Integrability of a linear center perturbed by a fourth degree homogeneous polynomial

In this work we study the integrability of a two-dimensional autonomous system in the plane with linear part of center type and non-linear part given by homogeneous polynomials of fourth degree. We give sufficient conditions for integrability in polar coordinates. Finally we establish a conjecture about the independence of the two classes of parameters which appear in the system; if this conjecture is true the integrable cases found will be the only possible ones

Integrability, degenerate centers, and limit cycles for a class of polynomial differential systems

AbstractWe consider the class of polynomial differential equations x˙ Pn(x,y)+Pn+1(x,y)+Pn+2(x,y), y˙=Qn(x,y)+Qn+1(x,y)+Qn+2(x,y), for n ≥ 1 and where Pi and Qi are homogeneous polynomials of degree i These systems have a linearly zero singular point at the origin if n > 2. Inside this class, we identify a new subclass of Darboux integrable systems, and some of them having a degenerate center, i.e., a center with linear part identically zero. Moreover, under additional conditions such Darboux integrable systems can have at most one limit cycle. We provide the explicit expression of this limit cycle

Integrability of a linear center perturbed by a fifth degree homogeneous polynomial

In this work we study the integrability of two-dimensional autonomous system in the plane with linear part of center type and non-linear part given by homogeneous polynomials of fifth degree. We give a simple characterisation for the integrable cases in polar coordinates. Finally we formulate a conjecture about the independence of the two classes of parameters which appear on the system; if this conjecture is true the integrable cases found will be the only possible ones

The null divergence factor

Let $(P, Q)$ be a $C^{1}$ vector field defined in a open subset $U \subset R^{2}$. We call a null divergence factor a $C^{1}$ solution $V(x, y)$ of the equation $P \frac{\partial V}{\partial x} + Q \frac{\partial V}{\partial y} = \left(\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y}\right) \, V$. In previous works it has been shown that this function plays a fundamental role in the problem of the center and in the determination of the limit cycles. In this paper we show how to construct systems with a given null divergence factor. The method presented in this paper is a generalization of the classical Darboux method to generate integrable systems

A Smirnov-Bickel-Rosenblatt theorem for compactly-supported wavelets

In nonparametric statistical problems, we wish to find an estimator of an unknown function f. We can split its error into bias and variance terms; Smirnov, Bickel and Rosenblatt have shown that, for a histogram or kernel estimate, the supremum norm of the variance term is asymptotically distributed as a Gumbel random variable. In the following, we prove a version of this result for estimators using compactly-supported wavelets, a popular tool in nonparametric statistics. Our result relies on an assumption on the nature of the wavelet, which must be verified by provably-good numerical approximations. We verify our assumption for Daubechies wavelets and symlets, with N = 6, ..., 20 vanishing moments; larger values of N, and other wavelet bases, are easily checked, and we conjecture that our assumption holds also in those cases

Exponential and moment inequalities for U-statistics

A Bernstein-type exponential inequality for (generalized) canonical U-statistics of order 2 is obtained and the Rosenthal and Hoffmann-J{\o}rgensen inequalities for sums of independent random variables are extended to (generalized) U-statistics of any order whose kernels are either nonnegative or canonicalComment: 22 page

Analysis of the production of ethylene in 2,4-D resistant and sensitive biotypes of "Papaver rhoeas L."

Hasta el momento no se ha realizado ningún estudio en España a fin de indagar en los mecanismos de resistencia de “Papaver Rhoeas” al 2,4-D. En otros trabajos se ha demostrado como la producción de etileno está involucrada en la respuesta resistente de ciertas malas hierbas a auxinas sintéticas. En el presente estudio se ha observado que las plantas sensibles de amapola producen más etileno que las resistentes después de la aplicación de 2,4-D. Estos resultados podrían ayudar a comprender de mejor manera como ciertos biotipos son capaces de resistir a este producto.So far there has been none study in Spain to investigate the 2,4-D resistance mechanism of Papaver rhoeas. Other studies have shown how ethylene production is involved in the response of certain resistant weeds when they are sprayed with synthetic auxins. In the present study it was observed that sensitive corn poppy plants produce more ethylene than resistant one after 2,4-D application. These findings could help to understanding how certain biotypes are able to resist this product

Fracturas abiertas de pelvis: Seguimiento a largo plazo

Presentamos una serie de fracturas abiertas de pelvis atendidas en nuestro centro. Para su estudio hemos revisado las historias clínicas de los pacientes, evaluando los resultados por medio de entrevista telefónica. La proporción de este tipo de fracturas, especialmente grave, con respecto a todas las fracturas de pelvis ingresadas es algo más del 2%. Con especial atención analizamos las lesiones asociadas, las complicaciones y la necesidad de transfusión. Así mismo presentamos el tratamiento efectuado. La mortalidad de nuestra serie es del 6.66%. Parece evidente la necesidad de un tratamiento agresivo, que incluya la temprana estabilización de la fractura, así como la colaboración con otros especialistas quirúrgicos.A series of pelvis fractures were retrospectively studied using the clinical records and telephone interview. This specially severe fractures account for a 2% of all pelvis fractures admitted to our hospital. We emphasised our review on associated injuries, complications and indication for transfusion. The treatment indicated is presented. The mortality was 6,7%. It is evident the need of an aggressive treatment including early fracture stabilization with a multidisciplinary approach

The Pioneer anomaly and the holographic scenario

In this paper we discuss the recently obtained relation between the Verlinde's holographic model and the first phenomenological Modified Newtonian dynamics. This gives also a promising possible explanation to the Pioneer anomaly.Comment: 5 pages, Accepted for publication in Astrophysics & Space Scienc

Analytic integrability of certain resonant saddle

We provide sufficient conditions of analytic integrability for a family of planar differential system with a p : −q resonant saddle at the origin. The conditions are then shown to be necessary in several finite dimensional families where the calculations can be performed explicitly. It is conjectured that the conditions are in fact complete for all families. Though the form of the equations is quite simple, their study merits further attention as they exhibit an interesting dichotomy between finite and arbitrary dimensional components of the center variety.The second author is partially supported by a MINECO/ FEDER grant number MTM2017-84383-P and an AGAUR (Generalitat de Catalunya) grant number 2017SGR-1276