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In applied and computational mathematics, quaternions are fundamental in representing three-dimensional rotations. However, specific types of quaternionic linear matrix equations remain few explored. This study introduces new quaternionic linear matrix equations and their necessary and sufficient conditions for solvability. We employ a methodology involving lemmas and ranks of coefficient matrices to develop a novel algorithm. This algorithm is validated through numerical examples, showing its applications in advanced fields. In control theory, these equations are used for analyzing control systems, particularly for spacecraft attitude control in aerospace engineering and for control of arms in robotics. In quantum computing, quaternionic equations model quantum gates and transformations, which are important for algorithms and error correction, contributing to the development of fault-tolerant quantum computers. In signal processing, these equations enhance multidimensional signal filtering and noise reduction, with applications in color image processing and radar signal analysis. We extend our study to include cases of $ \eta $-Hermitian and i-Hermitian solutions. Our work represents an advancement in applied mathematics, providing computational methods for solving quaternionic matrix equations and expanding their practical applications.

Many studies have been performed in different regions of the world as a result of the COVID-19 pandemic. In this work, we perform a statistical study related to the number of vaccinated cases and the number of deaths due to COVID-19 in ten South American countries. Our objective is to group countries according to the aforementioned variables. Once the groups of countries are built, they are characterized based on common properties of countries in the same group and differences between countries that are in different groups. Countries are grouped using principal component analysis and K-means analysis. These methods are combined in a single procedure that we propose for the classification of the countries. Regarding both variables, the countries were classified into three groups. Political decisions, availability of resources, bargaining power with suppliers and health infrastructure among others are some of the factors that can affect both the vaccination process and the timely care of infected people to avoid death. In general, the countries acted in a timely manner in relation to the vaccination of their citizens with the exception of two countries. Regarding the number of deaths, all countries reached peaks at some point in the study period.

Petrochemical and dairy industries, waste management, and paper manufacturing fall under the category of process industries where flow and liquid control are essential. Even when liquids are mixed or chemically treated in interconnected tanks, the fluid and flow should constantly be observed and controlled, especially when dealing with nonlinearity and imperfect plant models. In this study, we propose a nonlinear dynamic multiple-input multiple-output (MIMO) plant model. This model is then transformed through linearization, a technique frequently utilized in the analysis and modeling of fractional processes, and decoupling for decentralized fixed-structure H-infinity robust control design. Simulation tests based on MATLAB and SIMULINK are subsequently executed. Numerous assessments are conducted to evaluate tracking performance, external disturbance rejection, and plant parameter fluctuations to gauge the effectiveness of the proposed model. The objective of this work is to provide a framework that anticipates potential outcomes, paving the way for implementing a reliable controller synthesis for MIMO-connected tanks in real-world scenarios.

The continuous Bernoulli distribution is defined on the unit interval and has a unique property related to fractiles. A fractile is a position on a probability density function where the corresponding surface is a fixed proportion. This article presents the derivation of properties of the continuous Bernoulli distribution and formulates a fractile or quantile regression model for a unit response using the exponentiated continuous Bernoulli distribution. Monte Carlo simulation studies evaluate the performance of point and interval estimators for both the continuous Bernoulli distribution and the fractile regression model. Real-world datasets from science and education are analyzed to illustrate the modeling abilities of the continuous Bernoulli distribution and the exponentiated continuous Bernoulli quantile regression model.

In this paper, we introduce a SIVR model using the Laplace Adomian decomposition. This model focuses on a new trend in mathematical epidemiology dedicated to studying the characteristics of vaccination of infected communities. We analyze the epidemiological parameters using equilibrium stability and numerical analysis techniques. New mathematical strategies are also applied to establish our epidemic model, which is a pandemic model as well. In addition, we mathematically establish the chance for the next wave of any pandemic disease and show that a consistent vaccination strategy could control it. Our proposal is the first model introducing a vaccination strategy to actively infected cases. We are sure this work will serve as the basis for future research on COVID-19 and pandemic diseases since our study also considers the vaccinated population.

Source coding maps elements from an information source to a sequence of alphabetic symbols. Then, the source symbols can be recovered exactly from the binary units. In this paper, we derive an approach that includes information variation in the source coding. The approach is more realistic than its standard version. We employ the Shannon entropy for coding the sequences of a source. Our approach is also helpful for short sequences when the central limit theorem does not apply. We rely on a quantifier of the information variation as a source. This quantifier corresponds to the second central moment of a random variable that measures the information content of a source symbol; that is, considering the standard deviation. An interpretation of typical sequences is also provided through this approach. We show how to use a binary memoryless source as an example. In addition, Monte Carlo simulation studies are conducted to evaluate the performance of our approach. We apply this approach to two real datasets related to purity and wheat prices in Brazil.

The focus on renewable energy is increasing globally to lessen reliance on conventional sources and fossil fuels. For renewable energy systems to work at their best and produce the desired results, precise feedback control is required. Microbial electrochemical cells (MEC) are a relatively new technology for renewable energy. In this study, we design and implement a model-based robust controller for a continuous MEC reactor. We compare its performance with those of traditional methods involving a proportional integral derivative (PID), H-infinity (H∞) controller and PID controller tuned by intelligent genetic algorithms. Recently, a dynamic model of a MEC continuous reactor was proposed, which describes the complex dynamics of MEC through a set of nonlinear differential equations. Until now, no model-based control approaches for MEC have been proposed. For optimal and robust output control of a continuous-reactor MEC system, we linearize the model to state a linear time-invariant (LTI) state-space representation at the nominal operating point. The LTI model is used to design four different types of controllers. The designed controllers and systems are simulated, and their performances are evaluated and compared for various operating conditions. Our findings show that a structured linear fractional transformation (LFT)-based H∞ control approach is much better than the other approaches against various performance parameters. The study provides numerous possibilities for control applications of continuous MEC reactor processes.

Direct current (DC) servo motors are central to many complex systems, such as electrical, electro-mechanical, and electro-hydraulic frameworks. In practice, these systems can have nonlinear characteristics and parameter variations. Accurate model representation and position tracking of DC motors are the main issues in many real systems, such as twin rotors, aircraft, airships, and robot manipulators. The precise position tracking of these systems has already been achieved using conventional H-infinity (H∞) controllers. However, the order and structure become more intricate when employing complex weights to shape the closed-loop system, which limits the current proposals. To overcome the above-mentioned limitations, in this article, we provide a precise angular position tracking of a DC servo motor utilizing an intelligent, robust linear controller based on a fixed-structure linear fractional transformation. The conventional H∞ controllers are based on the minimization of an unstructured linear fractional transformation objective function that leads to a complex design of these controllers. The main advantage of the proposed intelligent H∞ synthesis is the fixed and simple structure that increases its practical implementation. The methodology is formulated in the MATLAB software for the robust design of the proposed synthesis based on an intelligent fixed-structure H∞ optimization. Simulation results are compared with conventional H∞ and proportional-integral-derivative controllers. The results are also validated experimentally.

In diverse branches of mathematics, several inequalities have been studied and applied. In this article, we improve Furuta’s inequality. Subsequently, we apply this improvement to obtain new radius inequalities that not been reported in the current literature. Numerical examples illustrate the main findings.

Determining success factors for managing supply chains is a relevant aspect for companies. Then, modeling the relationship between inventory cost savings and supply chain success factors is a route for stating such a determination. This is particularly important in pharmacies and food nutrition services (FNS), where the advances made on this topic are still scarce. In this article, we propose and formulate a robust compromise (RoCo) multi-criteria model based on non-linear programming and time-dependent demand. The novelty of our proposal is in defining a score that allows us to measure the mentioned success factors in a simple way, in meeting together all three elements (RoCo multi-criteria, non-linear programming, and time-dependent demand) to state a new model, and in applying it to pharmacies and FNS. This model relates inventory cost savings for pharmacy/FNS and success factors across their supply chains. Savings of inventory costs are predicted by lot sizes to be purchased and computed by comparing optimal and true inventory costs. We utilize a system that records the movements and costs of products to collect the data. Factors, such as purchasing organization, economies of scale, and synchronized supply, are assumed using the purchase system, with these factors ranked on a Likert scale. We consider multilevel relationships between savings obtained for 79 pharmacy/FNS products, and success factor scores according to these products. To deal with the endogeneity bias of the relationships proposed, internal instrumental variables are employed by utilizing generalized statistical moments. Among our main conclusions, we state that the greatest cost savings obtained from inventory models are directly associated with low-success supply chain factors. In this association, the success factors operate as endogenous variables, with respect to inventory cost savings, given the simultaneity of their relationship with cost savings when inventory decision-making.

Different types of mathematical inequalities have been largely analyzed and employed. In this paper, we introduce improvements to some Ostrowski type inequalities and present their corresponding proofs. The presented proofs are based on applying the celebrated Hayashi inequality to certain functions. We provide examples that show these improvements. Illustrations of the obtained results are stated in a probability framework.

In this paper, we study a type of disease that unknowingly spreads for a long time, but by default, spreads only to a minimal population. This disease is not usually fatal and often goes unnoticed. We propose and derive a novel epidemic mathematical model to describe such a disease, utilizing a fractional differential system under the Atangana–Baleanu–Caputo derivative. This model deals with the transmission between susceptible, exposed, infected, and recovered classes. After formulating the model, equilibrium points as well as stability and feasibility analyses are stated. Then, we present results concerning the existence of positivity in the solutions and a sensitivity analysis. Consequently, computational experiments are conducted and discussed via proper criteria. From our experimental results, we find that the loss and regain of immunity result in the gain and loss of infections. Epidemic models can be linked to symmetry and asymmetry from distinct points of view. By using our novel approach, much research may be expected in epidemiology and other areas, particularly concerning COVID-19, to state how immunity develops after being infected by this virus.

Governments must consider different issues when deciding on the location of healthcare centers. In addition to the costs of opening such centers, three further elements should be addressed: accessibility, demand, and equity. Such locations must be chosen to meet the corresponding demand, so that they guarantee a socially equitable distribution, and to ensure that they are accessible to a sufficient degree. The location of the centers must be chosen from a set of possible facilities to guarantee certain minimum standards for the operational viability of the centers. Since the set of potential locations does not necessarily cover the demand of all geographical zones, the efficiency criterion must be maximized. However, the efficient distribution of resources does not necessarily meet the equity criterion. Thus, decision-makers must consider the trade-off between these two criteria: efficiency and equity. The described problem corresponds to the challenge that governments face in seeking to minimize the impact of the pandemic on citizens, where healthcare centers may be either public hospitals that care for COVID-19 patients or vaccination points. In this paper, we focus on the problem of a zone-divided region requiring the localization of healthcare centers. We propose a non-linear programming model to solve this problem based on a coverage formula using the Gini index to measure equity and accessibility. Then, we consider an approach using epsilon constraints that makes this problem solvable with mixed integer linear computations at each iteration. A simulation algorithm is also considered to generate problem instances, while computational experiments are carried out to show the potential use of the proposed mathematical programming model. The results show that the spatial distribution influences the coverage level of the healthcare system. Nevertheless, this distribution does not reduce inequity at accessible healthcare centers, as the distribution of the supply of health centers must be incorporated into the decision-making process.

In this paper, we propose and derive a Birnbaum–Saunders distribution to model bimodal data. This new distribution is obtained using the product of the standard Birnbaum–Saunders distribution and a polynomial function of the fourth degree. We study the mathematical and statistical properties of the bimodal Birnbaum–Saunders distribution, including probabilistic features and moments. Inference on its parameters is conducted using the estimation methods of moments and maximum likelihood. Based on the acceptance–rejection criterion, an algorithm is proposed to generate values of a random variable that follows the new bimodal Birnbaum–Saunders distribution. We carry out a simulation study using the Monte Carlo method to assess the statistical performance of the parameter estimators. Illustrations with real-world data sets from environmental and medical sciences are provided to show applications that can be of potential use in real problems.

In this paper, we extend the use of disjoint orthogonal components to three-way table analysis with the parallel factor analysis model. Traditional methods, such as scaling, orthogonality constraints, non-negativity constraints, and sparse techniques, do not guarantee that interpretable loading matrices are obtained in this model. We propose a novel heuristic algorithm that allows simple structure loading matrices to be obtained by calculating disjoint orthogonal components. This algorithm is also an alternative approach for solving the well-known degeneracy problem. We carry out computational experiments by utilizing simulated and real-world data to illustrate the benefits of the proposed algorithm.

One of the main drawbacks of the traditional methods for computing components in the three-way Tucker model is the complex structure of the final loading matrices preventing an easy interpretation of the obtained results. In this paper, we propose a heuristic algorithm for computing disjoint orthogonal components facilitating the analysis of three-way data and the interpretation of results. We observe in the computational experiments carried out that our novel algorithm ameliorates this drawback, generating final loading matrices with a simple structure and then easier to interpret. Illustrations with real data are provided to show potential applications of the algorithm.