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A necessary and sufficient condition on a sequence ${\{{\mathcal{A}_{n}}\}_{n\in \mathbb{N}}}$ of σ-subalgebras which assures convergence almost everywhere of conditional expectations for functions in ${L^{\infty }}$ is given. It is proven that for $f\in {L^{\infty }}(\mathcal{A})$ \[ \mathsf{E}(f|{\mathcal{A}_{n}})\stackrel{a.e.}{\longrightarrow }\mathsf{E}(f|{\mathcal{A}_{\mu a.e.}}).\]

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 QR 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.

Nuestro trabajo plantea los aspectos que consideramos medulares de la Encíclica, en especial la unidad economía-sociedad-ecología, tomando como idea central la noción de “conversión ecológica” y, a partir de ésta, los límites de crecimiento de una civilización basada fundamentalmente en petróleo. Esto se debe a que los niveles de consumo encontrarán rápidamente la barrera de la propia naturaleza, pues la humanidad habrá consumido la mayor parte de los combustibles fósiles, a fines del siglo XXI. Se trata, en definitiva, del agotamiento de un ideal de ciencia y de progreso que no ha podido sostener las promesas del modelo ilustrado que las generó. En este contexto, presentamos las catástrofes ecológicas ocasionadas por derrames de petróleo en el último decenio y los aspectos positivos y negativos de los últimos procedimientos en extracción de petróleo; a pesar de estos esfuerzos y de la integración de energías renovables, la ecología en clave cristiana requiere de una renovación interior que lleve a una conciencia más plena del hombre como colaborar y custodio de la creación. Abstract: Our work raises the aspects that we consider to be central to the Encyclical, especially the unity between economy-society-ecology, taking as a central idea the notion of "ecological conversion" and, from this, the limits of growth of a civilization fundamentally based on oil. This is because consumption levels will quickly meet the barrier of nature itself, since humanity will have consumed most of the fossil fuels by the end of the 21st century. It is, in short, the exhaustion of an ideal of science and progress that has failed to keep the promises of the enlightened model in which it was born. In this context, we present the ecological catastrophes caused by oil spills in the last decade and the positive and negative aspects of the latest oil extraction procedures; Despite these efforts and the integration of renewable energies, ecology in a Christian key requires an interior renewal that leads to a fuller awareness of man as a collaborator and custodian of creation.

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.

The variation in moisture content between subsequent irrigations determines the use of infiltration equations that contain representative physical parameters of the soil when irrigation begins. This study analyzes the reliability of the hydrodynamic model to simulate the advanced phase in border irrigation. For the solution of the hydrodynamic model, a Lagrangian scheme in implicit finite differences is used, while for infiltration, the Kostiakov equation and the Green and Ampt equation are used and compared. The latter was solved using the Newton–Raphson method due to its implicit nature. The models were validated, and unknown parameters were optimized using experimental data available in the literature and the Levenberg–Marquardt method. The results show that it is necessary to use infiltration equations based on soil parameters, because in subsequent irrigations, the initial conditions change, modifying the advance curve in border irrigation. From the coupling of both equations, it is shown that the empirical Kostiakov equation is only representative for a specific irrigation event, while with the Green and Ampt equations, the subsequent irrigations can be modeled, and the advance/infiltration process can be observed in detail.

Due to the complexity imposed by all the attributes of the fracture network of many naturally fractured reservoirs, it has been observed that fluid flow does not necessarily represent a normal diffusion, i.e., Darcy’s law. Thus, to capture the sub-diffusion process, various tools have been implemented, from fractal geometry to characterize the structure of the porous medium to fractional calculus to include the memory effect in the fluid flow. Considering infinite naturally fractured reservoirs (Type I system of Nelson), a spatial fractional Darcy’s law is proposed, where the spatial derivative is replaced by the Weyl fractional derivative, and the resulting flow model also considers Caputo’s fractional derivative in time. The proposed model maintains its dimensional balance and is solved numerically. The results of analyzing the effect of the spatial fractional Darcy’s law on the pressure drop and its Bourdet derivative are shown, proving that two definitions of fractional derivatives are compatible. Finally, the results of the proposed model are compared with models that consider fractal geometry showing a good agreement. It is shown that modified Darcy’s law, which considers the dependency of the fluid flow path, includes the intrinsic geometry of the porous medium, thus recovering the heterogeneity at the phenomenological level.

There are a great many epidemiological models that have been implemented to describe COVID-19 data; however, few attempted to reproduce the entire phenomenon due to the complexity of modeling recurrent outbreaks. In this work a fractional growth model with delay is developed that implements the Caputo fractional derivative with 0<β≤1. Furthermore, in order to preserve the nature of the phenomenon and ensure continuity in the derivatives of the function, a method is proposed to construct an initial condition function to implement in the model with delay. This model is analyzed and generalized to model recurrent outbreaks. The model is applied to fit data of cumulative confirmed cases from Mexico, the United States, and Russia, obtaining excellent fitting corroborated by the coefficient of determination, where R2>0.9995 in all cases. Lastly, as a result of the implementation of the delay effect, the global phenomenon was decomposed into its local parts, allowing for directly comparing each outbreak and its different characteristics.

Growth models have been widely used to describe behavior in different areas of knowledge; among them the Logistics and Gompertz models, classified as models with a fixed inflection point, have been widely studied and applied. In the present work, a model is proposed that contains these growth models as extreme cases; this model is generalized by including the Caputo-type fractional derivative of order 0<β≤1, resulting in a Fractional Growth Model which could be classified as a growth model with non-fixed inflection point. Moreover, the proposed model is generalized to include multiple sigmoidal behaviors and thereby multiple inflection points. The models developed are applied to describe cumulative confirmed cases of COVID-19 in Mexico, US and Russia, obtaining an excellent adjustment corroborated by a coefficient of determination R2>0.999.

The present work proposes a new model to capture high heterogeneity of single phase flow in naturally fractured vuggy reservoirs. The model considers a three porous media reservoir; namely, fractured system, vugular system and matrix; the case of an infinite reservoir is considered in a full-penetrating wellbore. Furthermore, the model relaxes classic hypotheses considering that matrix permeability has a significant impact on the pressure deficit from the wellbore, reaching the triple permeability and triple porosity model wich allows the wellbore to be fed by all the porous media and not exclusively by the fractured system; where it is considered a pseudostable interporous flow. In addition, it is considered the anomalous flow phenomenon from the pressure of each independent porous medium and as a whole, through the temporal fractional derivative of Caputo type; the resulting phenomenon is studied for orders in the fractional derivatives in (0, 2), known as superdiffusive and subdiffusive phenomena. Synthetic results highlight the effect of anomalous flows throughout the entire transient behavior considering a significant permeability in the matrix and it is contrasted with the effect of an almost negligible matrix permeability. The model is solved analytically in the Laplace space, incorporating the Tartaglia–Cardano equations.

In the study of water transference in soil according to Darcy law, the knowledge of hydrodynamic characteristics, formed by the water retention curve θ(ψ), and the hydraulic conductivity curve K(ψ) are of great importance. The first one relates the water volumetric content (θ) with the water-soil pressure (ψ); the second one, the hydraulic conductivity (K) with the water-soil pressure. The objective of this work is to establish relationships between both curves using concepts of probability theory and fractal geometry in order to reduce the number of unknown functions. The introduction of four definitions used at the literature of the pore effective radius that is involve in the general model has permitted to establish four new specials models to predict the relative hydraulic conductivity. Some additional considerations related to the definitions of flow effective area and the tortuosity factor have allow us to deduce four classical models that are extensively used in different studies. In particular, we have given some interpretations of its empirical parameters in the fractal geometry context. The resulting functions for hydrodynamic characteristics can be utilized in many studies of water movement in the soil.