Simple maps, Hurwitz numbers, and Topological Recursion

We introduce the notion of fully simple maps, which are maps with non self-intersecting disjoint boundaries. In contrast, maps where such a restriction is not imposed are called ordinary. We study in detail the combinatorics of fully simple maps with topology of a disk or a cylinder. We show that the generating series of simple disks is given by the functional inversion of the generating series of ordinary disks. We also obtain an elegant formula for cylinders. These relations reproduce the relation between moments and free cumulants established by Collins et al. math.OA/0606431, and implement the symplectic transformation $x \leftrightarrow y$ on the spectral curve in the context of topological recursion. We conjecture that the generating series of fully simple maps are computed by the topological recursion after exchange of $x$ and $y$. We propose an argument to prove this statement conditionally to a mild version of symplectic invariance for the $1$-hermitian matrix model, which is believed to be true but has not been proved yet. Our argument relies on an (unconditional) matrix model interpretation of fully simple maps, via the formal hermitian matrix model with external field. We also deduce a universal relation between generating series of fully simple maps and of ordinary maps, which involves double monotone Hurwitz numbers. In particular, (ordinary) maps without internal faces -- which are generated by the Gaussian Unitary Ensemble -- and with boundary perimeters $(\lambda_1,\ldots,\lambda_n)$ are strictly monotone double Hurwitz numbers with ramifications $\lambda$ above $\infty$ and $(2,\ldots,2)$ above $0$. Combining with a recent result of Dubrovin et al. math-ph/1612.02333, this implies an ELSV-like formula for these Hurwitz numbers.Comment: 66 pages, 7 figure

Interfacial stability in binary liquid-liquid systems

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Relating ordinary and fully simple maps via monotone Hurwitz numbers

A direct relation between the enumeration of ordinary maps and that of fully simple maps first appeared in the work of the first and last authors. The relation is via monotone Hurwitz numbers and was originally proved using Weingarten calculus for matrix integrals. The goal of this paper is to present two independent proofs that are purely combinatorial and generalise in various directions, such as to the setting of stuffed maps and hypermaps. The main motivation to understand the relation between ordinary and fully simple maps is the fact that it could shed light on fundamental, yet still not well-understood, problems in free probability and topological recursion.Comment: 19 pages, 7 figure

Voluntary membership and cooperative identity in Mexican cooperative enterprises

El principio de asociación voluntaria es uno de los más importantes para que pueda existir una empresa cooperativa, pues esta sólo es dable cuando existe la voluntad de c/persona para ingresar a la organización, con el fin de resolver sus problemas y limitaciones mediante la ayuda mutua y el esfuerzo colectivo, éste principio está reconocido en la Declaración de Identidad Cooperativa de 1995. No obstante siguiendo algunas observaciones de la escritora Alicia Kaplan se analizan algunos problemas derivados de la asociación voluntaria, a efecto de conocer la forma cómo se proyectan en la Ley General de Sociedades Cooperativas en México

Half-Spin Tautological Relations and Faber's Proportionalities of Kappa Classes

We employ the $1/2$-spin tautological relations to provide a particular combinatorial identity. We show that this identity is a statement equivalent to Faber's formula for proportionalities of kappa-classes on $\mathcal{M}_g$, $g\geq 2$. We then prove several cases of the combinatorial identity, providing a new proof of Faber's formula for those cases

Between Southern Italy and Dalmatia Missal MR 166 of the Metropolitana Library, Zagreb

The Missal MR 166 from the Metropolitana Library, Zagreb, written in Beneventana script and dating back to the twelfth-thirteenth centuries, has long been considered a Dalmatian product, similar to the coeval illuminated manuscript in Beneventana script preserved in the Trogir Cathedral and originating in Zadar. Nevertheless, later studies - specifically based on the textual features of the manuscript - showed that it is undoubtedly a Southern Italian product, and a significant testimony of the uninterrupted book circulation that existed on both sides of the Adriatic for three centuries roughly from the eleventh to the thirteenth, thus influencing the activity of the Benedictine scriptoria on the Dalmatian coast. On the basis of the study that makes it possible to define more closely the group of manuscripts that make up the 'corpus of the illuminated manuscripts from Dalmatia', the paper aims to support the Southern Italian origin of the Missal by means of a critical analysis of the theories put forward so far about the 'typically Dalmatian' features of its Initialornamentik

Biología reproductiva de Nacella (P.) Deaurata (Gmelin, 1791) en Bahía Lapataia (Canal del Beagle)

The reproductive cycle of a Nacella (P.) deaurata population that inhabits the lower intertidal zone in Lapa-taia Bay (54°52´S; 68°29´W) was studied. Monthly samples were collected, and specimens were measured and weighed, fixing the gonads in Bouin´s fluid for histological analyses. The gonadosomatic index (GSI) was determined as a percentage of the ratio between gonadal and foot wet weight. Taking into account the presence and abundance of different cellular types, gonadal stages were established for males and females. The analyses of the variation of the gonadal stage percentages showed the first mature males and females in July. In 1989, mature females maintained a percentage higher than 30% until January. Spawning began in September and was massive in November. In 1990, the maximum percentage of mature females was found in September with all the specimens spawned in October. Nevertheless, the majority of the population was mature again in November. The male sexual cycle in 1989 showed the highest percentage of mature individuals in August, being high until October and slightly diminishing in November. The highest percentage of evacuated males was observed in November-December. In 1990, the cycle was similar to the one shown by the females, but without recovery in November. The GSI variability in males and females, and the adequate use of the GSI in determining the annual reproductive cycle were discussed. The biotic and environmental conditions that may act as a trigger for the spawning have been analyzed.Se estudió el ciclo reproductivo de Nacella (P.) deaurata en una población que habita el intermareal inferior en la costa sur de Bahía Lapataia (54°52’S; 68°29’O). Se realizaron muestreos mensuales, los ejemplares fueron medidos, pesándose las vísceras, pie y gónada, siendo esta última fijada en Bouin para su posterior procesado histológico. Se determinó el Índice Gonadosomático (IGS) como la expresión porcentual del peso gonadal sobre el peso húmedo del pie. Se establecieron histológicamente estadios gonadales tanto en hembras como en machos, de acuerdo con la presencia y abundancia de los diferentes tipos celulares. De acuerdo con el análisis de los porcentajes mensuales de los estadios gonadales, los individuos de ambos sexos se encontraron maduros a partir de julio. Las hembras maduras durante 1989 se presentaron en porcentajes superiores al 30% hasta enero. A partir de septiembre comenzó el desove, haciéndose masivo en la población en noviembre. En 1990 el pico máximo de maduración se alcanzó en septiembre. En octubre todos los individuos habian desovado, mientras que durante noviembre una gran proporción de la población se recuperó alcanzando nuevamente la madurez total. El ciclo sexual anual de los machos en 1989 mostró el mayor porcentaje de individuos maduros durante agosto, manteniéndose una alta proporción hasta octubre con un ligero descenso en noviembre. El máximo porcentaje de individuos evacuados se observó entre noviembre y diciembre. En 1990 el ciclo fué similar al que presentaron las hembras para el mismo año pero no se produjo recuperación de la maduración gonadal en noviembre. Se discute la variabilidad del IGS en ambos sexos y su validez para la determinación de los ciclos reproductivos anuales, analizándose las condiciones bióticas y ambientales que actuarían como desencadenantes del desove

Sarah Kane\u27s Post-Christian Spirituality in Cleansed

The existing scholarship on the work of British playwright Sarah Kane mostly focuses on exploring the use of extreme acts of violence in her plays. However, few scholars like Dr. Graham Saunders and Anabelle Singer can trace it back to the rejection of her Christian beliefs during her adolescence. This thesis explores how Kane used violence and images of impalement, dismemberment, and cross-dressing in her third play, Cleansed as a vehicle to examine and validate her notions of gender and sexuality as well as her developing post-Christian spirituality. Through the critical textual analysis of Cleansed in conjunction with scholarly literature regarding her work, I explore Kane’s life and spiritual background and tie them to the biblical allusions in the play and Kane’s adaptation thereof