Our work is aimed at modeling the American option price by combining the dynamic programming and the optimal stopping time under two asset price models. In doing so, we attempt to control the theoretical error and illustrate the asymptotic characteristics of each model; thus, using a numerical illustration of the convergence of the option price to an equilibrium price, we can notice its behavior when the number of paths tends to be a large number; therefore, we construct a simple estimator on each slice of the number of paths according to an upper and lower bound to control our error. Finally, to highlight our approach, we test it on different asset pricing models, in particular, the exponential Lévy model compared to the simple Black and Scholes model, and we will show how the latter outperforms the former in the real market (Microsoft “MSFT” put option as an example).

A bewildering large number of test statistics have been found for testing the presence of an outlier in multiple linear regression models. Exact critical values of these test statistics are not available, and approximate ones are usually obtained by the first-order Bonferroni upper bound or large-scale simulations. In this paper, we show that the upper bound values of two of these test statistics are algebraically the same. An application to real data for multiple linear regression is used to demonstrate the procedure.

The flexible link system (FLS) was a highly nonlinear model, multivariable and absolutely unstable dynamic system. In practice, it is common to integrate multiple subsystems into the main system with dynamically turned $\mathrm{k}$-input signals $\left(\mathrm{k}=1,2,\dots ,\mathrm{N}\right)$ to diversity the functionality of the main system. The flexibility of the division method to convert k-input MIMO system to SISOs system combined with the optimal algorithm creates a powerful tool that can be applied to many different MIMO nonlinear systems with high success rates. The optimal controllers can be created in the future for the flexible system is implemented on an experiment system using Arduino UNO micro-controller KIT. This paper describes division method to convert the k-input MIMO system to SISOs system, after that combined with the optimal algorithm to control for the flexible link system. Specifically, the author will conduct oscillating component analysis of a system with k input pairs $\left(\mathrm{k}=1,2\right)$ so that the author can better understand the nature of sub-components as they interact with the system.

The avoidance strategy of prey to predation and the predation strategy for predators are important topics in evolutionary biology. Both prey and predators adjust their behaviors in order to obtain the maximal benefits and to raise their biomass for each. Therefore, this paper is aimed at studying the impact of prey’s fear and group defense against predation on the dynamics of the food-web model. Consequently, in this paper, a mathematical model that describes a tritrophic Leslie-Gower food-web system is formulated. Sokol-Howell type of function response is adapted to describe the predation process due to the prey’s group defensive capability. The effects of fear due to the predation process are considered in the first two levels. It is assumed that the generalist predator grows logistically using the Leslie-Gower type of growth function. All the solution properties of the model are studied. Local dynamics behaviors are investigated. The basin of attraction for each equilibrium is determined using the Lyapunov function. The conditions of persistence of the model are specified. The study of local bifurcation in the model is done. Numerical simulations are implemented to show the obtained results. It is watched that the system is wealthy in its dynamics including chaos. The fear factor works as a stabilizing factor in the system up to a specific level; otherwise, it leads to the extinction of the predator. However, increasing the prey’s group defense leads to extinction in predator species.

This article develops the semianalytical analysis of couple stress hybrid nanofluid free stream past a rotating disc by applying the magnetic flux effects and radiation of thermal energy. The analysis of such kind of mixed convective flow is most important due to numerous industrial applications such as electronic devices, atomic reactors, central solar energy equipment, and heat transferring devices. The impact of variable permeability is also considered in the study. The permeability of the disc obeys the Darcy-Forchheimer model. The hybrid nanofluid is composed of water, titanium dioxide, and aluminum oxide. The set of governing equations in the PDE form are transformed to couple ODEs by applying similarity transformations. The ODE set are solved by applying the technique of HAM. The graphs of impacts of numerous physical parameters over momentum, energy, and concentration profiles are drawn in computer-based application Mathematica 11.0.1. In the sundry physical parameters, the porosity parameter, Reynolds number, inertial parameter, Prandtl number, Schmidt number, couple stress, and quotient of rotational momentum to elongating rate are included. During the analysis, it is found that the momentum profile of the couple stress hybrid nanofluid enhances with local inertial parameter, couple stress parameter, porosity parameter, and Reynolds number but declines for the growth in Hartmann number. Heat transfer rate enhances for radiation parameter but decreases in variable for temperature, thermal stratification parameter, thermophoresis parameter, and Brownian parameter.

This study is aimed at building and analysing a SIRS model and also simulating the model to predict the number of dengue fever cases. Methods applied for this model are building the SIRS model by modifying the SIR model, analysing the SIRS model using the Lyapunov function to prove three theorems (the existence, the free disease, and the endemic status of dengue fever), and simulating the SIRS model using the number of dengue case data in South Sulawesi by Maple. The results obtained are the SIRS model of dengue fever transmission, stability analysis, global stability, and the value of the basic reproduction number $R_0$. The simulation done for the dengue fever case in South Sulawesi found the basic reproduction number ; it means that South Sulawesi is in the endemic stage of transmission for dengue fever disease. Simulation of the SIRS model for dengue fever can predict the number of dengue cases in South Sulawesi that could be a recommendation for the government in an effort to prevent the number of dengue fever cases.

A three-compartmental delay model is formulated to describe the pharmacokinetics of drugs subjected to both intravenous and oral doses with reabsorptions by the central compartment. Model dynamics are analyzed rigorously, and two equilibrium points are obtained to be locally asymptotically stable under certain conditions. Time delays used as lags in reabsorption of drugs by central compartment from other two compartments caused rebounds or peaks and fluctuations in the time profiles for amounts of drug in all the compartments. Sensitivity analysis revealed that elimination rates decrease the amounts in all compartments. Furthermore, reabsorption rates cause superimposition at the initial phases of the drug amount profiles; subsequently, the quantities decrease in compartment one and increase in compartments two and three, respectively.

Extreme events in earthquakes, wind speed, among others are rare but may lead to catastrophic effects on humans and the environment. The primary parameter in the estimation of such rare events is the tail index which measures the tail heaviness of an underlying distribution. Since extreme events are rare, the presence of missing observations may further lead to flawed. In view of this, there is a growing effort by researchers to address this problem. However, the existing methods of estimating the tail index use only the available nonmissing data. Thus, if the missing observations are influential values, ignoring them could introduce more bias and higher mean square error (MSE) in the tail index estimation and subsequently other extreme event–estimators such as high quantiles and small exceedance probabilities. In this study, we propose imputation of the missing observations before applying some standard estimators (Hill and geometric-type) to estimate the tail index. Through a simulation study, we assess the performance of the standard estimators under the proposed data enhancement method and the existing modified estimators of the tail index. The results show that the enhanced estimators have relatively lower bias and MSE. The estimation method was illustrated with a practical dataset on wind speed with missing values. Therefore, we recommend imputation mechanism as viable for enhancing the performance of tail index estimators in the case where there is missingness.

The concepts of Hilbert implication algebra and generalized Hilbert implication algebra are introduced. The comparison theorem of Hilbert implication algebra and generalized Hilbert implication algebra is proved. In addition, the idea of groupoid and commutative Hilbert implication algebras is investigated. Ideals and filters in Hilbert implication algebras are also discussed. In general, different theorems which show different properties are proved.

Key $n$ -gram extraction can be seen as extracting $n$-grams which can distinguish different registers. Keyword (as $n=1$, 1-gram is the keyword) extraction models are generally carried out from two aspects, the feature extraction and the model design. By summarizing the advantages and disadvantages of existing models, we propose a novel key $n$-gram extraction model “attentive $n$-gram network” (ANN) based on the attention mechanism and multilayer perceptron, in which the attention mechanism scores each $n$-gram in a sentence by mining the internal semantic relationship between words, and their importance is given by the scores. Experimental results on the real corpus show that the key $n$-gram extracted from our model can distinguish a novel, news, and text book very well; the accuracy of our model is significantly higher than the baseline model. Also, we conduct experiments on key $n$-grams extracted from these registers, which turned out to be well clustered. Furthermore, we make some statistical analyses of the results of key $n$-gram extraction. We find that the key $n$-grams extracted by our model are very explanatory in linguistics.

In the paper, we have considered a nonlinear compartmental mathematical model that assesses the effect of treatment on the dynamics of HIV/AIDS and pneumonia coinfection in a human population at different infection stages. Our model revealed that the disease-free equilibrium points of the HIV/AIDS and pneumonia submodels are both locally and globally asymptotically stable whenever the associated basic reproduction numbers ( ${\mathcal{R}}_H$ and ${\mathcal{R}}_P$) are less than unity. Both the submodel endemic equilibrium points are locally and globally asymptotically stable whenever the associated basic reproduction numbers ( ${\mathcal{R}}_P$ and ${\mathcal{R}}_H$) are greater than unity. The full HIV/AIDS-pneumonia coinfection model has both locally and globally asymptotically stable disease-free equilibrium points whenever the basic reproduction number of the coinfection model $\left({\mathcal{R}}_{HP}\right)$ is less than unity. Using standard values of parameters collected from different kinds of literature, we found that the numerical values of the basic reproduction numbers of the HIV/AIDS-only submodel and pneumonia-only submodel are 17 and 7, respectively, and the basic reproduction number of the HIV/AIDS-pneumonia coinfection model is $\max \left\{7,17\right\}=17$. Applying sensitive analysis, we identified the most influential parameters to change the behavior of the solution of the considered coinfection dynamical system are the HIV/AIDS and pneumonia transmission rates ${\beta }_1$ and ${\beta }_2$, respectively. The coinfection model was numerically simulated to investigate the stability of the coinfection endemic equilibrium point, the impacts of transmission rates, and treatment strategies for HIV/AIDS-only, pneumonia-only, and HIV/AIDS-pneumonia coinfected individuals. Finally, we observed that numerical simulations indicate that treatment against infection at every stage lowers the rate of infection or disease prevalence.

The supra topological topic is of great importance in preserving some topological properties under conditions weaker than topology and constructing a suitable framework to describe many real-life problems. Herein, we introduce the version of complete Hausdorffness and complete regularity on supra topological spaces and discuss their fundamental properties. We show the relationships between them with the help of examples. In general, we study them in terms of hereditary and topological properties and prove that they are closed under the finite product space. One of the issues we are interested in is showing the easiness and diversity of constructing examples that satisfy supra $T_i$ spaces compared with their counterparts on general topology.

The biggest challenge of treating HIV is rampant liver-related morbidity and mortality. This is, to some extent, attributed to hepatocytes acting as viral reservoirs to both HIV and HBV. Viral reservoirs harbour latent provirus, rendering it inaccessible by combinational antiretroviral therapy (cART) that is specific to actively proliferating virus. Latency reversal agents (LRA) such as Shock and kill or lock and block, aiming at activating the latently infected cells, have been developed. However, they are CD4+ cell-specific only. There is evidence that the low replication level of HIV in hepatocytes is mainly due to the latency of the provirus in these cells. LRA are developed to reduce the number of latently infected cells; however, the impact of the period viral latency in hepatocytes especially, during HIV/HBV coinfection, needs to be investigated. Viral coinfection coupled with lifelong treatment of HIV/HBV necessitates investigation for the optimal control strategy. We propose a coinfection mathematical model with delay and use optimal control theory to analyse the effect of viral latency in hepatocytes on the dynamics of HIV/HBV coinfection. Analytical results indicate that HBV cannot take a competitive exclusion against HIV; thus, the coinfection endemic equilibrium implies chronic HBV in HIV-infected patients. Numerical and analytical results indicate that both HIV and HBV viral loads are higher with longer viral latency period in hepatocytes, which indicates the need to upgrade LRA to other non-CD4+ cell viral reservoirs. Higher viral load caused by viral latency coupled with the effects of cART partly explains why liver-related complications are the leading cause of mortality in HIV-infected persons.

Summarizing or averaging a sequential data set (i.e., a set of time series) can be comprehensively approached as a result of sophisticated computational tools. Averaging under Dynamic Time Warping (DTW) is one such tool that captures consensus patterns. DTW acts as a similarity measure between time series, and subsequently, an averaging method must be executed upon the behaviour of DTW. However, averaging under DTW somewhat neglects temporal aspect since it is on the search of similar appearances rather than stagnating on corresponding time-points. On the contrary, the mean series carrying point-wise averages provides only a weak consensus pattern as it may over-smooth important temporal variations. As a compromise, a pool of consensus series termed Ultimate Tamed Series (UTS) is studied here that adheres to temporal decomposition supported by the discrete Haar wavelet. We claim that UTS summarizes localized patterns, which would not be reachable via the series under DTW or the mean series. Neighbourhood of localization can be altered as a user can customize different levels of decomposition. In validation, comparisons are carried out with the series under DTW and the mean series via Euclidean distance and the distance resulted by DTW itself. Two sequential data sets are selected for this purpose from a standard repository.

To preserve crop production losses, monitoring of desert locust attacks is a significant feature of agriculture. In this paper, a mathematical model was formulated and analyzed to protect crops against desert locust attack via early intervention tactics. We consider a triple intervention approach, namely, proaction, reaction, and outbreak prevention. The model integrates a stage-structured locust population, logistics-based crop biomass, and blended early intervention via pesticide spray. We assume that the amount of pesticide spray is proportional to the density of the locust population in the infested area. Conventional short residual pesticides within ultralow volume formulation and equipment control operations are considered. The trivial and locust-free equilibrium of the model is unstable, whereas the interior equilibrium is asymptotically stable. Numerical simulations validate the theoretical results of the model. In the absence of intervention measures, desert locust losses are approximately 71% of expected crop production. The model projection shows that effective proactive early intervention on hopper stage locust contained locust infestation and subdued public health and environmental hazards. Relevant and up-to-date combined early interventions control desert locust aggression and crop production losses.

The face is the second most important biometric part of the human body, next to the finger print. Recognition of face image with partial occlusion (half image) is an intractable exercise as occlusions affect the performance of the recognition module. To this end, occluded images are sometimes reconstructed or completed with some imputation mechanism before recognition. This study assessed the performance of the principal component analysis and singular value decomposition algorithm using discrete wavelet transform (DWT-PCA/SVD) as preprocessing mechanism on the reconstructed face image database. The reconstruction of the half face images was done leveraging on the property of bilateral symmetry of frontal faces. Numerical assessment of the performance of the adopted recognition algorithm gave average recognition rates of 95% and 75% when left and right reconstructed face images were used for recognition, respectively. It was evident from the statistical assessment that the DWT-PCA/SVD algorithm gives relatively lower average recognition distance for the left reconstructed face images. DWT-PCA/SVD is therefore recommended as a suitable algorithm for recognizing face images under partial occlusion (half face images). The algorithm performs relatively better on left reconstructed face images.

Consider a closed cyclic queueing model that consists of two nodes and a total of $M$ customers. Each node buffer can accommodate all $M$ customers. Node 1 has $N\le M$ servers, each having an exponential service time with rate $\lambda$. The second node consists of a single server with a general service time distribution function $B\left(.\right)$. The well-known machine repair model with spares, where a set of identical machines, $N$, is served by a single repair person, is a key application of this model. This model has several other applications in performance evaluation, manufacturing, computer networks, and in reliability studies as it can be easily used to compute system availability. In this article, we give an efficient algorithm to derive an exact solution for the steady state system size probabilities. Our approach is based on developing an efficient polynomial convolution method to compute the transition probabilities of a birth process over node 2 service times and solving an imbedded Markov chain at node 2 service completion epochs. This is a significant improvement over the exponential algorithm given in an earlier paper. Numerical examples are given to demonstrate the performance of our method.

In this paper, under the effects of the largest primary radiation pressure, the elliptic restricted four-body problem is formulated in Hamiltonian form. Moreover, the canonical equations are obtained which are considered as the equations of motion. The Lagrangian points within the frame of the elliptic restricted four-body problem are obtained. The true anomalies are considered as independent variables. An analytical and numerical approach had been used. A code of Mathematica version 12 is constructed to truncate these considerations and is applied on the Earth-Moon-Sun system. In addition, the stability and periodicity of the motion about the equilibrium points are studied by using the Poincare maps. The motion about the collinear point L2 is presented as an example for the obtained results, and some families of periodic orbits are presented.

Novel coronavirus (COVID-19) has been spreading and wreaking havoc globally, despite massive efforts by the government and World Health Organization (WHO). Consideration of partially recovered carriers is hypothesized to play a leading role in the persistence of the disease and its introduction to new areas. A model for transmission of COVID-19 by symptomless partially recovered carriers is proposed and analysed. It is shown that key parameters can be identified such that below a threshold level, built on these parameters, the epidemic tends towards extinction, while above another threshold, it tends towards a nontrivial epidemic state. Moreover, optimal control analysis of the model, using Pontryagin’s maximum principle, is performed. The optimal controls are characterized in terms of the optimality system and solved numerically for several scenarios. Numerical simulations and sensitivity analysis of the basic reproduction number, $R_c$, indicate that the disease is mainly driven by parameters involving the partially recovered carriers rather than symptomatic ones. Moreover, optimal control analysis of the model, using Pontryagin’s maximum principle, is performed. The optimal controls were characterized in terms of the optimality system and solved numerically for several scenarios. Numerical simulations were explored to illustrate our theoretical findings, scenarios were built, and the model predicted that social distancing and treatment of the symptomatic will slow down the epidemic curve and reduce mortality of COVID-19 given that there is an average adherence to social distancing and effective treatment are administered.

Two-dimensional theta functions were found by the Borwein brothers to work on Gauss and Legendre’s arithmetic-geometric mean iteration. In this paper, some new Eisenstein series identities are obtained by using ($p, k$)-parametrization in terms of Borweins’ theta functions.

The Mumford-Shah model is extensively used in image segmentation. Its energy functional causes the content of the segments to remain homogeneous and the segment boundaries to become short. However, the problem is that optimization of the functional can be very slow. To attack this problem, we propose a reduced two-phase Mumford-Shah model to segment images having one prominent object. First, initial segmentation is obtained by the $\text{k}$-means clustering technique, further minimizing the Mumford-Shah functional by the Douglas-Rachford algorithm. Evaluation of segmentations with various error metrics shows that 70 percent of the segmentations keep the error values below 50%. Compared to the level set method to solve the Chan-Vese model, our algorithm is significantly faster. At the same time, it gives almost the same or better segmentation results. When compared to the recent $\text{k}$-means variant, it also gives much better segmentation with convex boundaries. The proposed algorithm balances well between time and quality of the segmentation. A crucial step in the design of machine vision systems is the extraction of discriminant features from the images, which is based on low-level segmentation which can be obtained by our approach.

In this work, we model the relationship between prey and predators by studying the interactive behavior of this prey-predator model and using the change of prey. The objective is to maximize the profit function of each predator by seeking the strategy provided by each predator to maximize its profit. To do so, we maximize this utility function being constrained by balance equations between biomass and trophic, and we show that this last problem is completely equivalent to finding the generalized Nash equilibrium point. To calculate it, we use the conditions of Karush-Kuhn-Tucker and we show that it is indeed a linear complementarity problem.

The main purpose of this paper is to solve the nonlinear Schrödinger equation using some suitable analytical and numerical methods such as Sumudu transform, Adomian Decomposition Method (ADM), and Padé approximation technique. In many literatures, we can see the Sumudu Adomian decomposition method (SADM) and the Laplace Adomian decomposition method (LADM); the SADM and LADM provide similar results. The SADM and LADM methods have been applied to solve nonlinear PDE, but the solution has small convergence radius for some PDE. We perform the SADM solution by using the function $P_{\left[L/M\right]}\left[·\right]$ called double Padé approximation. We will provide the graphical numerical simulations in 3D surface solutions of each application and the absolute error to illustrate the efficiency of the method. In our methods, the nonlinear terms are computed using Adomian polynomials, and the Padé approximation will be used to control the convergence of the series solutions. The suggested technique is successfully applied to nonlinear Schrödinger equations and proved to be highly accurate compared to the Sumudu Adomian decomposition method.

An eternal dominating set of a graph $G$ is a set of guards distributed on the vertices of a dominating set so that each vertex can be occupied by one guard only. These guards can defend any infinite series of attacks; an attack is defended by moving one guard along an edge from its position to the attacked vertex. We consider the “all guards move” of the eternal dominating set problem, in which one guard has to move to the attacked vertex and all the remaining guards are allowed to move to an adjacent vertex or stay in their current positions after each attack in order to form a dominating set on the graph and at each step can be moved after each attack. The “all guards move model” is called the $m$-eternal domination model. The size of the smallest $m$-eternal dominating set is called the $m$-eternal domination number and is denoted by ${\gamma }_m^{\infty }\left(G\right)$. In this paper, we find ${\gamma }_m^{\infty }\left(P\left(n,1\right)\right)$ and ${\gamma }_m^{\infty }\left(P\left(n,3\right)\right)$ for $n\equiv 0\left(\mathrm{mod}4\right)$. We also find upper bounds for ${\gamma }_m^{\infty }\left(P\left(n,2\right)\right)$ and ${\gamma }_m^{\infty }\left(P\left(n,3\right)\right)$ when $n$ is arbitrary.

This paper presents a mathematical model that describes the transmission dynamics of schistosomiasis for humans, snails, and the free living miracidia and cercariae. The model incorporates the treated compartment and a preventive factor due to water sanitation and hygiene (WASH) for the human subpopulation. A qualitative analysis was performed to examine the invariant regions, positivity of solutions, and disease equilibrium points together with their stabilities. The basic reproduction number, $R_0$, is computed and used as a threshold value to determine the existence and stability of the equilibrium points. It is established that, under a specific condition, the disease-free equilibrium exists and there is a unique endemic equilibrium when $R_0>1$. It is shown that the disease-free equilibrium point is both locally and globally asymptotically stable provided $R_0<1$, and the unique endemic equilibrium point is locally asymptotically stable whenever $R_0>1$ using the concept of the Center Manifold Theory. A numerical simulation carried out showed that at $R_0=1$, the model exhibits a forward bifurcation which, thus, validates the analytic results. Numerical analyses of the control strategies were performed and discussed. Further, a sensitivity analysis of $R_0$ was carried out to determine the contribution of the main parameters towards the die out of the disease. Finally, the effects that these parameters have on the infected humans were numerically examined, and the results indicated that combined application of treatment and WASH will be effective in eradicating schistosomiasis.

The objective of this paper is to compute the new modified method of variational iteration and the Laplace Adomian decomposition method for the solution of nonlinear fractional partial differential equations. We execute a comparatively newfangled analytical mechanism that is denoted by the modified Laplace variational iteration method (MLVIM) and Laplace Adomian decomposition method (LADM). The effect of the numerical results indicates that the double approximation is handy to execute and reliable when applied. It is shown that numerical solutions are gained in the form of approximately series which are facilely computable.

This paper proposes an algorithm for document plagiarism detection using the provided incremental knowledge construction with formal concept analysis (FCA). The incremental knowledge construction is presented to support document matching between the source document in storage and the suspect document. Thus, a new concept similarity measure is also proposed for retrieving formal concepts in the knowledge construction. The presented concept similarity employs appearance frequencies in the obtained knowledge construction. Our approach can be applied to retrieve relevant information because the obtained structure uses FCA in concept form that is definable by a conjunction of properties. This measure is mathematically proven to be a formal similarity metric. The performance of the proposed similarity measure is demonstrated in document plagiarism detection. Moreover, this paper provides an algorithm to build the information structure for document plagiarism detection. Thai text test collections are used for performance evaluation of the implemented web application.

Pollution has become an intense danger to our environment. The lake pollution model is formulated into the three-dimensional system of differential equations with three instances of input. In the present study, the new iterative method (NIM) was applied to the lake pollution model with three cases called impulse input, step input, and sinusoidal input for a longer time span. The main feature of the NIM is that the procedure is very simple, and it does not need to calculate any special type of polynomial or multipliers such as Adomian polynomials and Lagrange’s multipliers. Comparisons with the Adomian decomposition method (ADM) and the well-known purely numerical fourth-order Runge-Kutta method (RK4) suggest that the NIM is a powerful alternative for differential equations providing more realistic series solutions that converge very rapidly in real physical problems.

Determining the number of clusters in high-dimensional real-life datasets and interpreting the final outcome are among the challenging problems in data science. Discovering the number of classes in cancer and microarray data plays a vital role in the treatment and diagnosis of cancers and other related diseases. Nonnegative matrix factorization (NMF) plays a paramount role as an efficient data exploratory tool for extracting basis features inherent in massive data. Some algorithms which are based on incorporating sparsity constraints in the nonconvex NMF optimization problem are applied in the past for analyzing microarray datasets. However, to the best of our knowledge, none of these algorithms use block coordinate descent method which is known for providing closed form solutions. In this paper, we apply an algorithm developed based on columnwise partitioning and rank-one matrix approximation. We test this algorithm on two well-known cancer datasets: leukemia and multiple myeloma. The numerical results indicate that the proposed algorithm performs significantly better than related state-of-the-art methods. In particular, it is shown that this method is capable of robust clustering and discovering larger cancer classes in which the cluster splits are stable.

In this paper, the prey-predator model of five compartments is constructed with treatment given to infected prey and infected predator. We took predation incidence rates as functional response type II, and disease transmission incidence rates follow simple kinetic mass action function. The positivity, boundedness, and existence of the solution of the model are established and checked. Equilibrium points of the models are identified, and local stability analyses of trivial equilibrium, axial equilibrium, and disease-free equilibrium points are performed with the method of variation matrix and the Routh-Hurwitz criterion. It is found that the trivial equilibrium point $E_o$ is always unstable, and axial equilibrium point $E_A$ is locally asymptotically stable if $\beta k-\left(t_1+d_2\right)<0,\hspace{0.17em}q{p}_{1}k-d_3\left(s+k\right)<0$ and $q{p}_{3}k-\left(t_2+d_4\right)\left(s+k\right)<0$ conditions hold true. Global stability analysis of an endemic equilibrium point of the model has been proven by considering the appropriate Lyapunov function. The basic reproduction number of infected prey and infected predators are obtained as $R_{01}={\left(q{p}_1-d_3\right)}^{2}k\beta d_3{s}^2/\left(q{p}_1-d_3\right)\left\{{\left(q{p}_1-d_3\right)}^{2}ks\left(t_1+d_2\right)+rsq{p}_2\left(kq{p}_1-k{d}_3-d_{3}s\right)\right\}$ and $R_{02}=\left(q{p}_1-d_3\right)\left(q{p}_3{d}_3\right)k+\alpha rsq\left(kq{p}_1-k{d}_3-d_{3}s\right)/{\left(q{p}_1-d_3\right)}^2\left(t_2+d_4\right)k$, respectively. If the basic reproduction number is greater than one, then the disease will persist in the prey-predator system. If the basic reproduction number is one, then the disease is stable, and if the basic reproduction number is less than one, then the disease dies out from the prey-predator system. Finally, simulations are done with the help of DEDiscover software to clarify results.

Given a frequency assignment network model is a zero divisor graph $\Gamma =\left(V,E\right)$ of commutative ring $R_{\eta }$, in this model, each node is considered to be a channel and their labelings are said to be the frequencies, which are assigned by the $L\left(2,1\right)$ and $L\left(3,2,1\right)$ labeling constraints. For a graph $\Gamma$, $L\left(2,1\right)$ labeling is a nonnegative real valued function $f:V\left(G\right)⟶[0,\infty )$ such that $\mid f\left(x\right)-f\left(y\right)\mid \ge 2d$ if $d=1$ and $\mid f\left(x\right)-f\left(y\right)\mid \ge d$ if $d=2$ where $x$ and $y$ are any two vertices in $V$ and is a distance between $x$ and $y$. Similarly, one can extend this distance labeling terminology up to the diameter of a graph in order to enhance the channel clarity and to prevent the overlapping of signal produced with the minimum resource (frequency) provided. In general, this terminology is known as the $L\left(h,k\right)$ labeling where $h$ is the difference of any two vertex frequencies connected by a two length path. In this paper, our aim is to find the minimum spanning sharp upper frequency bound ${\lambda }_{\left(2,1\right)}$ and ${\lambda }_{\left(3,2,1\right)}$, within ${\Delta }^2$, in terms of maximum and minimum degree of $\Gamma$ by the distance labeling $L\left(2,1\right)$ and $L\left(3,2,1\right)$, respectively, for some order $\eta =p^{n}q,pqr,p^n$ where $p,q,r$ are distinct prime and $n$ is any positive integer.

In this article, a comparative study between optimal homotopy asymptotic method and multistage optimal homotopy asymptotic method is presented. These methods will be applied to obtain an approximate solution to the seventh-order Sawada-Kotera Ito equation. The results of optimal homotopy asymptotic method are compared with those of multistage optimal homotopy asymptotic method as well as with the exact solutions. The multistage optimal homotopy asymptotic method relies on optimal homotopy asymptotic method to obtain an analytic approximate solution. It actually applies optimal homotopy asymptotic method in each subinterval, and we show that it achieves better results than optimal homotopy asymptotic method over a large interval; this is one of the advantages of this method that can be used for long intervals and leads to more accurate results. As far as the authors are aware that multistage optimal homotopy asymptotic method has not been yet used to solve fractional partial differential equations of high order, we have shown that this method can be used to solve these problems. The convergence of the method is also addressed. The fractional derivatives are described in the Caputo sense.

Many architectures of face recognition modules have been developed to tackle the challenges posed by varying environmental constraints such as illumination, occlusions, pose, and expressions. These recognition systems have mainly focused on a single constraint at a time and have achieved remarkable successes. However, the presence of multiple constraints may deteriorate the performance of these face recognition systems. In this study, we assessed the performance of Principal Component Analysis and Singular Value Decomposition using Discrete Wavelet Transform (DWT-PCA/SVD) for preprocessing face recognition algorithm on multiple constraints (partially occluded face images acquired with varying expressions). Numerical evaluation of the study algorithm gave reasonably average recognition rates of 77.31% and 76.85% for left and right reconstructed face images with varying expressions, respectively. A statistically significant difference was established between the average recognition distance of the left and right reconstructed face images acquired with varying expressions using pairwise comparison test. The post hoc analysis using the Bonferroni simultaneous confidence interval revealed that the significant difference established through the pairwise comparison test was mainly due to the sad expressions. Although the performance of the DWT-PCA/SVD algorithm declined as compared to its performance on single constraints, the algorithm attained appreciable performance level under multiple constraints. The DWT-PCA/SVD recognition algorithm performs reasonably well for recognition when partial occlusion with varying expressions is the underlying constraint.

Aerodynamic heating due to turbulence significantly affects the operation of high-speed vehicles and the entrainment of fluid by turbulent plumes. In this paper, the heat generated and convected by fluid turbulence is examined by rearranging the Navier-Stokes equations into a single equation for the fluctuating dependent variables external to unsteady chaotic motions. This equation is similar to the nonhomogeneous heat equation where sources are terms resulting from this rearrangement. Mean and fluctuating quantities are introduced, and under certain circumstances, a heat equation for the fluctuating density results with corresponding mean and fluctuating source terms. The resultant equation is similar to Lighthill’s acoustic analogy and is a “heat analogy.” A solution is obtained with the use of Green’s function as long as the observer is located outside the region of chaotic motion. Predictions for the power spectrum are shown for high Reynolds number isotropic turbulence. The power spectrum decays as the inverse of the wavenumber of the turbulent velocity fluctuations. The developed theory can easily be applied to other turbulent flows if the statistics of unsteady motion can be estimated.

The present article analyzes the effect of viscous dissipations on natural convection heat transfer. The power law model for non-Newtonian fluid with heat generation or absorption effect along a sinusoidal wavy surface with isothermal boundary condition is investigated. A simple coordinate transform is employed to map the wavy surface into a flat surface, and also, the fully implicit finite difference method is incorporated for the numerical solution. The findings of this study can help better understand the effect of parameters such as the Brinkman number, heat generation/absorption, wave amplitude magnitude, and generalized Prandtl number on convective heat transfer in dilatant and pseudoplastic non-Newtonian. Results show that as the Brinkman number increases, the amount of heat transfer decreases. This is physically justifiable considering that the fluid becomes warmer due to the viscous dissipation, decreasing its temperature difference with the constant temperature surface. Also, the effect of the power law viscosity index is surveyed. It is demonstrated that the magnitude of the local Nusselt number in the plane leading edge has the smallest quantity for pseudoplastic fluids compared to dilatant Newtonian fluids. Additionally, as the distance from the plane leading edge increases, the heat transfer declines.

Over the past few years, hierarchical Bayesian models have been extensively used for modeling the joint spatial and temporal dependence of big spatio-temporal data which commonly involves a large number of missing observations. This article represented, assessed, and compared some recently proposed Bayesian and non-Bayesian models for predicting the daily average particulate matter with a diameter of less than 10 $\left({\mathrm{PM}}_{10}\right)$ measured in Qatar during the years 2016–2019. The disaggregating technique with a Markov chain Monte Carlo method with Gibbs sampler are used to handle the missing data. Based on the obtained results, we conclude that the Gaussian predictive processes with autoregressive terms of the latent underlying space-time process model is the best, compared with the Bayesian Gaussian processes and non-Bayesian generalized additive models.

Any bank’s financial management is essential to preparing the assets and liabilities for multiple goals. In this paper, we develop an optimal bank model for the financial management department in the Kingdom of Saudi Arabia. The lexicographic goal programming model was used to formulate the banks’ performance management. In this study, the six goals of one of the leading banks in Saudi Arabia, namely, maximize asset, minimize liability, maximize equity, maximize operating income, maximize net income, and maximizing total goal achievements in the financial statement, were studied. To illustrate the model, we have focused on Al Rajhi Bank’s financial statements as a case study. The data was obtained from the banks’ financial statements. The outcomes of the study exhibited that all goals were accomplished. This proposed model is dynamic because it will help examine the banks’ financial strengths located in the kingdom. As a result, the proposed model can guide banking firms in making decisions and developing strategies to deal with numerous monetary circumstances.

Asymptotic approximations of Tangent polynomials, Tangent-Bernoulli, and Tangent-Genocchi polynomials are derived using saddle point method and the approximations are expressed in terms of hyperbolic functions. For each polynomial there are two approximations derived with one having enlarged region of validity.

With the rapid development of the social economy, people’s demand for energy is increasing. Global warming has made people pay more attention to the importance of energy saving and emission reduction. The power grid is the system responsible for the transmission and distribution of electric energy. Therefore, in the context of energy saving and emission reduction, there are higher requirements for substations in the power grid. For this reason, an intelligent switch with built-in electronic voltage transformer is designed. The concept and key technical characteristics of the electronic voltage transformer intelligent switch are introduced; the overall design scheme of the built-in electronic voltage transformer intelligent switch is expounded; the hardware and software system of the device is designed; the management and control strategy of the design device is such as transformer tangent control, voltage and reactive control; and the operating system is ported. Finally, the simulation test of the system proves that the system has high effectiveness and stability. The research can provide reference for the realization of efficient, economic, stable, safe, and environmentally friendly distribution of power in a power grid.

A mathematical model for the transmission dynamics of pneumonia disease in the presence of drug resistance is formulated. Intervention strategies, namely, vaccination, public health education, and treatment are implemented. We compute the effective reproduction numbers and establish the local stability of the equilibria of the model. Global stability of the disease-free equilibrium is obtained through the comparison method. On the other hand, we apply the Lyapunov method to show that the drug-resistant equilibrium is globally asymptotically stable under some feasible biological conditions. Furthermore, we apply optimal control theory to the model aiming at minimizing the number of infections from drug-sensitive and drug-resistant strains. The necessary conditions for the optimal solutions of the model were derived by using Pontryagin’s Maximum Principle. The optimal controls are characterized in terms of the optimality system, which is solved numerically for several scenarios to investigate the best strategy. The incremental cost-effectiveness analysis technique is used to find the most cost-effective strategy, and it is observed that the vaccination program is the most cost-effective strategy in case of limited resources. However, results show that implementing the three strategies simultaneously provides the best results in controlling the disease.

An eternal dominating set of a graph $G$ is a set of guards distributed on the vertices of a dominating set so that each vertex can be occupied by one guard only. These guards can defend any infinite series of attacks, an attack is defended by moving one guard along an edge from its position to the attacked vertex. We consider the “all guards move” of the eternal dominating set problem, in which one guard has to move to the attacked vertex, and all the remaining guards are allowed to move to an adjacent vertex or stay in their current positions after each attack in order to form a dominating set on the graph and at each step can be moved after each attack. The “all guards move model” is called the $m$-eternal domination model. The size of the smallest $m$-eternal dominating set is called the $m$-eternal domination number and is denoted by ${\gamma }_m^{\infty }\left(G\right)$. In this paper, we find the domination number of Jahangir graph $J_{s,m}$ for $s\equiv 1,2\left(\mathrm{mod}3\right)$, and the $m$-eternal domination numbers of $J_{s,m}$ for $s,m$ are arbitraries.

In this paper, we develop and study a mathematical model for the dynamics of Scomber colias and Thunnus thynnus prey-predator with parasitic helminths. We search to analyze a bioeconomic model in which both susceptible and infected prey populations Scomber colias are exposed to the predator Thunnus thynnus, with varying degrees of exposure. However, the predator feeds preferentially on the most numerous prey types. This implies a kind of switching from the susceptible class to the infected class, and vice versa, as these two types of prey change in numerical superiority. So, the positivity, boundedness, equilibria, stability, and bioeconomic equilibrium are studied. Some numerical simulation of stability is cited. For giving a high yield and keeping the Scomber colias and Thunnus thynnus populations away from extension, we use the Maximum Principle of Pontryagin.

The dairy sector in Morocco is a sector that struggles to develop by facing several difficulties. Therefore, a significant gap has emerged between the main national producers and small farmers, while in the economy, the development of small economic actors presents an important lever of growth. If we assume that the consumer’s milk demand function is linear, that it depends only on the consumer’s price, and that production costs follow a decreasing return, and finally, by considering only two actors representing the different types of producers, we will show in this paper that a competitive production subsidy of 0.5 MAD can triple the market share of small farmers while being in a situation of Nash-Cournot’s equilibrium. In addition, the addition of a subsidy to production costs will reduce the price of milk and, thus, increase the annual milk consumption of Moroccans from 59 liters, which is below the world average consumption, to 94 liters annually, thus benefiting the consumer in addition to small farmers.

This article presents triple Laplace transform coupled with iterative method to obtain the exact solution of two-dimensional nonlinear sine-Gordon equation (NLSGE) subject to the appropriate initial and boundary conditions. The noise term in this equation is vanished by successive iterative method. The proposed technique has the advantage of producing exact solution, and it is easily applied to the given problems analytically. Four test problems from mathematical physics are taken to show the accuracy, convergence, and the efficiency of the proposed method. Furthermore, the results indicate that the introduced method is promising for solving other type systems of NLPDEs.

This study focuses on developing an efficient and easily implemented novel technique to solve the index-$k$ Hessenberg differential-algebraic equation (DAE) with input control. The implicit function theorem is first applied to solve the algebraic constraints of having unknown state differential variables to form a reduced state-space representation of an ordinary differential (control) system defined on smooth manifold with consistent initial conditions. The variational formulation is then developed for the reduced problem. A solution of the reduced problem is proven to be the critical point of the variational formulation, and the critical points of the variational formulation are the solutions of the reduced problem on the manifold. The approximate analytical solution of the equivalent variational formulation is represented as a finite number of basis functions with unknown parameters on a suitable separable Hilbert setting solution space. The unknown coefficients of the solution are obtained by solving a linear algebraic system. The different index problems of linear Hessenberg differential-algebraic control systems are approximately solved using this approach with comparisons. The numerical results reveal the good efficiency and accuracy of the proposed method. This technique is applicable for a large number of applications like linear quadratic optima, control problems, and constrained mechanical systems.

The natural mathematical abilities of humans have advanced civilizations. These abilities have been demonstrated in cultural heritage, especially traditional houses, which display evidence of an intuitive mathematics ability. Tribes around the world have built traditional houses with unique styles. The present study involved the collection of data from documentation, observation, and interview. The observations of several traditional buildings in Indonesia were based on camera images, aerial camera images, and documentation techniques. We first analyzed the images of some sample of the traditional houses in Indonesia using projective geometry and simple house theory and then formulated the definitions of building numbers and projective coordinates. The sample of the traditional houses is divided into two categories which are stilt houses and nonstilt house. The present article presents 7 types of simple houses, 21 building numbers, and 9 projective coordinates.