Trabajo de grado - Maestría en Matemáticas Aplicadashttp://hdl.handle.net/11232/17012024-03-29T13:54:59Z2024-03-29T13:54:59ZCover monoidal categories on species and free cover monoidsContreras Ordóñez, Luis Alfonsohttp://hdl.handle.net/11232/19242023-03-01T03:00:37Z2022-07-01T00:00:00ZCover monoidal categories on species and free cover monoids
Contreras Ordóñez, Luis Alfonso
We study a new product on the category of combinatorial species, which we call the cover Cauchy product. This product is defined as a relaxed version of the classical Cauchy product on species. We prove that the category of species together with the cover Cauchy product has the structure of a monoidal category. We define a cover Cauchy product on exponential generating functions and show that the exponential generating function of the cover Cauchy product of two species is precisely the cover Cauchy product of their exponential generating functions. Finally, we show that the free cover Cauchy monoid in one generator is given by the species of packed words, which are in bijective correspondence with surjections.
2022-07-01T00:00:00ZThe μ-polynomials of graph associahedra.Ávila Ramírez, Nicoláshttp://hdl.handle.net/11232/18212023-02-28T13:53:15Z2022-01-01T00:00:00ZThe μ-polynomials of graph associahedra.
Ávila Ramírez, Nicolás
We study two polynomials associated to a graph G that are of interest in the recent literature. The first one is the h-polynomial of the graph-associahedron of G defined by Carr and Devadoss. The second one is the μ-polynomial recently defined by González
D’León and Wachs, which in the case of trees the authors conjecture that is up to sign equal to its h-polynomial. We prove a more general relation between the h- and the μ-polynomial of G which in the special case of trees proves González D’León - Wachs’
conjecture. We give a new description of the μ-polynomials in terms of a family of forests that we call μ-forests. As applications of the tools developed, we compute the μ- polynomials of the families of cycle and kite-like graphs. These are related to the Narayana polynomials of type A and B. We also show that these families of polynomials are realrooted and form interlacing sequences giving support and extending previous conjectures to a general conjecture on real-rootedness and the interlacing property of the h- and the μ-polynomials of an arbitrary graph G.
2022-01-01T00:00:00ZSobre el método de reescritura en el estudio de operads no simétricos.Salamanca Valencia, Edward Julianhttp://hdl.handle.net/11232/18172022-08-26T21:31:55Z2022-01-01T00:00:00ZSobre el método de reescritura en el estudio de operads no simétricos.
Salamanca Valencia, Edward Julian
En este trabajo estudiamos la técnica combinatoria de reescritura, como está definida por Loday y Vallette, para el estudio de la propiedad de Koszul, definida por Ginzburg y Kapranov, en operads no simétrisus operads no simétricos duales de Koszulcos. En particular, nos centramos en los operads no simétricos conjuntistas binarios As, Dend y T ridend. Los operads no simétricos As, DiasAysT, Driasias, yy T para definir un nuevo operad no simétrico que llamamos rias pueden definirse de acuerdo a una estrategia de “transferencia de energía” que puede extenderse T etraas. Este operad permite modelar la categoría de álgebras tetra-asociativas. Mostramos como ejemplo que el álgebra de descensos de Solomon admite la estructura de álgebra tetra-asociativa. Mostramos también que el operad T etraas es conjuntista pero no es cancelativo. Usando código en SageMath/Python implementamos el método de reescritura para chequear la propiedad de Koszul en operads no simétricos conjuntistas. Con esto chequeamos los resultados positivos para esta propiedad sobre Dias y T rias, demostrados directamente por Loday y por Loday-Ronco usando técnicas homológicas. Adicionalmente, usamos esta misma técnica para demostrar que el operad T etraas, y su dual de Koszul T etradend, son operads no simétricos de Koszul. Finalmente,-arios, generalizamos la construcción por transferencia de energía de operads binarios a operads
(la propiedad de Koszul y dejamos como trabajo futuro estudiar dicha propiedad por medio del uso depara un entero positivok + 1) − T etraas. Mostramos que la técnica de reescritura para estos operads no es concluyente para k, introduciendo los operads no simétricos (k + 1) − Dias, (k ) T rias y técnicas homológicas.
2022-01-01T00:00:00ZThe Black Litterman Model: A Non-Normal Approach.Bedoya Riveros, Carlos Felipehttp://hdl.handle.net/11232/18002022-03-08T20:25:22Z2021-01-01T00:00:00ZThe Black Litterman Model: A Non-Normal Approach.
Bedoya Riveros, Carlos Felipe
The Black Litterman Model (BLM) is widely used tool for the estimation of the expected excess return of a given portfolio. It belongs to the field of Bayesian Statistics, and, assumptions on the prior and likelihood distributions are made. This work relaxes normality assumption on both distributions, and explores two scenarios, using Gamma and Skew Normal distributions. The posterior distribution is determined up to a constant and then data is sampled through Metropolis Hastings algorithms. A specific exercise is executed and its results are shown the end of the document.
2021-01-01T00:00:00Z