Related papers: Monodromy and Dulac's Problem for Piecewise analyt…

Xu introduced a system of partial differential equations to investigate singular vectors in the Verma modules of highest weight $\lambda$ over $\mathfrak{sl}(n,\mathbb{C})$. He proved that the solution space of this system in the space of…

Although the standard generally-covariant Dirac equation is unique in a topologically simple spacetime, it has been shown that it leads to non-uniqueness problems for the Hamiltonian and energy operators, including the non-uniqueness of the…

This paper investigates the geometry of smooth canonically polarized surfaces defined over a field of positive characteristic which have a nontrivial global vector field, and the implications that the existence of such surfaces has in the…

Consider a smooth action $\mathbf G\times M \rightarrow M$ of a compact connected Lie group $\mathbf G$ on a connected manifold $M$. Assume the existence of a point of $M$ whose isotropy group has a single element (free point). Then we…

The paper \cite{BM} proposed a construction of a twisted representation of the lattice vertex algebra corresponding to the Milnor lattice of a simple singularity. The main difficulty in extending the above construction to an arbitrary…

Let (V,0) be a germ of a complete intersection variety in \CC^{n+k}, n>0, having an isolated singularity at 0 and X be the germ of a holomorphic vector field on \CC^{n+k} tangent to V and having on V an isolated zero at 0. We show that in…

The vector transform operators are investigated; these operators are used at the solution of boundary value problems in piecewise homogeneous spherically symmetric areas. In particular, examples of transformation operators for vector…

The coexistence of singularities and regular orbits in chain transitive sets has been a major obstacle in understanding the hyperbolic/partial hyperbolic nature of robust dynamics. Notably, the vector fields with all periodic orbits…

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.

One of the various versions of the classical Lyapunov-Poincar\'e center theorem states that a nondegenerate real analytic center type planar vector field singularity admits an analytic first integral. In a more proof of this result, R.…

Given a graph $G$, an integer $k\geq 0$, and a non-negative integral function $f:V(G) \rightarrow \mathcal{N}$, the Vector Domination problem asks whether a set $S$ of vertices, of cardinality $k$ or less, exists in $G$ so that every vertex…

Given a lattice L in Z^m and a subset A of R^m, we say that a point in A is lonely if it is not equivalent modulo L to another point of A. We are interested in identifying lonely points for specific choices of L when A is a dilated standard…

There are some generalizations of the classical Eisenbud-Levine-Khimshashvili formula for the index of a singular point of an analytic vector field on $R^n$ for vector fields on singular varieties. We offer an alternative approach based on…

The singular complex analytic vector fields $X$ on the Riemann sphere $\widehat{\mathbb C}_z$ belonging to the family ${\mathscr E}(r,d)=\left\{ X(z)=\frac{1}{P(z)} e^{E(z)}\frac{\partial }{\partial z}\ \Big\vert \ P,…

When a singular point of a vector field passes through resonance, a formal invariant cone appears. In the seventies, Pyartli proved that for $(-1,1)$-resonance the cone is in fact analytic and is the degeneration of a family of invariant…

Let ${\mathbb F}$ be an arbitrary field. We consider a commutative, non-associative, $4$-dimensional algebra ${\mathfrak M}$ of the rock, the paper and the scissors with unit over ${\mathbb F}$ and we prove that the image over ${\mathfrak…

We describe all possible topological structures of codimension one gradient vector fields on the shpere with at most ten singular points. To describe structures, we use a graph whose edges are one-dimensional stable manifolds. The…

In this paper we study the equation $Lu=f$, where $L$ is a $\C$-valued vector field in $\R^2$ with a homogeneous singularity. The properties of the solutions are linked to the number theoretic properties of a pair of complex numbers…

For a complex polynomial or analytic function f, one has been studying intensively its so-called local zeta functions or complex powers; these are integrals of |f|^{2s}w considered as functions in s, where the w are differential forms with…

In this article it is proved that an analytical planar vector field with a non-degenerate center at $(0,0)$ is analytically conjugate, in a neighborhood of $(0,0)$, to a Hamiltonian vector field of the form $y\frac{\partial}{\partial…