Picard Sets for Entire Functions.
Polynomial bounds for the oscillation of solutions of Fuchsian systems
We study the problem of placing effective upper bounds for the number of zeroes of solutions of Fuchsian systems on the Riemann sphere. The principal result is an explicit (non-uniform) upper bound, polynomially growing on the frontier of the class of Fuchsian systems of a given dimension having singular points. As a function of , this bound turns out to be double exponential in the precise sense explained in the paper.As a corollary, we obtain a solution of the so-called restricted infinitesimal...
Polynomial solutions of a generalization of the first Painlevé differential equation.
Polynomial solutions to the Hele--Shaw problem.
Positive oriented periodic solutions of the first-order complex ODE with polynomial nonlinear part.
Prolongement des solutions d’une équation différentielle -adique
Properties of non-hermitian quantum field theories
In this paper I discuss quantum systems whose Hamiltonians are non-Hermitian but whose energy levels are all real and positive. Such theories are required to be symmetric under , but not symmetric under and separately. Recently, quantum mechanical systems having such properties have been investigated in detail. In this paper I extend the results to quantum field theories. Among the systems that I discuss are and theories. These theories all have unexpected and remarkable properties. I discuss...
Propriétés algébriques des opérateurs d'Airy de petit ordre
Pure imaginary solutions of the second Painlevé equation.
Quadratic transformations for the third and fifth Painlevé equations.
Quantum Painlevé equations: from continuous to discrete.
Quasialgebraic functions
We introduce and discuss a new class of (multivalued analytic) transcendental functions which still share with algebraic functions the property that the number of their isolated zeros can be explicitly counted. On the other hand, this class is sufficiently rich to include all periods (integral of rational forms over algebraic cycles).
Quiver varieties with multiplicities, Weyl groups of non-symmetric Kac-Moody algebras, and Painlevé equations.
Rational solutions of the Sasano system of type .
Recent work on differential Galois theory
Recherches analytiques sur la surface du troisième ordre qui est la réciproque de la surface de Steiner
Reducible functional differential equations.
Regular coordinates and reduction of deformation equations for Fuchsian systems
For a Fuchsian system , (F) being distinct points in ℂ and , the number α of accessory parameters is determined by the spectral types , where . We call the set of α parameters a regular coordinate if all entries of the are rational functions in z. It is not yet known that, for any irreducibly realizable set of spectral types, a regular coordinate does exist. In this paper we study a process of obtaining a new regular coordinate from a given one by a coalescence of eigenvalues of the matrices...
Relations de Fuchs pour les systèmes différentiels réguliers
Dans cet article, nous montrons que la notion analytique d’exposants développée par Levelt pour les systèmes différentiels linéaires en une singularité régulière s’interprète algébriquement en termes d’invariants de réseaux, relatifs à un réseau stable maximal que nous appelons « réseau de Levelt ». Nous obtenons en particulier un encadrement pour la somme des exposants des systèmes n’ayant que des singularités régulières sur ).
Remarks on inverse of matrix polynomials
Analysis of a non-classically damped engineering structure, which is subjected to an external excitation, leads to the solution of a system of second order ordinary differential equations. Although there exists a large variety of powerful numerical methods to accomplish this task, in some cases it is convenient to formulate the explicit inversion of the respective quadratic fundamental system. The presented contribution uses and extends concepts in matrix polynomial theory and proposes an implementation...