AppliedMath doi: 10.3390/appliedmath4040074

Authors: Gabriele Bonanno Valeria Morabito Donal O’Regan Bruno Vassallo

The existence of two positive solutions for an elliptic differential inclusion is established, assuming that the nonlinear term is an upper semicontinuous set-valued mapping with compact convex values having subcritical growth. Our approach is based on variational methods for locally Lipschitz functionals. As a consequence, a multiplicity result for elliptic Dirichlet problems having discontinuous nonlinearities is pointed out.

]]>AppliedMath doi: 10.3390/appliedmath4040073

Authors: Saeed Mirzajani Seyed Shahabeddin Moafimadani Majid Roohi

The computer network has fundamentally transformed modern interactions, enabling the effortless transmission of multimedia data. However, the openness of these networks necessitates heightened attention to the security and confidentiality of multimedia content. Digital images, being a crucial component of multimedia communications, require robust protection measures, as their security has become a global concern. Traditional color image encryption/decryption algorithms, such as DES, IDEA, and AES, are unsuitable for image encryption due to the diverse storage formats of images, highlighting the urgent need for innovative encryption techniques. Chaos-based cryptosystems have emerged as a prominent research focus due to their properties of randomness, high sensitivity to initial conditions, and unpredictability. These algorithms typically operate in two phases: shuffling and replacement. During the shuffling phase, the positions of the pixels are altered using chaotic sequences or matrix transformations, which are simple to implement and enhance encryption. However, since only the pixel positions are modified and not the pixel values, the encrypted image&rsquo;s histogram remains identical to the original, making it vulnerable to statistical attacks. In the replacement phase, chaotic sequences alter the pixel values. This research introduces a novel encryption technique for color images (RGB type) based on DNA subsequence operations to secure these images, which often contain critical information, from potential cyber-attacks. The suggested method includes two main components: a high-speed permutation process and adaptive diffusion. When implemented in the MATLAB software environment, the approach yielded promising results, such as NPCR values exceeding 98.9% and UACI values at around 32.9%, demonstrating its effectiveness in key cryptographic parameters. Security analyses, including histograms and Chi-square tests, were initially conducted, with passing Chi-square test outcomes for all channels; the correlation coefficient between adjacent pixels was also calculated. Additionally, entropy values were computed, achieving a minimum entropy of 7.0, indicating a high level of randomness. The method was tested on specific images, such as all-black and all-white images, and evaluated for resistance to noise and occlusion attacks. Finally, a comparison of the proposed algorithm&rsquo;s NPCR and UAC values with those of existing methods demonstrated its superior performance and suitability.

]]>AppliedMath doi: 10.3390/appliedmath4040072

Authors: Xuan Qiao Wei Yao

This study investigates the role of calcium ions in the release of action potentials by comparing two models based on the framework: the standard HH model and a HH + Ca model that incorporates calcium ion channels. Purkinje cells&rsquo; responses to four types of electrical current stimuli&mdash;constant direct current, step current, square wave current, and sine current&mdash;were simulated to analyze the impact of calcium on action potential characteristics. The results indicate that, under the constant direct current stimulation, the action potential firing frequency of both models increased with the escalating current intensity, while the delay time of the first action potential decreased. However, when the current intensity exceeded a specific threshold, the peak amplitude of the action potential gradually diminished. The HH + Ca model exhibited a longer delay in the first action potential compared to the HH model but maintained an action potential release under stronger currents. In response to the step current, both models showed an increased action potential frequency with a higher current, but the HH + Ca model generated subthreshold oscillations under weak currents. With the square wave current, the action potential frequency increased, though the HH + Ca model experienced suppression under high-frequency weak currents. Under the sine current, the action potential frequency rose, with the HH + Ca model showing less depression near the sine peak due to calcium&rsquo;s role in modulating membrane potential. These findings suggest that calcium ions contribute to a more stable action potential release under varying stimuli.

]]>AppliedMath doi: 10.3390/appliedmath4040071

Authors: Sofia Karakasidou Avraam Charakopoulos Loukas Zachilas

In the present study, we analyze the price time series behavior of selected vegetable products, using complex network analysis in two approaches: (a) correlation complex networks and (b) visibility complex networks based on transformed time series. Additionally, we apply time variability methods, including Hurst exponent and Hjorth parameter analysis. We have chosen products available throughout the year from the Central Market of Thessaloniki (Greece) as a case study. To the best of our knowledge, this kind of study is applied for the first time, both as a type of analysis and on the given dataset. Our aim was to investigate alternative ways of classifying products into groups that could be useful for management and policy issues. The results show that the formed groups present similarities related to their use as plates as well as price variation mode and variability depending on the type of analysis performed. The results could be of interest to government policies in various directions, such as products to develop greater stability, identify fluctuating prices, etc. This work could be extended in the future by including data from other central markets as well as with data with missing data, as is the case for products not available throughout the year.

]]>AppliedMath doi: 10.3390/appliedmath4040070

Authors: Melina Silva de Lima José Vicente Cardoso Santos José Humberto de Souza Prates Celso Barreto Silva Davidson Moreira Marcelo A. Moret

The objective of this study is to model astrophysical systems using the nonlinear Fokker&ndash;Planck equation, with the Adomian method chosen for its iterative and precise solutions in this context, applying boundary conditions relevant to data from the Rossi X-ray Timing Explorer (RXTE). The results include analysis of 156 X-ray intensity distributions from X-ray binaries (XRBs), exhibiting long-tail profiles consistent with Tsallis q-Gaussian distributions. The corresponding q-values align with the principles of Tsallis thermostatistics. Various diffusion hypotheses&mdash;classical, linear, nonlinear, and anomalous&mdash;are examined, with q-values further supporting Tsallis thermostatistics. Adjustments in the parameter &alpha; (related to the order of fractional temporal derivation) reveal the extent of the memory effect, strongly correlating with fractal properties in the diffusive process. Extending this research to other XRBs is both possible and recommended to generalize the characteristics of X-ray scattering and electromagnetic waves at different frequencies originating from similar astronomical objects.

]]>AppliedMath doi: 10.3390/appliedmath4040069

Authors: Carlo Bianca

The mathematical modeling of multicellular systems is an important branch of biophysics, which focuses on how the system properties emerge from the elementary interaction between the constituent elements. Recently, mathematical structures have been proposed within the thermostatted kinetic theory for the modeling of complex living systems and have been profitably employed for the modeling of various complex biological systems at the cellular scale. This paper deals with a class of generalized thermostatted kinetic theory frameworks that can stand in as background paradigms for the derivation of specific models in biophysics. Specifically, the fundamental homogeneous thermostatted kinetic theory structures of the recent literature are recovered and generalized in order to take into consideration further phenomena in biology. The generalizations concern the conservative, the nonconservative, and the mutative interactions between the inner system and the outer environment. In order to sustain the strength of the new structures, some specific models of the literature are reset into the style of the new frameworks of the thermostatted kinetic theory. The selected models deal with breast cancer, genetic mutations, immune system response, and skin fibrosis. Future research directions from the theoretical and modeling viewpoints are discussed in the whole paper and are mainly devoted to the well-posedness in the Hadamard sense of the related initial boundary value problems, to the spatial&ndash;velocity dynamics and to the derivation of macroscopic-scale dynamics.

]]>AppliedMath doi: 10.3390/appliedmath4040068

Authors: Chein-Shan Liu Chih-Wen Chang Chia-Cheng Tsai

For a two-block splitting iterative scheme to solve the complex linear equations system resulting from the complex Helmholtz equation, the iterative form using descent vector and residual vector is formulated. We propose splitting iterative schemes by considering the perpendicular property of consecutive residual vector. The two-block splitting iterative schemes are proven to have absolute convergence, and the residual is minimized at each iteration step. Single and double parameters in the two-block splitting iterative schemes are derived explicitly utilizing the orthogonality condition or the minimality conditions. Some simulations of complex Helmholtz equations are performed to exhibit the performance of the proposed two-block iterative schemes endowed with optimal values of parameters. The primary novelty and major contribution of this paper lies in using the orthogonality condition of residual vectors to optimize the iterative process. The proposed method might fill a gap in the current literature, where existing iterative methods either lack explicit parameter optimization or struggle with high wave numbers and large damping constants in the complex Helmholtz equation. The two-block splitting iterative scheme provides an efficient and convergent solution, even in challenging cases.

]]>AppliedMath doi: 10.3390/appliedmath4040067

Authors: Benkam Bobga Robert Gardner

A mixed graph has both edges and directed edges (or &ldquo;arcs&rdquo;). A complete mixed graph on v vertices, denoted Mv, has, for every pair of vertices u and v, an edge {u,v}, an arc (u,v), and an arc (v,u). A decomposition of the complete mixed graph on v vertices into a partial orientation of a three-cycle with one edge and two arcs (of which there are three types) is a mixed triple system of order v. Necessary and sufficient conditions for the existence of a mixed triple system of order v are well known. In this work packings and coverings of the complete mixed graph with mixed triples are considered. Necessary conditions are given for each of the three relevant mixed triples, and these conditions are shown to be sufficient for two of the relevant mixed triples. For the third mixed triple, a conjecture is given concerning the sufficient conditions. Applications of triple systems in general are discussed, as well as possible applications of mixed graphs, mixed triple systems, and packings and coverings with mixed triples.

]]>AppliedMath doi: 10.3390/appliedmath4040066

Authors: Victor Tebogo Monyayi Emile Franc Doungmo Goufo Ignace Tchangou Toudjeu

In this paper, we establish the existence and uniqueness results of the fractional Navier&ndash;Stokes (N-S) evolution equation using the Banach fixed-point theorem, where the fractional order &beta; is in the form of the Atangana&ndash;Baleanu&ndash;Caputo fractional order. The iterative method combined with the Laplace transform and Sumudu transform is employed to find the exact and approximate solutions of the fractional Navier&ndash;Stokes equation of a one-dimensional problem of unsteady flow of a viscous fluid in a tube. In the domains of science and engineering, these methods work well for solving a wide range of linear and nonlinear fractional partial differential equations and provide numerical solutions in terms of power series, with terms that are simple to compute and that quickly converge to the exact solution. After obtaining the solutions using these methods, we use Mathematica software Version 13.0.1.0 to present them graphically. We create two- and three-dimensional plots of the obtained solutions at various values of &beta; and manipulate other variables to visualize and model relationships between the variables. We observe that as the fractional order &beta; becomes closer to the integer order 1, the solutions approach the exact solution. Lastly, we plot a 2D graph of the first-, second-, third-, and fourth-term approximations of the series solution and observe from the graph that as the number of iterations increases, the approximate solutions become close to the series solution of the fourth-term approximation.

]]>AppliedMath doi: 10.3390/appliedmath4040065

Authors: Takashi Suzuki Takuya Tsuchiya

Here, we study Hadamard&rsquo;s variational formula for simple eigenvalues under dynamical and conformal deformations. Particularly, harmonic convexity of the first eigenvalue of the Laplacian under the mixed boundary condition is established for a two-dimensional domain, which implies several new inequalities.

]]>AppliedMath doi: 10.3390/appliedmath4040064

Authors: Luong Vuong Nguyen

Swarm Intelligence (SI) represents a paradigm shift in artificial intelligence, leveraging the collective behavior of decentralized, self-organized systems to solve complex problems. This study provides a comprehensive review of SI, focusing on its application to multi-robot systems. We explore foundational concepts, diverse SI algorithms, and their practical implementations by synthesizing insights from various reputable sources. The review highlights how principles derived from natural swarms, such as those of ants, bees, and birds, can be harnessed to enhance the efficiency, robustness, and scalability of multi-robot systems. We explore key advancements, ongoing challenges, and potential future directions. Through this extensive examination, we aim to provide a foundational understanding and a detailed taxonomy of SI research, paving the way for further innovation and development in theoretical and applied contexts.

]]>AppliedMath doi: 10.3390/appliedmath4030063

Authors: Alexander Robitzsch

The two-parameter logistic (2PL) item response theory (IRT) model is frequently applied to analyze group differences for multivariate binary random variables. The item parameters in the 2PL model are frequently fixed when estimating the mean and the standard deviation for a group of interest. This method is also called fixed item parameter calibration (FIPC). In this article, the bias and the linking error of the FIPC approach are analytically derived in the presence of random uniform differential item functioning (DIF). The adequacy of the analytical findings was validated in a simulation study. It turned out that the extent of the bias and the variance in distribution parameters increases with increasing variance of random DIF effects.

]]>AppliedMath doi: 10.3390/appliedmath4030062

Authors: Manabu Ichino Kadri Umbleja Hiroyuki Yaguchi

This paper introduces the Accumulated Concept Graph (ACG), a visualization tool based on the quantile method designed to analyze three-way data, including distributional data. Such data often have complex structures that make it difficult to identify patterns using conventional visualization techniques. The ACG represents each object with one or more monotonic line graphs. As a result, the three-way data are visualized as a set of parallel monotonic line graphs that never intersect. This non-intersecting property allows for the easy identification of both macroscopic and microscopic patterns within the data. We demonstrate the usefulness of ACGs and principal component analysis in the analysis of real three-way datasets.

]]>AppliedMath doi: 10.3390/appliedmath4030061

Authors: Ioannis G. Tsoulos Vasileios Charilogis Dimitrios Tsalikakis

In this paper, an innovative hybrid technique is proposed for the efficient training of artificial neural networks, which are used both in class learning problems and in data fitting problems. This hybrid technique combines the well-tested technique of Genetic Algorithms with an innovative variant of Simulated Annealing, in order to achieve high learning rates for the neural networks. This variant was applied periodically to randomly selected chromosomes from the population of the Genetic Algorithm in order to reduce the training error associated with these chromosomes. The proposed method was tested on a wide series of classification and data fitting problems from the relevant literature and the results were compared against other methods. The comparison with other neural network training techniques as well as the statistical comparison revealed that the proposed method is significantly superior, as it managed to significantly reduce the neural network training error in the majority of the used datasets.

]]>AppliedMath doi: 10.3390/appliedmath4030060

Authors: M. Fátima Brilhante M. Ivette Gomes Sandra Mendonça Dinis Pestana Rui Santos

The classical tests for combining p-values use suitable statistics T(P1,&hellip;,Pn), which are based on the assumption that the observed p-values are genuine, i.e., under null hypotheses, are observations from independent and identically distributed Uniform(0,1) random variables P1,&hellip;,Pn. However, the phenomenon known as publication bias, which generally results from the publication of studies that reject null hypotheses of no effect or no difference, can tempt researchers to replicate their experiments, generally no more than once, with the aim of obtaining &ldquo;better&rdquo; p-values and reporting the smallest of the two observed p-values, to increase the chances of their work being published. However, when such &ldquo;fake p-values&rdquo; exist, they tamper with the statistic T(P1,&hellip;,Pn) because they are observations from a Beta(1,2) distribution. If present, the right model for the random variables Pk is described as a tilted Uniform distribution, also called a Mendel distribution, since it was underlying Fisher&rsquo;s critique of Mendel&rsquo;s work. Therefore, methods for combining genuine p-values are reviewed, and it is shown how quantiles of classical combining test statistics, allowing a small number of fake p-values, can be used to make an informed decision when jointly combining fake (from Two P) and genuine (from not Two P) p-values.

]]>AppliedMath doi: 10.3390/appliedmath4030059

Authors: Wanyu Bian Yokhesh Krishnasamy Tamilselvam

Magnetic resonance imaging (MRI) is crucial for its superior soft tissue contrast and high spatial resolution. Integrating deep learning algorithms into MRI reconstruction has significantly enhanced image quality and efficiency. This paper provides a comprehensive review of optimization-based deep learning models for MRI reconstruction, focusing on recent advancements in gradient descent algorithms, proximal gradient descent algorithms, ADMM, PDHG, and diffusion models combined with gradient descent. We highlight the development and effectiveness of learnable optimization algorithms (LOAs) in improving model interpretability and performance. Our findings demonstrate substantial improvements in MRI reconstruction in handling undersampled data, which directly contribute to reducing scan times and enhancing diagnostic accuracy. The review offers valuable insights and resources for researchers and practitioners aiming to advance medical imaging using state-of-the-art deep learning techniques.

]]>AppliedMath doi: 10.3390/appliedmath4030058

Authors: Tristan Guillaume

This article provides an exact formula for the survival probability of Brownian motion with drift when the absorbing boundary is defined as an intermittent step barrier, i.e., an alternate sequence of time intervals when the boundary is piecewise constant, and time intervals without any defined boundary. Numerical implementation is dealt with by a simple and robust Monte Carlo integration algorithm directly derived from the formula, which compares favorably with conditional Monte Carlo simulation. Exact analytical benchmarks are also provided to assess the accuracy of the numerical implementation.

]]>AppliedMath doi: 10.3390/appliedmath4030057

Authors: Tahmineh Azizi

Cancer, a complex disease characterized by uncontrolled cell growth and metastasis, remains a formidable challenge to global health. Mathematical modeling has emerged as a critical tool to elucidate the underlying biological mechanisms driving tumor initiation, progression, and treatment responses. By integrating principles from biology, physics, and mathematics, mathematical oncology provides a quantitative framework for understanding tumor growth dynamics, microenvironmental interactions, and the evolution of cancer cells. This study explores the key applications of mathematical modeling in oncology, encompassing tumor growth kinetics, intra-tumor heterogeneity, personalized medicine, clinical trial optimization, and cancer immunology. Through the development and application of computational models, researchers aim to gain deeper insights into cancer biology, identify novel therapeutic targets, and optimize treatment strategies. Ultimately, mathematical oncology holds the promise of transforming cancer care by enabling more precise, personalized, and effective therapies.

]]>AppliedMath doi: 10.3390/appliedmath4030056

Authors: Robert de Sousa Marco António de Sales Monteiro Fernandes

This work considers the existence of solutions of the heteroclinic type in nonlinear second-order differential equations with &#981;-Laplacians, incorporating generalized impulsive conditions on the real line. For the construction of the results, it was only imposed that &#981; be a homeomorphism, using Schauder&rsquo;s fixed-point theorem, coupled with concepts of L1-Carath&eacute;odory sequences and functions along with impulsive points equiconvergence and equiconvergence at infinity. Finally, a practical part illustrates the main theorem and a possible application to bird population growth.

]]>AppliedMath doi: 10.3390/appliedmath4030055

Authors: Sofia Karakasidou Athanasios Fragkou Loukas Zachilas Theodoros Karakasidis

This study investigates the time-series behavior of vegetable prices in the Central Market of Thessaloniki, Greece, using Recurrence Plot (RP) analysis and Recurrence Quantification Analysis (RQA), which considers non-linearities and does not necessitate stationarity of time series. The period of study was 1999&ndash;2016 for practical and research reasons. In the present work, we focus on vegetables available throughout the year, exploring the dynamics and interrelationships between their prices to avoid missing data. The study applies RP visual inspection classification, a clustering based on RQA parameters, and a classification based on the RQA analysis graphs with epochs for the first time. The aim of the paper was to investigate the grouping of products based on their price dynamical behavior. The results show that the formed groups present similarities related to their use as dishes and their way of cultivation, which apparently affect the price dynamics. The results offer insights into market behaviors, helping to inform better management strategies and policymaking and offer a possibility to predict variability of prices. This information can interest government policies in various directions, such as what products to develop for greater stability, identity for fluctuating prices, etc. In future work, a larger dataset including missing data could be included, as well as a machine-learning algorithm to classify the products based on the RQA with epochs graphs.

]]>AppliedMath doi: 10.3390/appliedmath4030054

Authors: Tichaona Chikore Farai Nyabadza Maria Shaale

This paper explores the mathematical dynamics of consumer spending during a financial crisis using opponent process theory (OPT). Traditionally applied in psychology, OPT explains how initial emotional responses are followed by counteracting reactions to restore equilibrium. This study models the short-term boost in consumer spending and subsequent economic adjustments. Utilizing differential equations to represent these processes, this paper provides insights into the interplay between immediate policy effects and longer-term economic consequences. We focus on the United States (US) response to the 2008 Global Financial Crisis in this study. Results show evidence of diminishing response from prolonged stimuli due to demand saturation, resource allocation inefficiencies, and agent adaptation. Monetary stimuli may inflate debt/prices, outweighing benefits, and structural issues persist despite stimuli. Confidence and expectations impact response because perceived ineffectiveness weakens impact over time. Thus, while stimuli can initially boost activity, their sustained impact demands careful consideration of economic dynamics and agents&rsquo; responses.

]]>AppliedMath doi: 10.3390/appliedmath4030053

Authors: Kunle Adegoke Robert Frontczak

Multiple harmonic-like numbers are studied using the generating function approach. A closed form is stated for binomial sums involving these numbers and two additional parameters. Several corollaries and examples are presented which are immediate consequences of the main result. Finally, combinatorial identities involving harmonic-like numbers and other prominent sequences like hyperharmonic numbers and odd harmonic numbers are offered.

]]>AppliedMath doi: 10.3390/appliedmath4030052

Authors: Dimitrios Traperas Andreas Floros Nikolaos Grigorios Kanellopoulos

Mathematician and philosopher Charles Howard Hinton posited a plausible correlation between higher-dimensional spaces, also referred to as &lsquo;hyperspaces&rsquo;, and the allegorical concept articulated by the Ancient Greek philosopher Plato in his work, Republic, known as the &lsquo;Cave.&rsquo; In Plato&rsquo;s allegory, individuals find themselves situated in an underground &lsquo;Cave&rsquo;, constrained by chains on their legs and neck, perceiving shadows and sound reflections from the &lsquo;real&rsquo; world cast on the &lsquo;Cave&rsquo; wall as their immediate reality. Hinton extended the interpretation of these &lsquo;shadows&rsquo; through the induction method, asserting that, akin to a 3D object casting a 2D shadow, the &lsquo;shadow&rsquo; of a 4D hyper-object would exhibit one dimension less, manifesting as a 3D object. Building upon this conceptual framework, the authors posit a correlation between the perceived acoustic space of the bounded individuals within the &lsquo;Cave&rsquo; and the characteristics of a 4D acoustic space, a proposition substantiated mathematically by scientific inquiry. Furthermore, the authors introduce an interactive art application developed as a methodical approach to exploring the hypothetical 4D acoustic space within Plato&rsquo;s &lsquo;Cave&rsquo;, as perceived by the bounded individuals and someone liberated from his constraints navigating through the &lsquo;Cave.&rsquo;

]]>AppliedMath doi: 10.3390/appliedmath4030051

Authors: Shina Daniel Oloniiju Nancy Mukwevho Yusuf Olatunji Tijani Olumuyiwa Otegbeye

Fractional differential operators are inherently non-local, so global methods, such as spectral methods, are well suited for handling these non-local operators. Long-time integration of differential models such as chaotic dynamical systems poses specific challenges and considerations that make multi-domain numerical methods advantageous when dealing with such problems. This study proposes a novel multi-domain pseudospectral method based on the first kind of Chebyshev polynomials and the Gauss&ndash;Lobatto quadrature for fractional initial value problems.The proposed technique involves partitioning the problem&rsquo;s domain into non-overlapping sub-domains, calculating the fractional differential operator in each sub-domain as the sum of the &lsquo;local&rsquo; and &lsquo;memory&rsquo; parts and deriving the corresponding differentiation matrices to develop the numerical schemes. The linear stability analysis indicates that the numerical scheme is absolutely stable for certain values of arbitrary non-integer order and conditionally stable for others. Numerical examples, ranging from single linear equations to systems of non-linear equations, demonstrate that the multi-domain approach is more appropriate, efficient and accurate than the single-domain scheme, particularly for problems with long-term dynamics.

]]>AppliedMath doi: 10.3390/appliedmath4030050

Authors: Emilio Matricciani

Scholars of English Literature unanimously say that J.R.R. Tolkien influenced C.S. Lewis&rsquo;s writings. For the first time, we have investigated this issue mathematically by using an original multi-dimensional analysis of linguistic parameters, based on surface deep language variables and linguistic channels. To set our investigation in the framework of English Literature, we have considered some novels written by earlier authors, such as C. Dickens, G. MacDonald and others. The deep language variables and the linguistic channels, discussed in the paper, are likely due to writers&rsquo; unconscious design and reveal connections between texts far beyond the writers&rsquo; awareness. In summary, the capacity of the extended short-term memory required to readers, the universal readability index of texts, the geometrical representation of texts and the fine tuning of linguistic channels within texts&mdash;all tools largely discussed in the paper&mdash;revealed strong connections between The Lord of the Rings (Tolkien), The Chronicles of Narnia, The Space Trilogy (Lewis) and novels by MacDonald, therefore agreeing with what the scholars of English Literature say.

]]>AppliedMath doi: 10.3390/appliedmath4030049

Authors: Steven Prestwich

Most optimisation research focuses on relatively simple cases: one decision maker, one objective, and possibly a set of constraints. However, real-world optimisation problems often come with complications: they might be multi-objective, multi-agent, multi-stage or multi-level, and they might have uncertainty, partial knowledge or nonlinear objectives. Each has led to research areas with dedicated solution methods. However, when new hybrid problems are encountered, there is typically no solver available. We define a broad class of discrete optimisation problem called an influence program, and describe a lightweight algorithm based on multi-agent multi-objective reinforcement learning with sampling. We show that it can be used to solve problems from a wide range of literatures: constraint programming, Bayesian networks, stochastic programming, influence diagrams (standard, limited memory and multi-objective), and game theory (multi-level programming, Bayesian games and level-k reasoning). We expect it to be useful for the rapid prototyping of solution methods for new hybrid problems.

]]>AppliedMath doi: 10.3390/appliedmath4030048

Authors: Christian Bipongo Ndeke Marco Adonis Ali Almaktoof

Voltage source converters (VSCs) have emerged as the key components in modern power systems, facilitating efficient energy conversion and flexible power flow control. Understanding the fundamental circuit model of VSCs is essential for their accurate modeling and analysis in power system studies. A basic voltage source converter circuit model connected to an LC filter is essential because it lowers the harmonic distortions and enhances the overall power quality of the micro-grid. This guarantees a clean and steady power supply, which is necessary for the integration of multiple renewable energy sources and sensitive loads. A comprehensive methodology for developing a basic circuit model of VSCs, focusing on the key components and principals involved, is presented in this paper. The methodology includes the modeling of space vector pulse-width modulation (SVPWM) as well as the direct quadrature zero synchronous reference frame. Different design controls, including the design of current control loop in the S-domain, the design of the direct current (DC) bus voltage control loop in the S-domain, and the design of the alternating current (AC) voltage control loop in the S-domain, are explored to capture the dynamic behavior and control strategies of VSCs accurately. The proposed methodology provides a systematic framework for modeling VSCs, enabling engineers and researchers to analyze their performance and assess their impact on power system stability and operation. Future studies can be conducted by using case studies and simulation scenarios to show the efficiency and applicability of the developed models in analyzing VSC-based power electronics applications, including high-voltage direct current (HVDC) transmission systems and flexible alternating current transmission systems (FACTS). The significance of this work lies in its potential to advance the understanding and application of VSCs, contributing to more resilient and efficient power systems. By providing a solid foundation for future research and development, this study supports the ongoing integration of renewable energy sources and the advancement of modern electrical infrastructure.

]]>AppliedMath doi: 10.3390/appliedmath4030047

Authors: Sanjar M. Abrarov Rehan Siddiqui Rajinder Kumar Jagpal Brendan M. Quine

In this work, we consider the properties of the two-term Machin-like formula and develop an algorithm for computing digits of &pi; by using its rational approximation. In this approximation, both terms are constructed by using a representation of 1/&pi; in the binary form. This approach provides the squared convergence in computing digits of &pi; without any trigonometric functions and surd numbers. The Mathematica codes showing some examples are presented.

]]>AppliedMath doi: 10.3390/appliedmath4030046

Authors: Pierre Gaillard

We know that the degeneracy of solutions to PDEs, given in terms of theta functions on Riemann surfaces, provides important results about particular solutions, as in the case of the NLS equation. Here, we degenerate the so called finite gap solutions of the Toda lattice equation from the general formulation in terms of abelian functions when the gaps tend to points. This degeneracy allows us to recover the Sato formulas without using inverse scattering theory or geometric or representation theoretic methods.

]]>AppliedMath doi: 10.3390/appliedmath4030045

Authors: Karl Lewis Mark Anthony Caruana David Paul Suda

Devising a financial trading strategy that allows for long-term gains is a very common problem in finance. This paper aims to formulate a mathematically rigorous framework for the problem and compare and contrast the results obtained. The main approach considered is based on Dynamic Bayesian Networks (DBNs). Within the DBN setting, a long-term as well as a short-term trading strategy are considered and applied on twelve equities obtained from developed and developing markets. It is concluded that both the long-term and the medium-term strategies proposed in this paper outperform the benchmark buy-and-hold (B&amp;H) trading strategy. Despite the clear advantages of the former trading strategies, the limitations of this model are discussed along with possible improvements.

]]>AppliedMath doi: 10.3390/appliedmath4030044

Authors: Marina Bershadsky Božidar Ivanković Marko Pušić

We coin the term &ldquo;network goodness&rdquo; for a value we define for a network embedded in a given environment as a metric that describes the suitability of that network for meeting a demand. Three formulas are proposed to calculate the metric from three variable values. The first variable considers parts of the environment gravitated by the network. For these parts of the environment, we define a value that measures user costs refusing them the use of the network. Last but not least, the network maintenance costs are considered. The results are obtained after focusing on infrastructure and transport networks, but can be used for other types of networks as well.

]]>AppliedMath doi: 10.3390/appliedmath4030043

Authors: Elizabeth A. Stoll

Cortical neurons integrate upstream signals and random electrical noise to gate signaling outcomes, leading to statistically random patterns of activity. Yet classically, the neuron is modeled as a binary computational unit, encoding Shannon entropy. Here, the neuronal membrane potential is modeled as a function of inherently probabilistic ion behavior. In this new model, each neuron computes the probability of transitioning from an off-state to an on-state, thereby encoding von Neumann entropy. Component pure states are integrated into a physical quantity of information, and the derivative of this high-dimensional probability distribution yields eigenvalues across the multi-scale quantum system. In accordance with the Hellman&ndash;Feynman theorem, the resolution of the system state is paired with a spontaneous shift in charge distribution, so this defined system state instantly becomes the past as a new probability distribution emerges. This mechanistic model produces testable predictions regarding the wavelength of free energy released upon information compression and the temporal relationship of these events to physiological outcomes. Overall, this model demonstrates how cortical neurons might achieve non-deterministic signaling outcomes through a computational process of noisy coincidence detection.

]]>AppliedMath doi: 10.3390/appliedmath4020042

Authors: Santosh Kumar Elias Munapo

This paper reviews some recent contributions by the authors and their associates and highlights a few innovative ideas, which led them to address some hard combinatorial network routing and ordered optimisation problems. The travelling salesman, which is in the NP hard category, has been reviewed and solved as an index-restricted shortest connected graph, and therefore, it opens a question about its &lsquo;NP Hard&rsquo; category. The routing problem through &lsquo;K&rsquo; specified nodes and ordered optimum solutions are computationally demanding but have been made computationally feasible. All these approaches are based on the strategic creation and use of an alternative solution in that situation. The efficiency of these methods requires further investigation.

]]>AppliedMath doi: 10.3390/appliedmath4020041

Authors: Peter D. Neilson Megan D. Neilson

At every point p on a smooth n-manifold M there exist n+1 skew-symmetric tensor spaces spanning differential r-forms &omega; with r=0,1,&#8943;,n. Because d&#8728;d is always zero where d is the exterior differential, it follows that every exact r-form (i.e., &omega;=d&lambda; where &lambda; is an r&minus;1-form) is closed (i.e., d&omega;=0) but not every closed r-form is exact. This implies the existence of a third type of differential r-form that is closed but not exact. Such forms are called harmonic forms. Every smooth n-manifold has an underlying topological structure. Many different possible topological structures exist. What distinguishes one topological structure from another is the number of holes of various dimensions it possesses. De Rham&rsquo;s theory of differential forms relates the presence of r-dimensional holes in the underlying topology of a smooth n-manifold M to the presence of harmonic r-form fields on the smooth manifold. A large amount of theory is required to understand de Rham&rsquo;s theorem. In this paper we summarize the differential geometry that links holes in the underlying topology of a smooth manifold with harmonic fields on the manifold. We explore the application of de Rham&rsquo;s theory to (i) visual, (ii) mechanical, (iii) electrical and (iv) fluid flow systems. In particular, we consider harmonic flow fields in the intracellular aqueous solution of biological cells and we propose, on mathematical grounds, a possible role of harmonic flow fields in the folding of protein polypeptide chains.

]]>AppliedMath doi: 10.3390/appliedmath4020040

Authors: Roy M. Howard

The relationship between the inverse Langevin function and the proposed r0-r1-Lambert W function is defined. The derived relationship leads to new approximations for the inverse Langevin function with lower relative error bounds than comparable published approximations. High accuracy approximations, based on Schr&ouml;der&rsquo;s root approximations of the first kind, are detailed. Several applications are detailed.

]]>AppliedMath doi: 10.3390/appliedmath4020039

Authors: Daniel A. Griffith

This paper describes various selected properties and features of negative binomial (NB) random variables, with special reference to NB2 (i.e., p = 2), and some generalizations to NBp (i.e., p &ge; 2), specifications. It presents new results (e.g., the NBp moment-generating function) with regard to the relationship between a sample mean and its accompanying variance, as well as spatial statistical/econometric numerical and empirical examples, whose parameter estimators are maximum likelihood or method of moment ones. Finally, it highlights the Moran eigenvector spatial filtering methodology within the context of generalized linear modeling, demonstrating it in terms of spatial negative binomial regression. Its overall conclusion is a bolstering of important findings the literature already reports with a newly recognized empirical example of an NB3 phenomenon.

]]>AppliedMath doi: 10.3390/appliedmath4020038

Authors: Vasileios Charilogis Ioannis G. Tsoulos

The topic of efficiently finding the global minimum of multidimensional functions is widely applicable to numerous problems in the modern world. Many algorithms have been proposed to address these problems, among which genetic algorithms and their variants are particularly notable. Their popularity is due to their exceptional performance in solving optimization problems and their adaptability to various types of problems. However, genetic algorithms require significant computational resources and time, prompting the need for parallel techniques. Moving in this research direction, a new global optimization method is presented here that exploits the use of parallel computing techniques in genetic algorithms. This innovative method employs autonomous parallel computing units that periodically share the optimal solutions they discover. Increasing the number of computational threads, coupled with solution exchange techniques, can significantly reduce the number of calls to the objective function, thus saving computational power. Also, a stopping rule is proposed that takes advantage of the parallel computational environment. The proposed method was tested on a broad array of benchmark functions from the relevant literature and compared with other global optimization techniques regarding its efficiency.

]]>AppliedMath doi: 10.3390/appliedmath4020037

Authors: Huizeng Qin Youmin Lu

If the matrix function f(At) posses the properties of f(At)=gf(tkA, then the recurrence formula fi&minus;1=gfi,i=N,N&minus;1,&#8943;,1,f(tA)=f0, can be established. Here, fN=f(AN)=&sum;j=0majANj,AN=tkNA. This provides an algorithm for computing the matrix function f(At). By specifying the calculation accuracy p, a method is presented to determine m and N in a way that minimizes the time of the above algorithm, thus providing a fast algorithm for f(At). It is important to note that m only depends on the calculation accuracy p and is independent of the matrix A and t. Therefore, f(AN) has a fixed calculation format that is easily computed. On the other hand, N depends not only on A, but also on t. This provides a means to select t such that N is equal to 0, a property of significance. In summary, the algorithm proposed in this article enables users to establish a desired level of accuracy and then utilize it to select the appropriate values for m and N to minimize computation time. This approach ensures that both accuracy and efficiency are addressed concurrently. We develop a general algorithm, then apply it to the exponential, trigonometric, and logarithmic matrix functions, and compare the performance with that of the internal system functions of Mathematica and Pade approximation. In the last section, an example is provided to illustrate the rapid computation of numerical solutions for linear differential equations.

]]>AppliedMath doi: 10.3390/appliedmath4020036

Authors: Benito Chen-Charpentier

The basic reproduction, or reproductive number, is a useful index that indicates whether or not there will be an epidemic. However, it is also very important to determine whether an epidemic will eventually decrease and disappear or persist as an endemic. Different infectious diseases have different behaviors and mathematical models used to simulated them should capture the most important processes; however, the models also involve simplifications. Influenza epidemics are usually short-lived and can be modeled with ordinary differential equations without considering demographics. Delays such as the infection time can change the behavior of the solutions. The same is true if there is permanent or temporary immunity, or complete or partial immunity. Vaccination, isolation and the use of antivirals can also change the outcome. In this paper, we introduce several new models and use them to find the effects of all the above factors paying special attention to whether the model can represent an infectious process that eventually disappears. We determine the equilibrium solutions and establish the stability of the disease-free equilibrium using various methods. We also show that many models of influenza or other epidemics with a short duration do not have solutions with a disappearing epidemic. The main objective of the paper is to introduce different ways of modeling immunity in epidemic models. Several scenarios with different immunities are studied since a person may not be re-infected because he/she has total or partial immunity or because there were no close contacts. We show that some relatively small changes, such as in the vaccination rate, can significantly change the dynamics; for example, the existence and number of the disease-free equilibria. We also illustrate that while introducing delays makes the models more realistic, the dynamics have the same qualitative behavior.

]]>AppliedMath doi: 10.3390/appliedmath4020035

Authors: Fadal Abdullah Ali Aldhufairi Jungsywan H. Sepanski

We develop a framework for creating distortion functions that are used to construct new bivariate copulas. It is achieved by transforming non-negative random variables with Lomax-related distributions. In this paper, we apply the distortions to the base copulas of independence, Clayton, Frank, and Gumbel copulas. The properties of the tail dependence coefficient, tail order, and concordance ordering are explored for the new families of distorted copulas. We conducted an empirical study using the daily net returns of Amazon and Google stocks from January 2014 to December 2023. We compared the popular Clayton, Gumbel, Frank, and Gaussian copula models to their corresponding distorted copula models induced by the unit-Lomax and unit-inverse Pareto distortions. The new families of distortion copulas are equipped with additional parameters inherent in the distortion function, providing more flexibility, and are demonstrated to perform better than the base copulas. After analyzing the data, we have found that the joint extremes of Amazon and Google stocks are more likely for high daily net returns than for low daily net returns.

]]>AppliedMath doi: 10.3390/appliedmath4020034

Authors: Vasil Georgiev Angelov

The main purpose of the present paper is to prove the existence of periodic solutions of the three-body problem in the 3D Kepler formulation. We have solved the same problem in the case when the three particles are considered in an external inertial system. We start with the three-body equations of motion, which are a subset of the equations of motion (previously derived by us) for any number of bodies. In the Minkowski space, there are 12 equations of motion. It is proved that three of them are consequences of the other nine, so their number becomes nine, as much as the unknown trajectories are. The Kepler formulation assumes that one particle (the nucleus) is placed at the coordinate origin. The motion of the other two particles is described by a neutral system with respect to the unknown velocities. The state-dependent delays arise as a consequence of the finite vacuum speed of light. We obtain the equations of motion in spherical coordinates and split them into two groups. In the first group all arguments of the unknown functions are delays. We take their solutions as initial functions. Then, the equations of motion for the remaining two particles must be solved to the right of the initial point. To prove the existence&ndash;uniqueness of a periodic solution, we choose a space consisting of periodic infinitely smooth functions satisfying some supplementary conditions. Then, we use a suitable operator which acts on these spaces and whose fixed points are periodic solutions. We apply the fixed point theorem for the operators acting on the spaces of periodic functions. In this manner, we show the stability of the He atom in the frame of classical electrodynamics. In a previous paper of ours, we proved the existence of spin functions for plane motion. Thus, we confirm the Bohr and Sommerfeld&rsquo;s hypothesis for the He atom.

]]>AppliedMath doi: 10.3390/appliedmath4020033

Authors: Chao-Kang Feng Jyh-Haw Tang

The infinite series solution to the boundary-value problems of Laplace&rsquo;s equation with discontinuous Dirichlet boundary conditions was found by using the basic method of separation of variables. The merit of this paper is that the closed-form solution, or the singular similarity solution in the semi-infinite strip domain and the first-quadrant domain, can be generated from the basic infinite series solution in the rectangular domain. Moreover, based on the superposition principle, the infinite series solution in the rectangular domain can be related to the singular similarity solution in the semi-infinite strip domain. It is proven that the analytical source-type singular behavior in the infinite series solution near certain singular points in the rectangular domain can be revealed from the singular similarity solution in the semi-infinite strip domain. By extending the boundary of the rectangular domain, the infinite series solution to Laplace&rsquo;s equation in the first-quadrant domain can be derived to obtain the analytical singular similarity solution in a direct and much easier way than by using the methods of Fourier transform, images, and conformal mapping.

]]>AppliedMath doi: 10.3390/appliedmath4020032

Authors: Gerassimos Manoussakis

The G-modified Helmholtz equation is a partial differential equation that enables us to express gravity intensity g as a series of spherical harmonics having radial distance r in irrational powers. The Laplace equation in three-dimensional space (in Cartesian coordinates, is the sum of the second-order partial derivatives of the unknown quantity equal to zero) is used to express the Earth&rsquo;s gravity potential (disturbing and normal potential) in order to represent other useful quantities&mdash;which are also known as functionals of the disturbing potential&mdash;such as gravity disturbance, gravity anomaly, and geoid undulation as a series of spherical harmonics. We demonstrate that by using the G-modified Helmholtz equation, not only gravity intensity but also disturbing potential and its functionals can be expressed as a series of spherical harmonics. Having gravity intensity represented as a series of spherical harmonics allows us to create new Global Gravity Models. Furthermore, a more detailed examination of the Earth&rsquo;s isogravitational surfaces is conducted. Finally, we tabulate our results, which makes it clear that new Global Gravity Models for gravity intensity g will be very useful for many geophysical and geodetic applications.

]]>AppliedMath doi: 10.3390/appliedmath4020031

Authors: Diana Monteoliva Angelo Plastino Angel Ricardo Plastino

In our study, we investigate the phenomenon of information loss, as measured by the Kullback&ndash;Leibler divergence, in a many-fermion system, such as the Lipkin model. Information loss is introduced as the number N of particles increases, particularly when the system is in a mixed state. We find that there is a significant loss of information under these conditions. However, we observe that this loss nearly disappears when the system is in a pure state. Our analysis employs tools from information theory to quantify and understand these effects.

]]>AppliedMath doi: 10.3390/appliedmath4020030

Authors: Qusay Muzaffar David Levin Michael Werman

This paper presents a novel deep-learning network designed to detect intervals of jump discontinuities in single-variable piecewise smooth functions from their noisy samples. Enhancing the accuracy of jump discontinuity estimations can be used to find a more precise overall approximation of the function, as traditional approximation methods often produce significant errors near discontinuities. Detecting intervals of discontinuities is relatively straightforward when working with exact function data, as finite differences in the data can serve as indicators of smoothness. However, these smoothness indicators become unreliable when dealing with highly noisy data. In this paper, we propose a deep-learning network to pinpoint the location of a jump discontinuity even in the presence of substantial noise.

]]>AppliedMath doi: 10.3390/appliedmath4020029

Authors: Ziyi Su Ephraim Agyingi

The threat posed by the COVID-19 pandemic has been accompanied by an epidemic of misinformation, causing confusion and mistrust among the public. Misinformation about COVID-19 whether intentional or unintentional takes many forms, including conspiracy theories, false treatments, and inaccurate information about the origins and spread of the virus. Though the pandemic has brought to light the significant impact of misinformation on public health, mathematical modeling emerged as a valuable tool for understanding the spread of COVID-19 and the impact of public health interventions. However, there has been limited research on the mathematical modeling of the spread of misinformation related to COVID-19. In this paper, we present a mathematical model of the spread of misinformation related to COVID-19. The model highlights the challenges posed by misinformation, in that rather than focusing only on the reproduction number that drives new infections, there is an additional threshold parameter that drives the spread of misinformation. The equilibria of the model are analyzed for both local and global stability, and numerical simulations are presented. We also discuss the model&rsquo;s potential to develop effective strategies for combating misinformation related to COVID-19.

]]>AppliedMath doi: 10.3390/appliedmath4020028

Authors: Edoardo Ballico

We study properties of the minimal Terracini loci, i.e., families of certain zero-dimensional schemes, in a projective plane. Among the new results here are: a maximality theorem and the existence of arbitrarily large gaps or non-gaps for the integers x for which the minimal Terracini locus in degree d is non-empty. We study similar theorems for the critical schemes of the minimal Terracini sets. This part is framed in a more general framework.

]]>AppliedMath doi: 10.3390/appliedmath4020027

Authors: Mumuni Amadu Adango Miadonye

The transition zone (TZ) of hydrocarbon reservoirs is an integral part of the hydrocarbon pool which contains a substantial fraction of the deposit, particularly in carbonate petroleum systems. Consequently, knowledge of its thickness and petrophysical properties, viz. its pore size distribution and wettability characteristic, is critical to optimizing hydrocarbon production in this zone. Using classical formation evaluation techniques, the thickness of the transition zone has been estimated, using well logging methods including resistivity and Nuclear Magnetic Resonance, among others. While hydrocarbon fluids&rsquo; accumulation in petroleum reservoirs occurs due to the migration and displacement of originally water-filled potential structural and stratigraphic traps, the development of their TZ integrates petrophysical processes that combine spontaneous capillary imbibition and wettability phenomena. In the literature, wettability phenomena have been shown to also be governed by electrostatic phenomena. Therefore, given that reservoir rocks are aggregates of minerals with ionizable surface groups that facilitate the development of an electric double layer, a definite theoretical relationship between the TZ and electrostatic theory must be feasible. Accordingly, a theoretical approach to estimating the TZ thickness, using the electrostatic theory and based on the electric double layer theory, is attractive, but this is lacking in the literature. Herein, we fill the knowledge gap by using the interfacial electrostatic theory based on the fundamental tenets of the solution to the Poisson&ndash;Boltzmann mean field theory. Accordingly, we have used an existing model of capillary rise based on free energy concepts to derive a capillary rise equation that can be used to theoretically predict observations based on the TZ thickness of different reservoir rocks, using well-established formation evaluation methods. The novelty of our work stems from the ability of the model to theoretically and accurately predict the TZ thickness of the different lithostratigraphic units of hydrocarbon reservoirs, because of the experimental accessibility of its model parameters.

]]>AppliedMath doi: 10.3390/appliedmath4020026

Authors: Juan José Fernández-Durán María Mercedes Gregorio-Domínguez

The sum of independent circular uniformly distributed random variables is also circular uniformly distributed. In this study, it is shown that a family of circular distributions based on nonnegative trigonometric sums (NNTS) is also closed under summation. Given the flexibility of NNTS circular distributions to model multimodality and skewness, these are good candidates for use as alternative models to test for circular uniformity to detect different deviations from the null hypothesis of circular uniformity. The circular uniform distribution is a member of the NNTS family, but in the NNTS parameter space, it corresponds to a point on the boundary of the parameter space, implying that the regularity conditions are not satisfied when the parameters are estimated by using the maximum likelihood method. Two NNTS tests for circular uniformity were developed by considering the standardised maximum likelihood estimator and the generalised likelihood ratio. Given the nonregularity condition, the critical values of the proposed NNTS circular uniformity tests were obtained via simulation and interpolated for any sample size by the fitting of regression models. The validity of the proposed NNTS circular uniformity tests was evaluated by generating NNTS models close to the circular uniformity null hypothesis.

]]>AppliedMath doi: 10.3390/appliedmath4020025

Authors: David Ellerman

The new approach to quantum mechanics (QM) is that the mathematics of QM is the linearization of the mathematics of partitions (or equivalence relations) on a set. This paper develops those ideas using vector spaces over the field Z2={0.1} as a pedagogical or toy model of (finite-dimensional, non-relativistic) QM. The 0,1-vectors are interpreted as sets, so the model is &ldquo;quantum mechanics over sets&rdquo; or QM/Sets. The key notions of partitions on a set are the logical-level notions to model distinctions versus indistinctions, definiteness versus indefiniteness, or distinguishability versus indistinguishability. Those pairs of concepts are the key to understanding the non-classical &lsquo;weirdness&rsquo; of QM. The key non-classical notion in QM is the notion of superposition, i.e., the notion of a state that is indefinite between two or more definite- or eigen-states. As Richard Feynman emphasized, all the weirdness of QM is illustrated in the double-slit experiment, so the QM/Sets version of that experiment is used to make the key points.

]]>AppliedMath doi: 10.3390/appliedmath4020024

Authors: Ghazanfar Shahgholian Arman Fathollahi

The frequency deviation from the nominal working frequency in power systems is a consequence of the imbalance between total electrical loads and the aggregate power supplied by production units. The sensitivity of energy system frequency to both minor and major load variations underscore the need for effective frequency load control mechanisms. In this paper, frequency load control in single-area power system with multi-source energy is analysed and simulated. Also, the effect of the photovoltaic system on the frequency deviation changes in the energy system is shown. In the single area energy system, the dynamics of thermal turbine with reheat, thermal turbine without reheat and hydro turbine are considered. The simulation results using Simulink/Matlab and model analysis using eigenvalue analysis show the dynamic behaviour of the power system in response to changes in the load.

]]>AppliedMath doi: 10.3390/appliedmath4020023

Authors: John Constantine Venetis

In this paper, an analytical exact form of the ramp function is presented. This seminal function constitutes a fundamental concept of the digital signal processing theory and is also involved in many other areas of applied sciences and engineering. In particular, the ramp function is performed in a simple manner as the pointwise limit of a sequence of real and continuous functions with pointwise convergence. This limit is zero for strictly negative values of the real variable x, whereas it coincides with the independent variable x for strictly positive values of the variable x. Here, one may elucidate beforehand that the pointwise limit of a sequence of continuous functions can constitute a discontinuous function, on the condition that the convergence is not uniform. The novelty of this work, when compared to other research studies concerning analytical expressions of the ramp function, is that the proposed formula is not exhibited in terms of miscellaneous special functions, e.g., gamma function, biexponential function, or any other special functions, such as error function, hyperbolic function, orthogonal polynomials, etc. Hence, this formula may be much more practical, flexible, and useful in the computational procedures, which are inserted into digital signal processing techniques and other engineering practices.

]]>AppliedMath doi: 10.3390/appliedmath4020022

Authors: Liang Kong Yanhui Guo Chung-wei Lee

Accurate forecasting of the coronavirus disease 2019 (COVID-19) spread is indispensable for effective public health planning and the allocation of healthcare resources at all levels of governance, both nationally and globally. Conventional prediction models for the COVID-19 pandemic often fall short in precision, due to their reliance on homogeneous time-dependent transmission rates and the oversight of geographical features when isolating study regions. To address these limitations and advance the predictive capabilities of COVID-19 spread models, it is imperative to refine model parameters in accordance with evolving insights into the disease trajectory, transmission rates, and the myriad economic and social factors influencing infection. This research introduces a novel hybrid model that combines classic epidemic equations with a recurrent neural network (RNN) to predict the spread of the COVID-19 pandemic. The proposed model integrates time-dependent features, namely the numbers of individuals classified as susceptible, infectious, recovered, and deceased (SIRD), and incorporates human mobility from neighboring regions as a crucial spatial feature. The study formulates a discrete-time function within the infection component of the SIRD model, ensuring real-time applicability while mitigating overfitting and enhancing overall efficiency compared to various existing models. Validation of the proposed model was conducted using a publicly available COVID-19 dataset sourced from Italy. Experimental results demonstrate the model&rsquo;s exceptional performance, surpassing existing spatiotemporal models in three-day ahead forecasting. This research not only contributes to the field of epidemic modeling but also provides a robust tool for policymakers and healthcare professionals to make informed decisions in managing and mitigating the impact of the COVID-19 pandemic.

]]>AppliedMath doi: 10.3390/appliedmath4010021

Authors: Yuhlong Lio Ding-Geng Chen Tzong-Ru Tsai Liang Wang

The reliability of the multicomponent stress&ndash;strength system was investigated under the two-parameter Burr X distribution model. Based on the structure of the system, the type II censored sample of strength and random sample of stress were obtained for the study. The maximum likelihood estimators were established by utilizing the type II censored Burr X distributed strength and complete random stress data sets collected from the multicomponent system. Two related approximate confidence intervals were achieved by utilizing the delta method under the asymptotic normal distribution theory and parametric bootstrap procedure. Meanwhile, point and confidence interval estimators based on alternative generalized pivotal quantities were derived. Furthermore, a likelihood ratio test to infer the equality of both scalar parameters is provided. Finally, a practical example is provided for illustration.

]]>AppliedMath doi: 10.3390/appliedmath4010020

Authors: Lucian Trifina Daniela Tărniceriu Ana-Mirela Rotopănescu

In this paper, we address the inverse of a true fourth-degree permutation polynomial (4-PP), modulo a positive integer of the form&nbsp;32kL&Psi;, where&nbsp;kL&isin;{1,3}&nbsp;and&nbsp;&Psi;&nbsp;is a product of different prime numbers greater than three. Some constraints are considered for the 4-PPs to avoid some complicated coefficients&rsquo; conditions. With the fourth- and third-degree coefficients of the form&nbsp;k4,f&Psi;&nbsp;and&nbsp;k3,f&Psi;, respectively, we prove that the inverse PP is (I) a 4-PP when&nbsp;k4,f&isin;{1,3}&nbsp;and&nbsp;k3,f&isin;{1,3,5,7}&nbsp;or when&nbsp;k4,f=2&nbsp;and (II) a 5-PP when&nbsp;k4,f&isin;{1,3}&nbsp;and&nbsp;k3,f&isin;{0,2,4,6}.

]]>AppliedMath doi: 10.3390/appliedmath4010019

Authors: Michel Adès Serge B. Provost Yishan Zang

Four measures of association, namely, Spearman&rsquo;s &rho;, Kendall&rsquo;s &tau;, Blomqvist&rsquo;s &beta; and Hoeffding&rsquo;s &Phi;2, are expressed in terms of copulas. Conveniently, this article also includes explicit expressions for their empirical counterparts. Moreover, copula representations of the four coefficients are provided for the multivariate case, and several specific applications are pointed out. Additionally, a numerical study is presented with a view to illustrating the types of relationships that each of the measures of association can detect.

]]>AppliedMath doi: 10.3390/appliedmath4010018

Authors: Alexander Melnikov Pouneh Mohammadi Nejad

This paper investigates a financial market where asset prices follow a multi-dimensional Brownian motion process and a multi-dimensional Poisson process characterized by diverse credit and deposit rates where the credit rate is higher than the deposit rate. The focus extends to evaluating European options by establishing upper and lower hedging prices through a transition to a suitable auxiliary market. Introducing a lemma elucidates the same solution to the pricing problem in both markets under specific conditions. Additionally, we address the minimization of shortfall risk and determine no-arbitrage price bounds within the framework of incomplete markets. This study provides a comprehensive understanding of the challenges posed by the multi-dimensional jump-diffusion model and varying interest rates in financial markets.

]]>AppliedMath doi: 10.3390/appliedmath4010017

Authors: Salma A. A. Ahmedai Abd Allah Precious Sibanda Sicelo P. Goqo Uthman O. Rufai Hloniphile Sithole Mthethwa Osman A. I. Noreldin

In this paper, we extend the block hybrid method with equally spaced intra-step points to solve linear and nonlinear third-order initial value problems. The proposed block hybrid method uses a simple iteration scheme to linearize the equations. Numerical experimentation demonstrates that equally spaced grid points for the block hybrid method enhance its speed of convergence and accuracy compared to other conventional block hybrid methods in the literature. This improvement is attributed to the linearization process, which avoids the use of derivatives. Further, the block hybrid method is consistent, stable, and gives rapid convergence to the solutions. We show that the simple iteration method, when combined with the block hybrid method, exhibits impressive convergence characteristics while preserving computational efficiency. In this study, we also implement the proposed method to solve the nonlinear Jerk equation, producing comparable results with other methods used in the literature.

]]>AppliedMath doi: 10.3390/appliedmath4010016

Authors: Marco Antonio Montufar Benítez Jaime Mora Vargas José Raúl Castro Esparza Héctor Rivera Gómez Oscar Montaño Arango

The main purpose of this paper is to implement a simulation model in @RISKTM and study the impact of incorporating random variables, such as the degree days in a traditional deterministic model, for calculating the optimum thickness of thermal insulation in walls. Currently, green buildings have become important because of the increasing worldwide interest in the reduction of environmental pollution. One method of saving energy is to use thermal insulation. The optimum thickness of these insulators has traditionally been calculated using deterministic models. With the information generated from real data using the degree days required in a certain zone in Palestine during winter, random samples of the degree days required annually in this town were generated for periods of 10, 20, 50, and 70 years. The results showed that the probability of exceeding the net present value of the cost calculated using deterministic analysis ranges from 0% to 100%, without regard to the inflation rate. The results also show that, for design lifetimes greater than 40 years, the risk of overspending is lower if the building lasts longer than the period for which it was designed. Moreover, this risk is transferred to whomever will pay the operating costs of heating the building. The contribution of this research is twofold: (a) a stochastic approach is incorporated into the traditional models that determine the optimum thickness of thermal insulation used in buildings, by introducing the variability of the degree days required in a given region; (b) a measure of the economic risk incurred by building heating is established as a function of the years of use for which the building is designed and the number of years it is actually used.

]]>AppliedMath doi: 10.3390/appliedmath4010015

Authors: Constantin Fetecau Costică Moroşanu Shehraz Akhtar

In this work, we investigate isothermal MHD motions of a large class of rate type fluids through a porous medium between two infinite horizontal parallel plates when a differential expression of the non-trivial shear stress is prescribed on the boundary. Exact expressions are provided for the dimensionless steady state velocities, shear stresses and Darcy&rsquo;s resistances. Obtained solutions can be used to find the necessary time to touch the steady state or to bring to light certain characteristics of the fluid motion. Graphical representations showed the fluid moves slower in presence of a magnetic field or porous medium. In addition, contrary to our expectations, the volume flux across a plane orthogonal to the velocity vector per unit width of this plane is zero. Finally, based on a simple remark regarding the governing equations of velocity and shear stress for MHD motions of incompressible generalized Burgers&rsquo; fluids between infinite parallel plates, provided were the first exact solutions for MHD motions of these fluids when the two plates apply oscillatory or constant shear stresses to the fluid. This important remark offers the possibility to solve any isothermal MHD motion of these fluids between infinite parallel plates or over an infinite plate when the non-trivial shear stress is prescribed on the boundary. As an application, steady state solutions for MHD motions of same fluids have been developed when a differential expression of the fluid velocity is prescribed on the boundary.

]]>AppliedMath doi: 10.3390/appliedmath4010014

Authors: Manabu Ichino

The quantile method transforms each complex object described by different histogram values to a common number of quantile vectors. This paper retraces the authors&rsquo; research, including a principal component analysis, unsupervised feature selection using hierarchical conceptual clustering, and lookup table regression model. The purpose is to show that this research is essentially based on the monotone property of quantile vectors and works cooperatively in the exploratory analysis of the given distributional data.

]]>AppliedMath doi: 10.3390/appliedmath4010013

Authors: Vladimir Volenec Marija Šimić Horvath Ema Jurkin

In this paper, we study the properties of a complete quadrangle in the Euclidean plane. The proofs are based on using rectangular coordinates symmetrically on four vertices and four parameters a,b,c,d. Here, many properties of the complete quadrangle known from earlier research are proved using the same method, and some new results are given.

]]>AppliedMath doi: 10.3390/appliedmath4010012

Authors: Elisabetta Barletta Sorin Dragomir Francesco Esposito

We study the random flow, through a thin cylindrical tube, of a physical quantity of random density, in the presence of random sinks and sources. We model convection in terms of the expectations of the flux and density and solve the initial value problem for the resulting convection equation. We propose a difference scheme for the convection equation, that is both stable and satisfies the Courant&ndash;Friedrichs&ndash;Lewy test, and estimate the difference between the exact and approximate solutions.

]]>AppliedMath doi: 10.3390/appliedmath4010011

Authors: Robert Gardner Kazeem Kosebinu

Graph and digraph decompositions are a fundamental part of design theory. Probably the best known decompositions are related to decomposing the complete graph into 3-cycles (which correspond to Steiner triple systems), and decomposing the complete digraph into orientations of a 3-cycle (the two possible orientations of a 3-cycle correspond to directed triple systems and Mendelsohn triple systems). Decompositions of the &lambda;-fold complete graph and the &lambda;-fold complete digraph have been explored, giving generalizations of decompositions of complete simple graphs and digraphs. Decompositions of the complete mixed graph (which contains an edge and two distinct arcs between every two vertices) have also been explored in recent years. Since the complete mixed graph has twice as many arcs as edges, an isomorphic decomposition of a complete mixed graph into copies of a sub-mixed graph must involve a sub-mixed graph with twice as many arcs as edges. A partial orientation of a 6-star with two edges and four arcs is an example of such a mixed graph; there are five such mixed stars. In this paper, we give necessary and sufficient conditions for a decomposition of the &lambda;-fold complete mixed graph into each of these five mixed stars for all &lambda;&gt;1.

]]>AppliedMath doi: 10.3390/appliedmath4010010

Authors: Frederika Rentzeperis Benjamin Coleman Dorothy Wallace

Radiotherapy can differentially affect the phases of the cell cycle, possibly enhancing suppression of tumor growth, if cells are synchronized in a specific phase. A model is designed to replicate experiments that synchronize cells in the S phase using gemcitabine before radiation at various doses, with the goal of quantifying this effect. The model is used to simulate a clinical trial with a cohort of 100 individuals receiving only radiation and another cohort of 100 individuals receiving radiation after cell synchronization. The simulations offered in this study support the statement that, at suitably high levels of radiation, synchronizing melanoma cells with gemcitabine before treatment substantially reduces the final tumor size. The improvement is statistically significant, and the effect size is noticeable, with the near suppression of growth at 8 Gray and 92% synchronization.

]]>AppliedMath doi: 10.3390/appliedmath4010009

Authors: Benito Chen-Charpentier

Hepatitis B is a liver disease caused by the human hepatitis B virus (HBV). Mathematical models help further the understanding of the processes involved and help make predictions. The basic reproduction number, R0, is an index that predicts whether the disease will be chronic or not. This is the single most-important information that a mathematical model can give. Within-host virus processes involve delays. We study two within-host hepatitis B virus infection models without and with delay. One is a standard one, and the other considering additional processes and with two delays is new. We analyze the basic reproduction number and alternative threshold indices. The values of R0 and the alternative indices change depending on the model. All these indices predict whether the infection will persist or not, but they do not give the same rate of growth of the infection when it is starting. Therefore, the choice of the model is very important in establishing whether the infection is chronic or not and how fast it initially grows. We analyze these indices to see how to decrease their value. We study the effect of adding delays and how the threshold indices depend on how the delays are included. We do this by studying the local asymptotic stability of the disease-free equilibrium or by using an equivalent method. We show that, for some models, the indices do not change by introducing delays, but they change when the delays are introduced differently. Numerical simulations are presented to confirm the results. Finally, some conclusions are presented.

]]>AppliedMath doi: 10.3390/appliedmath4010008

Authors: Peter Berzi

A system of simultaneous multi-variable nonlinear equations can be solved by Newton&rsquo;s method with local q-quadratic convergence if the Jacobian is analytically available. If this is not the case, then quasi-Newton methods with local q-superlinear convergence give solutions by approximating the Jacobian in some way. Unfortunately, the quasi-Newton condition (Secant equation) does not completely specify the Jacobian approximate in multi-dimensional cases, so its full-rank update is not possible with classic variants of the method. The suggested new iteration strategy (&ldquo;T-Secant&rdquo;) allows for a full-rank update of the Jacobian approximate in each iteration by determining two independent approximates for the solution. They are used to generate a set of new independent trial approximates; then, the Jacobian approximate can be fully updated. It is shown that the T-Secant approximate is in the vicinity of the classic quasi-Newton approximate, providing that the solution is evenly surrounded by the new trial approximates. The suggested procedure increases the superlinear convergence of the Secant method &phi;S=1.618&hellip; to super-quadratic &phi;T=&phi;S+1=2.618&hellip; and the quadratic convergence of the Newton method &phi;N=2 to cubic &phi;T=&phi;N+1=3 in one-dimensional cases. In multi-dimensional cases, the Broyden-type efficiency (mean convergence rate) of the suggested method is an order higher than the efficiency of other classic low-rank-update quasi-Newton methods, as shown by numerical examples on a Rosenbrock-type test function with up to 1000 variables. The geometrical representation (hyperbolic approximation) in single-variable cases helps explain the basic operations, and a vector-space description is also given in multi-variable cases.

]]>AppliedMath doi: 10.3390/appliedmath4010007

Authors: Emilio Matricciani

The purpose of the present paper is to further investigate the mathematical structure of sentences&mdash;proposed in a recent paper&mdash;and its connections with human short&ndash;term memory. This structure is defined by two independent variables which apparently engage two short&ndash;term memory buffers in a series. The first buffer is modelled according to the number of words between two consecutive interpunctions&mdash;variable referred to as the word interval, IP&mdash;which follows Miller&rsquo;s 7&plusmn;2 law; the second buffer is modelled by the number of word intervals contained in a sentence, MF, ranging approximately for one to seven. These values result from studying a large number of literary texts belonging to ancient and modern alphabetical languages. After studying the numerical patterns (combinations of IP and MF) that determine the number of sentences that theoretically can be recorded in the two memory buffers&mdash;which increases with the use of IP and MF&mdash;we compare the theoretical results with those that are actually found in novels from Italian and English literature. We have found that most writers, in both languages, write for readers with small memory buffers and, consequently, are forced to reuse sentence patterns to convey multiple meanings.

]]>AppliedMath doi: 10.3390/appliedmath4010006

Authors: Kabiru Michael Adeyemo Kayode Oshinubi Umar Muhammad Adam Adejimi Adeniji

A co-infection model for onchocerciasis and Lassa fever (OLF) with periodic variational vectors and optimal control is studied and analyzed to assess the impact of controls against incidence infections. The model is qualitatively examined in order to evaluate its asymptotic behavior in relation to the equilibria. Employing a Lyapunov function, we demonstrated that the disease-free equilibrium (DFE) is globally asymptotically stable; that is, the related basic reproduction number is less than unity. When it is bigger than one, we use a suitable nonlinear Lyapunov function to demonstrate the existence of a globally asymptotically stable endemic equilibrium (EE). Furthermore, the necessary conditions for the presence of optimum control and the optimality system for the co-infection model are established using Pontryagin&rsquo;s maximum principle. The model is quantitatively analyzed by studying how sensitive the basic reproduction number is to the model parameters and the model simulation using Runge&ndash;Kutta technique of order 4 is also presented to study the effects of the treatments. We deduced from the quantitative analysis that, if there is an effective treatment and diagnosis of those exposed to and infected with the disease, the spread of the viral disease can be effectively managed. The results presented in this work will be useful for the proper mitigation of the disease.

]]>AppliedMath doi: 10.3390/appliedmath4010005

Authors: Ayan Bhattacharya

It is common in financial markets for market makers to offer prices on derivative instruments even though they are uncertain about the underlying asset&rsquo;s value. This paper studies the mathematical problem that arises as a result. Derivatives are priced in the risk-neutral framework, so as the market maker acquires more information about the underlying asset, the change of measure for transition to the risk-neutral framework (the pricing kernel) evolves. This evolution takes a precise form when the market maker is Bayesian. It is shown that Bayesian updates can be characterized as additional informational drift in the underlying asset&rsquo;s stochastic process. With Bayesian updates, the change of measure needed for pricing derivatives is two-fold: the first change is from the prior probability measure to the posterior probability measure, and the second change is from the posterior probability measure to the risk-neutral measure. The relation between the regular pricing kernel and the pricing kernel under this two-fold change of measure is characterized.

]]>AppliedMath doi: 10.3390/appliedmath4010004

Authors: Joan-Carles Artés Jaume Llibre Nicolae Vulpe

The following differential quadratic polynomial differential system &nbsp;dxdt=y&minus;x,&nbsp;dydt=2y&minus;y&gamma;&minus;12&minus;&gamma;y&minus;5&gamma;&minus;4&gamma;&minus;1x, when the parameter &gamma;&isin;(1,2] models the structure equations of an isotropic star having a linear barotropic equation of state, being x=m(r)/r where m(r)&ge;0 is the mass inside the sphere of radius r of the star, y=4&pi;r2&rho; where &rho; is the density of the star, and t=ln(r/R) where R is the radius of the star. First, we classify all the topologically non-equivalent phase portraits in the Poincar&eacute; disc of these quadratic polynomial differential systems for all values of the parameter &gamma;&isin;R&#8726;{1}. Second, using the information of the different phase portraits obtained we classify the possible limit values of m(r)/r and 4&pi;r2&rho; of an isotropic star when r decreases.

]]>AppliedMath doi: 10.3390/appliedmath4010003

Authors: Paul Romatschke

If a quantum field theory has a Landau pole, the theory is usually called &lsquo;sick&rsquo; and dismissed as a candidate for an interacting UV-complete theory. In a recent study on the interacting 4d O(N) model at large N, it was shown that at the Landau pole, observables remain well-defined and finite. In this work, I investigate both relevant and irrelevant deformations of the said model at the Landau pole, finding that physical observables remain unaffected. Apparently, the Landau pole in this theory is benign. As a phenomenological application, I compare the O(N) model to QCD by identifying &Lambda;MS&macr; with the Landau pole in the O(N) model.

]]>AppliedMath doi: 10.3390/appliedmath4010002

Authors: Maria de Fátima Brilhante Dinis Pestana Pedro Pestana Maria Luísa Rocha

Modeling the vulnerabilities lifecycle and exploitation frequency are at the core of security of networks evaluation. Pareto, Weibull, and log-normal models have been widely used to model the exploit and patch availability dates, the time to compromise a system, the time between compromises, and the exploitation volumes. Random samples (systematic and simple random sampling) of the time from publication to update of cybervulnerabilities disclosed in 2021 and in 2022 are analyzed to evaluate the goodness-of-fit of the traditional Pareto and log-normal laws. As censoring and thinning almost surely occur, other heavy-tailed distributions in the domain of attraction of extreme value or geo-extreme value laws are investigated as suitable alternatives. Goodness-of-fit tests, the Akaike information criterion (AIC), and the Vuong test, support the statistical choice of log-logistic, a geo-max stable law in the domain of attraction of the Fr&eacute;chet model of maxima, with hyperexponential and general extreme value fittings as runners-up. Evidence that the data come from a mixture of differently stretched populations affects vulnerabilities scoring systems, specifically the common vulnerabilities scoring system (CVSS).

]]>AppliedMath doi: 10.3390/appliedmath4010001

Authors: Loukas Zachilas Christos Benos

Our aim is to provide an insight into the procedures and the dynamics that lead the spread of contagious diseases through populations. Our simulation tool can increase our understanding of the spatial parameters that affect the diffusion of a virus. SIR models are based on the hypothesis that populations are &ldquo;well mixed&rdquo;. Our model constitutes an attempt to focus on the effects of the specific distribution of the initially infected individuals through the population and provide insights, considering the stochasticity of the transmission process. For this purpose, we represent the population using a square lattice of nodes. Each node represents an individual that may or may not carry the virus. Nodes that carry the virus can only transfer it to susceptible neighboring nodes. This important revision of the common SIR model provides a very realistic property: the same number of initially infected individuals can lead to multiple paths, depending on their initial distribution in the lattice. This property creates better predictions and probable scenarios to construct a probability function and appropriate confidence intervals. Finally, this structure permits realistic visualizations of the results to understand the procedure of contagion and spread of a disease and the effects of any measures applied, especially mobility restrictions, among countries and regions.

]]>AppliedMath doi: 10.3390/appliedmath3040052

Authors: Roberto Herrero Joan Nieves Augusto Gonzalez

The innate immune system is the first line of defense against pathogens. Its composition includes barriers, mucus, and other substances as well as phagocytic and other cells. The purpose of the present paper is to compare tissues with regard to their immune response to infections and to cancer. Simple ideas and the qualitative theory of differential equations are used along with general principles such as the minimization of the pathogen load and economy of resources. In the simplest linear model, the annihilation rate of pathogens in any tissue should be greater than the pathogen&rsquo;s average replication rate. When nonlinearities are added, a stability condition emerges, which relates the strength of regular threats, barrier height, and annihilation rate. The stability condition allows for a comparison of immunity in different tissues. On the other hand, in cancer immunity, the linear model leads to an expression for the lifetime risk, which accounts for both the effects of carcinogens (endogenous or external) and the immune response. The way the tissue responds to an infection shows a correlation with the way it responds to cancer. The results of this paper are formulated in the form of precise statements in such a way that they could be checked by present-day quantitative immunology.

]]>AppliedMath doi: 10.3390/appliedmath3040051

Authors: Ekta Sharma Shubham Kumar Mittal J. P. Jaiswal Sunil Panday

New three-step with-memory iterative methods for solving nonlinear equations are presented. We have enhanced the convergence order of an existing eighth-order memory-less iterative method by transforming it into a with-memory method. Enhanced acceleration of the convergence order is achieved by introducing two self-accelerating parameters computed using the Hermite interpolating polynomial. The corresponding R-order of convergence of the proposed uni- and bi-parametric with-memory methods is increased from 8 to 9 and 10, respectively. This increase in convergence order is accomplished without requiring additional function evaluations, making the with-memory method computationally efficient. The efficiency of our with-memory methods NWM9 and NWM10 increases from 1.6818 to 1.7320 and 1.7783, respectively. Numeric testing confirms the theoretical findings and emphasizes the superior efficacy of suggested methods when compared to some well-known methods in the existing literature.

]]>AppliedMath doi: 10.3390/appliedmath3040050

Authors: Alex Santana dos Santos Marcos Eduardo Valle

Max-C and min-D projection auto-associative fuzzy morphological memories (max-C and min-D PAFMMs) are two-layer feedforward fuzzy morphological neural networks designed to store and retrieve finite fuzzy sets. This paper addresses the main features of these auto-associative memories: unlimited absolute storage capacity, fast retrieval of stored items, few spurious memories, and excellent tolerance to either dilative or erosive noise. Particular attention is given to the so-called Zadeh&rsquo; PAFMM, which exhibits the most significant noise tolerance among the max-C and min-D PAFMMs besides performing no floating-point arithmetic operations. Computational experiments reveal that Zadeh&rsquo;s max-C PFAMM, combined with a noise masking strategy, yields a fast and robust classifier with a strong potential for face recognition tasks.

]]>AppliedMath doi: 10.3390/appliedmath3040049

Authors: Jochen Staudacher Tim Pollmann

Computing Shapley values for large cooperative games is an NP-hard problem. For practical applications, stochastic approximation via permutation sampling is widely used. In the context of machine learning applications of the Shapley value, the concept of antithetic sampling has become popular. The idea is to employ the reverse permutation of a sample in order to reduce variance and accelerate convergence of the algorithm. We study this approach for the Shapley and Banzhaf values, as well as for the Owen value which is a solution concept for games with precoalitions. We combine antithetic samples with established stratified sampling algorithms. Finally, we evaluate the performance of these algorithms on four different types of cooperative games.

]]>AppliedMath doi: 10.3390/appliedmath3040048

Authors: Isaac Elishakoff Nicolas Yvain

In this study, we tackle the subject of interval quadratic equations and we aim to accurately determine the root enclosures of quadratic equations, whose coefficients constitute interval variables. This study focuses on interval quadratic equations that contain only one coefficient considered as an interval variable. The four methods reviewed here in order to solve this problem are: (i) the method of classic interval analysis used by Elishakoff and Daphnis, (ii) the direct method based on minimizations and maximizations also used by the same authors, (iii) the method of quantifier elimination used by Ioakimidis, and (iv) the interval parametrization method suggested by Elishakoff and Miglis and again based on minimizations and maximizations. We will also compare the results yielded by all these methods by using the computer algebra system Mathematica for computer evaluations (including quantifier eliminations) in order to conclude which method would be the most efficient way to solve problems relevant to interval quadratic equations.

]]>AppliedMath doi: 10.3390/appliedmath3040047

Authors: Ivan Arraut Ka-I Lei

We review some general aspects about the Black&ndash;Scholes equation, which is used for predicting the fair price of an option inside the stock market. Our analysis includes the symmetry properties of the equation and its solutions. We use the Hamiltonian formulation for this purpose. Taking into account that the volatility inside the Black&ndash;Scholes equation is a parameter, we then introduce the Merton&ndash;Garman equation, where the volatility is stochastic, and then it can be perceived as a field. We then show how the Black&ndash;Scholes equation and the Merton&ndash;Garman one are locally equivalent by imposing a gauge symmetry under changes in the prices over the Black&ndash;Scholes equation. This demonstrates that the stochastic volatility emerges naturally from symmetry arguments. Finally, we analyze the role of the volatility on the decisions taken by the holders of the options when they use the solution of the Black&ndash;Scholes equation as a tool for making investment decisions.

]]>AppliedMath doi: 10.3390/appliedmath3040046

Authors: Kunle Adegoke Robert Frontczak Taras Goy

In this paper, we provide a first systematic treatment of binomial sum relations involving (generalized) Fibonacci and Lucas numbers. The paper introduces various classes of relations involving (generalized) Fibonacci and Lucas numbers and different kinds of binomial coefficients. We also present some novel relations between sums with two and three binomial coefficients. In the course of exploration, we rediscover a few isolated results existing in the literature, commonly presented as problem proposals.

]]>AppliedMath doi: 10.3390/appliedmath3040045

Authors: J. Leonel Rocha Sónia Carvalho Beatriz Coimbra

This paper introduces the mathematical formalization of two probabilistic procedures for susceptible-infected-recovered (SIR) and susceptible-infected-susceptible (SIS) infectious diseases epidemic models, over Erd&ouml;s-R&eacute;nyi contact networks. In our approach, we consider the epidemic threshold, for both models, defined by the inverse of the spectral radius of the associated adjacency matrices, which expresses the network topology. The epidemic threshold dynamics are analyzed, depending on the global dynamics of the network structure. The main contribution of this work is the relationship established between the epidemic threshold and the topological entropy of the Erd&ouml;s-R&eacute;nyi contact networks. In addition, a relationship between the basic reproduction number and the topological entropy is also stated. The trigger of the infectious state is studied, where the probability value of the stability of the infected state after the first instant, depending on the degree of the node in the seed set, is proven. Some numerical studies are included and illustrate the implementation of the probabilistic procedures introduced, complementing the discussion on the choice of the seed set.

]]>AppliedMath doi: 10.3390/appliedmath3040044

Authors: Sara Mollaeivaneghi Allan Santos Florian Steinke

For linear optimization problems with a parametric objective, so-called parametric linear programs (PLP), we show that the optimal decision values are, under few technical restrictions, unimodal functions of the parameter, at least in the two-degrees-of-freedom case. Assuming that the parameter is random and follows a known probability distribution, this allows for an efficient algorithm to determe the quantiles of linear combinations of the optimal decisions. The novel results are demonstrated with probabilistic economic dispatch. For an example setup with uncertain fuel costs, quantiles of the resulting inter-regional power flows are computed. The approach is compared against Monte Carlo and piecewise computation techniques, proving significantly reduced computation times for the novel procedure. This holds especially when the feasible set is complex and/or extreme quantiles are desired. This work is limited to problems with two effective degrees of freedom and a one-dimensional uncertainty. Future extensions to higher dimensions could yield a key tool for the analysis of probabilistic PLPs and, specifically, risk management in energy systems.

]]>AppliedMath doi: 10.3390/appliedmath3040043

Authors: Alexander Uzhinskiy

According to the Food and Agriculture Organization, the world&rsquo;s food production needs to increase by 70 percent by 2050 to feed the growing population. However, the EU agricultural workforce has declined by 35% over the last decade, and 54% of agriculture companies have cited a shortage of staff as their main challenge. These factors, among others, have led to an increased interest in advanced technologies in agriculture, such as IoT, sensors, robots, unmanned aerial vehicles (UAVs), digitalization, and artificial intelligence (AI). Artificial intelligence and machine learning have proven valuable for many agriculture tasks, including problem detection, crop health monitoring, yield prediction, price forecasting, yield mapping, pesticide, and fertilizer usage optimization. In this scoping mini review, scientific achievements regarding the main directions of agricultural technologies will be explored. Successful commercial companies, both in the Russian and international markets, that have effectively applied these technologies will be highlighted. Additionally, a concise overview of various AI approaches will be presented, and our firsthand experience in this field will be shared.

]]>AppliedMath doi: 10.3390/appliedmath3040042

Authors: Daniel A. Griffith

Two linear algebra problems implore a solution to them, creating the themes pursued in this paper. The first problem interfaces with graph theory via binary 0-1 adjacency matrices and their Laplacian counterparts. More contemporary spatial statistics/econometrics applications motivate the second problem, which embodies approximating the eigenvalues of massively large versions of these two aforementioned matrices. The proposed solutions outlined in this paper essentially are a reformulated multiple linear regression analysis for the first problem and a matrix inertia refinement adapted to existing work for the second problem.

]]>AppliedMath doi: 10.3390/appliedmath3040041

Authors: Yiqiao Wang Guanghong Ding Wei Yao

Based on the Hodgkin&ndash;Huxley theory, this paper establishes several nonlinear system models, analyzes the models&rsquo; stability, and studies the conditions for repetitive discharge of neuronal membrane potential. Our dynamic analysis showed that the main channel currents (the fast transient sodium current, the potassium delayed rectifier current, and the fixed leak current) of a neuron determine its dynamic properties and that the GHK formula will greatly widen the stimulation current range of the repetitive discharge condition compared with the Nernst equation. The model including the change in ion concentration will lead to spreading depression (SD)-like depolarization, and the inclusion of a Na-K pump will weaken the current stimulation effect by decreasing the extracellular K accumulation. The results indicate that the Hodgkin&ndash;Huxley model is suitable for describing the response to initial stimuli, but due to changes in ion concentration, it is not suitable for describing the response to long-term stimuli.

]]>AppliedMath doi: 10.3390/appliedmath3040040

Authors: Muhsin Tamturk Marco Carenzo

In this study, we design an algorithm to work on gate-based quantum computers. Based on the algorithm, we construct a quantum circuit that represents the surplus process of a cedant under a reinsurance agreement. This circuit takes into account a variety of factors: initial reserve, insurance premium, reinsurance premium, and specific amounts related to claims, retention, and deductibles for two different non-proportional reinsurance contracts. Additionally, we demonstrate how to perturb the actuarial stochastic process using Hadamard gates to account for unpredictable damage. We conclude by presenting graphs and numerical results to validate our capital modelling approach.

]]>AppliedMath doi: 10.3390/appliedmath3040039

Authors: Aghalaya S. Vatsala Govinda Pageni

Computing the solution of the Caputo fractional differential equation plays an important role in using the order of the fractional derivative as a parameter to enhance the model. In this work, we developed a power series solution method to solve a linear Caputo fractional differential equation of the order q,0&lt;q&lt;1, and this solution matches with the integer solution for q=1. In addition, we also developed a series solution method for a linear sequential Caputo fractional differential equation with constant coefficients of order 2q, which is sequential for order q with Caputo fractional initial conditions. The advantage of our method is that the fractional order q can be used as a parameter to enhance the mathematical model, compared with the integer model. The methods developed here, namely, the series solution method for solving Caputo fractional differential equations of constant coefficients, can be extended to Caputo sequential differential equation with variable coefficients, such as fractional Bessel&rsquo;s equation with fractional initial conditions.

]]>AppliedMath doi: 10.3390/appliedmath3040038

Authors: Robert Gardner Matthew Gladin

Motivated by results on the location of the zeros of a complex polynomial with monotonicity conditions on the coefficients (such as the classical Enestr&ouml;m&ndash;Kakeya theorem and its recent generalizations), we impose similar conditions and give bounds on the number of zeros in certain regions. We do so by introducing a reversal in monotonicity conditions on the real and imaginary parts of the coefficients and also on their moduli. The conditions imposed are less restrictive than many of those in the current literature and hence apply to polynomials not covered by previous results. The results presented naturally apply to certain classes of lacunary polynomials. In particular, the results apply to certain polynomials with two gaps in their coefficients.

]]>AppliedMath doi: 10.3390/appliedmath3040037

Authors: Oluwatosin Babasola Evans Otieno Omondi Kayode Oshinubi Nancy Matendechere Imbusi

Mathematical models have been of great importance in various fields, especially for understanding the dynamical behaviour of biosystems. Several models, based on classical ordinary differential equations, delay differential equations, and stochastic processes are commonly employed to gain insights into these systems. However, there is potential to extend such models further by combining the features from the classical approaches. This work investigates stochastic delay differential equations (SDDEs)-based models to understand the behaviour of biosystems. Numerical techniques for solving these models that demonstrate a more robust representation of real-life scenarios are presented. Additionally, quantitative roles of delay and noise to gain a deeper understanding of their influence on the system&rsquo;s overall behaviour are analysed. Subsequently, numerical simulations that illustrate the model&rsquo;s robustness are provided and the results suggest that SDDEs provide a more comprehensive representation of many biological systems, effectively accounting for the uncertainties that arise in real-life situations.

]]>AppliedMath doi: 10.3390/appliedmath3030036

Authors: Edoardo Ballico

Let X be a smooth projective variety and f:X&rarr;Pr a morphism birational onto its image. We define the Terracini loci of the map f. Most results are only for the case dimX=1. With this new and more flexible definition, it is possible to prove strong nonemptiness results with the full classification of all exceptional cases. We also consider Terracini loci with restricted support (solutions not intersecting a closed set B&#8842;X or solutions containing a prescribed p&isin;X). Our definitions work both for the Zariski and the euclidean topology and we suggest extensions to the case of real varieties. We also define Terracini loci for joins of two or more subvarieties of the same projective space. The proofs use algebro-geometric tools.

]]>AppliedMath doi: 10.3390/appliedmath3030035

Authors: Jose Pablo Rodriguez David F. Muñoz

The Mexico City Metrobus is one of the most popular forms of public transportation inside the city, and since its opening in 2005, it has become a vital piece of infrastructure for the city; this is why the optimal functioning of the system is of key importance to Mexico City, as it plays a crucial role in moving millions of passengers every day. This paper presents a model to simulate Line 1 of the Mexico City Metrobus, which can be adapted to simulate other bus rapid transit (BRT) systems. We give a detailed description of the model development so that the reader can replicate our model. We developed various response variables in order to evaluate the system&rsquo;s performance, which focused on passenger satisfaction and measured the maximum occupancy that a passenger experiences inside the buses, as well as the time that he spends in the queues at the stations. The results of the experiments show that it is possible to increase passenger satisfaction by considering different combinations of routes while maintaining the same fuel consumption. It was shown that, by considering an appropriate combination of routes, the average passenger satisfaction could surpass the satisfaction levels obtained by a 10% increase in total fuel consumption.

]]>AppliedMath doi: 10.3390/appliedmath3030034

Authors: Nicola Cufaro Petroni

In this article, some prescriptions to define a distribution on the set Q0 of all rational numbers in [0,1] are outlined. We explored a few properties of these distributions and the possibility of making these rational numbers asymptotically equiprobable in a suitable sense. In particular, it will be shown that in the said limit&mdash;albeit no absolutely continuous uniform distribution can be properly defined in Q0&mdash;the probability allotted to every single q&isin;Q0 asymptotically vanishes, while that of the subset of Q0 falling in an interval [a,b]&sube;Q0 goes to b&minus;a. We finally present some hints to complete sequencing without repeating the numbers in Q0 as a prerequisite to laying down more distributions on it.

]]>AppliedMath doi: 10.3390/appliedmath3030033

Authors: Ayokunle J. Tadema Micheal O. Ogundiran

This paper is concerned with the existence of solutions of a class of Cauchy problems for hyperbolic partial fractional differential inclusions (HPFD) involving the Caputo fractional derivative with an impulse whose right hand side is convex and non-convex valued. Our results are achieved within the framework of the nonlinear alternative of Leray-Schauder type and contraction multivalued maps. A detailed example was provided to support the theorem.

]]>AppliedMath doi: 10.3390/appliedmath3030032

Authors: Fateh Mohamed Ali Adhnouss Husam M. Ali El-Asfour Kenneth McIsaac Idris El-Feghi

Artificial Intelligence (AI) systems are increasingly being deployed in decentralized environments where they interact with other AI systems and humans. In these environments, each participant may have different ways of expressing the same semantics, leading to challenges in communication and collaboration. To address these challenges, this paper presents a novel hybrid model for shared conceptualization in decentralized AI systems. This model integrates ontology, epistemology, and epistemic logic, providing a formal framework for representing and reasoning about shared conceptualization. It captures both the intensional and extensional components of the conceptualization structure and incorporates epistemic logic to capture knowledge and belief relationships between agents. The model&rsquo;s unique contribution lies in its ability to handle different perspectives and beliefs, making it particularly suitable for decentralized environments. To demonstrate the model&rsquo;s practical application and effectiveness, it is applied to a scenario in the healthcare sector. The results show that the model has the potential to improve AI system performance in a decentralized context by enabling efficient communication and collaboration among agents. This study fills a gap in the literature concerning the representation of shared conceptualization in decentralized environments and provides a foundation for future research in this area.

]]>AppliedMath doi: 10.3390/appliedmath3030031

Authors: David Fernando Muñoz

When there is uncertainty in the value of parameters of the input random components of a stochastic simulation model, two-level nested simulation algorithms are used to estimate the expectation of performance variables of interest. In the outer level of the algorithm n observations are generated for the parameters, and in the inner level m observations of the simulation model are generated with the values of parameters fixed at the values generated in the outer level. In this article, we consider the case in which the observations at both levels of the algorithm are independent and show how the variance of the observations can be decomposed into the sum of a parametric variance and a stochastic variance. Next, we derive central limit theorems that allow us to compute asymptotic confidence intervals to assess the accuracy of the simulation-based estimators for the point forecast and the variance components. Under this framework, we derive analytical expressions for the point forecast and the variance components of a Bayesian model to forecast sporadic demand, and we use these expressions to illustrate the validity of our theoretical results by performing simulation experiments with this forecast model. We found that, given a fixed number of total observations nm, the choice of only one replication in the inner level (m=1) is recommended to obtain a more accurate estimator for the expectation of a performance variable.

]]>AppliedMath doi: 10.3390/appliedmath3030030

Authors: Luis A. Moncayo-Martínez Elias H. Arias-Nava

The simple assembly line balancing (SALB) problem is a significant challenge faced by industries across various sectors aiming to optimise production line efficiency and resource allocation. One important issue when the decision-maker balances a line is how to keep the cycle time under a given time across all cells, even though there is variability in some parameters. When there are stochastic elements, some approaches use constraint relaxation, intervals for the stochastic parameters, and fuzzy numbers. In this paper, a three-part algorithm is proposed that first solves the balancing problem without considering stochastic parameters; then, using simulation, it measures the effect of some parameters (in this case, the inter-arrival time, processing times, speed of the material handling system which is manually performed by the workers in the cell, and the number of workers who perform the tasks on the machines); finally, the add-on OptQuest in SIMIO solves an optimisation problem to constrain the cycle time using the stochastic parameters as decision variables. A Gearbox instance from literature is solved with 15 tasks and 14 precedence rules to test the proposed approach. The deterministic balancing problem is solved optimally using the open solver GLPK and the Pyomo programming language, and, with simulation, the proposed algorithm keeps the cycle time less than or equal to 70 s in the presence of variability and deterministic inter-arrival time. Meanwhile, with stochastic inter-arrival time, the maximum cell cycle is 72.04 s. The reader can download the source code and the simulation models from the GitHub page of the authors.

]]>AppliedMath doi: 10.3390/appliedmath3030029

Authors: Sangeeta Yadav

We propose a Quantum Neural Network (QNN) for predicting stabilization parameter for solving Singularly Perturbed Partial Differential Equations (SPDE) using the Streamline Upwind Petrov Galerkin (SUPG) stabilization technique. SPDE-Q-Net, a QNN, is proposed for approximating an optimal value of the stabilization parameter for SUPG for 2-dimensional convection-diffusion problems. Our motivation for this work stems from the recent progress made in quantum computing and the striking similarities observed between neural networks and quantum circuits. Just like how weight parameters are adjusted in traditional neural networks, the parameters of the quantum circuit, specifically the qubits&rsquo; degrees of freedom, can be fine-tuned to learn a nonlinear function. The performance of SPDE-Q-Net is found to be at par with SPDE-Net, a traditional neural network-based technique for stabilization parameter prediction in terms of the numerical error in the solution. Also, SPDE-Q-Net is found to be faster than SPDE-Net, which projects the future benefits which can be earned from the speed-up capabilities of quantum computing.

]]>AppliedMath doi: 10.3390/appliedmath3030028

Authors: Jianying Zhang

As a class of non-Newtonian fluids with yield stresses, Bingham fluids possess both solid and liquid phases separated by implicitly defined non-physical yield surfaces, which makes the standard numerical discretization challenging. The variational reformulation established by Duvaut and Lions, coupled with an augmented Lagrange method (ALM), brings about a finite element approach, whereas the inevitable local mesh refinement and preconditioning of the resulting large-scaled ill-conditioned linear system can be involved. Inspired by the mesh-free feature and architecture flexibility of physics-informed neural networks (PINNs), an ALM-PINN approach to steady-state Bingham fluid flow simulation, with dynamically adaptable weights, is developed and analyzed in this work. The PINN setting enables not only a pointwise ALM formulation but also the learning of families of (physical) parameter-dependent numerical solutions through one training process, and the incorporation of ALM into a PINN induces a more feasible loss function for deep learning. Numerical results obtained via the ALM-PINN training on one- and two-dimensional benchmark models are presented to validate the proposed scheme. The efficacy and limitations of the relevant loss formulation and optimization algorithms are also discussed to motivate some directions for future research.

]]>AppliedMath doi: 10.3390/appliedmath3030027

Authors: Polychronis Manousopoulos Vasileios Drakopoulos Efstathios Polyzos

Time series of financial data are both frequent and important in everyday practice. Numerous applications are based, for example, on time series of asset prices or market indices. In this article, the application of fractal interpolation functions in modelling financial time series is examined. Our motivation stems from the fact that financial time series often present fluctuations or abrupt changes which the fractal interpolants can inherently model. The results indicate that the use of fractal interpolation in financial applications is promising.

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