Almost everywhere continuity of conditional expectations

A necessary and sufficient condition on a sequence $\{\mathfrak{A}_n\}_{n\in \mathbb{N}}$ of $\sigma$-subalgebras that assures convergence almost every where of conditional expectations is given

Impuesto a las aguas saborizadas (refrescos): una alternativa para financiar el combate a la diabetes en México

Diabetes is the number one cause of death in people of productiveage in México, generating high economic and social costs at themicro and macroeconomic levels. In 90% of diabetes cases, theyare attributed to obesity, which is directly related to imbalancesin the diet and to a sedentary lifestyle. The average expenditurein soda consumption is equivalent to 12 and 7.5% of the basicfood basket, rural and urban, respectively. The objective ofthis study was to propose an alternative for financing diabetestreatment by imposing a tax on soda consumption. Themethodology consists in estimating the expenditure in diabetestreatment and soda consumption using information from theENSANUT 2006 databases, and performing a sensibility analysiswith different tax sums, and in view of different scenarios forproduct price elasticities. Results indicate that the incomegenerated from a tax of two to three pesos per liter, taking intoaccount a price elasticity between -0.5 and 1.0, might coverthe total cost generated by diabetes treatment in México, anddecrease its consumption in 13% to 19%.La diabetes es la primera causa de muerte en personas en edadproductiva en México, lo que genera elevados costos económicos ysociales a niveles micro y macroeconómicos. El 90% de los casos dediabetes se atribuyen a la obesidad, que se relaciona directamentecon desequilibrios en la dieta y sedentarismo. El gasto promedioen consumo de refresco equivale a 12 y 7.5% de la canasta básicaalimentaria rural y urbana respectivamente. El objetivo de estetrabajo fue proponer una alternativa para financiar el tratamientode diabetes imponiendo un impuesto al consumo de aguas saborizadas.La metodología consiste en estimar el gasto en el tratamientode diabetes y consumo de refresco a partir de las bases de datos dela ENSANUT 2006, y realizar un análisis de sensibilidad con distintosmontos de impuesto y ante distintos escenarios de elasticidadesprecio de los productos. Los resultados indican que el ingreso generadoa partir de un impuesto de entre dos y tres pesos por litro,contemplando una elasticidad precio entre -0.5 y 1.0, permitiríacubrir el gasto total generado por el tratamiento de diabetes enMéxico, y disminuir entre 13% y 19% su consumo

Fractional Newton-Raphson method

The Newton-Raphson (N-R) method is useful to find the roots of a polynomial of degree n, with n ∈ N. However, this method is limited since it diverges for the case in which polynomials only have complex roots if a real initial condition is taken. In the present work, we explain an iterative method that is created using the fractional calculus, which we will call the Fractional Newton-Raphson (F N-R) Method, which has the ability to enter the space of complex numbers given a real initial condition, which allows us to find both the real and complex roots of a polynomial unlike the classical Newton-Raphson method

Abelian groups of fractional operators

Comunicació presentada a la The 5th Mexican Workshop on Fractional Calculus, celebrada del 5 al 7 d'octubre de 2022 a Monterrey, Mèxic.Taking into count the large number of fractional operators that have been generated over the years, and considering that their number is unlikely to stop increasing at the time of writing this paper due to the recent boom of fractional calculus, everything seems to indicate that an alternative that allows to fully characterize some elements of fractional calculus is through the use of sets. Therefore, this paper presents a recapitulation of some fractional derivatives, fractional integrals, and local fractional operators that may be found in the literature, as well as a summary of how to define sets of fractional operators that allow to fully characterize some elements of fractional calculus, such as the Taylor series expansion of a scalar function in multi-index notation. In addition, it is presented a way to define finite and infinite Abelian groups of fractional operators through a family of sets of fractional operators and two different internal operations. Finally, using the above results, it is shown one way to define commutative and unitary rings of fractional operators

Proposal for use of the fractional derivative of radial functions in interpolation problems

This paper presents the construction of a family of radial functions aimed at emulating the behavior of the radial basis function known as thin plate spline (TPS). Additionally, a method is proposed for applying fractional derivatives, both partially and fully, to these functions for use in interpolation problems. Furthermore, a technique is employed to precondition the matrices generated in the presented problems through decomposition. Similarly, a method is introduced to define two different types of abelian groups for any fractional operator defined in the interval [0,1) , among which the Riemann–Liouville fractional integral, Riemann–Liouville fractional derivative, and Caputo fractional derivative are worth mentioning. Finally, a form of radial interpolant is suggested for application in solving fractional differential equations using the asymmetric collocation method, and examples of its implementation in differential operators utilizing the aforementioned fractional operators are shown

Acceleration of the order of convergence of a family of fractional fixed-point methods and its implementation in the solution of a nonlinear algebraic system related to hybrid solar receivers

This paper presents one way to define an uncountable family of fractional fixed-point methods through a set of matrices that can generate a group of fractional matrix operators, as well as one way to define groups of fractional operators that are isomorphic to the group of integers under the addition, and shows one way to classify and accelerate the order of convergence of the family of proposed iterative methods, which may be useful to continue expanding the applications of the fractional operators. The proposed method to accelerate the order of convergence is used in a fractional iterative method, and with the obtained method are solved simultaneously two nonlinear algebraic systems that depend on time-dependent parameters, and that allow obtaining the temperatures and efficiencies of a hybrid solar receiver. Finally, two uncountable families of fractional fixed-point methods are presented, in which the proposed method to accelerate convergence can be implemented

Sets of fractional operators and some of their applications

This chapter presents one way to define Abelian groups of fractional operators isomorphic to the group of integers under addition through a family of sets of fractional operators and a modified Hadamard product, as well as one way to define finite Abelian groups of fractional operators through sets of positive residual classes less than a prime number. Furthermore, it is presented one way to define sets of fractional operators which allow generalizing the Taylor series expansion of a vector-valued function in multi-index notation, as well as one way to define a family of fractional fixed-point methods and determine their order of convergence analytically through sets

Exploración de la geometría de la estimación en un modelo de regresión con variables instrumentales

En el modelo de regresión ortodoxo en el cual las variables predictoras son no aleatorias y los errores aleatorios son independientes e idénticamente distribuidos, el estimador de mínimos cuadrados ordinarios para los parámetros del modelo posee excelentes propiedades, como consistencia e insesgamiento, además es el mejor estimador lineal insesgado. Sin embargo, cuando las variables predictoras son aleatorias y además correlacionadas con los errores aleatorios, los estimadores de mínimos cuadrados ordinarios pueden perder propiedades muy deseables como la consistencia. En la investigación que se presenta se pretende explorar la geometría de métodos como el de variables instrumentales que pretende resolver algunas de las limitaciones planteadas para método de mínimos cuadrados ordinarios

Numerical solution using radial basis functions for multidimensional fractional partial differential equations of type Black–Scholes

In this paper, as far as the authors know, for the first time, a one-dimensional partial differential model is generalized using fractional differential operators and the same principle that provides the dimensional invariance of the radial basis functions methodology, resulting in a multidimensional fractional partial differential model that can be solved using a numerical scheme of radial basis functions. A radial basis functions scheme is proposed to solve numerically, on different node configurations, multidimensional fractional partial differential equations, both in space and in time. Using the QR factorization, a way to reduce the condition number of the interpolation matrices of the proposed scheme is presented, the resulting scheme is used to numerically solve the diffusion equation that may be obtained from the Black–Scholes model, as well as some generalizations of this diffusion model with fractional differential operators and multiple dimensions. The Caputo fractional derivative is discretized with an order error, with . The examples of fractional partial differential equations that are presented involve the Caputo fractional operator in the temporal part due to the memory phenomenon, and the Riemann–Liouville fractional operator in the spatial part due to the property of nonlocality