631 research outputs found
On characterisations of the input to state stability properties for conformable fractional order bilinear systems
This paper proposes for the first time the theoretical requirements that a fractional-order bilinear system with conformable derivative has to fulfil in order to satisfy different input-to-state stability (ISS) properties. Variants of ISS, namely ISS itself, integral ISS, exponential integral ISS, small-gain ISS, and strong integral ISS for the general class of conformable fractional-order bilinear systems are investigated providing a set of necessary and sufficient conditions for their existence and then compared. Finally, the correctness of the obtained theoretical results is verified by numerical example
Gaussian lower bounds for the density via Malliavin calculus
In this paper, based on a known formula, we use a simple idea to get a new
representation for the density of Malliavin differentiable random variables.
This new representation is particularly useful for finding lower bounds for the
density.Comment: To appear in Comptes Rendus Mathematique, 10 page
Generalized Schwarzschild's method
We describe a new finite element method (FEM) to construct continuous
equilibrium distribution functions of stellar systems. The method is a
generalization of Schwarzschild's orbit superposition method from the space of
discrete functions to continuous ones. In contrast to Schwarzschild's method,
FEM produces a continuous distribution function (DF) and satisfies the intra
element continuity and Jeans equations. The method employs two finite-element
meshes, one in configuration space and one in action space. The DF is
represented by its values at the nodes of the action-space mesh and by
interpolating functions inside the elements. The Galerkin projection of all
equations that involve the DF leads to a linear system of equations, which can
be solved for the nodal values of the DF using linear or quadratic programming,
or other optimization methods. We illustrate the superior performance of FEM by
constructing ergodic and anisotropic equilibrium DFs for spherical stellar
systems (Hernquist models). We also show that explicitly constraining the DF by
the Jeans equations leads to smoother and/or more accurate solutions with both
Schwarzschild's method and FEM.Comment: 14 pages, 7 Figures, Submitted to MNRA
The discontinuous Galerkin method for fractional degenerate convection-diffusion equations
We propose and study discontinuous Galerkin methods for strongly degenerate
convection-diffusion equations perturbed by a fractional diffusion (L\'evy)
operator. We prove various stability estimates along with convergence results
toward properly defined (entropy) solutions of linear and nonlinear equations.
Finally, the qualitative behavior of solutions of such equations are
illustrated through numerical experiments
Isotropic Gaussian random fields on the sphere: Regularity, fast simulation and stochastic partial differential equations
Isotropic Gaussian random fields on the sphere are characterized by
Karhunen-Lo\`{e}ve expansions with respect to the spherical harmonic functions
and the angular power spectrum. The smoothness of the covariance is connected
to the decay of the angular power spectrum and the relation to sample
H\"{o}lder continuity and sample differentiability of the random fields is
discussed. Rates of convergence of their finitely truncated Karhunen-Lo\`{e}ve
expansions in terms of the covariance spectrum are established, and algorithmic
aspects of fast sample generation via fast Fourier transforms on the sphere are
indicated. The relevance of the results on sample regularity for isotropic
Gaussian random fields and the corresponding lognormal random fields on the
sphere for several models from environmental sciences is indicated. Finally,
the stochastic heat equation on the sphere driven by additive, isotropic Wiener
noise is considered, and strong convergence rates for spectral discretizations
based on the spherical harmonic functions are proven.Comment: Published at http://dx.doi.org/10.1214/14-AAP1067 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths
We consider differential equations driven by rough paths and study the
regularity of the laws and their long time behavior. In particular, we focus on
the case when the driving noise is a rough path valued fractional Brownian
motion with Hurst parameter . Our contribution
in this work is twofold. First, when the driving vector fields satisfy
H\"{o}rmander's celebrated "Lie bracket condition," we derive explicit
quantitative bounds on the inverse of the Malliavin matrix. En route to this,
we provide a novel "deterministic" version of Norris's lemma for differential
equations driven by rough paths. This result, with the added assumption that
the linearized equation has moments, will then yield that the transition laws
have a smooth density with respect to Lebesgue measure. Our second main result
states that under H\"{o}rmander's condition, the solutions to rough
differential equations driven by fractional Brownian motion with
enjoy a suitable version of the strong Feller
property. Under a standard controllability condition, this implies that they
admit a unique stationary solution that is physical in the sense that it does
not "look into the future."Comment: Published in at http://dx.doi.org/10.1214/12-AOP777 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
(R1497) On the Invariant Subspaces of the Fractional Integral Operator
In operator theory, there is an important problem called the invariant subspace problem. This important problem of mathematics has been clear for more than half a century. However the solution seems to be nowhere in sight. With this motivation, we investigate the invariant subspaces of the fractional integral operator in the Banach space with certain conditions in this paper. Also by using the Duhamel product method, unicellularity of the fractional integral operator on some space is obtained and the description of the invariant subspaces is given
Generalized powers of strongly dependent random variables
Generalized powers of strongly dependent random variablesDobrushin, Major and Taqqu have studied the weak
convergence of normalized sums of Hm(Yk) where Hm is the Hermite
polynomial of order m and where {Yk} is a strongly dependent
stationary Gaussian sequence. The limiting process Zm(t) is
non-Gaussian when m > l.
We study here the weak convergence to Zm(t) of normalized sums
of stationary sequences {Uk}. These Uk can be off-diagonal multilinear
forms or they can be of the form Uk = pm(\) where the
polynomial pm is a generalized power and where \ is a strongly
dependent non-Gaussian finite variance moving average.Research supported by the National Science Foundation grant ECS-84-08524 at Cornell Universit
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