The inhomogeneous Suslov problem

We consider the Suslov problem of nonholonomic rigid body motion with inhomogeneous constraints. We show that if the direction along which the Suslov constraint is enforced is perpendicular to a principal axis of inertia of the body, then the reduced equations are integrable and, in the generic case, possess a smooth invariant measure. Interestingly, in this generic case, the first integral that permits integration is transcendental and the density of the invariant measure depends on the angular velocities. We also study the Painlev\'e property of the solutions.Comment: 10 pages, 5 figure

Extension of maps defined on many fibres

Let $S$ be a fibred surface. We prove that the existence of morphisms from non countably many fibres to curves implies, up to base change, the existence of a rational map from $S$ to another surface fibred over the same base reflecting the properties of the original morphisms. Under some conditions of unicity base change is not needed and one recovers exactly the initial maps

Diseño y Optimización del tamaño de grano triturado para alimentación avícola para criaderos y suministro de balanceado/Design and Optimization of crushed grain size for poultry feed for hatcheries and supply of balanced

En la investigación se diseñó, modeló y mejoró un molino de martillo económico y accesible aplicando métodos lógicos deductivos, la máquina provee un tamaño de grano ideal, que sirve como alimentación directa para aves, o también se puede utilizar como un componente para balanceado en la producción avícola, acorde a los requerimientos de las distintas pymes ya sea para consumo cárnico, producción de huevos, entre otros. Por esta razón es necesario definir distintos parámetros como el tamaño adecuado del tamiz teniendo una variedad desde los 2mm hasta los 5 mm, cuchillas, selección adecuada del motor dependiendo de la cantidad de horas por día, para esto se realizó un análisis mecánico y dinámico de los distintos elementos de la máquina mediante simulación en el software Solidworks free students, luego de tener el dimensionamiento óptimo bajo parámetros ingenieriles de cada una de las partes de la máquina, se realizó una cotización de los materiales bajo consideración económica en el que fue necesario realizar una correcta selección de los mismo elementos basados en catálogos nacionales para que la maquina sea rentable y accesible a los micro productores así como a los mayoristas, una vez considerado todo el procedimiento se construyó la máquina con un costo final de 355,39 dólares americanos, siendo esta de fácil desmontaje, mantenimiento y aplicación para la sociedad, el cual permite la reducción de costos, tiempos de producción y aumenta la eficiencia en la obtención grano triturado para el consumo el sector avícola. In the research an economical and accessible hammer mill was designed, modeled and improved by applying deductive logical methods, the machine provides an ideal grain size, which serves as direct feed for birds, or it can also be used as a component for balanced in the poultry production, according to the requirements of the different SMEs, whether for meat consumption, egg production, among others. For this reason, it is necessary to define different parameters such as the appropriate size of the sieve having a variety from 2mm to 5 mm, blades, suitable engine selection depending on the number of hours per day, for this a mechanical and dynamic analysis of the different elements of the machine by means of simulation in the software Solidworks free students, after having the optimal dimensioning under engineering parameters of each one of the parts of the machine, a quotation of the materials was made under economic consideration in which it was necessary to carry out a correct selection of the same elements based on national catalogs so that the machine is profitable and accessible to the micro producers as well as to the wholesalers. Once the entire procedure was considered, the machine was built with a final cost of US $ 355.39. being this easy to disassembly, maintenance and application for society, which allows the reduction of costs, production times and increases the efficiency in obtaining crushed grain for consumption in the poultry sector. Palabras clave: diseño, producción avícola, grano, balanceado, molino, grano triturado. Keywords: design, poultry production, grain, balanced, mill, crushed grain

Metaplectic and spin representations: a parallel treatment

The analogies between symplectic and orthogonal groups, regarded as symmetries of real bilinear forms, are manifest in their (metaplectic and spin) projective representations. In finite dimensions, those are true representations of doubly covering groups; but one may also use group extensions by a circle. Here we lay out a parallel treatment of of the Mp$^\mathrm{c}$ and Spin$^\mathrm{c}$ covering groups, acting on the respective Fock spaces by permuting certain Gaussian vectors. The cocycles of these extensions exhibit interesting similarities.Comment: Latex, 38 page

Cyclic Coverings of genus 2 curves of Sophie Germain type

We consider cyclic unramified coverings of degree d of irreducible complex smooth genus 2 curves and their corresponding Prym varieties. They provide natural examples of polarized abelian varieties with automorphisms of order d. The rich geometry of the associated Prym map, has been studied in several papers, and the cases d=2, 3, 5, 7 are quite well-understood. Nevertheless, very few is known for higher values of d. In this article we investigate if the covering can be reconstructed from its Prym variety, that is, if the generic Prym Torelli Theorem holds for these coverings. We prove this is so for the so-called Sophie Germain prime numbers, that is, for $d\ge 11$ prime such that $(d-1)/2$ is also prime. We use results of arithmetic nature on $GL_2$-type abelian varieties combined with theta-duality techniques

On the dimension of Voisin sets in the moduli space of abelian varieties

We study the subsets $V_k(A)$ of a complex abelian variety $A$ consisting in the collection of points $x\in A$ such that the zero-cycle $\{x\}-\{0_A\}$ is $k$-nilpotent with respect to the Pontryagin product in the Chow group. These sets were introduced recently by Voisin and she showed that $\dim V_k(A) \leq k-1$ and $V_k(A)$ is countable for a very general abelian variety of dimension at least $2k-1$. We study in particular the locus $\mathcal V_{g,2}$ in the moduli space of abelian varieties of dimension $g$ with a fixed polarization, where $V_2(A)$ is positive dimensional. We prove that an irreducible subvariety $\mathcal Y \subset \mathcal V_{g,2}$, $g\ge 3$, such that for a very general $y \in \mathcal Y$ there is a curve in $V_2(A_y)$ generating $A$ satisfies $\dim \mathcal Y \le 2g - 1.$ The hyperelliptic locus shows that this bound is sharp.Comment: Final version to appear in Mathematische Annale