On the Perron-Frobenius-ruelle operator for rational maps on the riemann sphere and for Hölder continuous functions

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On three-term recurrence and Christoffel–Darboux identity for orthogonal rational functions on the real line

First, we give an algebraic proof to the Christoffel–Darboux identity of formal orthogonal rational functions on the real line by exposing some underlying algebraic properties. This proof does not involve the three-term recurrence relationship. Besides, it is shown that if a family of rational functions satisfies the Christoffel–Darboux relation, then it also admits a three-term recurrence relationship. Thus, the equivalence between both relations is revealed.     

Riemann–Hilbert approach for the Camassa–Holm equation on the line

We present a Riemann–Hilbert problem formalism for the initial value problem for the Camassa–Holm equation $u_t?u_{txx}+2\omega u_x+3u{u}_x=2u_x{u}_{xx}+u{u}_{xxx}$ on the line (CH). We show that: (i) for all $\omega >0$, the solution of this problem can be obtained in a parametric form via the solution of some associated Riemann–Hilbert problem; (ii) for large time, it develops into a train of smooth solitons; (iii) for small ω, this soliton train is close to a train of peakons, which are piecewise smooth solutions of the CH equation for $\omega =0$. To cite this article: A. Boutet de Monvel, D. Shepelsky, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Holonomic gradient descent for the Fisher–Bingham distribution on the d-dimensional sphere

We propose an accelerated version of the holonomic gradient descent and apply it to calculating the maximum likelihood estimate (MLE) of the Fisher–Bingham distribution on a \(d\) -dimensional sphere. We derive a Pfaffian system (an integrable connection) and a series expansion associated with the normalizing constant with an error estimation. These enable us to solve some MLE problems up to dimension \(d=7\) with a specified accuracy.     

The Riemann–Roch theorem for the Dynnikov–Novikov discrete complex analysis

We prove an analog of the Riemann–Roch Theorem for the Dynnikov–Novikov discrete complex analysis.     

A black-box rational Arnoldi variant for Cauchy–Stieltjes matrix functions

Rational Arnoldi is a powerful method for approximating functions of large sparse matrices times a vector. The selection of asymptotically optimal parameters for this method is crucial for its fast convergence. We present and investigate a novel strategy for the automated parameter selection when the function to be approximated is of Cauchy–Stieltjes (or Markov) type, such as the matrix square root or the logarithm. The performance of this approach is demonstrated by numerical examples involving symmetric and nonsymmetric matrices. These examples suggest that our black-box method performs at least as well, and typically better, as the standard rational Arnoldi method with parameters being manually optimized for a given matrix.     

The rational approximation of convex functions of the class Lip α

It is proved that if a function f(x) is convex on [a, b] and f ∈ Lip_K(f)α, 0<α<1, then the least uniform deviation of this function from rational functions of degree not higher than n does not exceed (v is a natural number; C(α, v) depends only onα andv; K(f) is a Lipschitz constant; and      

The Cauchy–Riemann equations on product domains

We establish the L ² theory for the Cauchy–Riemann equations on product domains provided that the Cauchy–Riemann operator has closed range on each factor. We deduce regularity of the canonical solution on (p, 1)-forms in special Sobolev spaces represented as tensor products of Sobolev spaces on the factors of the product. This leads to regularity results for smooth data.     

A Petrov–Galerkin RBF method for diffusion equation on the unit sphere

This paper concerns a numerical solution for the diffusion equation on the unit sphere. The given method is based on the spherical basis function approximation and the Petrov–Galerkin test discretization. The method is meshless because spherical triangulation is not required neither for approximation nor for numerical integration. This feature is achieved through the spherical basis function approximation and the use of local weak forms instead of a global variational formulation. The local Petrov–Galerkin formulation allows to compute the integrals on small independent spherical caps without any dependence on a connected background mesh. Experimental results show the accuracy and the efficiency of the new method.     

The Riemann–Hilbert Problem for Analytic Description of the DM Solitons

A simple exact formula is derived for the profile of the optical pulse propagating over a DM fiber with zero mean dispersion. The dissipation is neglected, and dispersion is assumed to be constant along the adjacent legs of the waveguide, thus providing the applicability of the integrable NLS models within each leg. The formula describes a class of solutions called dispersion-managed solitons (DM solitons), which are periodic along the waveguide and exponentially localized in time. The DM solitons are parameterized by a certain class of spectral data, specified from numerical simulations. Using a related Riemann–Hilbert problem, we reconstruct a profile of the DM soliton from the given spectral data. For sufficiently long legs, the leading term of DM soliton is found in explicit form by asymptotic undressing of the Riemann–Hilbert problem. The analytic results are compared with numerical simulations.     

A note on the Ambrosetti–Rabinowitz condition for an elliptic system

In this note, we study the existence and multiplicity of solutions for a system of coupled elliptic equations. We introduce a revised Ambrosetti–Rabinowitz condition, and show that the system has a nontrivial solution or even infinitely many solutions.     

Riemann–Hilbert formulation for the KdV equation on a finite interval

The initial-boundary value problem for the KdV equation on a finite interval is analyzed in terms of a singular Riemann–Hilbert problem for a matrix-valued function in the complex k-plane which depends explicitly on the space–time variables. For an appropriate set of initial and boundary data, we derive the k-dependent “spectral functions” which guarantee the uniqueness of Riemann–Hilbert problem's solution. The latter determines a solution of the initial-boundary value problem for KdV equation, for which an integral representation is given. To cite this article: I. Hitzazis, D. Tsoubelis, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

The initial-boundary value for the combined Schrödinger and Gerdjikov–Ivanov equation on the half-line via the Riemann–Hilbert approach

Theoretical and Mathematical Physics - The Fokas method is used to study the initial-boundary value problem for the combined Schrödinger and Gerdjikov–Ivanov equation on the half-line....