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AppliedMath
Volume 3
Issue 4
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AppliedMath, Volume 3, Issue 4 (December 2023) – 16 articles

Cover Story (view full-size image): Zero eigenvalues coincide with linear combinations of some matrix column subsets being identical to other columns. Likewise, symmetry also transfers this relationship to matrix rows. Here, the surface partitioning square-star grid overlay on the Fajardo, Puerto Rico, region satellite image portrayal generates a matrix with three zero eigenvalues, identified in its labels and accompanying spatial weights matrix by red, blue, and green. A linear regression of Column A as a response variable and columns B through K as covariates produces the three linear combinations affiliated with these zero eigenvalues, and all three points are coincident in a rank order scatterplot, which distorts their alignment with the graph’s near-linear trendline. View this paper
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11 pages, 341 KiB  
Article
First-Stage Dynamics of the Immune System and Cancer
by Roberto Herrero, Joan Nieves and Augusto Gonzalez
AppliedMath 2023, 3(4), 1034-1044; https://doi.org/10.3390/appliedmath3040052 - 12 Dec 2023
Viewed by 725
Abstract
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 [...] Read more.
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’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. Full article
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15 pages, 488 KiB  
Article
An Efficient Bi-Parametric With-Memory Iterative Method for Solving Nonlinear Equations
by Ekta Sharma, Shubham Kumar Mittal, J. P. Jaiswal and Sunil Panday
AppliedMath 2023, 3(4), 1019-1033; https://doi.org/10.3390/appliedmath3040051 - 11 Dec 2023
Viewed by 706
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Contemporary Iterative Methods with Applications in Applied Sciences)
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30 pages, 1226 KiB  
Article
Max-C and Min-D Projection Auto-Associative Fuzzy Morphological Memories: Theory and an Application for Face Recognition
by Alex Santana dos Santos and Marcos Eduardo Valle
AppliedMath 2023, 3(4), 989-1018; https://doi.org/10.3390/appliedmath3040050 - 08 Dec 2023
Viewed by 720
Abstract
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 [...] Read more.
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’ 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’s max-C PFAMM, combined with a noise masking strategy, yields a fast and robust classifier with a strong potential for face recognition tasks. Full article
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32 pages, 451 KiB  
Article
Assessing Antithetic Sampling for Approximating Shapley, Banzhaf, and Owen Values
by Jochen Staudacher and Tim Pollmann
AppliedMath 2023, 3(4), 957-988; https://doi.org/10.3390/appliedmath3040049 - 06 Dec 2023
Viewed by 750
Abstract
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 [...] Read more.
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. Full article
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48 pages, 663 KiB  
Review
Interval Quadratic Equations: A Review
by Isaac Elishakoff and Nicolas Yvain
AppliedMath 2023, 3(4), 909-956; https://doi.org/10.3390/appliedmath3040048 - 01 Dec 2023
Viewed by 654
Abstract
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 [...] Read more.
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. Full article
27 pages, 542 KiB  
Review
The Role of the Volatility in the Option Market
by Ivan Arraut and Ka-I Lei
AppliedMath 2023, 3(4), 882-908; https://doi.org/10.3390/appliedmath3040047 - 01 Dec 2023
Viewed by 602
Abstract
We review some general aspects about the Black–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 [...] Read more.
We review some general aspects about the Black–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–Scholes equation is a parameter, we then introduce the Merton–Garman equation, where the volatility is stochastic, and then it can be perceived as a field. We then show how the Black–Scholes equation and the Merton–Garman one are locally equivalent by imposing a gauge symmetry under changes in the prices over the Black–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–Scholes equation as a tool for making investment decisions. Full article
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31 pages, 367 KiB  
Article
Binomial Sum Relations Involving Fibonacci and Lucas Numbers
by Kunle Adegoke, Robert Frontczak and Taras Goy
AppliedMath 2023, 3(4), 851-881; https://doi.org/10.3390/appliedmath3040046 - 30 Nov 2023
Viewed by 588
Abstract
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 [...] Read more.
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. Full article
23 pages, 4338 KiB  
Article
Probabilistic Procedures for SIR and SIS Epidemic Dynamics on Erdös-Rényi Contact Networks
by J. Leonel Rocha, Sónia Carvalho and Beatriz Coimbra
AppliedMath 2023, 3(4), 828-850; https://doi.org/10.3390/appliedmath3040045 - 16 Nov 2023
Viewed by 838
Abstract
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 Erdös-Rényi contact networks. In our approach, we consider the epidemic threshold, for both models, defined by the inverse of the spectral radius [...] Read more.
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 Erdös-Rényi 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 Erdös-Rényi 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. Full article
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14 pages, 495 KiB  
Article
Unimodality of Parametric Linear Programming Solutions and Efficient Quantile Estimation
by Sara Mollaeivaneghi, Allan Santos and Florian Steinke
AppliedMath 2023, 3(4), 814-827; https://doi.org/10.3390/appliedmath3040044 - 07 Nov 2023
Viewed by 657
Abstract
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 [...] Read more.
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. Full article
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15 pages, 7258 KiB  
Review
Advanced Technologies and Artificial Intelligence in Agriculture
by Alexander Uzhinskiy
AppliedMath 2023, 3(4), 799-813; https://doi.org/10.3390/appliedmath3040043 - 01 Nov 2023
Viewed by 2380
Abstract
According to the Food and Agriculture Organization, the world’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 [...] Read more.
According to the Food and Agriculture Organization, the world’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. Full article
(This article belongs to the Special Issue Application of Machine Learning and Deep Learning Methods in Science)
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28 pages, 8158 KiB  
Article
Some Comments about Zero and Non-Zero Eigenvalues from Connected Undirected Planar Graph Adjacency Matrices
by Daniel A. Griffith
AppliedMath 2023, 3(4), 771-798; https://doi.org/10.3390/appliedmath3040042 - 01 Nov 2023
Viewed by 1132
Abstract
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 [...] Read more.
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. Full article
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13 pages, 2406 KiB  
Article
Dynamic Analysis of Neuron Models
by Yiqiao Wang, Guanghong Ding and Wei Yao
AppliedMath 2023, 3(4), 758-770; https://doi.org/10.3390/appliedmath3040041 - 30 Oct 2023
Viewed by 837
Abstract
Based on the Hodgkin–Huxley theory, this paper establishes several nonlinear system models, analyzes the models’ 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 [...] Read more.
Based on the Hodgkin–Huxley theory, this paper establishes several nonlinear system models, analyzes the models’ 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–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. Full article
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17 pages, 1953 KiB  
Article
Quantum Computing in Insurance Capital Modelling under Reinsurance Contracts
by Muhsin Tamturk and Marco Carenzo
AppliedMath 2023, 3(4), 741-757; https://doi.org/10.3390/appliedmath3040040 - 26 Oct 2023
Viewed by 942
Abstract
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: [...] Read more.
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. Full article
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11 pages, 275 KiB  
Article
Series Solution Method for Solving Sequential Caputo Fractional Differential Equations
by Aghalaya S. Vatsala and Govinda Pageni
AppliedMath 2023, 3(4), 730-740; https://doi.org/10.3390/appliedmath3040039 - 20 Oct 2023
Cited by 1 | Viewed by 758
Abstract
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 [...] Read more.
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<q<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’s equation with fractional initial conditions. Full article
8 pages, 250 KiB  
Article
The Number of Zeros in a Disk of a Complex Polynomial with Coefficients Satisfying Various Monotonicity Conditions
by Robert Gardner and Matthew Gladin
AppliedMath 2023, 3(4), 722-729; https://doi.org/10.3390/appliedmath3040038 - 17 Oct 2023
Viewed by 583
Abstract
Motivated by results on the location of the zeros of a complex polynomial with monotonicity conditions on the coefficients (such as the classical Eneström–Kakeya theorem and its recent generalizations), we impose similar conditions and give bounds on the number of zeros in certain [...] Read more.
Motivated by results on the location of the zeros of a complex polynomial with monotonicity conditions on the coefficients (such as the classical Eneström–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. Full article
20 pages, 662 KiB  
Article
Stochastic Delay Differential Equations: A Comprehensive Approach for Understanding Biosystems with Application to Disease Modelling
by Oluwatosin Babasola, Evans Otieno Omondi, Kayode Oshinubi and Nancy Matendechere Imbusi
AppliedMath 2023, 3(4), 702-721; https://doi.org/10.3390/appliedmath3040037 - 09 Oct 2023
Cited by 1 | Viewed by 1522
Abstract
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 [...] Read more.
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’s overall behaviour are analysed. Subsequently, numerical simulations that illustrate the model’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. Full article
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