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On movable singularities of self-similar solutions of semilinear wave equations

Radosław A. Kycia (2012)

Banach Center Publications

In this paper we analyze movable singularities of the solutions of the equation for self-similar profiles resulting from semilinear wave equation. We study local analytic solutions around two fixed singularity points of this equation- ρ = 0 and ρ = 1. The movable singularities of local analytic solutions at the origin will be connected with those of the Lane-Emden equation. The function describing approximately their position on the complex plane will be derived. For ρ > 1 some topological considerations...

On the exact WKB analysis of microdifferential operators of WKB type

Takashi Aoki, Takahiro Kawai, Tatsuya Koike, Yoshitsugu Takei (2004)

Annales de l’institut Fourier

We first introduce the notion of microdifferential operators of WKB type and then develop their exact WKB analysis using microlocal analysis; a recursive way of constructing a WKB solution for such an operator is given through the symbol calculus of microdifferential operators, and their local structure near their turning points is discussed by a Weierstrass-type division theorem for such operators. A detailed study of the Berk-Book equation is given in Appendix.

Quasialgebraic functions

G. Binyamini, D. Novikov, S. Yakovenko (2011)

Banach Center Publications

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).

Semicompleteness of homogeneous quadratic vector fields

Adolfo Guillot (2006)

Annales de l’institut Fourier

We investigate the quadratic homogeneous holomorphic vector fields on  C n that are semicomplete, this is, those whose solutions are single-valued in their maximal definition domain. To a generic quadratic vector field we rationally associate some complex numbers that turn out to be integers in the semicomplete case, thus showing that the linear equivalence classes of semicomplete vector fields are contained in some sort of lattice in the space of linear equivalence classes of quadratic ones. We prove...

Singular sets of holonomy maps for algebraic foliations

Gabriel Calsamiglia, Bertrand Deroin, Sidney Frankel, Adolfo Guillot (2013)

Journal of the European Mathematical Society

In this article we investigate the natural domain of definition of a holonomy map associated to a singular holomorphic foliation of the complex projective plane. We prove that germs of holonomy between algebraic curves can have large sets of singularities for the analytic continuation. In the Riccati context we provide examples with natural boundary and maximal sets of singularities. In the generic case we provide examples having at least a Cantor set of singularities and even a nonempty open set...

Some addition to the generalized Riemann-Hilbert problem

R.R. Gontsov, I.V. Vyugin (2009)

Annales de la faculté des sciences de Toulouse Mathématiques

We consider the generalized Riemann-Hilbert problem for linear differential equations with irregular singularities. If one weakens the conditions by allowing one of the Poincaré ranks to be non-minimal, the problem is known to have a solution. In this article we give a bound for the possibly non-minimal Poincaré rank. We also give a bound for the number of apparent singularities of a scalar equation with prescribed generalized monodromy data.

Some Examples of Rigid Representations

Kostov, Vladimir (2000)

Serdica Mathematical Journal

*Research partially supported by INTAS grant 97-1644.Consider the Deligne-Simpson problem: give necessary and sufficient conditions for the choice of the conjugacy classes Cj ⊂ GL(n,C) (resp. cj ⊂ gl(n,C)) so that there exist irreducible (p+1)-tuples of matrices Mj ∈ Cj (resp. Aj ∈ cj) satisfying the equality M1 . . .Mp+1 = I (resp. A1+. . .+Ap+1 = 0). The matrices Mj and Aj are interpreted as monodromy operators and as matrices-residua of fuchsian systems on Riemann’s sphere. We give new examples...

Tempered solutions of 𝒟 -modules on complex curves and formal invariants

Giovanni Morando (2009)

Annales de l’institut Fourier

Let X be a complex analytic curve. In this paper we prove that the subanalytic sheaf of tempered holomorphic solutions of 𝒟 -modules on X induces a fully faithful functor on a subcategory of germs of formal holonomic 𝒟 -modules. Further, given a germ ℳ of holonomic 𝒟 -module, we obtain some results linking the subanalytic sheaf of tempered solutions of ℳ and the classical formal and analytic invariants of ℳ .

The number of absorbed individuals in branching brownian motion with a barrier

Pascal Maillard (2013)

Annales de l'I.H.P. Probabilités et statistiques

We study supercritical branching Brownian motion on the real line starting at the origin and with constant drift c . At the point x g t ; 0 , we add an absorbing barrier, i.e. individuals touching the barrier are instantly killed without producing offspring. It is known that there is a critical drift c 0 , such that this process becomes extinct almost surely if and only if c ≥ c 0 . In this case, if Z x denotes the number of individuals absorbed at the barrier, we give an asymptotic for P ( Z x = n ) as n goes to infinity. If c = c 0 ...

Un exemple de feuilletage modulaire déduit d’une solution algébrique de l’équation de Painlevé VI

Gaël Cousin (2014)

Annales de l’institut Fourier

On peut construire facilement des exemples de connexions plates de rang 2 sur ℙ 2 comme tirés en arrière de connexions sur ℙ 1 . On donne un exemple de connexion qui ne peut être obtenue de cette manière. Cet exemple est construit à partir d’une solution algébrique de l’équation de Painlevé VI. On en déduit un feuilletage modulaire. La preuve de ce fait repose sur la classification des feuilletages sur les surfaces projectives par leurs dimensions de Kodaira, fruit du travail de Brunella, McQuillan et...

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