AppliedMath, Volume 5, Issue 2 (June 2025) – 42 articles

The aim of this paper is to propose mathematical models to predict and optimize the cost of wastewater treatment using bacteria and oxygen under fluctuating resource and cultivation conditions. We have thus developed deterministic mathematical models based on dynamic systems and applied optimal control theory to reduce treatment costs. Two wastewater treatment models are proposed: one using only one type of aerobic bacteria, thermophilic bacteria; and the second using two types of aerobic bacteria, thermophilic and mesophilic bacteria. For each model, an optimal control problem is solved and numerical simulations illustrate the theoretical results. Full article

In this study, we introduce a novel application of the Smooth Overlap of Atomic Positions (SOAP) descriptor for pixel-wise image feature extraction and classification as a benchmark for SOAP point cloud feature extraction, using MNIST handwritten digits as a benchmark. By converting 2D images into 3D point sets, we compute pixel-centered SOAP vectors that are intrinsically invariant to translation, rotation, and mirror symmetry. We demonstrate how the descriptor’s hyperparameters—particularly the cutoff radius—significantly influence classification accuracy, and show that the high-dimensional SOAP vectors can be efficiently compressed using PCA or autoencoders with minimal loss in predictive performance. Our experiments also highlight the method’s robustness to positional noise, exhibiting graceful degradation even under substantial Gaussian perturbations. Overall, this approach offers an effective and flexible pipeline for extracting rotationally and translationally invariant image features, potentially reducing reliance on extensive data augmentation and providing a robust representation for further machine learning tasks. Full article

This article aims to limit the rule explosion problem affecting market basket analysis (MBA) algorithms. More specifically, it is shown how, if the minimum support threshold is not specified explicitly, but in terms of the number of items to consider, it is possible to compute an upper bound for the number of generated association rules. Moreover, if the results of previous analyses (with different thresholds) are available, this information can also be taken into account, hence refining the upper bound and also being able to compute lower bounds. The support determination technique is implemented as an extension to the Apriori algorithm but may be applied to any other MBA technique. Tests are executed on benchmarks and on a real problem provided by one of the major Italian supermarket chains, regarding more than $500,000$ transactions. Experiments show, on these benchmarks, that the rate of growth in the number of rules between tests with increasingly more permissive thresholds ranges, with the proposed method, is from $21.4$ to $31.8$, while it would range from $39.6$ to $3994.3$ if the traditional thresholding method were applied. Full article

Despite its significance, hysteresis remains underrepresented in mainstream models of plasticity. In this work, we propose a novel framework that explicitly models hysteresis in simple one- and two-neuron models. Our models capture key feedback-dependent phenomena such as bistability, multistability, periodicity, quasi-periodicity, and chaos, offering a more accurate and general representation of neural adaptation. This opens the door to new insights in computational neuroscience and neuromorphic system design. Synaptic weights change in several contexts or mechanisms including, Bienenstock–Cooper–Munro (BCM) synaptic modification, where synaptic changes depend on the level of post-synaptic activity; homeostatic plasticity, where all of a neuron synapses simultaneously scale up or down to maintain stability; metaplasticity, or plasticity of plasticity; neuromodulation, where neurotransmitters influence synaptic weights; developmental processes, where synaptic connections are actively formed, pruned and refined; disease or injury; for example, neurological conditions can induce maladaptive synaptic changes; spike-time dependent plasticity (STDP), where changes depend on the precise timing of pre- and postsynaptic spikes; and structural plasticity, where changes in dendritic spines and axonal boutons can alter synaptic strength. The ability of synapses and neurons to change in response to activity is fundamental to learning, memory formation, and cognitive adaptation. This paper presents simple continuous and discrete neuro-modules with adapting feedback synapses which in turn are subject to feedback. The dynamics of continuous periodically driven Hopfield neural networks with adapting synapses have been investigated since the 1990s in terms of periodicity and chaotic behaviors. For the first time, one- and two-neuron models are considered as parameters are varied using a feedback mechanism which more accurately represents real-world simulation, as explained earlier. It is shown that these models are history dependent. A simple discrete two-neuron model with adapting feedback synapses is analyzed in terms of stability and bifurcation diagrams are plotted as parameters are increased and decreased. This work has the potential to improve learning algorithms, increase understanding of neural memory formation, and inform neuromorphic engineering research. Full article

This article studies the location–price Hotelling game. Numerous studies have been conducted on the Hotelling game with simultaneous decisions; however, in real-life scenarios, decisions are frequently sequential. Unfortunately, studies on the sequential Hotelling (SHOT) game are quite scarce. This article contributes to the study of the SHOT game by considering the case in which the location of one of the players, either the leader or the follower, is externally fixed. The game is studied analytically and by numerical simulation to address scenarios where mathematical analysis is cumbersome due to the discontinuous nature of the game. Simulation is found to be particularly useful in evaluating the subgame perfect equilibrium (SPE) solution of these SHOT games, where the follower outperforms the leader as a very general rule, with very few exceptions. This article complements a previous study of the SHOT game where the two locations are parameterized and paves the way to address the analysis of more sophisticated formulations of the SHOT game, such as those with reservation cost and with elastic demand. Full article

After linear differential systems in the plane, the easiest systems are quadratic polynomial differential systems in the plane. Due to their nonlinearity and their many applications, these systems have been studied by many authors. Such quadratic polynomial differential systems have been divided into ten families. Here, for two of these families, we classify all topologically distinct phase portraits in the Poincaré disc. These two families have already been studied previously, but several mistakes made there are repaired here thanks to the use of a more powerful technique. This new technique uses the invariant theory developed by the Sibirskii School, applied to differential systems, which allows to determine all the algebraic bifurcations in a relatively easy way. Even though the goal of obtaining all the phase portraits of quadratic systems for each of the ten families is not achievable using only this method, the coordination of different approaches may help us reach this goal. Full article

For a traditional assignment problem with the same number of objects and agents, we introduce a new assignment lottery based on the notion of rank and analyze some of its properties. In particular, we prove that, like the Random Serial Dictatorship, it is ex post efficient and guarantees positive probability to each Pareto optimal deterministic assignment; moreover, the expected rank of this new assignment lottery, which is a measure of the social welfare, cannot be greater than the Random Serial Dictatorship’s one and there exist assignment problems where it is strictly lower. Full article

Schröder based series for the principal and negative one branches of the Lambert W function are defined; the series are generic and are in terms of an initial, arbitrary, approximating function. Upper and lower bounds for the initial approximating functions, consistent with convergence, are determined. Approximations for both branches of the Lambert W function are proposed which have modest relative error bounds over their domains of definition and which are suitable as initial approximation functions for a convergent Schröder series. For the principal branch, a proposed approximation yields, for a series of order 128, a relative error bound below 10⁻¹³⁶. For the negative one branch, a proposed approximation yields, for a series of order 128, a relative error bound below 10⁻¹⁴³. Applications of the approximations for the principal and negative one branches include new approximations for the Lambert W function, analytical approximations for the integral of the Lambert W function, upper and lower bounded functions for the Lambert W function, approximations for the power of the Lambert W function and approximations to the solution of the equations c^c = y and C^C = e^v, respectively, for c and C. Full article

This study presents the soft limit and upper (lower) soft limit proposed by Molodtsov, with several theoretical contributions. It investigates some of their basic properties, such as some fundamental soft limit rules, the relation between soft limit and boundedness, and the sandwich/squeeze theorem. Moreover, the paper proposes left and right soft limits and studies some of their main properties. Furthermore, it defines the soft limit at infinity and explores some of its basic properties. Additionally, the present study exemplifies these concepts and their properties to better understand them. The paper then compares the aforesaid concepts with their classical forms. Afterward, this paper presents soft continuity and upper (lower) soft continuity, proposed by Molodtsov, theoretically contributes to these concepts, and investigates some of their key properties, such as some fundamental soft continuity rules, the relation between soft continuity and boundedness, Bolzano’s theorem, and the intermediate value theorem. Moreover, it defines left and right soft continuity and studies some of their basic properties. The present study exemplifies soft continuity types and their properties. In addition, it compares them with their classical forms. Finally, this study discusses whether the aspects should be further analyzed. Full article

In the modern era of informatics, where data are very important, efficient management of data is necessary and critical. Two of the most important data management techniques are searching and data ordering (technically sorting). Traditional sorting algorithms work in quadratic time $\left(\mathcal{O}\left(x^2\right)\right)$, and in the optimized cases, they take linearithmic time $\left(\mathcal{O}\left(x·\log x\right)\right)$, with no existing method surpass this lower bound, given arbitrary data, i.e., ordering a list of cardinality x in $\left(\mathcal{O}\left(x·\log x\right)-ϵ\left(x\right)\right)\forall ϵ\left(x\right)>0$. This research proposes Search-o-Sort, which reinterprets sorting in terms of searching, thereby offering a new framework for ordering arbitrary data. The framework is applied to classical search algorithms,–Linear Search, Binary Search (in general, k-ary Search), and extended to more optimized methods such as Interpolation and Jump Search. The analysis suggests theoretical pathways to reduce the computational complexity of sorting algorithms, thus enabling algorithmic development based on the proposed viewpoint. Full article

This study innovatively explores vibrational control with reference to elliptical thickness variation. Traditionally, plate vibrations have been analysed by incorporating circular, linear, parabolic, and exponential thickness variations. However, these variations often fall short in optimizing vibrational characteristics. So, we develop a new formula specifically for orthotropic as well as isotropic plates with elliptical thickness profiles and employ the Rayleigh–Ritz method to calculate the vibrational frequencies of the plate. This research demonstrates that elliptical variation significantly reduces vibrational frequencies compared to conventional thickness profiles. The findings indicate that this unique configuration enhances vibrational control, offering potential applications in engineering fields where vibration reduction is essential. The results provide a foundation for further exploration of non-standard thickness variations in the design of advanced structural components. The study reveals that the elliptical variation in tapering parameter is a much better choice than other variation parameters studied in the literature for the purpose of optimizing the vibrational frequency of plates. Full article

The tumor microenvironment is a highly dynamic and complex system where cellular interactions evolve over time, influencing tumor growth, immune response, and treatment resistance. In this study, we develop a graph-theoretic framework to model the tumor microenvironment, where nodes represent different cell types, and edges denote their interactions. The temporal evolution of the tumor microenvironment is governed by fundamental biological processes, including proliferation, apoptosis, migration, and angiogenesis, which we model using differential equations with stochastic effects. Specifically, we describe tumor cell population dynamics using a logistic growth model incorporating both apoptosis and random fluctuations. Additionally, we construct a dynamic network to represent cellular interactions, allowing for an analysis of structural changes over time. Through numerical simulations, we investigate how key parameters such as proliferation rates, apoptosis thresholds, and stochastic fluctuations influence tumor progression and network topology. Our findings demonstrate that graph theory provides a powerful mathematical tool to analyze the spatiotemporal evolution of tumors, offering insights into potential therapeutic strategies. This approach has implications for optimizing cancer treatments by targeting critical network structures within the tumor microenvironment. Full article

In this paper, we scrutinize a generalized (2+1)-dimensional nonlinear wave equation (NWE) which describes the waves propagation in plasma physics by utilizing Lie group analysis, Lie point symmetry are obtained and thereafter symmetry reductions are performed which lead to nonlinear ordinary differential equations (NODEs). These NODEs are then solved using various methods that includes the direct integration method. This then leads us to explicit exact solutions of NWE. Graphical representation of the achieved results is given to have a good understanding of the nature of solutions obtained. In conclusion, we construct conserved vectors of the NWE by invoking Ibragimov’s theorem. Full article

Dynamic mode decomposition with control (DMDc) is a widely used technique for analyzing dynamic systems influenced by external control inputs. It is a recent development and an extension of dynamic mode decomposition (DMD) tailored for input–output systems. In this work, we investigate and analyze an alternative approach for computing DMDc. Compared to the traditional formulation, the proposed method restructures the computation by decoupling the influence of the state and control components, allowing for a more modular and interpretable implementation. The algorithm avoids compound operator approximations typical of standard approaches, which makes it potentially more efficient in real-time applications or systems with streaming data. The new scheme aims to improve computational efficiency while maintaining the reliability and accuracy of the decomposition. We provide a theoretical proof that the dynamic modes produced by the proposed method are exact eigenvectors of the corresponding Koopman operator. Compared to the standard DMDc approach, the new algorithm is shown to be more efficient, requiring fewer calculations and less memory. Numerical examples are presented to demonstrate the theoretical results and illustrate potential applications of the modified approach. The results highlight the promise of this alternative formulation for advancing data-driven modeling and control in various engineering and scientific domains. Full article

We investigate the application of the Shapley value in addressing risk-related challenges, focusing on two primary areas. The first area explores the role of the Shapley value in the financial sector, specifically in managing portfolio risk. By conceptualizing a portfolio of assets as a cooperative game, we analyze the contribution of individual securities to the reduction in overall portfolio risk. The second area addresses emergency facility logistics, where the Shapley value is utilized to optimize the selection of potential facility locations and mitigate the risks associated with the storage and transportation of hazardous materials. Using Markowitz’s mean-variance framework, the Shapley value facilitates a fair and efficient allocation of risk across portfolio assets, identifying both risk-increasing and risk-reducing assets. Through numerical experiments, we demonstrate that the Shapley value offers valuable insights into the equitable distribution of financial resources and the strategic placement of facilities to manage systemic risks. These findings highlight the practical advantages of integrating game-theoretic approaches into risk management strategies to enhance fairness, efficiency, and the robustness of decision-making processes. Full article

Over the past decade, the field of object detection has advanced remarkably, especially in the accurate recognition of medium- and large-sized objects. Nevertheless, detecting small objects is still difficult because their low-resolution appearance provides insufficient discriminative features, and they often suffer severe occlusions, particularly in the safety-critical context of autonomous driving. Conventional detectors often fail to extract sufficient information from shallow feature maps, which limits their ability to detect small objects with high precision. To address this issue, we propose the ECAN-Detector, an efficient context-aggregation method designed to enrich the feature representation of shallow layers, which are particularly beneficial for small-object detection. The model first employs an additional shallow detection layer to extract high-resolution features that provide more detailed information for subsequent stages of the network, and then incorporates a dynamic scaled transformer (DST) that enriches spatial perception by adaptively fusing global semantics and local context. Concurrently, a context-augmentation module (CAM) embedded in the shallow layer complements both global and local features relevant to small objects. To further boost the average precision of small-object detection, we implement a faster method utilizing two reparametrized convolutions in the detection head. Finally, extensive experiments conducted on the VisDrone2012-DET and VisDrone2021-DET datasets verified that our proposed method surpasses the baseline model, and achieved a significant improvement of 3.1% in AP and 3.5% in $A{P}_s$. Compared with recent state-of-the-art (SOTA) detectors, ECAN Detector delivers comparable accuracy yet preserves real-time throughput, reaching 54.3 FPS. Full article

In this article, we propose a nonlinear mathematical model that incorporates a discrete time delay. The model is used to analyze the dynamics of a socioeconomic system that includes economic growth, corruption, and unemployment. We introduce the time delay in the logistic economic growth term due to the effect of the previous state of the economic growth on its current state. A local stability analysis is performed to investigate the dynamics of the socioeconomic system. We established conditions for the existence of Hopf bifurcations and the appearance of economic limit cycles. We found threshold values for the discrete-time delay in which these Hopf bifurcations occur. We corroborate the theoretical findings by performing numerical simulations for a variety of scenarios. We find various interesting socioeconomic situations where different socioeconomic limit cycles occur. Finally, we present a discussion and future directions of research. Full article

The method of particular solutions using polynomial basis functions (MPS-PBF) has been extensively used to solve various types of partial differential equations. Traditional methods employing radial basis functions (RBFs)—such as Gaussian, multiquadric, and Matérn functions—often suffer from accuracy issues due to their dependence on a shape parameter, which is very difficult to select optimally. In this study, we adopt the MPS-PBF to solve the time-dependent Schrödinger equation in two dimensions. By utilizing polynomial basis functions, our approach eliminates the need to determine a shape parameter, thereby simplifying the solution process. Spatial discretization is performed using the MPS-PBF, while temporal discretization is handled via the backward Euler and Crank–Nicolson methods. To address the ill conditioning of the resulting system matrix, we incorporate a multi-scale technique. To validate the efficacy of the proposed scheme, we present four numerical examples and compare the results with known analytical solutions, demonstrating the accuracy and robustness of the scheme. Full article

Although fractional calculus has evolved significantly since its origin in the 1695 correspondence between Leibniz and L’Hôpital, the numerical treatment of multi-order fractional differential equations remains a challenge. Existing methods are often either computationally expensive or reliant on complex operational frameworks that hinder their broader applicability. In the present study, a novel numerical algorithm is proposed based on orthogonal hybrid functions (HFs), which were constructed as linear combinations of piecewise constant sample-and-hold functions and piecewise linear triangular functions. These functions, belonging to the class of degree-1 orthogonal polynomials, were employed to obtain the numerical solution of multi-order fractional differential equations defined in the Caputo sense. A generalized one-shot operational matrix was derived to explicitly express the Riemann–Liouville fractional integral of HFs in terms of the HFs themselves. This allowed the original multi-order fractional differential equation to be transformed directly into a system of algebraic equations, thereby simplifying the solution process. The developed algorithm was then applied to a range of benchmark problems, including both linear and nonlinear multi-order FDEs with constant and variable coefficients. Numerical comparisons with well-established methods in the literature revealed that the proposed approach not only achieved higher accuracy but also significantly reduced computational effort, demonstrating its potential as a reliable and efficient numerical tool for fractional-order modeling. Full article

Understanding the time dynamics of systemic risk in banking networks is crucial for preventing financial crises and ensuring economic stability. This paper aims to quantify key transition times in the evolution of distress within a banking system using a mathematical framework. We investigate the dynamics of systemic risk in a hypothetical, homogeneous banking network using the Undistressed–Exposed–Distressed–Recovered (UEDR) model. The UEDR model, inspired by compartmental epidemic frameworks, captures how financial distress propagates and recedes through interactions between banks. It is selected because of its tractability and its ability to distinguish between different stages of bank vulnerability. We focus on two critical times, denoted as $t_1$ and $t_2$, which play a fundamental role in understanding the behavior of the distressed compartment (representing the number of distressed banks) over time. The time $t_1$ represents the first instance of a decrease in the number of distressed banks, indicating the containment of systemic risk. On the other hand, the time $t_2$ marks the onset when the number of undistressed banks falls below a specified threshold, signifying the restoration of financial stability. We examine these time dependencies by considering the initial conditions of the UEDR model and assess their characteristics using partial differential equations. We establish the continuity, smoothness, and uniqueness of solutions for $t_1$ and $t_2$, along with their corresponding boundary conditions. Furthermore, we provide explicit representation formulas for $t_1$ and $t_2$, allowing for precise estimation when the initial population compartments are large. Our results provide practical insights for financial regulators and policymakers in determining time-sensitive interventions for mitigating systemic risk and accelerating recovery in banking systems. The findings highlight how mathematical modeling can inform real-time risk management strategies in financial networks. Full article

Background: The issue of digital piracy is increasingly prevalent, with its proliferation further fueled by the widespread use of social media outlets such as WhatsApp, Snapchat, Instagram, Pinterest, and X. These platforms have become hotspots for the unauthorized sharing of copyrighted materials without due recognition to the original creators. Current techniques for digital watermarking are inadequate; they frequently choose less-than-ideal locations for embedding watermarks. This often results in a compromise on maintaining critical relationships within the data. Purpose: This research aims to tackle the growing problem of digital piracy, which represents a major risk to rights holders in various sectors, most notably those involved in entertainment. The goal is to devise a robust watermarking approach that effectively safeguards intellectual property rights and guarantees rightful earnings for those who create content. Approach: To address the issues at hand, this study presents an innovative technique for digital video watermarking. Utilizing the 2D-DWT along with the KAZE feature detection algorithm, which incorporates the Accelerated Segment Test with Zero Eigenvalue, scrutinize and pinpoint data points that exhibit circular symmetry. The KAZE algorithm pinpoints a quintet of stable features within the brightness aspect of video frames to act as central embedding sites. This research selects the chief embedding site by identifying the point of greatest intensity on a specific arc segment on a circle’s edge, while three other sites are chosen based on principles of circular symmetry. Following these procedures, the proposed method subjects videos to several robustness tests to simulate potential disturbances. The efficacy of the proposed approach is quantified using established objective metrics that confirm strong correlation and outstanding visual fidelity in watermarked videos. Moreover, statistical validation through t-tests corroborates the effectiveness of the watermarking strategy in maintaining integrity under various types of assaults. This fortifies the team’s confidence in its practical deployment. Full article

Planning a multi-joint minimum-effort coordinated human movement to achieve a visual goal is computationally difficult: (i) The number of anatomical elemental movements of the human body greatly exceeds the number of degrees of freedom specified by visual goals; and (ii) the mass–inertia mechanical load about each elemental movement varies not only with the posture of the body but also with the mechanical interactions between the body and the environment. Given these complications, the amount of nonlinear dynamical computation needed to plan visually-guided movement is far too large for it to be carried out within the reaction time needed to initiate an appropriate response. Consequently, we propose that, as part of motor and visual development, starting with bootstrapping by fetal and neonatal pattern-generator movements and continuing adaptively from infancy to adulthood, most of the computation is carried out in advance and stored in a motor association memory network. From there it can be quickly retrieved by a selection process that provides the appropriate movement synergy compatible with the particular visual goal. We use theorems of Riemannian geometry to describe the large amount of nonlinear dynamical data that have to be pre-computed and stored for retrieval. Based on that geometry, we argue that the logical mathematical sequence for the acquisition of these data parallels the natural development of visually- guided human movement. Full article

We show how the dynamics of inflation as represented by the Phillips curve follow from a response formalism suggested by Phillips and motivated by the Keynesian notion that it takes time for an economy to respond to an economic shock. The resulting expressions for the Phillips curve are isomorphic to those of anelasticity—a result that provides a straightforward and parsimonious approach to macroeconomic-model construction. Our approach unifies forms of the Phillips curve that are used to account for time dependence of the Phillips curve, expands the possible microeconomic explanations of this time dependence, and broadens the reach of this formalism in economics. Full article

This model extends the classical economic production quantity (EPQ) model to address the complexities within a two-echelon supply chain system. The model integrates the cost of raw materials necessary for production and takes into account the presence of imperfect quality items within the acquired raw materials. Upon receipt of the raw material, a thorough screening process is conducted to identify imperfect quality items. Combining imperfect raw material and the concept of shifting production rate, two different inventory models for deteriorating products are formulated under imperfect production with demand dependent on the stock level. In the first model, the imperfect raw materials are sold at a discounted price at the end of the screening period, whereas in the second one, imperfect items are kept in stock until the end of the inventory cycle and then returned to the supplier. Numerical analysis reveals that selling imperfect raw materials yields a favourable financial outcome, with an optimal inventory level $I_1$ = 11,774 units, optimal cycle time $T=2140$ h, and a total profit per hour of USD 183, while keeping the imperfect raw materials to return them to the supplier results in a negative profit of USD $-4.44\times {10}^3$ per hour, indicating an unfavourable financial outcome with the optimal inventory level $I_1$ and optimal cycle time T of 26,349 units and 4702.6 h, respectively. The findings show the importance of selling imperfect raw materials rather than returning them and provide valuable insights for inventory management in systems with deteriorating products and imperfect production processes. Sensitivity analysis further demonstrates the robustness of the model. This study contributes to satisfying the need for inventory models that consider both the procurement of imperfect raw materials, stock-dependent demand, and deteriorating products, along with shifts in production rates in a multi-echelon supply chain. Full article

It is widely believed that one of the main causes of the replication crisis in scientific research is some of the most commonly used statistical methods, such as null hypothesis significance testing (NHST). This has prompted many scientists to call for statistics reform. As a practitioner in hydraulics and measurement science, the author extensively used statistical methods in environmental engineering and hydrological survey projects. The author strongly concurs with the need for statistics reform. This paper offers a practitioner’s perspective on statistics reform. In the author’s view, some statistical methods are good and should withstand statistics reform, while others are flawed and should be abandoned and removed from textbooks and software packages. This paper focuses on the following two methods derived from the t-distribution: the two-sample t-test and the t-interval method for calculating measurement uncertainty. We demonstrate why both methods should be abandoned. We recommend using “advanced estimation statistics” in place of the two-sample t-test and an unbiased estimation method in place of the t-interval method. Two examples are presented to illustrate the recommended approaches. Full article

It has been shown that at the present stage of the evolution of the universe, cosmological acceleration is an inevitable kinematical consequence of quantum theory in semiclassical approximation. Quantum theory does not involve such classical concepts as Minkowski or de Sitter spaces. In classical theory, when choosing Minkowski space, a vacuum catastrophe occurs, while when choosing de Sitter space, the value of the cosmological constant can be arbitrary. On the contrary, in quantum theory, there is no uncertainties in view of the following: (1) the de Sitter algebra is the most general ten-dimensional Lie algebra; (2) the Poincare algebra is a special degenerate case of the de Sitter algebra in the limit $R\to \infty$ where R is the contraction parameter for the transition from the de Sitter to the Poincare algebra and R has nothing to do with the radius of de Sitter space; (3) R is fundamental to the same extent as c and ℏ: c is the contraction parameter for the transition from the Poincare to the Galilean algebra and ℏ is the contraction parameter for the transition from quantum to classical theory; (4) as a consequence, the question (why the quantities (c, ℏ, R) have the values which they actually have) does not arise. The solution to the problem of cosmological acceleration follows on from the results of irreducible representations of the de Sitter algebra. This solution is free of uncertainties and does not involve dark energy, quintessence, and other exotic mechanisms, the physical meaning of which is a mystery. Full article

After an infectious contact, there is a time lapse before the individual actually becomes infected. In this paper, we present different ways of incorporating this incubation or infection time into epidemic mathematical models. For simplicity, we consider the Susceptible–Infective–Susceptible and the Susceptible–Infective–Recovered–Susceptible models with no demographic effects, so we can concentrate on the infection process. We study the different methods from a modeling point of view to determine their biological validity and find that not all methods presented in the literature are valid. Specifically, the infection part of the model should only move individuals from one compartment to another but should not change the total population. We consider models with no delay, discrete delay, distributed delay, exposed populations, and fractional derivatives. We analyze the methods that are realistic and find their steady solutions, stability, and bifurcations. We also investigate the effect of the duration of the infection time on the solutions. Numerical simulations are conducted and guidelines on how to chose a method are presented. Full article

In this paper, we analyze an optimal control problem with a mixed cost function, which combines a terminal cost at the final state and an integral term involving the state and control variables. The problem includes both state and control constraints, which adds complexity to the analysis. We establish a necessary optimality condition in the form of the maximum principle, where the adjoint equation is an integral equation involving the Riemann and Stieltjes integrals with respect to a Borel measure. Our approach is based on the Dubovitskii–Milyutin theory, which employs conic approximations to efficiently manage state constraints. To illustrate the applicability of our results, we consider two examples related to epidemiological models, specifically the SIR model. These examples demonstrate how the developed framework can inform optimal control strategies to mitigate disease spread. Furthermore, we explore the implications of our findings in broader contexts, emphasizing how mixed cost functions manifest in various applied settings. Incorporating state constraints requires advanced mathematical techniques, and our approach provides a structured way to address them. The integral nature of the adjoint equation highlights the role of measure-theoretic tools in optimal control. Through our examples, we demonstrate practical applications of the proposed methodology, reinforcing its usefulness in real-life situations. By extending the Dubovitskii–Milyutin framework, we contribute to a deeper understanding of constrained control problems and their solutions. Full article

The drying process is widely used in the food industry for its ability to remove water, provide microbial stability, and reduce spoilage reactions, as well as storage and transportation costs. In particular, rotary drum drying becomes important when it is applied to liquid and pasty foods because of the desire to maintain defined characteristics in terms of product moisture. This drying process is characterized by the existence of many linearities; therefore, different strategies for controlling this process have been proposed. This work focuses on the design of a hybrid PI–fuzzy control scheme for the rotary drum drying process; the idea is to use the advantages of fuzzy logic to obtain a robust monitoring and control system. A pilot plant rotary drum dryer was used to tune the PI control part. Then, the proposed scheme was programmed and tested at the simulation level, comparing it with a classical PI control algorithm. Full article

We analyze a discrete-time random walk on the vertices of an unbounded two-dimensional L-lattice. We determine the probability generating function, and we prove the independence of the coordinates. In particular, we find a relation of each component with a one-dimensional biased random walk with time changing. Therefore, the transition probabilities and the main moments of the random walk can be obtained. The asymptotic behavior of the process is studied, both in the classical sense and involving the large deviations theory. We investigate first-passage-time problems of the random walk through certain straight lines, and we determine the related probabilities in closed form and other features of interest. Finally, we develop a simulation approach to study the first-exit problem of the process thought ellipses. Full article