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Mathematics

Selected Works

2011

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Articles 1 - 30 of 114

Full-Text Articles in Physical Sciences and Mathematics

Multiple Soliton Solutions Of (2+1)-Dimensional Potential Kadomtsev-Petviashvili Equation, Mohammad Najafi M.Najafi, Ali Jamshidi Dec 2011

Multiple Soliton Solutions Of (2+1)-Dimensional Potential Kadomtsev-Petviashvili Equation, Mohammad Najafi M.Najafi, Ali Jamshidi

mohammad najafi

We employ the idea of Hirota’s bilinear method, to obtain some new exact soliton solutions for high nonlinear form of (2+1)-dimensional potential Kadomtsev-Petviashvili equation. Multiple singular soliton solutions were obtained by this method. Moreover, multiple singular soliton solutions were also derived.

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Nonparametric Copula Density Estimation In Sensor Networks, Leming Qu, Hao Chen, Yicheng Tu Dec 2011

Nonparametric Copula Density Estimation In Sensor Networks, Leming Qu, Hao Chen, Yicheng Tu

Leming Qu

Statistical and machine learning is a fundamental task in sensor networks. Real world data almost always exhibit dependence among different features. Copulas are full measures of statistical dependence among random variables. Estimating the underlying copula density function from distributed data is an important aspect of statistical learning in sensor networks. With limited communication capacities or privacy concerns, centralization of the data is often impossible. By only collecting the ranks of the data observed by different sensors, we estimate and evaluate the copula density on an equally spaced grid after binning the standardized ranks at the fusion center. Without assuming any …

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Cardinal Invariants And The Borel Tukey Order, Samuel Coskey Dec 2011

Cardinal Invariants And The Borel Tukey Order, Samuel Coskey

Samuel Coskey

Many proofs of inequalities between cardinal characteristics of the continuum are combinatorial in nature. These arguments can be carried out in any model of set theory, even a model of CH where the inequalities themselves are trivial. Thus, such arguments appear to establish a stronger relationship than a mere inequality. The Borel Tukey order was introduced by Blass in a 1996 article to address just this. Specifically, he observed that the combinatorial information linking two cardinal characteristics is often captured by a pair of Borel maps called a Borel Tukey morphism. The existence of a Borel Tukey morphisms between …

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Mixed Discriminants, Eduardo Cattani, Maria Angelica Cueto, Alicia Dickenstein, Sandra Di Rocco, Bernd Strumfels Dec 2011

Mixed Discriminants, Eduardo Cattani, Maria Angelica Cueto, Alicia Dickenstein, Sandra Di Rocco, Bernd Strumfels

Eduardo Cattani

No abstract provided.

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Symmetry-Breaking Bifurcation In The Nonlinear Schrödinger Equation With Symmetric Potentials, E. Kirr, Panos Kevrekidis, D. E. Pelinovsky Dec 2011

Symmetry-Breaking Bifurcation In The Nonlinear Schrödinger Equation With Symmetric Potentials, E. Kirr, Panos Kevrekidis, D. E. Pelinovsky

Panos Kevrekidis

We consider the focusing (attractive) nonlinear Schr\"odinger (NLS) equation with an external, symmetric potential which vanishes at infinity and supports a linear bound state. We prove that the symmetric, nonlinear ground states must undergo a symmetry breaking bifurcation if the potential has a non-degenerate local maxima at zero. Under a generic assumption we show that the bifurcation is either subcritical or supercritical pitchfork. In the particular case of double-well potentials with large separation, the power of nonlinearity determines the subcritical or supercritical character of the bifurcation. The results are obtained from a careful analysis of the spectral properties of the …

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Differential Geometry Based Solvation Model Ii: Lagrangian Formulation, Zhan Chen, Nathan A. Baker, Guo-Wei Wei Dec 2011

Differential Geometry Based Solvation Model Ii: Lagrangian Formulation, Zhan Chen, Nathan A. Baker, Guo-Wei Wei

Zhan Chen

Solvation is an elementary process in nature and is of paramount importance to more sophisticated chemical, biological and biomolecular processes. The understanding of solvation is an essential prerequisite for the quantitative description and analysis of biomolecular systems. This work presents a Lagrangian formulation of our differential geometry based solvation models. The Lagrangian representation of biomolecular surfaces has a few utilities/advantages. First, it provides an essential basis for biomolecular visualization, surface electrostatic potential map and visual perception of biomolecules. Additionally, it is consistent with the conventional setting of implicit solvent theories and thus, many existing theoretical algorithms and computational software packages …

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Multiple Dark-Bright Solitons In Atomic Bose-Einstein Condensates, D. Yan, J. J. Chang, C. Hamner, Panos Kevrekidis, P. Engels, V. Achilleos, D. J. Frantzeskakis, R. Carretero-Gonz´Alez, P. Schmelcher Nov 2011

Multiple Dark-Bright Solitons In Atomic Bose-Einstein Condensates, D. Yan, J. J. Chang, C. Hamner, Panos Kevrekidis, P. Engels, V. Achilleos, D. J. Frantzeskakis, R. Carretero-Gonz´Alez, P. Schmelcher

Panos Kevrekidis

Motivated by recent experimental results, we present a systematic theoretical analysis of dark-bright-soliton interactions and multiple-dark-bright-soliton complexes in atomic two-component Bose-Einstein condensates. We study analytically the interactions between two dark-bright solitons in a homogeneous condensate and then extend our considerations to the presence of the trap. We illustrate the existence of robust stationary dark-bright-soliton “molecules,” composed of two or more solitons, which are formed due to the competition of the interaction forces between the dark- and bright-soliton components and the trap force. Our analysis is based on an effective equation of motion, derived for the distance between two dark-bright solitons. …

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Los Modelos De Equilibrio General: La Revisión De Chancelier Y Una Crítica A Debreu Y Mckenzie, Rodrigo Lopez-Pablos Nov 2011

Los Modelos De Equilibrio General: La Revisión De Chancelier Y Una Crítica A Debreu Y Mckenzie, Rodrigo Lopez-Pablos

Lopez-Pablos, Rodrigo

A revision on general equilibrium theory from an entropic perspective. JEL CLASSIFICATION: D50, O21, Z19

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Highly Connected Multicoloured Subgraphs Of Multicoloured Graphs, H. Liu, R. Morris, N. Prince Nov 2011

Highly Connected Multicoloured Subgraphs Of Multicoloured Graphs, H. Liu, R. Morris, N. Prince

Hua Kun Liu

Suppose the edges of the complete graph on n vertices, E(Kn), are coloured using r colours; how large a k-connected subgraph are we guaranteed to find, which uses only at most s of the colours? This question is due to Bollobás, and the case s=1 was considered in Liu et al. [Highly connected monochromatic subgraphs of multicoloured graphs, J. Graph Theory, to appear]. Here we shall consider the case s is greater than or equal to 2, proving in particular that when s=2 and r+1 is a power of 2 then the answer lies between 4n/(r+1)-17kr(r+2k+1) and 4n/(r+1)+4 and that …

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Breathers In Oscillator Chains With Hertzian Interactions, Guillaume James, Panos Kevrekidis, Jesus Cuevas Nov 2011

Breathers In Oscillator Chains With Hertzian Interactions, Guillaume James, Panos Kevrekidis, Jesus Cuevas

Panos Kevrekidis

We prove nonexistence of breathers (spatially localized and time-periodic oscillations) for a class of Fermi-Pasta-Ulam lattices representing an uncompressed chain of beads interacting via Hertz's contact forces. We then consider the setting in which an additional on-site potential is present, motivated by the Newton's cradle under the effect of gravity. Using both direct numerical computations and a simplified asymptotic model of the oscillator chain, the so-called discrete p-Schr\"odinger (DpS) equation, we show the existence of discrete breathers and study their spectral properties and mobility. Due to the fully nonlinear character of Hertzian interactions, breathers are found to be much more …

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Defeating The Kalka–Teicher–Tsaban Linear Algebra Attack On The Algebraic Eraser, Dorian Goldfeld, Paul E. Gunnells Nov 2011

Defeating The Kalka–Teicher–Tsaban Linear Algebra Attack On The Algebraic Eraser, Dorian Goldfeld, Paul E. Gunnells

Paul Gunnells

The Algebraic Eraser (AE) is a public key protocol for shar- ing information over an insecure channel using commutative and non- commutative groups; a concrete realization is given by Colored Burau Key Agreement Protocol (CBKAP). In this paper, we describe how to choose data in CBKAP to thwart an attack by Kalka–Teicher–Tsaban.

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Rabi Flopping Induces Spatial Demixing Dynamics, E. Nicklas, H. Strobel, T. Zibold, C. Gross, B. A. Malomed, Panos Kevrekidis, M. K. Oberthaler Nov 2011

Rabi Flopping Induces Spatial Demixing Dynamics, E. Nicklas, H. Strobel, T. Zibold, C. Gross, B. A. Malomed, Panos Kevrekidis, M. K. Oberthaler

Panos Kevrekidis

We experimentally investigate the mixing and demixing dynamics of Bose-Einstein condensates in the presence of a linear coupling between two internal states. The observed amplitude reduction of the Rabi oscillations can be understood as a result of demixing dynamics of dressed states as experimentally confirmed by reconstructing the spatial profile of dressed state amplitudes. The observations are in quantitative agreement with numerical integration of coupled Gross-Pitaevskii equations without free parameters, which also reveals the criticality of the dynamics on the symmetry of the system. Our observations demonstrate new possibilities for changing effective atomic interactions and studying critical phenomena.

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Fluctuating And Dissipative Dynamics Of Dark Solitons In Quasicondensates S., S. P. Cockburn, H. E. Nistazakis, T. P. Horikis, Panos Kevrekidis, N. P. Proukakis, D. J. Frantzeskakis Oct 2011

Fluctuating And Dissipative Dynamics Of Dark Solitons In Quasicondensates S., S. P. Cockburn, H. E. Nistazakis, T. P. Horikis, Panos Kevrekidis, N. P. Proukakis, D. J. Frantzeskakis

Panos Kevrekidis

The fluctuating and dissipative dynamics of matter-wave dark solitons within harmonically trapped, partially condensed Bose gases is studied both numerically and analytically. A study of the stochastic Gross-Pitaevskii equation, which correctly accounts for density and phase fluctuations at finite temperatures, reveals dark-soliton decay times to be lognormally distributed at each temperature, thereby characterizing the previously predicted long-lived soliton trajectories within each ensemble of numerical realizations [ S. P. Cockburn et al. Phys. Rev. Lett. 104 174101 (2010)]. Expectation values for the average soliton lifetimes extracted from these distributions are found to agree well with both numerical and analytic predictions based …

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Counter-Propagating Two-Soliton Solutions In The Fermi–Pasta–Ulam Lattice, Aaron Hoffman, C.E. Wayne Oct 2011

Counter-Propagating Two-Soliton Solutions In The Fermi–Pasta–Ulam Lattice, Aaron Hoffman, C.E. Wayne

Aaron Hoffman

We study the interaction of small amplitude, long-wavelength solitary wavesin the Fermi–Pasta–Ulam model with general nearest-neighbour interactionpotential. We establish global-in-time existence and stability of counterpropagatingsolitary wave solutions. These solutions are close to the linearsuperposition of two solitary waves for large positive and negative values oftime; for intermediate values of time these solutions describe the interactionof two counter-propagating pulses. These solutions are stable with respectto perturbations in L2 and asymptotically stable with respect to perturbationswhich decay exponentially at spatial ±∞.

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Resolutions Of The Steinberg Module For Gl(N), Avner Ash, Paul E. Gunnells, Mark Mcconnell Oct 2011

Resolutions Of The Steinberg Module For Gl(N), Avner Ash, Paul E. Gunnells, Mark Mcconnell

Paul Gunnells

We give several resolutions of the Steinberg representation St_n for the general linear group over a principal ideal domain, in particular over Z. We compare them, and use these results to prove that the computations in [AGM4] are definitive. In particular, in [AGM4] we use two complexes to compute certain cohomology groups of congruence subgroups of SL(4,Z). One complex is based on Voronoi's polyhedral decomposition of the symmetric space for SL(n,R), whereas the other is a larger complex that has an action of the Hecke operators. We prove that both complexes allow us to compute the relevant cohomology groups, and …

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Boundary Effects In Large-Aspect-Ratio Lasers, G.K. Harkness, W.J. Firth, John B. Geddes, J.V Moloney, E.M. Wright Oct 2011

Boundary Effects In Large-Aspect-Ratio Lasers, G.K. Harkness, W.J. Firth, John B. Geddes, J.V Moloney, E.M. Wright

John B. Geddes

We study theoretically the effect of transverse boundary conditions on the traveling waves foundin infinitely extended and positively detuned laser systems. We find that for large-aspect-ratiosystems, well above threshold and away from the boundaries, the traveling waves persist. Sourceand sink defects are observed on the boundaries, and in very-large-aspect-ratio systems these defectscan also exist away from the boundaries. The transverse size of the sink defect, relative to the sizeof the transverse domain, is important in determining the final pattern observed, and so, close tothreshold, standing waves are always observed.

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Constructing Large Numbers With Cheap Computers, Andrew Shallue, Steven Hayman Oct 2011

Constructing Large Numbers With Cheap Computers, Andrew Shallue, Steven Hayman

Andrew Shallue

No abstract provided.

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How To Study Mathematics, Lawrence N. Stout Sep 2011

How To Study Mathematics, Lawrence N. Stout

Lawrence N. Stout

In high school mathematics much of your time was spent learning algorithms and manipulative techniques which you were expected to be able to apply in certain well-defined situations. This limitation of material and expectations for your performance has probably led you to develop study habits which were appropriate for high school mathematics but may be insufficient for college mathematics. This can be a source of much frustration for you and for your instructors. My object in writing this essay is to help ease this frustration by describing some study strategies which may help you channel your abilities and energies in …

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A High-Resolution Finite-Difference Method For Simulating Two-Fluid, Viscoelastic Gel Dynamics, Grady Wright, Robert D. Guy, Jian Du, Aaron L. Fogelson Sep 2011

A High-Resolution Finite-Difference Method For Simulating Two-Fluid, Viscoelastic Gel Dynamics, Grady Wright, Robert D. Guy, Jian Du, Aaron L. Fogelson

Grady Wright

An important class of gels are those composed of a polymer network and fluid solvent. The mechanical and rheological properties of these two-fluid gels can change dramatically in response to temperature, stress, and chemical stimulus. Because of their adaptivity, these gels are important in many biological systems, e.g. gels make up the cytoplasm of cells and the mucus in the respiratory and digestive systems, and they are involved in the formation of blood clots. In this study we consider a mathematical model for gels that treats the network phase as a viscoelastic fluid with spatially and temporally varying material parameters …

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Operation Comics: Making Math Fun, Bruce Kessler Sep 2011

Operation Comics: Making Math Fun, Bruce Kessler

Bruce Kessler

This talk gives a background on the Operation Comics series, which integrates mathematics into a comic book storyline, as an example of how creativity is not exclusive to the traditional arts, like music and dance, but is a vital part of math, science, and engineering.

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Dark–Bright Ring Solitons In Bose–Einstein Condensates, J. Stockhofe, Panos Kevrekidis, D. J. Frantzeskakis, P. Schmelcher Sep 2011

Dark–Bright Ring Solitons In Bose–Einstein Condensates, J. Stockhofe, Panos Kevrekidis, D. J. Frantzeskakis, P. Schmelcher

Panos Kevrekidis

We study dark–bright (DB) ring solitons in two-component Bose–Einstein condensates. In the limit of large densities of the dark component, we describe the soliton dynamics by means of an equation of motion for the ring radius. The presence of the bright, 'filling' species is demonstrated to have a stabilizing effect on the ring dark soliton. Near the linear limit, we discuss the symmetry-breaking bifurcations of DB soliton stripes and vortex-bright soliton clusters from the DB ring and relate the stabilizing effect of filling to changes in the bifurcation diagram. Finally, we show that the stabilization by means of a second …

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Stationary States Of A Nonlinear Schrödinger Lattice With A Harmonic Trap, V. Achilleos, G. Theocharis, Panos Kevrekidis, N. I. Karachalios, F. K. Diakonos, D. J. Frantzeskakis Sep 2011

Stationary States Of A Nonlinear Schrödinger Lattice With A Harmonic Trap, V. Achilleos, G. Theocharis, Panos Kevrekidis, N. I. Karachalios, F. K. Diakonos, D. J. Frantzeskakis

Panos Kevrekidis

We study a discrete nonlinear Schrödinger lattice with a parabolic trapping potential. The model, describing, e.g., an array of repulsive Bose-Einstein condensate droplets confined in the wells of an optical lattice, is analytically and numerically investigated. Starting from the linear limit of the problem, we use global bifurcation theory to rigorously prove that – in the discrete regime – all linear states lead to nonlinear generalizations thereof, which assume the form of a chain of discrete dark solitons (as the density increases). The stability of the ensuing nonlinear states is studied and it is found that the ground state is …

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Modeling Of The Gastric Mucus Gel On The Gastric Epithelium, Frank Lynch, J.P. Keener Sep 2011

Modeling Of The Gastric Mucus Gel On The Gastric Epithelium, Frank Lynch, J.P. Keener

Frank Lynch

The mucus lining of the mammalian stomach has a thickness on the order of hundreds of microns. Across this mucus layer there is a large change in pH, from pH ∼ 2 inside the stomach to pH ∼ 7 at the intersection of the mucus lining with the stomach wall. The gel layer is thought to simultaneously protect the stomach lining from the acidity of the lumen while transporting gastric acid and other important digestive enzymes toward the lumen. The mechanisms by which these phenomena occur are not well understood. We use partial differential equations to model the protective features …

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Wavelet Thresholding In Partially Linear Models: A Computation And Simulation, Leming Qu Sep 2011

Wavelet Thresholding In Partially Linear Models: A Computation And Simulation, Leming Qu

Leming Qu

Partially linear models have a linear part as in the linear regression and a non-linear part similar to that in the non-parametric regression. The estimates in Partially Linear Models have been studied previously using traditional smoothing methods such as smoothing spline, kernel and piecewise polynomial smoothers. In this paper, a wavelet thresholding method for estimating the corresponding parameters in Partially Linear Models is presented. Extensive simulation results shows that wavelet smoothing approach is comparable to traditional smoothing methods when their assumptions are satisfied. But wavelet smoothing is often superior when assumptions about the smoothness of the underlying function of non-parametric …

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Change Point Estimation Of Bilevel Functions, Leming Qu, Yi-Cheng Tu Sep 2011

Change Point Estimation Of Bilevel Functions, Leming Qu, Yi-Cheng Tu

Leming Qu

Reconstruction of a bilevel function such as a bar code signal in a partially blind deconvolution problem is an important task in industrial processes. Existing methods are based on either the local approach or the regularization approach with a total variation penalty. This article reformulated the problem explicitly in terms of change points of the 0-1 step function. The bilevel function is then reconstructed by solving the nonlinear least squares problem subject to linear inequality constraints, with starting values provided by the local extremas of the derivative of the convolved signal from discrete noisy data. Simulation results show a considerable …

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Inversion For Non-Smooth Models With Physical Bounds, Partha S. Routh, Leming Qu, Mrinal K. Sen, Phil D. Anno Sep 2011

Inversion For Non-Smooth Models With Physical Bounds, Partha S. Routh, Leming Qu, Mrinal K. Sen, Phil D. Anno

Leming Qu

Geological processes produce structures at multiple scales. A discontinuity in the subsurface can occur due to layering, tectonic activities such as faulting, folding and fractures. Traditional approaches to invert geophysical data employ smoothness constraints. Such methods produce smooth models and thefore sharp contrasts in the medium such as lithological boundaries are not easily discernible. The methods that are able to produce non-smooth models, can help interpret the geological discontinuity. In this paper we examine various approaches to obtain non-smooth models from a finite set of noisy data. Broadly they can be categorized into approaches: (1) imposing non-smooth regularization in the …

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Rayleigh Wave Dispersion Curve Inversion: Occam Versus The L1-Norm, Matthew M. Haney, Leming Qu Sep 2011

Rayleigh Wave Dispersion Curve Inversion: Occam Versus The L1-Norm, Matthew M. Haney, Leming Qu

Leming Qu

We compare inversions of Rayleigh wave dispersion curves for shear wave velocity depth profiles based on the L2-norm (Occam's Inversion) and L1-norm (TV Regularization). We forward model Rayleigh waves using a finite-element method instead of the conventional technique based on a recursion formula and root-finding. The forward modeling naturally leads to an inverse problem that is overparameterized in depth. Solving the inverse problem with Occam's Inversion gives the smoothest subsurface model that satisfies the data. However, the subsurface need not be smooth and we therefore also solve the inverse problem with TV Regularization, a procedure that does not penalize discontinuities. …

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Bayesian Wavelet Estimation Of Long Memory Parameter, Leming Qu Sep 2011

Bayesian Wavelet Estimation Of Long Memory Parameter, Leming Qu

Leming Qu

A Bayesian wavelet estimation method for estimating parameters of a stationary I(d) process is represented as an useful alternative to the existing frequentist wavelet estimation methods. The effectiveness of the proposed method is demonstrated through Monte Carlo simulations. The sampling from the posterior distribution is through the Markov Chain Monte Carlo (MCMC) easily implemented in the WinBUGS software package.

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Copula Density Estimation By Total Variation Penalized Likelihood With Linear Equality Constraints, Leming Qu, Wotao Yin Sep 2011

Copula Density Estimation By Total Variation Penalized Likelihood With Linear Equality Constraints, Leming Qu, Wotao Yin

Leming Qu

A copula density is the joint probability density function (PDF) of a random vector with uniform marginals. An approach to bivariate copula density estimation is introduced that is based on a maximum penalized likelihood estimation (MPLE) with a total variation (TV) penalty term. The marginal unity and symmetry constraints for copula density are enforced by linear equality constraints. The TV-MPLE subject to linear equality constraints is solved by an augmented Lagrangian and operator-splitting algorithm. It offers an order of magnitude improvement in computational efficiency over another TV-MPLE method without constraints solved by log-barrier method for second order cone program. A …

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Wavelet Image Restoration And Regularization Parameters Selection, Leming Qu Sep 2011

Wavelet Image Restoration And Regularization Parameters Selection, Leming Qu

Leming Qu

For the restoration of an image based on its noisy distorted observations, we propose wavelet domain restoration by scale-dependent ∫1 penalized regularization method (WaveRSL1). The data adaptive choice of the regularization parameters is based on the Akaike Information Criterion (AIC) and the degrees of freedom (df) is estimated by the number of nonzero elements in the solution. Experiments on some commonly used testing images illustrate that the proposed method possesses good empirical properties.

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