Pub Date : 2023-04-06DOI: 10.3389/fams.2023.1064130
M. Mulk, Kazi Nusrat Islam, M. H. A. Biswas
Tissue-mimicking materials [e.g., polyvinyl alcohol cryogel (PVA-C)] are extensively used in clinical applications such as tissue repair and tissue engineering. Various mechanical testing techniques have been used to assess the biomechanical compatibility of tissue-mimicking materials. This article presents the development of inverse finite element (FE) techniques that are solved using numerical optimization to characterize the mechanical properties of PVA-C specimens. In this study, a numerical analysis where the displacement influence factor was employed in conjunction with a linear elastic model of finite thickness was performed. In the analysis, the effects of Poisson's ratio, specimen aspect ratio, and relative indentation depth were investigated, and a novel mathematical term was introduced to Sneddon's equation. In addition, a robust optimization algorithm was developed in MATLAB that utilized FE modeling for parameter estimation before it was rigorously validated.
{"title":"Modeling and numerical analysis for mechanical characterization of soft tissue mechanism applying inverse finite element technique","authors":"M. Mulk, Kazi Nusrat Islam, M. H. A. Biswas","doi":"10.3389/fams.2023.1064130","DOIUrl":"https://doi.org/10.3389/fams.2023.1064130","url":null,"abstract":"Tissue-mimicking materials [e.g., polyvinyl alcohol cryogel (PVA-C)] are extensively used in clinical applications such as tissue repair and tissue engineering. Various mechanical testing techniques have been used to assess the biomechanical compatibility of tissue-mimicking materials. This article presents the development of inverse finite element (FE) techniques that are solved using numerical optimization to characterize the mechanical properties of PVA-C specimens. In this study, a numerical analysis where the displacement influence factor was employed in conjunction with a linear elastic model of finite thickness was performed. In the analysis, the effects of Poisson's ratio, specimen aspect ratio, and relative indentation depth were investigated, and a novel mathematical term was introduced to Sneddon's equation. In addition, a robust optimization algorithm was developed in MATLAB that utilized FE modeling for parameter estimation before it was rigorously validated.","PeriodicalId":36662,"journal":{"name":"Frontiers in Applied Mathematics and Statistics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47046692","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-05DOI: 10.3389/fams.2023.1090753
A. Hodgkinson, Aisha Tursynkozha, D. Trucu
The eukaryotic cell cycle comprises 4 phases (G1, S, G2, and M) and is an essential component of cellular health, allowing the cell to repair damaged DNA prior to division. Facilitating this processes, p53 is activated by DNA-damage and arrests the cell cycle to allow for the repair of this damage, while mutations in the p53 gene frequently occur in cancer. As such, this process occurs on the cell-scale but affects the organism on the cell population-scale. Here, we present two models of cell cycle progression: The first of these is concerned with the cell-scale process of cell cycle progression and the temporal biochemical processes, driven by cyclins and underlying progression from one phase to the next. The second of these models concerns the cell population-scale process of cell-cycle progression and its arrest under the influence of DNA-damage and p53-activation. Both systems take advantage of structural modeling conventions to develop novels methods for describing and exploring cell-cycle dynamics on these two divergent scales. The cell-scale model represents the accumulations of cyclins across an internal cell space and demonstrates that such a formalism gives rise to a biological clock system, with definite periodicity. The cell population-scale model allows for the exploration of interactions between various regulating proteins and the DNA-damage state of the system and quantitatively demonstrates the structural dynamics which allow p53 to regulate the G2- to M-phase transition and to prevent the mitosis of genetically damaged cells. A divergent periodicity and clear distribution of transition times is observed, as compared with the single-cell system. Comparison to a system with a reduced genetic repair rate shows a greater delay in cell cycle progression and an increased accumulation of cell in the G2-phase, as a result of the p53 biochemical feedback mechanism.
{"title":"Structured dynamics of the cell-cycle at multiple scales","authors":"A. Hodgkinson, Aisha Tursynkozha, D. Trucu","doi":"10.3389/fams.2023.1090753","DOIUrl":"https://doi.org/10.3389/fams.2023.1090753","url":null,"abstract":"The eukaryotic cell cycle comprises 4 phases (G1, S, G2, and M) and is an essential component of cellular health, allowing the cell to repair damaged DNA prior to division. Facilitating this processes, p53 is activated by DNA-damage and arrests the cell cycle to allow for the repair of this damage, while mutations in the p53 gene frequently occur in cancer. As such, this process occurs on the cell-scale but affects the organism on the cell population-scale. Here, we present two models of cell cycle progression: The first of these is concerned with the cell-scale process of cell cycle progression and the temporal biochemical processes, driven by cyclins and underlying progression from one phase to the next. The second of these models concerns the cell population-scale process of cell-cycle progression and its arrest under the influence of DNA-damage and p53-activation. Both systems take advantage of structural modeling conventions to develop novels methods for describing and exploring cell-cycle dynamics on these two divergent scales. The cell-scale model represents the accumulations of cyclins across an internal cell space and demonstrates that such a formalism gives rise to a biological clock system, with definite periodicity. The cell population-scale model allows for the exploration of interactions between various regulating proteins and the DNA-damage state of the system and quantitatively demonstrates the structural dynamics which allow p53 to regulate the G2- to M-phase transition and to prevent the mitosis of genetically damaged cells. A divergent periodicity and clear distribution of transition times is observed, as compared with the single-cell system. Comparison to a system with a reduced genetic repair rate shows a greater delay in cell cycle progression and an increased accumulation of cell in the G2-phase, as a result of the p53 biochemical feedback mechanism.","PeriodicalId":36662,"journal":{"name":"Frontiers in Applied Mathematics and Statistics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43775831","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-05DOI: 10.3389/fams.2023.1165371
Bernardo Cockburn, Shukai Du, M. Sánchez
We provide a short introduction to the devising of a special type of methods for numerically approximating the solution of Hamiltonian partial differential equations. These methods use Galerkin space-discretizations which result in a system of ODEs displaying a discrete version of the Hamiltonian structure of the original system. The resulting system of ODEs is then discretized by a symplectic time-marching method. This combination results in high-order accurate, fully discrete methods which can preserve the invariants of the Hamiltonian defining the ODE system. We restrict our attention to linear Hamiltonian systems, as the main results can be obtained easily and directly, and are applicable to many Hamiltonian systems of practical interest including acoustics, elastodynamics, and electromagnetism. After a brief description of the Hamiltonian systems of our interest, we provide a brief introduction to symplectic time-marching methods for linear systems of ODEs which does not require any background on the subject. We describe then the case in which finite-difference space-discretizations are used and focus on the popular Yee scheme (1966) for electromagnetism. Finally, we consider the case of finite-element space discretizations. The emphasis is placed on the conservation properties of the fully discrete schemes. We end by describing ongoing work.
{"title":"Combining finite element space-discretizations with symplectic time-marching schemes for linear Hamiltonian systems","authors":"Bernardo Cockburn, Shukai Du, M. Sánchez","doi":"10.3389/fams.2023.1165371","DOIUrl":"https://doi.org/10.3389/fams.2023.1165371","url":null,"abstract":"We provide a short introduction to the devising of a special type of methods for numerically approximating the solution of Hamiltonian partial differential equations. These methods use Galerkin space-discretizations which result in a system of ODEs displaying a discrete version of the Hamiltonian structure of the original system. The resulting system of ODEs is then discretized by a symplectic time-marching method. This combination results in high-order accurate, fully discrete methods which can preserve the invariants of the Hamiltonian defining the ODE system. We restrict our attention to linear Hamiltonian systems, as the main results can be obtained easily and directly, and are applicable to many Hamiltonian systems of practical interest including acoustics, elastodynamics, and electromagnetism. After a brief description of the Hamiltonian systems of our interest, we provide a brief introduction to symplectic time-marching methods for linear systems of ODEs which does not require any background on the subject. We describe then the case in which finite-difference space-discretizations are used and focus on the popular Yee scheme (1966) for electromagnetism. Finally, we consider the case of finite-element space discretizations. The emphasis is placed on the conservation properties of the fully discrete schemes. We end by describing ongoing work.","PeriodicalId":36662,"journal":{"name":"Frontiers in Applied Mathematics and Statistics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41932541","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-04DOI: 10.3389/fams.2023.1100147
Absana Tarammim, M. T. Akter
This research study inspects the effectiveness of synchronization methods such as active control and backstepping control from systematic design procedures of a synchronized Shimizu–Morioka system for the same parameter. It aimed to achieve synchronization between the state variables of two identical Shimizu–Morioka chaotic systems by defining the proposed varieties of the error dynamics coefficient matrix. Furthermore, this study also aimed to designed an active controller that enables the synchronization of these systems. The use of designed recursive backstepping nonlinear controllers was based on the Lyapunov function. Furthermore, it also demonstrated the stability of the synchronization of the nonlinear identical Shimizu–Morioka system. The new virtual state variable and establishment of Lyapunov functionals are used in the backstepping controller to stabilize and reduce errors between the Master (MS)/Drive (DS) systems. For comparison, the complexity of active controllers is verified to be such that the designed controller's effectiveness based on backstepping is attainable in engineering applications. Finally, numerical simulations are performed to demonstrate the effectiveness of the proposed synchronization strategy with the Runge–Kutta (RK-4) algorithm of fourth order through MatLab Simulink.
{"title":"Shimizu–Morioka's chaos synchronization: An efficacy analysis of active control and backstepping methods","authors":"Absana Tarammim, M. T. Akter","doi":"10.3389/fams.2023.1100147","DOIUrl":"https://doi.org/10.3389/fams.2023.1100147","url":null,"abstract":"This research study inspects the effectiveness of synchronization methods such as active control and backstepping control from systematic design procedures of a synchronized Shimizu–Morioka system for the same parameter. It aimed to achieve synchronization between the state variables of two identical Shimizu–Morioka chaotic systems by defining the proposed varieties of the error dynamics coefficient matrix. Furthermore, this study also aimed to designed an active controller that enables the synchronization of these systems. The use of designed recursive backstepping nonlinear controllers was based on the Lyapunov function. Furthermore, it also demonstrated the stability of the synchronization of the nonlinear identical Shimizu–Morioka system. The new virtual state variable and establishment of Lyapunov functionals are used in the backstepping controller to stabilize and reduce errors between the Master (MS)/Drive (DS) systems. For comparison, the complexity of active controllers is verified to be such that the designed controller's effectiveness based on backstepping is attainable in engineering applications. Finally, numerical simulations are performed to demonstrate the effectiveness of the proposed synchronization strategy with the Runge–Kutta (RK-4) algorithm of fourth order through MatLab Simulink.","PeriodicalId":36662,"journal":{"name":"Frontiers in Applied Mathematics and Statistics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42216777","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-03DOI: 10.3389/fams.2023.1151270
Abdulai Kailan Suhuyini, Baba Seidu
Typhoid fever is a potentially fatal illness that is caused by the bacteria Salmonella typhi. In this study, a deterministic mathematical model was formulated to look into transmission dynamics of typhoid fever with treatment and booster vaccination. The reproduction number R0 is calculated using the next-generation matrix approach. Then, a stability analysis on the equilibrium points was performed using Routh–Hurwitz criteria. It was revealed that the disease-free equilibrium point is locally asymptotically stable whenever R0 is less than 1 together with other conditions. We also showed that R0≤1 does not guarantee global stability of the typhoid-free equilibrium point and corroborated the result by showing the possible existence of backward bifurcation at R0=1. The model parameters in R0 were also subjected to sensitivity analysis, which revealed that the transmission rate, infection through an exposed person, and bacteria are the most influential parameters of the reproduction number R0. Numerical simulations were run to determine the impact of various parameters on the dynamics of typhoid.
{"title":"A mathematical model on the transmission dynamics of typhoid fever with treatment and booster vaccination","authors":"Abdulai Kailan Suhuyini, Baba Seidu","doi":"10.3389/fams.2023.1151270","DOIUrl":"https://doi.org/10.3389/fams.2023.1151270","url":null,"abstract":"Typhoid fever is a potentially fatal illness that is caused by the bacteria Salmonella typhi. In this study, a deterministic mathematical model was formulated to look into transmission dynamics of typhoid fever with treatment and booster vaccination. The reproduction number R0 is calculated using the next-generation matrix approach. Then, a stability analysis on the equilibrium points was performed using Routh–Hurwitz criteria. It was revealed that the disease-free equilibrium point is locally asymptotically stable whenever R0 is less than 1 together with other conditions. We also showed that R0≤1 does not guarantee global stability of the typhoid-free equilibrium point and corroborated the result by showing the possible existence of backward bifurcation at R0=1. The model parameters in R0 were also subjected to sensitivity analysis, which revealed that the transmission rate, infection through an exposed person, and bacteria are the most influential parameters of the reproduction number R0. Numerical simulations were run to determine the impact of various parameters on the dynamics of typhoid.","PeriodicalId":36662,"journal":{"name":"Frontiers in Applied Mathematics and Statistics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42396810","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-31DOI: 10.3389/fams.2023.1000785
M. Vinyals, R. Sabbadin, S. Couture, L. Sadou, R. Thomopoulos, Kevin Chapuis, Baptiste Lesquoy, P. Taillandier
In this paper, we tackle innovation diffusion from the perspective of an institution which aims to encourage the adoption of a new product (i.e., an innovation) with mostly social rather than individual benefits. Designing such innovation adoption policies is a very challenging task because of the difficulty to quantify and predict its effect on the behaviors of non-adopters and the exponential size of the space of possible policies. To solve these issues, we propose an approach that uses agent-based modeling to simulate in a credible way the behaviors of possible adopters and (deep) reinforcement learning to efficiently explore the policy search space. An application of our approach is presented for the question of the use of digital technologies in agriculture. Empirical results on this case study validate our scheme and show the potential of our approach to learn effective innovation diffusion policies.
{"title":"Toward AI-designed innovation diffusion policies using agent-based simulations and reinforcement learning: The case of digital tool adoption in agriculture","authors":"M. Vinyals, R. Sabbadin, S. Couture, L. Sadou, R. Thomopoulos, Kevin Chapuis, Baptiste Lesquoy, P. Taillandier","doi":"10.3389/fams.2023.1000785","DOIUrl":"https://doi.org/10.3389/fams.2023.1000785","url":null,"abstract":"In this paper, we tackle innovation diffusion from the perspective of an institution which aims to encourage the adoption of a new product (i.e., an innovation) with mostly social rather than individual benefits. Designing such innovation adoption policies is a very challenging task because of the difficulty to quantify and predict its effect on the behaviors of non-adopters and the exponential size of the space of possible policies. To solve these issues, we propose an approach that uses agent-based modeling to simulate in a credible way the behaviors of possible adopters and (deep) reinforcement learning to efficiently explore the policy search space. An application of our approach is presented for the question of the use of digital technologies in agriculture. Empirical results on this case study validate our scheme and show the potential of our approach to learn effective innovation diffusion policies.","PeriodicalId":36662,"journal":{"name":"Frontiers in Applied Mathematics and Statistics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45598758","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-30DOI: 10.3389/fams.2023.1126952
M. Sedghizadeh, M. Van den Berghe, R. Shcherbakov
Microseismicity is expected in potash mining due to the associated rock-mass response. This phenomenon is known, but not fully understood. To assess the safety and efficiency of mining operations, producers must quantitatively discern between normal and abnormal seismic activity. In this work, statistical aspects and clustering of microseismicity from a Saskatchewan, Canada, potash mine are analyzed and quantified. Specifically, the frequency-magnitude statistics display a rich behavior that deviates from the standard Gutenberg-Richter scaling for small magnitudes. To model the magnitude distribution, we consider two additional models, i.e., the tapered Pareto distribution and a mixture of the tapered Pareto and Pareto distributions to fit the bi-modal catalog data. To study the clustering aspects of the observed microseismicity, the nearest-neighbor distance (NND) method is applied. This allowed the identification of potential cluster characteristics in time, space, and magnitude domains. The implemented modeling approaches and obtained results will be used to further advance strategies and protocols for the safe and efficient operation of potash mines.
{"title":"Statistical and clustering analysis of microseismicity from a Saskatchewan potash mine","authors":"M. Sedghizadeh, M. Van den Berghe, R. Shcherbakov","doi":"10.3389/fams.2023.1126952","DOIUrl":"https://doi.org/10.3389/fams.2023.1126952","url":null,"abstract":"Microseismicity is expected in potash mining due to the associated rock-mass response. This phenomenon is known, but not fully understood. To assess the safety and efficiency of mining operations, producers must quantitatively discern between normal and abnormal seismic activity. In this work, statistical aspects and clustering of microseismicity from a Saskatchewan, Canada, potash mine are analyzed and quantified. Specifically, the frequency-magnitude statistics display a rich behavior that deviates from the standard Gutenberg-Richter scaling for small magnitudes. To model the magnitude distribution, we consider two additional models, i.e., the tapered Pareto distribution and a mixture of the tapered Pareto and Pareto distributions to fit the bi-modal catalog data. To study the clustering aspects of the observed microseismicity, the nearest-neighbor distance (NND) method is applied. This allowed the identification of potential cluster characteristics in time, space, and magnitude domains. The implemented modeling approaches and obtained results will be used to further advance strategies and protocols for the safe and efficient operation of potash mines.","PeriodicalId":36662,"journal":{"name":"Frontiers in Applied Mathematics and Statistics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45981691","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-30DOI: 10.3389/fams.2023.1083410
M. Cadena, M. Méndez
Here we assess countries' management of the coronavirus 2019 (COVID-19) pandemic using the reliability measure P(X ≤ Y). In this management, all kind of strategies as interventions deployed by governments as well individuals' initiatives to prevent, mitigate, and reduce the contagion of this disease are taken into account. Also, typical customs practiced locally and influencing contagion are included. Regarding a number of countries and rates associated to deaths and incidence, orderings of countries about such management are established, by using the measure of reliability indicated above. In this way, countries are distinguished from each other depending on how they managed this pandemic. This kind of analysis may be extended to the management of other diseases.
{"title":"Ordering countries when managing COVID-19","authors":"M. Cadena, M. Méndez","doi":"10.3389/fams.2023.1083410","DOIUrl":"https://doi.org/10.3389/fams.2023.1083410","url":null,"abstract":"Here we assess countries' management of the coronavirus 2019 (COVID-19) pandemic using the reliability measure P(X ≤ Y). In this management, all kind of strategies as interventions deployed by governments as well individuals' initiatives to prevent, mitigate, and reduce the contagion of this disease are taken into account. Also, typical customs practiced locally and influencing contagion are included. Regarding a number of countries and rates associated to deaths and incidence, orderings of countries about such management are established, by using the measure of reliability indicated above. In this way, countries are distinguished from each other depending on how they managed this pandemic. This kind of analysis may be extended to the management of other diseases.","PeriodicalId":36662,"journal":{"name":"Frontiers in Applied Mathematics and Statistics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42367612","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-30DOI: 10.3389/fams.2023.1086240
Stefanie Hittmeyer, B. Krauskopf, H. Osinga, Katsutoshi Shinohara
Introduction The extension of the Smale horseshoe construction for diffeomorphisms in the plane to those in spaces of at least dimension three may result in a hyperbolic invariant set referred to as a blender. The defining property of a blender is that it has a stable or unstable invariant manifold that appears to have a dimension larger than expected. In this study, we consider a Hénon-like family in ℝ3, which is the only explicitly given example of a system known to feature a blender in a certain range of a parameter (corresponding to an expansion or contraction rate). More specifically, as part of its hyperbolic set, this family has a pair of saddle fixed points with one-dimensional stable or unstable manifolds. When there is a blender, the closure of these manifolds cannot be avoided by one-dimensional curves coming from an appropriate direction. This property has been checked for the Hénon-like family by the method of computing extremely long pieces of global one-dimensional manifolds to determine the parameter range over which a blender exists. Methods In this study, we take the complimentary and local point of view of constructing an actual three-dimensional box (a parallelopiped) that acts as an outer cover of the hyperbolic set. The successive forward or backward images of this box form a nested sequence of sub-boxes that contains the hyperbolic set, as well as its respective local invariant manifold. Results This constitutes a three-dimensional horseshoe that, in contrast to the idealized affine construction, is quite general and features sub-boxes with curved edges. The initial box is defined in a parameter-dependent way, and this allows us to characterize properties of the hyperbolic set intuitively. Discussion In particular, we trace relevant edges of sub-boxes as a function of the parameter to provide additional geometric insight into when the hyperbolic set may or may not be a blender.
{"title":"Boxing-in of a blender in a Hénon-like family","authors":"Stefanie Hittmeyer, B. Krauskopf, H. Osinga, Katsutoshi Shinohara","doi":"10.3389/fams.2023.1086240","DOIUrl":"https://doi.org/10.3389/fams.2023.1086240","url":null,"abstract":"Introduction The extension of the Smale horseshoe construction for diffeomorphisms in the plane to those in spaces of at least dimension three may result in a hyperbolic invariant set referred to as a blender. The defining property of a blender is that it has a stable or unstable invariant manifold that appears to have a dimension larger than expected. In this study, we consider a Hénon-like family in ℝ3, which is the only explicitly given example of a system known to feature a blender in a certain range of a parameter (corresponding to an expansion or contraction rate). More specifically, as part of its hyperbolic set, this family has a pair of saddle fixed points with one-dimensional stable or unstable manifolds. When there is a blender, the closure of these manifolds cannot be avoided by one-dimensional curves coming from an appropriate direction. This property has been checked for the Hénon-like family by the method of computing extremely long pieces of global one-dimensional manifolds to determine the parameter range over which a blender exists. Methods In this study, we take the complimentary and local point of view of constructing an actual three-dimensional box (a parallelopiped) that acts as an outer cover of the hyperbolic set. The successive forward or backward images of this box form a nested sequence of sub-boxes that contains the hyperbolic set, as well as its respective local invariant manifold. Results This constitutes a three-dimensional horseshoe that, in contrast to the idealized affine construction, is quite general and features sub-boxes with curved edges. The initial box is defined in a parameter-dependent way, and this allows us to characterize properties of the hyperbolic set intuitively. Discussion In particular, we trace relevant edges of sub-boxes as a function of the parameter to provide additional geometric insight into when the hyperbolic set may or may not be a blender.","PeriodicalId":36662,"journal":{"name":"Frontiers in Applied Mathematics and Statistics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47336905","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-30DOI: 10.3389/fams.2023.1072447
Qusain Haider, A. Hassan, S. M. Eldin
This article aims to describe the simulation studies of the hepatitis B virus non-linear system using supervised neural networks procedures supported by Levenberg-Marquardt back propagation methodology. The proposed strategy has five distinct quantities: susceptible X(t), symptomatic infections Y(t), chronic infections W(t), recovered population R(t), and a population that has received vaccinations Z(t). The reference data set for all three distinct cases has been obtained utilizing the ND-Solver and Adams method in Mathematica software. The outcomes have been validated with performance plots for all cases. To check the accuracy and effectiveness of proposed methodology mean square error has are presented. State transition, and regression plots are illustrated to elaborated the testing, training, and validation methodology. Additionally, absolute errors for different components of hepatitis B virus model are demonstrated to depict the error occurring during distinct cases. Whereas the data assigned to training is 81%, and 9% for each testing and validation. The mean square error for all three cases is 10−12 this show the accuracy and correctness of proposed methodology.
{"title":"Artificial neural network scheme to solve the hepatitis B virus model","authors":"Qusain Haider, A. Hassan, S. M. Eldin","doi":"10.3389/fams.2023.1072447","DOIUrl":"https://doi.org/10.3389/fams.2023.1072447","url":null,"abstract":"This article aims to describe the simulation studies of the hepatitis B virus non-linear system using supervised neural networks procedures supported by Levenberg-Marquardt back propagation methodology. The proposed strategy has five distinct quantities: susceptible X(t), symptomatic infections Y(t), chronic infections W(t), recovered population R(t), and a population that has received vaccinations Z(t). The reference data set for all three distinct cases has been obtained utilizing the ND-Solver and Adams method in Mathematica software. The outcomes have been validated with performance plots for all cases. To check the accuracy and effectiveness of proposed methodology mean square error has are presented. State transition, and regression plots are illustrated to elaborated the testing, training, and validation methodology. Additionally, absolute errors for different components of hepatitis B virus model are demonstrated to depict the error occurring during distinct cases. Whereas the data assigned to training is 81%, and 9% for each testing and validation. The mean square error for all three cases is 10−12 this show the accuracy and correctness of proposed methodology.","PeriodicalId":36662,"journal":{"name":"Frontiers in Applied Mathematics and Statistics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48261986","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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