1,029 research outputs found
Singular value decay of operator-valued differential Lyapunov and Riccati equations
We consider operator-valued differential Lyapunov and Riccati equations,
where the operators and may be relatively unbounded with respect to
(in the standard notation). In this setting, we prove that the singular values
of the solutions decay fast under certain conditions. In fact, the decay is
exponential in the negative square root if generates an analytic semigroup
and the range of has finite dimension. This extends previous similar
results for algebraic equations to the differential case. When the initial
condition is zero, we also show that the singular values converge to zero as
time goes to zero, with a certain rate that depends on the degree of
unboundedness of . A fast decay of the singular values corresponds to a low
numerical rank, which is a critical feature in large-scale applications. The
results reported here provide a theoretical foundation for the observation
that, in practice, a low-rank factorization usually exists.Comment: Corrected some misconceptions, which lead to more general results
(e.g. exponential stability is no longer required). Also fixed some
off-by-one errors, improved the presentation, and added/extended several
remarks on possible generalizations. Now 22 pages, 8 figure
A new two--parameter family of isomonodromic deformations over the five punctured sphere
The object of this paper is to describe an explicit two--parameter family of
logarithmic flat connections over the complex projective plane. These
connections have dihedral monodromy and their polar locus is a prescribed
quintic composed of a circle and three tangent lines. By restricting them to
generic lines we get an algebraic family of isomonodromic deformations of the
five--punctured sphere. This yields new algebraic solutions of a Garnier
system. Finally, we use the associated Riccati one--forms to construct an
interesting non--generic family of transversally projective Lotka--Volterra
foliations.Comment: English text, 30 page
Fuchs versus Painlev\'e
We briefly recall the Fuchs-Painlev\'e elliptic representation of Painlev\'e
VI. We then show that the polynomiality of the expressions of the correlation
functions (and form factors) in terms of the complete elliptic integral of the
first and second kind,
and , is a straight consequence of the fact that the differential
operators corresponding to the entries of Toeplitz-like determinants, are
equivalent to the second order operator which has as solution (or,
for off-diagonal correlations to the direct sum of and ). We show
that this can be generalized, mutatis mutandis, to the anisotropic Ising model.
The singled-out second order linear differential operator being replaced
by an isomonodromic system of two third-order linear partial differential
operators associated with , the Jacobi's form of the complete elliptic
integral of the third kind (or equivalently two second order linear partial
differential operators associated with Appell functions, where one of these
operators can be seen as a deformation of ). We finally explore the
generalizations, to the anisotropic Ising models, of the links we made, in two
previous papers, between Painlev\'e non-linear ODE's, Fuchsian linear ODE's and
elliptic curves. In particular the elliptic representation of Painlev\'e VI has
to be generalized to an ``Appellian'' representation of Garnier systems.Comment: Dedicated to the : Special issue on Symmetries and Integrability of
Difference Equations, SIDE VII meeting held in Melbourne during July 200
Computing stability of multi-dimensional travelling waves
We present a numerical method for computing the pure-point spectrum
associated with the linear stability of multi-dimensional travelling fronts to
parabolic nonlinear systems. Our method is based on the Evans function shooting
approach. Transverse to the direction of propagation we project the spectral
equations onto a finite Fourier basis. This generates a large, linear,
one-dimensional system of equations for the longitudinal Fourier coefficients.
We construct the stable and unstable solution subspaces associated with the
longitudinal far-field zero boundary conditions, retaining only the information
required for matching, by integrating the Riccati equations associated with the
underlying Grassmannian manifolds. The Evans function is then the matching
condition measuring the linear dependence of the stable and unstable subspaces
and thus determines eigenvalues. As a model application, we study the stability
of two-dimensional wrinkled front solutions to a cubic autocatalysis model
system. We compare our shooting approach with the continuous orthogonalization
method of Humpherys and Zumbrun. We then also compare these with standard
projection methods that directly project the spectral problem onto a finite
multi-dimensional basis satisfying the boundary conditions.Comment: 23 pages, 9 figures (some in colour). v2: added details and other
changes to presentation after referees' comments, now 26 page
Supersymmetric methods in the traveling variable: inside neurons and at the brain scale
We apply the mathematical technique of factorization of differential
operators to two different problems. First we review our results related to the
supersymmetry of the Montroll kinks moving onto the microtubule walls as well
as mentioning the sine-Gordon model for the microtubule nonlinear excitations.
Second, we find analytic expressions for a class of one-parameter solutions of
a sort of diffusion equation of Bessel type that is obtained by supersymmetry
from the homogeneous form of a simple damped wave equations derived in the
works of P.A. Robinson and collaborators for the corticothalamic system. We
also present a possible interpretation of the diffusion equation in the brain
contextComment: 14 pages, 1 figur
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