Some Applications of the Extended Bendixson-Dulac Theorem

During the last years the authors have studied the number of limit cycles of several families of planar vector fields. The common tool has been the use of an extended version of the celebrated Bendixson-Dulac Theorem. The aim of this work is to present an unified approach of some of these results, together with their corresponding proofs. We also provide several applications.Comment: 19 pages, 3 figure

Periodic solutions with nonconstant sign in Abel equations of the second kind

The study of periodic solutions with constant sign in the Abel equation of the second kind can be made through the equation of the first kind. This is because the situation is equivalent under the transformation $x\mapsto x^{-1}$, and there are many results available in the literature for the first kind equation. However, the equivalence breaks down when one seeks for solutions with nonconstant sign. This note is devoted to periodic solutions with nonconstant sign in Abel equations of the second kind. Specifically, we obtain sufficient conditions to ensure the existence of a periodic solution that shares the zeros of the leading coefficient of the Abel equation. Uniqueness and stability features of such solutions are also studied.Comment: 10 page

Ciència i tecnologia de la carn i dels productes carnis: passat i present

La carn ha estat, des de sempre, un aliment clau en la dieta dels éssers humans per les seves propietats nutritives i les seves característiques sensorials. Els productes carnis han estat una forma de conservar la carn, que tradicionalment es feia per l'experiència acumulada i que actualment es basa en coneixements científics. En el decurs del segle xx es va produir un avanç extraordinari en la ciència de la carn, en la gestió de la seguretat i qualitat de la carn i productes carnis, en la disseminació dels coneixements generats i en la innovació en la indústria càrnia. En un futur proper es preveu que la carn seguirà essent un element clau en la dieta humana i que l'aplicació de les noves tecnologies tindran un impacte important en els processos de transformació. No obstant això, caldrà esmerçar esforços per minimitzar l'impacte en el medi ambient pel que fa a la cria i sacrifici dels animals i la transformació de la carn en productes carnis per aconseguir que tot el procés sigui més sostenible.La carne ha sido, desde siempre, un alimento clave en la dieta de los seres humanos por sus propiedades nutritivas y sus características sensoriales. Los productos cárnicos han sido una forma de conservar la carne, que tradicionalmente se hacía por la experiencia acumulada y que actualmente se basa en conocimientos científicos. En el transcurso del siglo xx se produjo un avance extraordinario en la ciencia de la carne, en la gestión de la seguridad y calidad de la carne y productos cárnicos, en la diseminación de los conocimientos generados y en la innovación en la industria cárnica. En un futuro próximo se prevé que la carne seguirá siendo un elemento clave en la dieta humana y que la aplicación de las nuevas tecnologías tendrá un impacto importante en los procesos de transformación. Sin embargo, habrá que invertir esfuerzos para minimizar el impacto en el medio ambiente en relación con la cría y sacrificio de los animales y la transformación de la carne en productos cárnicos para conseguir que todo el proceso sea más sostenible

Ciència i tecnologia de la carn i dels productes carnis: passat i present

La carne ha sido, desde siempre, un alimento clave en la dieta de los seres humanos por sus propiedades nutritivas y sus características sensoriales. Los productos cárnicos han sido una forma de conservar la carne, que tradicionalmente se hacía por la experiencia acumulada y que actualmente se basa en conocimientos científicos. En el transcurso del siglo XX se produjo un avance extraordinario en la ciencia de la carne, en la gestión de la seguridad y calidad de la carne y productos cárnicos, en la diseminación de los conocimientos generados y en la innovación en la industria cárnica. En un futuro próximo se prevé que la carne seguirá siendo un elemento clave en la dieta humana y que la aplicación de las nuevas tecnologías tendrá un impacto importante en los procesos de transformación. Sin embargo, habrá que invertir esfuerzos para minimizar el impacto en el medio ambiente en relación con la cría y sacrificio de los animales y la transformación de la carne en productos cárnicos para conseguir que todo el proceso sea más sostenible.La carn ha estat, des de sempre, un aliment clau en la dieta dels éssers humans per les seves propietats nutritives i les seves característiques sensorials. Els productes carnis han estat una forma de conservar la carn, que tradicionalment es feia per l'experiència acumulada i que actualment es basa en coneixements científics. En el decurs del segle XX es va produir un avanç extraordinari en la ciència de la carn, en la gestió de la seguretat i qualitat de la carn i productes carnis, en la disseminació dels coneixements generats i en la innovació en la indústria càrnia. En un futur proper es preveu que la carn seguirà essent un element clau en la dieta humana i que l'aplicació de les noves tecnologies tindran un impacte important en els processos de transformació. No obstant això, caldrà esmerçar esforços per minimitzar l'impacte en el medi ambient pel que fa a la cria i sacrifici dels animals i la transformació de la carn en productes carnis per aconseguir que tot el procés sigui més sostenible

Global periodicity conditions for maps and recurrences via Normal Forms

We face the problem of characterizing the periodic cases in parametric families of (real or complex) rational diffeomorphisms having a fixed point. Our approach relies on the Normal Form Theory, to obtain necessary conditions for the existence of a formal linearization of the map, and on the introduction of a suitable rational parametrization of the parameters of the family. Using these tools we can find a finite set of values p for which the map can be p-periodic, reducing the problem of finding the parameters for which the periodic cases appear to simple computations. We apply our results to several two and three dimensional classes of polynomial or rational maps. In particular we find the global periodic cases for several Lyness type recurrences.Comment: 25 page

Some results on homoclinic and heteroclinic connections in planar systems

Consider a family of planar systems depending on two parameters $(n,b)$ and having at most one limit cycle. Assume that the limit cycle disappears at some homoclinic (or heteroclinic) connection when $\Phi(n,b)=0.$ We present a method that allows to obtain a sequence of explicit algebraic lower and upper bounds for the bifurcation set ${\Phi(n,b)=0}.$ The method is applied to two quadratic families, one of them is the well-known Bogdanov-Takens system. One of the results that we obtain for this system is the bifurcation curve for small values of $n$, given by $b=\frac5 7 n^{1/2}+{72/2401}n- {30024/45294865}n^{3/2}- {2352961656/11108339166925} n^2+O(n^{5/2})$. We obtain the new three terms from purely algebraic calculations, without evaluating Melnikov functions

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

Remotely Piloted Aircraft Systems and a Wireless Sensors Network for Radiological Accidents

In critical radiological situations, the real time information that we could get from the disaster area becomes of great importance. However, communication systems could be affected after a radiological accident. The proposed network in this research consists of distributed sensors in charge of collecting radiological data and ground vehicles that are sent to the nuclear plant at the moment of the accident to sense environmental and radiological information. Afterwards, data would be analyzed in the control center. Collected data by sensors and ground vehicles would be delivered to a control center using Remotely Piloted Aircraft Systems (RPAS) as a message carrier. We analyze the pairwise contacts, as well as visiting times, data collection, capacity of the links, size of the transmission window of the sensors, and so forth. All this calculus was made analytically and compared via network simulations

Worldwide trends in diabetes since 1980: a pooled analysis of 751 population-based studies with 4.4 million participants

BACKGROUND: One of the global targets for non-communicable diseases is to halt, by 2025, the rise in the age-standardised adult prevalence of diabetes at its 2010 levels. We aimed to estimate worldwide trends in diabetes, how likely it is for countries to achieve the global target, and how changes in prevalence, together with population growth and ageing, are affecting the number of adults with diabetes. METHODS: We pooled data from population-based studies that had collected data on diabetes through measurement of its biomarkers. We used a Bayesian hierarchical model to estimate trends in diabetes prevalence—defined as fasting plasma glucose of 7·0 mmol/L or higher, or history of diagnosis with diabetes, or use of insulin or oral hypoglycaemic drugs—in 200 countries and territories in 21 regions, by sex and from 1980 to 2014. We also calculated the posterior probability of meeting the global diabetes target if post-2000 trends continue. FINDINGS: We used data from 751 studies including 4 372 000 adults from 146 of the 200 countries we make estimates for. Global age-standardised diabetes prevalence increased from 4·3% (95% credible interval 2·4–7·0) in 1980 to 9·0% (7·2–11·1) in 2014 in men, and from 5·0% (2·9–7·9) to 7·9% (6·4–9·7) in women. The number of adults with diabetes in the world increased from 108 million in 1980 to 422 million in 2014 (28·5% due to the rise in prevalence, 39·7% due to population growth and ageing, and 31·8% due to interaction of these two factors). Age-standardised adult diabetes prevalence in 2014 was lowest in northwestern Europe, and highest in Polynesia and Micronesia, at nearly 25%, followed by Melanesia and the Middle East and north Africa. Between 1980 and 2014 there was little change in age-standardised diabetes prevalence in adult women in continental western Europe, although crude prevalence rose because of ageing of the population. By contrast, age-standardised adult prevalence rose by 15 percentage points in men and women in Polynesia and Micronesia. In 2014, American Samoa had the highest national prevalence of diabetes (>30% in both sexes), with age-standardised adult prevalence also higher than 25% in some other islands in Polynesia and Micronesia. If post-2000 trends continue, the probability of meeting the global target of halting the rise in the prevalence of diabetes by 2025 at the 2010 level worldwide is lower than 1% for men and is 1% for women. Only nine countries for men and 29 countries for women, mostly in western Europe, have a 50% or higher probability of meeting the global target. INTERPRETATION: Since 1980, age-standardised diabetes prevalence in adults has increased, or at best remained unchanged, in every country. Together with population growth and ageing, this rise has led to a near quadrupling of the number of adults with diabetes worldwide. The burden of diabetes, both in terms of prevalence and number of adults affected, has increased faster in low-income and middle-income countries than in high-income countries. FUNDING: Wellcome Trust

Non-autonomous 2-periodic Gumovski-Mira difference equations

We consider two types of non-autonomous 2-periodic Gumovski-Mira difference equations. We show that while the corresponding autonomous recurrences are conjugated, the behavior of the sequences generated by the 2-periodic ones differ dramatically: in one case the behavior of the sequences is simple (integrable) and in the other case it is much more complicated (chaotic). We also present a global study of the integrable case that includes which periods appear for the recurrence.Comment: 20 pages, 11 figure