Multiplicative relations among singular moduli

We consider some Diophantine problems of mixed modular-multiplicative type associated with the Zilber-Pink conjecture. In particular, we prove a finiteness statement for the number of multiplicative relations between singular moduli (j-invariants of elliptic curves with complex multiplication.)Comment: Comments Welcome

System Dynamics Simulation to Test Operational Policies in the Milk-Cheese Supply Chain Case study: Piar Municipality, Bolivar State, Venezuela.

With the purpose of detecting the impact that variations of demand cause in the milk-cheese supply chain, and determining how the operational policies of capacity, inventories or labor force can mitigate this impact, a system dynamics simulation model has been designed based on a survey conducted on a sample of cheese manufacturers and their links with milk farms, transportation companies and cheese distributors. This supply chain will be consolidated when a milk center that will collect the raw milk is completed. From this center, and after adequate treatment, milk will be distributed to the different cheese manufacturers in the supply chain. Managing adequately the milk-cheese supply chain represents an important challenge due to the short life of these products. Although this study was done in a region in Latin America, its results can be applicable to food supply chains by introducing some modifications. The milk-cheese supply chain in this case study contemplates three milk producers, one milk center, five cheese producers and several distributing agents. These companies operate individually under normal conditions, but they have understood that their integration in a supply chain improves the competitiveness of all its members. That is to say, the sum is greater than the parts. For its initial design a simulation software model is used in which the resources of the supply chain are optimized. Later the product of this optimization facilitates some initial values to be used in the system dynamics model in which causeeffect or influence relationships have been previously established considering the most representative variables. Finally, changes in operational policies that can reduce the level of pending orders in the supply chain are tested using other simulation software. The main contribution of this research is that it can serve as support or contribute to reduce the uncertainty in the decision making process of the supply chain management due to the speed with which individual or combined policies can be analyzed. In response to a variation of demand the most adequate policy may be selected and that can be done before the policy is implemented

Atención educativa al alumnado inmigrante

Treball de Final de Grau en Mestre o Mestra d'Educació Infantil. Codi: MI1040. Curs acadèmic 2014-2015En esta investigación se ofrece una reflexión sobre la importancia que tiene la Educación Intercultural en la escuela, ya que en la actualidad nos encontramos con una gran influencia de población inmigrante en el contexto educativo. Esto ha creado situaciones de contacto con otras culturas y ha provocado que por parte del Sistema Educativo español se deban de dar una respuesta al alumnado extranjero. De esta manera ha surgido la necesidad de unos programas que dieran un tipo de atención educativa al alumnado inmigrante en la escuela. La incorporación de este tipo de alumnado hace que los docentes deban de plantearse nuevas estrategias metodológicas ante su alumnado, evitando de esta forma situaciones de discriminación o desventajas educativas. El principal objetivo de esta investigación es el conocimiento de cómo se debe atender al alumnado inmigrante en su nueva incorporación en el sistema educativo, destacando el Plan de Acogida al alumnado inmigrante, las Aulas Temporales de Adaptación Lingüística o también conocidas como ATAL y concluyendo la investigación con las Estrategias Didácticas, que en ella podremos conocer cómo debe de actuar el profesorado y las cualidades que se deberán de llevar a cabo por parte del docente

CM relations in fibered powers of elliptic families

Let $E_\lambda$ be the Legendre family of elliptic curves. Given $n$ linearly independent points $P_1,\dots , P_n \in E_\lambda\left(\overline{\mathbb{Q}(\lambda)}\right)$ we prove that there are at most finitely many complex numbers $\lambda_0$ such that $E_{\lambda_0}$ has complex multiplication and $P_1(\lambda_0), \dots ,P_n(\lambda_0)$ are dependent over $End(E_{\lambda_0})$. This implies a positive answer to a question of Bertrand and, combined with a previous work in collaboration with Capuano, proves the Zilber-Pink conjecture for a curve in a fibered power of an elliptic scheme when everything is defined over $\overline{\mathbb{Q}}$.Comment: The formulation of Theorem 2.1 is now correc

Counting lattice points and o-minimal structures

Let $\Lambda$ be a lattice in $\R^n$, and let $Z\subseteq \R^{m+n}$ be a definable family in an o-minimal structure over $\R$. We give sharp estimates for the number of lattice points in the fibers $Z_T={x\in \R^n: (T,x)\in Z}$. Along the way we show that for any subspace $\Sigma\subseteq\R^n$ of dimension $j>0$ the $j$-volume of the orthogonal projection of $Z_T$ to $\Sigma$ is, up to a constant depending only on the family $Z$, bounded by the maximal $j$-dimensional volume of the orthogonal projections to the $j$-dimensional coordinate subspaces.Comment: Revised versio

Heights of pre-special points of Shimura varieties

Let s be a special point on a Shimura variety, and x a pre-image of s in a fixed fundamental set of the associated Hermitian symmetric domain. We prove that the height of x is polynomially bounded with respect to the discriminant of the centre of the endomorphism ring of the corresponding ZZ -Hodge structure. Our bound is the final step needed to complete a proof of the André–Oort conjecture under the conjectural lower bounds for the sizes of Galois orbits of special points, using a strategy of Pila and Zannier

Rational points on Grassmannians and unlikely intersections in tori

In this paper, we present an alternative proof of a finiteness theorem due to Bombieri, Masser and Zannier concerning intersections of a curve in the multiplicative group of dimension n with algebraic subgroups of dimension n-2. The proof uses a method introduced for the first time by Pila and Zannier to give an alternative proof of Manin-Mumford conjecture and a theorem to count points that satisfy a certain number of linear conditions with rational coefficients. This method has been largely used in many different problems in the context of "unlikely intersections".Comment: 16 page