Bounding the number of zeros of certain Abelian integrals

AbstractIn this paper we prove a criterion that provides an easy sufficient condition in order for any nontrivial linear combination of n Abelian integrals to have at most n+k−1 zeros counted with multiplicities. This condition involves the functions in the integrand of the Abelian integrals and it can be checked, in many cases, in a purely algebraic way

Note on the Markus–Yamabe conjecture for gradient dynamical systems

AbstractLet v:Rn→Rn be a C1 vector field which has a singular point O and its linearization is asymptotically stable at every point of Rn. We say that the vector field v satisfies the Markus–Yamabe conjecture if the critical point O is a global attractor of the dynamical system x˙=v(x). In this note we prove that if v is a gradient vector field, i.e. v=∇f (f∈C2), then the basin of attraction of the critical point O is the whole Rn, thus implying the Markus–Yamabe conjecture for this class of vector fields. An analogous result for discrete dynamical systems of the form xm+1=∇f(xm) is proved

Entropy stability and Milnor-Thurston invariants for Bowen-Series-like maps

We define a family of discontinuous maps on the circle, called Bowen-Series-like maps, for geometric presentations of surface groups. The family has $2N$ parameters, where $2N$ is the number of generators of the presentation. We prove that all maps in the family have the same topological entropy, which coincides with the volume entropy of the group presentation. This approach allows a simple algorithmic computation of the volume entropy from the presentation only, using the Milnor-Thurston theory for one dimensional maps

Substrate-selective repair and restart of replication forks by DNA translocases

Stalled replication forks are sources of genetic instability. Multiple fork-remodeling enzymes are recruited to stalled forks, but how they work to promote fork restart is poorly understood. By combining ensemble biochemical assays and single-molecule studies with magnetic tweezers, we show that SMARCAL1 branch migration and DNA-annealing activities are directed by the single-stranded DNA-binding protein RPA to selectively regress stalled replication forks caused by blockage to the leading-strand polymerase and to restore normal replication forks with a lagging-strand gap. We unveil the molecular mechanisms by which RPA enforces SMARCAL1 substrate preference. E. coli RecG acts similarly to SMARCAL1 in the presence of E. coli SSB, whereas the highly related human protein ZRANB3 has different substrate preferences. Our findings identify the important substrates of SMARCAL1 in fork repair, suggest that RecG and SMARCAL1 are functional orthologs, and provide a comprehensive model of fork repair by these DNA translocases

Some properties of the k-dimensional Lyness' map

This paper is devoted to study some properties of the k-dimensional Lyness' map. Our main result presentes a rational vector field that gives a Lie symmetry for F. This vector field is used, for k less or equal to 5 to give information about the nature of the invariant sets under F. When k is odd, we also present a new (as far as we know) first integral for F^2 which allows to deduce in a very simple way several properties of the dynamical system generated by F. In particular for this case we prove that, except on a given codimension one algebraic set, none of the positive initial conditions can be a periodic point of odd period.Comment: 22 pages; 3 figure

On the number of polynomial solutions of Bernoulli and Abel polynomial differential equations

In this paper we determine the maximum number of polynomial solutions of Bernoulli differential equations and of some integrable polynomial Abel differential equations. As far as we know, the tools used to prove our results have not been utilized before for studying this type of questions. We show that the addressed problems can be reduced to know the number of polynomial solutions of a related polynomial equation of arbitrary degree. Then we approach to these equations either applying several tools developed to study extended Fermat problems for polynomial equations, or reducing the question to the computation of the genus of some associated planar algebraic curves