We provide an axiomatic base for the set of natural numbers, that has been proposed as a canonical construction, and use this definition of $\mathbb N$ to find several results on finite group theory. Every finite group $G$, is well represented with a natural number $N_G$; if $N_G=N_H$ then $H,G$ are in the same isomorphism class. We have a linear order on all finite groups, that is well behaved with respect to cardinality. In fact, if $H,G$ are two finite groups such that $|H|=m<n=|G|$, then $H<\mathbb Z_n\leq G$. There is also a canonical order for the elements of $G$ and we can define equivalent objects of $G$. This allows us to find the automorphisms of $G$. The Cayley table of $G$ takes canonical block form, and a minimal set of independent equations that define the group is obtained. We show how to find all groups of order $n$, and order them. Examples are given using all groups with order smaller than $10$. The canonical block form of the symmetry group $\Delta_4$ is given and we find its automorphisms. These results are extended to the infinite case. A real number is an infinite set of natural numbers. A real function is a set of real numbers, and a sequence of real functions $f_1,f_2,\ldots$ is well represented by a set of real numbers, as well. We make brief comments on treating the calculus of real numbers. In general, we represent mathematical objects using the smallest possible data-type. In the last section, mathematical objects are well assigned to tree structures. We conclude with brief comments on type theory and future work on computational and physical aspects of these representations.

We present the real number system as a natural generalization of the natural numbers. First, we prove the co-finite topology, $Cof(\mathbb N)$, is isomorphic to the natural numbers. Then, we generalize these results to describe the continuum $[0,1]$. Then we prove the power set $2^{\mathbb Z}$ contains a subset isomorphic to the non-negative real numbers, with all its defining structure of operations and order. Finally, we provide two different constructions of the entire real number line. We see that the power set $2^{\mathbb N}$ can be given the defining structure of $\mathbb R$. The constructions here provided give simple rules for calculating addition, multiplication, subtraction, division, powers and rational powers of real numbers, and logarithms. The supremum and infimum are explicitly constructed by means of a well defined algorithm that ends in denumerable steps. In section 5 we give evidence our construction of $\mathbb N$ and $\mathbb R$ are canonical; these constructions are as natural as possible. In the same section, we propose a new axiomatic basis for analysis. In the last section we provide a series of graphic representations and physical models that can be used to represent the real number system. We conclude that the system of real numbers is completely defined by the order structure of $\mathbb N$.}

This work focused on the evaluation of some machine learning (ML) models and their application in e-health, using intermediate nodes within an Internet of Things (IoT) platform used for heartbeat anomaly detection. For the evaluation of ML models, a set of statistical validation metrics was selected. These metrics were applied in the training, testing and validation phases of the models. The results obtained can determine relevant factors for the selection of ML models, either based on the statistical and intrinsic efficiency of the ML models, or on their suitability to be implemented in intermediate nodes within an IoT platform. the more Lightweight models such as Simple Linear Regression, Logistic Regression, and K Nearest Neighbors, could easily operate in intermediate nodes, and they are models that require low processing and storage to work. In conclusion, the approach for intermediate nodes of Internet of Things platforms using cognitive networks decreases the processing cost in cloud computing and transfers it to the fog layer.

Urban planning is recognized as an interaction between the state and society, which aims to articulate public policies in the territory, facilitating their administration in favor of greater development and well-being of society. However, this interaction becomes complex because consumption demands increase, and the needs of the community exceed the capacity of the urban ecosystem to supply them, hindering its sustainable functionality. With this panorama, it becomes relevant to study urban planning from a sustainable environmental planning perspective, based on four topics: urban planning, sustainability, resilience and smart cities. The methodology used is based on a bibliometric study through a PRISMA adjustment to 87 articles, supported by VOSviewer® to construct and visualize the co-occurrence networks of important terms extracted from a body of scientific literature. The main result is to consider cities with a complex systems approach that works like a gear, that is, there is a connective element between inter- and intra-urban processes. This relationship is the key factor that allows understanding their synchronization, stating that the deepening of each of these topics is crucial to the ideal of a territorial administration through time scales, by means of adaptive cycles, allowing to provide new tools to concepts such as carrying capacity and the measurement of the environmental footprint.

The management of large medical files is a critical challenge in the health sector. Conventional systems have security, scalability, and efficiency deficiencies. This project proposes a hybrid architecture between blockchain and cloud computing to address these challenges. Blockchain ensures the immutability and traceability of medical records, while the cloud allows scalable and efficient storage. Together, these technologies can transform data management in electronic health record applications. The essential objective of this research is to thoroughly look at the existing strategies, devices, and conventions that encourage a secure interaction between blockchain applications and the cloud. Special emphasis is set on maintaining the integrity and security of the blockchain while tackling the potential and efficiency of cloud infrastructures. Through a meticulous investigation of these approaches, along with their successful usage and encountered challenges, this paper offers a comprehensive overview of the present state of this integration. Altogether, this paper does not propose new concepts or developments. Its central focus is to provide a comprehensive and insightful examination of the modern landscape concerning the integration of blockchain and cloud advances. By shedding light on the existing approaches and assessing their viability, this work aims to highlight the current challenges and build up a solid establishment for future development. This, in turn, will cultivate a consistent integration of blockchain security with the dynamic potential of cloud computing while guaranteeing information integrity and security remain uncompromised. In conclusion, this paper serves as a important resource for analysts, specialists, and partners looking for to dig into and development the integration of blockchain and cloud innovations, subsequently contributing to a more secure and productive cloud environment.