Related papers: Monodromic nilpotent singular points with odd Andr…
We explore the convergence/divergence of the normal form for a singularity of a vector field on $\C^n$ with nilpotent linear part. We show that a Gevrey-$\alpha$ vector field $X$ with a nilpotent linear part can be reduced to a normal form…
This work establishes a strong uniqueness property for a class of planar locally integrable vector fields. A result on pointwise convergence to the boundary value is also proved for bounded solutions.
The aim of this paper is to investigate two classical problems related to nilpotent center conditions and bifurcation of limit cycles in switching polynomial systems. Due to the difficulty in calculating the Lyapunov constants of switching…
Classifications of irreducible components of the set of polynomial differential equations with a fixed degree and with at least one center singularity lead to some other new problems on Picard-Lefschetz theory and Brieskorn modules of…
We consider the problem of minimizing variational integrals defined on \cc{nonlinear} Sobolev spaces of competitors taking values into the sphere. The main novelty is that the underlying energy features a non-uniformly elliptic integrand…
We consider manifolds with isolated singularities, i.e., topological spaces which are manifolds (say, $C^\infty$--) outside discrete subsets (sets of singular points). For (germs of) manifolds with, so called, cone--like singularities, a…
It is known that for $n \geq 3$ centres and positive energies the $n$-centre problem of celestial mechanics leads to a flow with a strange repellor and positive topological entropy. Here we consider the energies above some threshold and…
In planar analytic vector fields, a monodromic singularity can be distinguished between a focus or a center by means of the Lyapunov coefficients, which are given in terms of the power series coefficients of the first-return map defined…
We study the non-selfadjoint Dirac system on a finite interval having non-integrable regular singularities in interior points with additional matching conditions at these points. Properties of spectral characteristics are established, and…
Let $p$ be an odd prime number. We show that the modular isomorphism problem has a positive answer for finite $p$-groups whose center has index $p^3$, which is a strong contrast to the analogous situation for $p = 2$.
In this paper, we study the isomonodromy systems associated with the Garnier systems of type 9/2 and type 5/2+3/2. We show that the both of isomonodromy systems admit the singularity reduction (restriction to a movable pole), and the…
We give a precise description of how the class group of a number field measures the failure of unique factorization in its ring of integers. Specifically, following ideas of Kummer, we determine the structure of all irreducible…
Consider the problem, usually called the P\'olya-Chebotarev problem, of finding a continuum in the complex plane including some given points such that the logarithmic capacity of this continuum is minimal. We prove that each connected…
We first generalize a classical iteration formula for one variable holomorphic mappings to a formula for higher dimensional holomorphic mappings. Then, as an application, we give a short and intuitive proof of a classical theorem, due to H.…
Theorems on the existence of vector fields with given sets of Indexes of isolated Singular points are proved for the cases of closed manifolds, pairs of manifolds, manifolds with boundary, and gradient fields. It is proved that, on a…
The integrability (solvability via an associated single-valued linear problem) of a differential equation is closely related to the singularity structure of its solutions. In particular, there is strong evidence that all integrable…
We study movable singularities of Garnier systems using the connection of the latter with isomonodromic deformations of Fuchsian systems. Questions on the existence of solutions for some inverse monodromy problems are also considered.
The longstanding nonnegative inverse eigenvalue problem (NIEP) is to determine which multisets of complex numbers occur as the spectrum of an entry-wise nonnegative matrix. Although there are some well-known necessary conditions, a solution…
We study the conjecture of Jarque and Villadelprat stating that every center of a planar polynomial Hamiltonian system of even degree is nonisochronous. This conjecture is proved for quadratic and quartic systems. Using the correction of a…
Under special conditions the Painlev\'e V equation has more than one rational solution solving it with the same parameters. In the setting of formalism that identifies points on orbits of the fundamental shift operators of $A^{(1)}_{3}$…