On the dynamics of the Szekeres system

The gravitational Szekeres differential system is completely integrable with two rational first integrals and an additional analytical first integral. We describe the dynamics of the Szekeres system when one of these two rational first integrals is negative, showing that all the orbits come from the infinity of ${\mathbb{R}}^4$ and go to infinity.     

Hjelmslev quadrilateral central configurations

A Hjelmslev quadrilateral is a quadrilateral with two right angles at opposite vertices. Using mutual distances as coordinates, we show that any four-body central configuration forming a Hjelmslev quadrilateral must be a right kite configuration.     

Limit Cycles of Piecewise Smooth Differential Equations on Two Dimensional Torus

In this paper we study the limit cycles of some classes of piecewise smooth vector fields defined in the two dimensional torus. The piecewise smooth vector fields that we consider are composed by linear, Ricatti with constant coefficients and perturbations of these one, which are given in (3). Considering these piecewise smooth vector fields we characterize the global dynamics, studying the upper bound of number of limit cycles, the existence of non-trivial recurrence and a continuum of periodic orbits. We also present a family of piecewise smooth vector fields that posses a finite number of fold points and, for this family we prove that for any 2k number of limit cycles there exists a piecewise smooth vector fields in this family that presents k number of limit cycles and prove that some classes of piecewise smooth vector fields presents a non-trivial recurrence or a continuum of periodic orbits.     

Topological entropy of continuous self-maps on closed surfaces

The objective of this work is to present sufficient conditions for having positive topological entropy for continuous self-maps defined on a closed surface by using the action of this map on the homological groups of the closed surface.     

Null decomposition of trees

Let $T$ be a tree. We show that the null space of the adjacency matrix of $T$ has relevant information about the structure of $T$. We introduce the Null Decomposition of trees, which is a decomposition into two different types of trees: N-trees and S-trees. N-trees are the trees that have a unique maximum (perfect) matching. S-trees are the trees with a unique maximum independent set. We obtain formulas for the independence number and the matching number of a tree using this decomposition. We also show how the number of maximum matchings and the number of maximum independent sets in a tree are related to its null decomposition.     

Nucleophilic β‐Carbon Activation of Propionic Acid as a 3‐Carbon Synthon by Carbene Organocatalysis

Direct β‐carbon activation of propionic acid (C₂H₅CO₂H) by carbene organocatalysis has been developed. This activation affords the smallest azolium homoenolate intermediate (without any substituent) as a 3‐carbon nucleophile for enantioselective reactions. Propionic acid is an excellent raw material because it is cheap, stable, and safe. This approach provides a much better solution to azolium homoenolate synthesis than the previously established use of acrolein (enal without any substituent), which is expensive, unstable, and toxic.     

Centers for the Kukles homogeneous systems with even degree

For the polynomial differential system $\dot{x}=-y$, $\dot{y}=x +Q_n(x,y)$, where $Q_n(x,y)$ is a homogeneous polynomial of degree $n$ there are the following two conjectures done in 1999. (1) Is it true that the previous system for $n \ge 2$ has a center at the origin if and only if its vector field is symmetric about one of the coordinate axes? (2) Is it true that the origin is an isochronous center of the previous system with the exception of the linear center only if the system has even degree? We give a step forward in the direction of proving both conjectures for all $n$ even. More precisely, we prove both conjectures in the case $n = 4$ and for $n\ge 6$ even under the assumption that if the system has a center or an isochronous center at the origin, then it is symmetric with respect to one of the coordinate axes, or it has a local analytic first integral which is continuous in the parameters of the system in a neighborhood of zero in the parameters space. The case of $n$ odd was studied in [8].     

Alkaloids Analysis of Habranthus cardenasianus (Amaryllidaceae), Anti-Cholinesterase Activity and Biomass Production by Propagation Strategies

Plants in the Amaryllidaceae family synthesize a diversity of bioactive alkaloids. Some of these plant species are not abundant and have a low natural multiplication rate. The aims of this work were the alkaloids analysis of a Habranthus cardenasianus bulbs extract, the evaluation of its inhibitory activity against cholinesterases, and to test several propagation strategies for biomass production. Eleven compounds were characterized by GC-MS in the alkaloid extract, which showed a relatively high proportion of tazettine. The known alkaloids tazettine, haemanthamine, and the epimer mixture haemanthidine/6-epi-haemanthidine were isolated and identified by spectroscopic methods. Inhibitory cholinesterases activity was not detected. Three forms of propagation were performed: bulb propagation from seed, cut-induced bulb division, and micropropagated bulbs. Finally, different imbibition and post-collection times were evaluated in seed germination assays. The best propagation method was cut-induced bulb division with longitudinal cuts into quarters (T1) while the best conditions for seed germination were 0-day of post-collection and two days of imbibition. The alkaloids analyses of the H. cardenasianus bulbs showed that they are a source of anti-tumoral alkaloids, especially pretazettine (tazettine) and T1 is a sustainable strategy for its propagation and domestication to produce bioactive alkaloids.     

Regularization of Discontinuous Vector Fields on $${\mathbb{R}^3}$$ via Singular Perturbation

Singular perturbations problems in dimension three which are approximations of discontinuous vector fields are studied in this paper. The main result states that the regularization process developed by Sotomayor and Teixeira produces a singular problem for which the discontinuous set is a center manifold. Moreover, the definition of sliding vector field coincides with the reduced problem of the corresponding singular problem for a class of vector fields.