Canonical characters on quasi-symmetric functions and bivariate Catalan numbers

Every character on a graded connected Hopf algebra decomposes uniquely as a product of an even character and an odd character (Aguiar, Bergeron, and Sottile, math.CO/0310016). We obtain explicit formulas for the even and odd parts of the universal character on the Hopf algebra of quasi-symmetric functions. They can be described in terms of Legendre's beta function evaluated at half-integers, or in terms of bivariate Catalan numbers: $$C(m,n)=\frac{(2m)!(2n)!}{m!(m+n)!n!}.$$ Properties of characters and of quasi-symmetric functions are then used to derive several interesting identities among bivariate Catalan numbers and in particular among Catalan numbers and central binomial coefficients

Quantum Analogy of Poisson Geometry, Related Dendriform Algebras and Rota-Baxter Operators

We will introduce an associative (or quantum) version of Poisson structure tensors. This object is defined as an operator satisfying a "generalized" Rota-Baxter identity of weight zero. Such operators are called generalized Rota-Baxter operators. We will show that generalized Rota-Baxter operators are characterized by a cocycle condition so that Poisson structures are so. By analogy with twisted Poisson structures, we propose a new operator "twisted Rota-Baxter operators" which is a natural generalization of generalized Rota-Baxter operators. It is known that classical Rota-Baxter operators are closely related with dendriform algebras. We will show that twisted Rota-Baxter operators induce NS-algebras which is a twisted version of dendriform algebra. The twisted Poisson condition is considered as a Maurer-Cartan equation up to homotopy. We will show the twisted Rota-Baxter condition also is so. And we will study a Poisson-geometric reason, how the twisted Rota-Baxter condition arises.Comment: 18 pages. Final versio

Free Rota-Baxter algebras and rooted trees

A Rota-Baxter algebra, also known as a Baxter algebra, is an algebra with a linear operator satisfying a relation, called the Rota-Baxter relation, that generalizes the integration by parts formula. Most of the studies on Rota-Baxter algebras have been for commutative algebras. Two constructions of free commutative Rota-Baxter algebras were obtained by Rota and Cartier in the 1970s and a third one by Keigher and one of the authors in the 1990s in terms of mixable shuffles. Recently, noncommutative Rota-Baxter algebras have appeared both in physics in connection with the work of Connes and Kreimer on renormalization in perturbative quantum field theory, and in mathematics related to the work of Loday and Ronco on dendriform dialgebras and trialgebras. This paper uses rooted trees and forests to give explicit constructions of free noncommutative Rota--Baxter algebras on modules and sets. This highlights the combinatorial nature of Rota--Baxter algebras and facilitates their further study. As an application, we obtain the unitarization of Rota-Baxter algebras.Comment: 23 page

Deterministic and Stochastic Dynamics of COVID-19: The Case Study of Italy and Spain

In December 2019, a severe respiratory syndrome (COVID-19) caused by a new coronavirus (SARS-CoV-2) was identified in China and spread rapidly around the globe. COVID-19 was declared a pandemic by the World Health Organization (WHO) in March 2020. With eventually substantial global underestimation, more than 225 million cases were confirmed by the end of August 2021, counting more than 4.5 million deaths. COVID-19 symptoms range from mild (or no symptoms) to severe illness, with disease severity and death occurring according to a hierarchy of risks, with age and preexisting health conditions enhancing the risks of disease severity manifestation. In this paper, a mathematical model for COVID-19 transmission is proposed and analyzed. The model stratifies the studied population into two groups, older and younger. Applied to the COVID-19 outbreaks in Spain and in Italy, we find the disease-free equilibrium and the basic reproduction number for each case study. A sensitivity analysis to identify the key parameters which influence the basic reproduction number, and hence regulate the transmission dynamics of COVID-19, is also performed. Finally, the model is extended to its stochastic counterpart to encapsulate the variation or uncertainty found in the transmissibility of the disease. We observe the variability of the infectious population finding its distribution at a given time, demonstrating that for small populations, stochasticity will play an important role.Marie Skłodowska-Curie grant agreement No. 79249

Predictors of educational aspirations of Roma mothers in Czech Republic, Greece, and Portugal

Roma communities are a disadvantaged minority in Europe which is particularly underrepresented in social and educational research. This study aimed to investigate the predictors of Roma mothers’ educational aspirations for their children in the Czech Republic, Greece, and Portugal. Participants included 461 mothers with a Roma background (135 from the Czech Republic, 130 from Greece, and 196 from Portugal), with a child between 3 to 6 (n = 181) or 9 to 12 (n = 280) years old. Data were based on mothers’ reports, obtained during a structured in-person interview. Material deprivation (microsystem level), frequency and quality of interactions with non-Roma parents, as well as the quality of parent-teacher interactions (mesosystem level), predicted Roma mothers’ educational aspirations. Findings suggest that, in addition to microsystemic variables such as material deprivation, mesosystemic predictors such as those examining contact with non-Roma parents may play an important role in shaping Roma mothers’ educational aspirations and need to be further examined.info:eu-repo/semantics/publishedVersio

The MobyDick Project: A Mobile Heterogeneous All-IP Architecture

Proceedings of Advanced Technologies, Applications and Market Strategies for 3G (ATAMS 2001). Cracow, Poland: 17-20 June, 2001.This paper presents the current stage of an IP-based architecture for heterogeneous environments, covering UMTS-like W-CDMA wireless access technology, wireless and wired LANs, that is being developed under the aegis of the IST Moby Dick project. This architecture treats all transmission capabilities as basic physical and data-link layers, and attempts to replace all higher-level tasks by IP-based strategies. The proposed architecture incorporates aspects of mobile-IPv6, fast handover, AAA-control, and Quality of Service. The architecture allows for an optimised control on the radio link layer resources. The Moby dick architecture is currently under refinement for implementation on field trials. The services planned for trials are data transfer and voice-over-IP.Publicad

Birkhoff type decompositions and the Baker-Campbell-Hausdorff recursion

We describe a unification of several apparently unrelated factorizations arisen from quantum field theory, vertex operator algebras, combinatorics and numerical methods in differential equations. The unification is given by a Birkhoff type decomposition that was obtained from the Baker-Campbell-Hausdorff formula in our study of the Hopf algebra approach of Connes and Kreimer to renormalization in perturbative quantum field theory. There we showed that the Birkhoff decomposition of Connes and Kreimer can be obtained from a certain Baker-Campbell-Hausdorff recursion formula in the presence of a Rota-Baxter operator. We will explain how the same decomposition generalizes the factorization of formal exponentials and uniformization for Lie algebras that arose in vertex operator algebra and conformal field theory, and the even-odd decomposition of combinatorial Hopf algebra characters as well as to the Lie algebra polar decomposition as used in the context of the approximation of matrix exponentials in ordinary differential equations.Comment: accepted for publication in Comm. in Math. Phy

Efeito de diferentes solventes orgânicos sobre a viabilidade do pólen de jerivá (Syagrus romanzoffiana S. CHAM).

EVINCI. Resumo