1,029 research outputs found

    Singular value decay of operator-valued differential Lyapunov and Riccati equations

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    We consider operator-valued differential Lyapunov and Riccati equations, where the operators BB and CC may be relatively unbounded with respect to AA (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 AA generates an analytic semigroup and the range of CC 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 CC. 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

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    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

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    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, K K and E E, 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 LE L_E which has E E as solution (or, for off-diagonal correlations to the direct sum of LE L_E and d/dt d/dt). We show that this can be generalized, mutatis mutandis, to the anisotropic Ising model. The singled-out second order linear differential operator LE L_E being replaced by an isomonodromic system of two third-order linear partial differential operators associated with Π1 \Pi_1, 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 LE L_E). 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

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    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

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    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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