use methods of harmonic analysis to compute the global Holder regularity of the parametrization for different maps and rotation numbers. We obtain that the regularity of the boundaries and the scaling exponents are universal numbers in the sense of renormalization theory (i.e., they do not depend on the map when the map ranges in an open set), and only depend on the order of the critical point of the map in the boundary of the Siegel disk and the tail of the continued function expansion of the rotation number. We also discuss some possible relations between the regularity of the parametrization of the boundaries and the corresponding scaling exponents. (C) 2008 American Institute of Physics.
Characterizing Curvilinear Features Using The Localized Normal-Score Ensemble Kalman Filter
(2012-03) Zhou, Haiyan; Li, Liangping; Gomez-Hernandez, J. Jaime; Zhou, Haiyan
The localized normal-score ensemble Kalman filter is shown to work for the characterization of non-multi-Gaussian distributed hydraulic conductivities by assimilating state observation data. The influence of type of flow regime, number of observation piezometers, and the prior model structure are evaluated in a synthetic aquifer. Steady-state observation data are not sufficient to identify the conductivity channels. Transient-state data are necessary for a good characterization of the hydraulic conductivity curvilinear patterns. Such characterization is very good with a dense network of observation data, and it deteriorates as the number of observation piezometers decreases. It is also remarkable that, even when the prior model structure is wrong, the localized normal-score ensemble Kalman filter can produce acceptable results for a sufficiently dense observation network.
Combinatorial Games with a Pass: A Dynamical Systems Approach
(2011-12) Morrison, Rebecca E.; Friedman, Eric J.; Landsberg, Adam S.; Morrison, Rebecca E.
By treating combinatorial games as dynamical systems, we are able to address a longstanding open question in combinatorial game theory, namely, how the introduction of a "pass" move into a game affects its behavior. We consider two well known combinatorial games, 3-pile Nim and 3-row Chomp. In the case of Nim, we observe that the introduction of the pass dramatically alters the game's underlying structure, rendering it considerably more complex, while for Chomp, the pass move is found to have relatively minimal impact. We show how these results can be understood by recasting these games as dynamical systems describable by dynamical recursion relations. From these recursion relations, we are able to identify underlying structural connections between these "games with passes" and a recently introduced class of "generic (perturbed) games." This connection, together with a (non-rigorous) numerical stability analysis, allows one to understand and predict the effect of a pass on a game. (C) 2011 American Institute of Physics. [doi:10.1063/1.3650234]
A Dynamical Systems Approach to Spiral Wave Dynamics
(1994-09) Barkley, Dwight; Kevrekidis, Ioannis G.; Barkley, Dwight
A simple system of five nonlinear ordinary differential equations is shown to reproduce many dynamical features of spiral waves in two-dimensional excitable media.
Early History Of ISNA
(2012-09) Hamilton, M. F.; Muir, T. G.; Blackstock, D. T.; Hamilton, Mark F.; Muir, Thomas G.; Blackstock, David T.
The International Symposia on Nonlinear Acoustics, now referred to as ISNA, have convened regularly since 1968, bringing together scientists and engineers to report and discuss the latest developments in this branch of nonlinear physics. The fact that this series of symposia is still going strong after more than four decades is testimony that nonlinear acoustics has established itself as a distinct, important, and vibrant field of research. In this paper we take a look back at the early years of ISNA to recall how it all began and trace the evolution of the symposia into their current form.
Fast Hybrid Algorithms For High Frequency Scattering
(2009-03) Engquist, B.; Tran, K.; Ying, L. X.; Engquist, Björn
This paper deals with numerical methods for high frequency wave scattering. It introduces a new hybrid technique that couples a directional fast multipole method for a subsection of it scattering surface to an asymptotic formulation over the rest of the scattering domain. The directional fast multipole method is new and highly efficient for the solution of the boundary integral formulation of a general scattering problem but it requires at least a few unknowns per wavelength on the boundary. The asymptotic method that was introduced by Bruno and collaborators requires much fewer unknowns. Oil the other hand the scattered field must have a simple structure. Hybridization of these two methods retains their best properties for the solution of the full problem, Numerical examples are given for the solution of the Helmholtz equation in two space dimensions.
Instability Criteria for Steady Flows of a Perfect Fluid
(1992-07) Friedlander, Susan; Vishik, Misha M.; Vishik, Misha M.
An instability criterion based on the positivity of a Lyapunov-type exponent is used to study the stability of the Euler equations governing the motion of an inviscid incompressible fluid. It is proved that any flow with exponential stretching of the fluid particles is unstable. In the case of an arbitrary axisymmetric steady integrable flow, a sufficient condition for instability is exhibited in terms of the curvature and the geodesic torsion of a stream line and the helicity of the flow.
Lagrangian Based Methods for Coherent Structure Detection
(2015-09) Allshouse, Michael R.; Peacock, Thomas; Allshouse, Michael R.
There has been a proliferation in the development of Lagrangian analytical methods for detecting coherent structures in fluid flow transport, yielding a variety of qualitatively different approaches. We present a review of four approaches and demonstrate the utility of these methods via their application to the same sample analytic model, the canonical double-gyre flow, highlighting the pros and cons of each approach. Two of the methods, the geometric and probabilistic approaches, are well established and require velocity field data over the time interval of interest to identify particularly important material lines and surfaces, and influential regions, respectively. The other two approaches, implementing tools from cluster and braid theory, seek coherent structures based on limited trajectory data, attempting to partition the flow transport into distinct regions. All four of these approaches share the common trait that they are objective methods, meaning that their results do not depend on the frame of reference used. For each method, we also present a number of example applications ranging from blood flow and chemical reactions to ocean and atmospheric flows. (C) 2015 AIP Publishing LLC.
Model For The Dynamics Of A Bubble Undergoing Small Shape Oscillations Between Elastic Layers
(2012) Ilinskii, Y. A.; Hay, T. A.; Zabolotskaya, Evgenia A.; Hamilton, M. F.; Ilinskii, Yurii A.; Hay, Todd A.; Zabolotskaya, Evgenia A.; Hamilton, Mark F.
A model is presented for a pulsating and translating gas bubble in a channel formed by two soft elastic parallel layers. The bubble is free to undergo small shape deformations. Coupled nonlinear second-order differential equations are obtained for the shape and position of the bubble, and numerical integration of an expression for the liquid velocity at the layer interfaces yields an estimate of their displacement. Simulations reveal behavior consistent with laboratory observations.
Propagation of a Solitary Fission Wave
(2012-06) Osborne, A. G.; Recktenwald, G. D.; Deinert, M. R.; Osborne, A. G.; Recktenwald, G. D.; Deinert, M. R.; Osborne, A. G.; Recktenwald, G. D.; Deinert, M. R.
Reaction-diffusion phenomena are encountered in an astonishing array of natural systems. Under the right conditions, self stabilizing reaction waves can arise that will propagate at constant velocity. Numerical studies have shown that fission waves of this type are also possible and that they exhibit soliton like properties. Here, we derive the conditions required for a solitary fission wave to propagate at constant velocity. The results place strict conditions on the shapes of the flux, diffusive, and reactive profiles that would be required for such a phenomenon to persist, and this condition would apply to other reaction diffusion phenomena as well. Numerical simulations are used to confirm the results and show that solitary fission waves fall into a bistable class of reaction diffusion phenomena. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4729927]
Quasi-Two-Dimensional Dynamics of Plasmas and Fluids
(1994-06) Horton, Wendell; Hasegawa, Akira; Horton, Wendell
In the lowest order of approximation quasi-twa-dimensional dynamics of planetary atmospheres and of plasmas in a magnetic field can be described by a common convective vortex equation, the Charney and Hasegawa-Mirna (CHM) equation. In contrast to the two-dimensional Navier-Stokes equation, the CHM equation admits "shielded vortex solutions" in a homogeneous limit and linear waves ("Rossby waves" in the planetary atmosphere and "drift waves" in plasmas) in the presence of inhomogeneity. Because of these properties, the nonlinear dynamics described by the CHM equation provide rich solutions which involve turbulent, coherent and wave behaviors. Bringing in non ideal effects such as resistivity makes the plasma equation significantly different from the atmospheric equation with such new effects as instability of the drift wave driven by the resistivity and density gradient. The model equation deviates from the CHM equation and becomes coupled with Maxwell equations. This article reviews the linear and nonlinear dynamics of the quasi-two-dimensional aspect of plasmas and planetary atmosphere starting from the introduction of the ideal model equation (CHM equation) and extending into the most recent progress in plasma turbulence.
Refining Finite-Time Lyapunov Exponent Ridges and the Challenges of Classifying Them
(2015-08) Allshouse, Michael R.; Peacock, Thomas; Allshouse, Michael R.
While more rigorous and sophisticated methods for identifying Lagrangian based coherent structures exist, the finite-time Lyapunov exponent (FTLE) field remains a straightforward and popular method for gaining some insight into transport by complex, time-dependent two-dimensional flows. In light of its enduring appeal, and in support of good practice, we begin by investigating the effects of discretization and noise on two numerical approaches for calculating the FTLE field. A practical method to extract and refine FTLE ridges in two-dimensional flows, which builds on previous methods, is then presented. Seeking to better ascertain the role of a FTLE ridge in flow transport, we adapt an existing classification scheme and provide a thorough treatment of the challenges of classifying the types of deformation represented by a FTLE ridge. As a practical demonstration, the methods are applied to an ocean surface velocity field data set generated by a numerical model. (C) 2015 AIP Publishing LLC.
Regularity Criterion For 3D Navier-Stokes Equations In Terms Of The Direction Of The Velocity
(2009-02) Vasseur, Alexis; Vasseur, Alexis
In this short note we give a link between the regularity of the solution u to the 3D Navier-Stokes equation and the behavior of the direction of the velocity u/|u|. It is shown that the control of div(u/vertical bar u vertical bar) in a suitable L (t/p) (L (x/q) ) norm is enough to ensure global regularity. The result is reminiscent of the criterion in terms of the direction of the vorticity, introduced first by Constantin and Fefferman. However, in this case the condition is not on the vorticity but on the velocity itself. The proof, based on very standard methods, relies on a straightforward relation between the divergence of the direction of the velocity and the growth of energy along streamlines.
Rhombic Patterns: Broken Hexagonal Symmetry
(1993-10) Ouyang, Qi; Gunaratne, Gemunu H.; Swinney, Harry L.; Ouyang, Qi; Swinney, Harry L.
Landau-Ginzburg equations derived to conserve two-dimensional spatial symmetries lead to the prediction that rhombic arrays with characteristic angles slightly differ from 60 degrees should form in many systems. Beyond the bifurcation from the uniform state to patterns, rhombic patterns are linearly stable for a band of angles near the 60 degrees angle of regular hexagons. Experiments conducted on a reaction-diffusion system involving a chlorite-iodide-malonic acid reaction yield rhombic patterns in good accord with the theory.
Scaling of Geochemical Reaction Rates via Advective Solute Transport
(2015-07) Hunt, A. G.; Ghanbarian, B.; Skinner, T. E.; Ewing, R. P.; Ghanbarian, B.
Transport in porous media is quite complex, and still yields occasional surprises. In geological porous media, the rate at which chemical reactions (e.g., weathering and dissolution) occur is found to diminish by orders of magnitude with increasing time or distance. The temporal rates of laboratory experiments and field observations differ, and extrapolating from laboratory experiments (in months) to field rates (in millions of years) can lead to order-of-magnitude errors. The reactions are transport-limited, but characterizing them using standard solute transport expressions can yield results in agreement with experiment only if spurious assumptions and parameters are introduced. We previously developed a theory of non-reactive solute transport based on applying critical path analysis to the cluster statistics of percolation. The fractal structure of the clusters can be used to generate solute distributions in both time and space. Solute velocities calculated from the temporal evolution of that distribution have the same time dependence as reaction-rate scaling in a wide range of field studies and laboratory experiments, covering some 10 decades in time. The present theory thus both explains a wide range of experiments, and also predicts changes in the scaling behavior in individual systems with increasing time and/or length scales. No other theory captures these variations in scaling by invoking a single physical mechanism. Because the successfully predicted chemical reactions include known results for silicate weathering rates, our theory provides a framework for understanding changes in the global carbon cycle, including its effects on extinctions, climate change, soil production, and denudation rates. It further provides a basis for understanding the fundamental time scales of hydrology and shallow geochemistry, as well as the basis of industrial agriculture. (C) 2015 AIP Publishing LLC.
Some Generalizations And Modifications Of Iterative Methods For Solving Large Sparse Symmetric Indefinite Linear Systems
(2014-04) Li, Yu-Chien; Chen, Jen-Yuan; Kincaid, David R.; Kincaid, David R.
We discuss a variety of iterative methods that are based on the Arnoldi process for solving large sparse symmetric indefinite linear systems. We describe the SYMMLQ and SYMMQR methods, as well as generalizations and modifications of them. Then, we cover the Lanczos/MSYMMLQ and Lanczos/MSYMMQR methods, which arise from a double linear system. We present pseudocodes for these algorithms.
Transition to Chemical Turbulence
(1991-12) Ouyang, Q.; Swinney, Harry L.; Ouyang, Q.; Swinney, Harry L.
Experiments have been conducted on Turing-type chemical spatial patterns and their variants in a quasi-two-dimensional open spatial reactor with a chlorite-iodide-malonic acid reaction. A variety of stationary spatial structures-hexagons, stripes, and mixed states-were observed, and transitions to these states were studied. For conditions beyond those corresponding to the emergence of patterns, a transition was observed from stationary spatial patterns to chemical turbulence, which is marked by a continuous motion of the pattern within a domain and of the grain boundaries between domains. The transition to chemical turbulence was analyzed by measuring the correlation length, the average pattern speed, and the total length of the domain boundaries. The emergence of chemical turbulence is accompanied by a large increase in the defects in the pattern, which suggests that this is an example of defect-mediated turbulence.