Abstract:
The surface corresponding to the moduli space of quadratic endomorphisms of $\mathbb{P}^1$ with a marked periodic point of order $n$ is studied. It is shown that the surface is rational over $\mathbb{Q}$ when $n\le 5$ and is of general type for $n=6$. An explicit description of the $n=6$ surface lets us find several infinite families of quadratic endomorphisms $f: \mathbb{P}^1 \to \mathbb{P}^1$ defined over $\mathbb{Q}$ with a rational periodic point of order $6$. In one of these families, $f$ also has a rational fixed point, for a total of at least $7$ periodic and $7$ preperiodic points. This is in contrast with the polynomial case, where it is conjectured that no polynomial endomorphism defined over $\mathbb{Q}$ admits rational periodic points of order $n>3$.

Abstract:
this study aims to analyze the contribution that the written press makes to the visibility of the possible improvements which might lead to the implementation of the law autonomy promotion and care for people in a situation of dependency,for the elderly person and his family. several authors note that the press is a good resource to publicize, promote health, create public opinion and influence in the conduct of the people. for this work, were checked five newspapers for 239 days to collect related news to the law. the largest number of news found alludes to the portfolio of services and their cost. stress the large number of texts that are more about political posturing than specific aspects of the law.

Abstract:
El presente trabajo tiene como objetivo analizar la aportación que la prensa escrita hace durante un período concreto a la visibilidad de las posibles mejoras que supondrá la implementación de la Ley de Promoción de Autonomía Personal y atención a las personas en situación de Dependencia, para el mayor y su familia. Diversos autores constatan que la prensa escrita es buen recurso para divulgar, promover la salud, crear opinión pública e influenciar en las conductas de las personas. Para la realización de este trabajo se revisaron 5 periódicos durante 239 días recopilando noticias relacionadas con la Ley. El mayor número de noticias encontradas hace alusión a la cartera de servicios y su coste. Conclusión principal: destaca el elevado número de textos que hacen referencia más a gestos políticos que a aspectos concretos de la ley. This study aims to analyze the contribution that the written press makes to the visibility of the possible improvements which might lead to the implementation of the Law Autonomy Promotion and care for people in a situation of dependency,for the elderly person and his family. Several authors note that the press is a good resource to publicize, promote health, create public opinion and influence in the conduct of the people. For this work, were checked five newspapers for 239 days to collect related news to the Law. The largest number of news found alludes to the portfolio of services and their cost. Stress the large number of texts that are more about political posturing than specific aspects of the law.

Abstract:
Let $K$ be a number field. Let $S$ be a finite set of places of $K$ containing all the archimedean ones. Let $R_S$ be the ring of $S$-integers of $K$. In the present paper we consider endomorphisms of $\pro$ of degree 2, defined over $K$, with good reduction outside $S$. We prove that there exist only finitely many such endomorphisms, up to conjugation by ${\rm PGL}_2(R_S)$, admitting a periodic point in $\po$ of order $>3$. Also, all but finitely many classes with a periodic point in $\po$ of order 3 are parametrized by an irreducible curve.

Abstract:
Let $K$ be a number field and $\phi\in K(z)$ a rational function. Let $S$ be the set of all archimedean places of $K$ and all non-archimedean places associated to the prime ideals of bad reduction for $\phi$. We prove an upper bound for length of finite orbits of $\phi$ in $\mathbb{P}_1(K)$ depending only on the cardinality of $S$.

Abstract:
Let $k$ be an algebraic closed field of characteristic zero. Let $K$ be the rational function field $K=k(t)$. Let $\phi$ be a non isotrivial rational function in $K(z)$. We prove a bound for the cardinality of the set of $K$--rational preperiodic points for $\phi$ in terms of the number of places of bad reduction and the degree $d$ of $\phi$.

Abstract:
Let $K$ be a number field and $v$ a non archimedean valuation on $K$. We say that an endomorphism $\Phi\colon \mathbb{P}_1\to \mathbb{P}_1$ has good reduction at $v$ if there exists a model $\Psi$ for $\Phi$ such that $\deg\Psi_v$, the degree of the reduction of $\Psi$ modulo $v$, equals $\deg\Psi$ and $\Psi_v$ is separable. We prove a criterion for good reduction that is the natural generalization of a result due to Zannier in \cite{Uz3}. Our result is in connection with other two notions of good reduction, the simple and the critically good reduction. The last part of our article is dedicated to prove a characterization of the maps whose iterates, in a certain sense, preserve the critically good reduction.

Abstract:
Let $K$ be a number field and $S$ a fixed finite set of places of $K$ containing all the archimedean ones. Let $R_S$ be the ring of $S$-integers of $K$. In the present paper we study the cycles for rational maps of $\mathbb{P}_1(K)$ of degree $\geq2$ with good reduction outside $S$. We say that two ordered $n$-tuples $(P_0,P_1,...,P_{n-1})$ and $(Q_0,Q_1,...,Q_{n-1})$ of points of $\mathbb{P}_1(K)$ are equivalent if there exists an automorphism $A\in{\rm PGL}_2(R_S)$ such that $P_i=A(Q_i)$ for every index $i\in\{0,1,...,n-1\}$. We prove that if we fix two points $P_0,P_1\in\mathbb{P}_1(K)$, then the number of inequivalent cycles for rational maps of degree $\geq2$ with good reduction outside $S$ which admit $P_0,P_1$ as consecutive points is finite and depends only on $S$. We also prove that this result is in a sense best possible.

Abstract:
The spectrophotometric characterization of high efficiency, optically-active samples such as light-emitting organic bulks and thin films can be problematic because their broad-band luminescence is detected together with the monochromatic transmitted and reflected signals, hence perturbing measurements of optical transmittance and reflectance at wavelengths within the photoexcitation band. As a matter of fact, most commercial spectrophotometers apply spectral filtering before the light beam reaches the sample, not after it. In this Report, we introduce and discuss the method we have developed to correct photometric spectra that are perturbed by photoluminescence.

Abstract:
We provide a complete classification of possible graphs of rational preperiodic points of endomorphisms of the projective line of degree 2 defined over the rationals with a rational periodic critical point of period 2, under the assumption that these maps have no periodic points of period at least 7. We explain how this extends results of Poonen on quadratic polynomials. We show that there are 13 possible graphs, and that such maps have at most 9 rational preperiodic points. We provide data related to the analogous classification of graphs of endomorphisms of degree 2 with a rational periodic critical point of period 3 or 4.