Displaying similar documents to “Uniform Gaussian Bounds for Subelliptic Heat Kernels and an Application to the Total Variation Flow of Graphs over Carnot Groups”

Genetic Exponentially Fitted Method for Solving Multi-dimensional Drift-diffusion Equations

M. R. Swager, Y. C. Zhou (2013)

Molecular Based Mathematical Biology

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A general approach was proposed in this article to develop high-order exponentially fitted basis functions for finite element approximations of multi-dimensional drift-diffusion equations for modeling biomolecular electrodiffusion processes. Such methods are highly desirable for achieving numerical stability and efficiency. We found that by utilizing the one-to-one correspondence between the continuous piecewise polynomial space of degree k + 1 and the divergencefree vector space of...

On the derivation and mathematical analysis of some quantum–mechanical models accounting for Fokker–Planck type dissipation: Phase space, Schrödinger and hydrodynamic descriptions

José Luis López, Jesús Montejo–Gámez (2013)

Nanoscale Systems: Mathematical Modeling, Theory and Applications

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This paper is intended to provide the reader with a review of the authors’ latest results dealing with the modeling of quantum dissipation/diffusion effects at the level of Schrödinger systems, in connection with the corresponding phase space and fluid formulations of such kind of phenomena, especially in what concerns the role of the Fokker–Planck mechanism in the description of open quantum systems and the macroscopic dynamics associated with some viscous hydrodynamic models of Euler...

Fully implicit ADI schemes for solving the nonlinear Poisson-Boltzmann equation

Weihua Geng, Shan Zhao (2013)

Molecular Based Mathematical Biology

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The Poisson-Boltzmann (PB) model is an effective approach for the electrostatics analysis of solvated biomolecules. The nonlinearity associated with the PB equation is critical when the underlying electrostatic potential is strong, but is extremely difficult to solve numerically. In this paper, we construct two operator splitting alternating direction implicit (ADI) schemes to efficiently and stably solve the nonlinear PB equation in a pseudo-transient continuation approach. The operator...

A Formula for Popp’s Volume in Sub-Riemannian Geometry

Davide Barilari, Luca Rizzi (2013)

Analysis and Geometry in Metric Spaces

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For an equiregular sub-Riemannian manifold M, Popp’s volume is a smooth volume which is canonically associated with the sub-Riemannian structure, and it is a natural generalization of the Riemannian one. In this paper we prove a general formula for Popp’s volume, written in terms of a frame adapted to the sub-Riemannian distribution. As a first application of this result, we prove an explicit formula for the canonical sub- Laplacian, namely the one associated with Popp’s volume. Finally,...

The Lusin Theorem and Horizontal Graphs in the Heisenberg Group

Piotr Hajłasz, Jacob Mirra (2013)

Analysis and Geometry in Metric Spaces

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In this paper we prove that every collection of measurable functions fα , |α| = m, coincides a.e. withmth order derivatives of a function g ∈ Cm−1 whose derivatives of order m − 1 may have any modulus of continuity weaker than that of a Lipschitz function. This is a stronger version of earlier results of Lusin, Moonens-Pfeffer and Francos. As an application we construct surfaces in the Heisenberg group with tangent spaces being horizontal a.e.

Mathematical modeling of semiconductor quantum dots based on the nonparabolic effective-mass approximation

Jinn-Liang Liu (2012)

Nanoscale Systems: Mathematical Modeling, Theory and Applications

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Within the effective mass and nonparabolic band theory, a general framework of mathematical models and numerical methods is developed for theoretical studies of semiconductor quantum dots. It includes single-electron models and many-electron models of Hartree-Fock, configuration interaction, and current-spin density functional theory approaches. These models result in nonlinear eigenvalue problems from a suitable discretization. Cubic and quintic Jacobi-Davidson methods of block or nonblock...

On certain transformations of Archimedean copulas: Application to the non-parametric estimation of their generators

Elena Di Bernardino, Didier Rullière (2013)

Dependence Modeling

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We study the impact of certain transformations within the class of Archimedean copulas. We give some admissibility conditions for these transformations, and define some equivalence classes for both transformations and generators of Archimedean copulas. We extend the r-fold composition of the diagonal section of a copula, from r ∈ N to r ∈ R. This extension, coupled with results on equivalence classes, gives us new expressions of transformations and generators. Estimators deriving directly...

Quantum optimal control using the adjoint method

Alfio Borzì (2012)

Nanoscale Systems: Mathematical Modeling, Theory and Applications

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Control of quantum systems is central in a variety of present and perspective applications ranging from quantum optics and quantum chemistry to semiconductor nanostructures, including the emerging fields of quantum computation and quantum communication. In this paper, a review of recent developments in the field of optimal control of quantum systems is given with a focus on adjoint methods and their numerical implementation. In addition, the issues of exact controllability and optimal...

Bounds on Capital Requirements For Bivariate Risk with Given Marginals and Partial Information on the Dependence

Carole Bernard, Yuntao Liu, Niall MacGillivray, Jinyuan Zhang (2013)

Dependence Modeling

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Nelsen et al. [20] find bounds for bivariate distribution functions when there are constraints on the values of its quartiles. Tankov [25] generalizes this work by giving explicit expressions for the best upper and lower bounds for a bivariate copula when its values on a compact subset of [0; 1]2 are known. He shows that they are quasi-copulas and not necessarily copulas. Tankov [25] and Bernard et al. [3] both give sufficient conditions for these bounds to be copulas. In this note we...

Dependence of Stock Returns in Bull and Bear Markets

Jadran Dobric, Gabriel Frahm, Friedrich Schmid (2013)

Dependence Modeling

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Despite of its many shortcomings, Pearson’s rho is often used as an association measure for stock returns. A conditional version of Spearman’s rho is suggested as an alternative measure of association. This approach is purely nonparametric and avoids any kind of model misspecification. We derive hypothesis tests for the conditional rank-correlation coefficients particularly arising in bull and bear markets and study their finite-sample performance by Monte Carlo simulation. Further,...

Fractional Maximal Functions in Metric Measure Spaces

Toni Heikkinen, Juha Lehrbäck, Juho Nuutinen, Heli Tuominen (2013)

Analysis and Geometry in Metric Spaces

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We study the mapping properties of fractional maximal operators in Sobolev and Campanato spaces in metric measure spaces. We show that, under certain restrictions on the underlying metric measure space, fractional maximal operators improve the Sobolev regularity of functions and map functions in Campanato spaces to Hölder continuous functions. We also give an example of a space where fractional maximal function of a Lipschitz function fails to be continuous.