James Dalby, Yucen Han, Apala Majumdar, Lidia Mrad
SIAM Journal on Applied Mathematics, Volume 83, Issue 6, Page 2284-2309, December 2023. Abstract. We study order reconstruction (OR) solutions in the Beris–Edwards framework for nematodynamics, for both passive and active nematic flows in a microfluidic channel. OR solutions exhibit polydomains and domain walls, and as such, are of physical interest. We show that OR solutions exist for passive flows with constant velocity and pressure, but only for specific boundary conditions. We prove the existence of unique, symmetric, and nonsingular nematic profiles for boundary conditions that do not allow for OR solutions. We compute asymptotic expansions for OR-type solutions for passive flows with nonconstant velocity and pressure, and active flows, which shed light on the internal structure of domain walls. The asymptotics are complemented by numerical studies that demonstrate the universality of OR-type structures in static and dynamic scenarios.
{"title":"A Multifaceted Study of Nematic Order Reconstruction in Microfluidic Channels","authors":"James Dalby, Yucen Han, Apala Majumdar, Lidia Mrad","doi":"10.1137/22m1490909","DOIUrl":"https://doi.org/10.1137/22m1490909","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 83, Issue 6, Page 2284-2309, December 2023. <br/> Abstract. We study order reconstruction (OR) solutions in the Beris–Edwards framework for nematodynamics, for both passive and active nematic flows in a microfluidic channel. OR solutions exhibit polydomains and domain walls, and as such, are of physical interest. We show that OR solutions exist for passive flows with constant velocity and pressure, but only for specific boundary conditions. We prove the existence of unique, symmetric, and nonsingular nematic profiles for boundary conditions that do not allow for OR solutions. We compute asymptotic expansions for OR-type solutions for passive flows with nonconstant velocity and pressure, and active flows, which shed light on the internal structure of domain walls. The asymptotics are complemented by numerical studies that demonstrate the universality of OR-type structures in static and dynamic scenarios.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":"157 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2023-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138528003","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Badal Joshi, Nidhi Kaihnsa, Tung D. Nguyen, Anne Shiu
SIAM Journal on Applied Mathematics, Volume 83, Issue 6, Page 2260-2283, December 2023. Abstract. For reaction networks arising in systems biology, the capacity for two or more steady states, that is, multistationarity, is an important property that underlies biochemical switches. Another property receiving much attention recently is absolute concentration robustness (ACR), which means that some species concentration is the same at all positive steady states. In this work, we investigate the prevalence of each property while paying close attention to when the properties occur together. Specifically, we consider a stochastic block framework for generating random networks and prove edge-probability thresholds at which, with high probability, multistationarity appears and ACR becomes rare. We also show that the small window in which both properties occur only appears in networks with many species. Taken together, our results confirm that, in random reversible networks, ACR and multistationarity together, or even ACR on its own, is highly atypical. Our proofs rely on two prior results, one pertaining to the prevalence of networks with deficiency zero and the other “lifting” multistationarity from small networks to larger ones.
{"title":"Prevalence of Multistationarity and Absolute Concentration Robustness in Reaction Networks","authors":"Badal Joshi, Nidhi Kaihnsa, Tung D. Nguyen, Anne Shiu","doi":"10.1137/23m1549316","DOIUrl":"https://doi.org/10.1137/23m1549316","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 83, Issue 6, Page 2260-2283, December 2023. <br/> Abstract. For reaction networks arising in systems biology, the capacity for two or more steady states, that is, multistationarity, is an important property that underlies biochemical switches. Another property receiving much attention recently is absolute concentration robustness (ACR), which means that some species concentration is the same at all positive steady states. In this work, we investigate the prevalence of each property while paying close attention to when the properties occur together. Specifically, we consider a stochastic block framework for generating random networks and prove edge-probability thresholds at which, with high probability, multistationarity appears and ACR becomes rare. We also show that the small window in which both properties occur only appears in networks with many species. Taken together, our results confirm that, in random reversible networks, ACR and multistationarity together, or even ACR on its own, is highly atypical. Our proofs rely on two prior results, one pertaining to the prevalence of networks with deficiency zero and the other “lifting” multistationarity from small networks to larger ones.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":"16 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138528019","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Applied Mathematics, Volume 83, Issue 6, Page 2237-2259, December 2023. Abstract. In a reaction network, the concentration of a species with the property of dynamic absolute concentration robustness (dynamic ACR) converges to the same value independent of the overall initial values. This property endows a biochemical network with output robustness and therefore is essential for its functioning in a highly variable environment. It is important to identify the structure of the dynamical system as well as the constraints required for dynamic ACR. We propose a power-engine-load form of dynamic ACR and obtain results regarding convergence to the ACR value based on this form.
{"title":"Power-Engine-Load Form for Dynamic Absolute Concentration Robustness","authors":"Badal Joshi, Gheorghe Craciun","doi":"10.1137/22m1535450","DOIUrl":"https://doi.org/10.1137/22m1535450","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 83, Issue 6, Page 2237-2259, December 2023. <br/> Abstract. In a reaction network, the concentration of a species with the property of dynamic absolute concentration robustness (dynamic ACR) converges to the same value independent of the overall initial values. This property endows a biochemical network with output robustness and therefore is essential for its functioning in a highly variable environment. It is important to identify the structure of the dynamical system as well as the constraints required for dynamic ACR. We propose a power-engine-load form of dynamic ACR and obtain results regarding convergence to the ACR value based on this form.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":"2 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138528001","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Applied Mathematics, Ahead of Print. Abstract. Cell polarity and movement are fundamental to many biological functions. Experimental and theoretical studies have indicated that interactions of certain proteins lead to the cell polarization which plays a key role in controlling the cell movement. We study the cell polarity and movement based on a class of biophysical models that consist of reaction-diffusion equations for different proteins and the dynamics of a moving cell boundary. Such a moving boundary is often simulated by a phase-field model. We first apply the matched asymptotic analysis to give a rigorous derivation of the sharp-interface model of the cell boundary from a phase-field model. We then develop a robust numerical approach that combines the level-set method to track the sharp boundary of a moving cell and accurate discretization techniques for solving the reaction-diffusion equations on the moving cell region. Our extensive numerical simulations predict the cell polarization under various kinds of stimuli and capture both the linear and the circular trajectories of a moving cell for a long period of time. In particular, we have identified some key parameters controlling different cell trajectories that are less accurately predicted by reduced models. Our work has linked different models and also developed tools that can be adapted for the challenging three-dimensional simulations.
{"title":"Cell Polarity and Movement with Reaction-Diffusion and Moving Boundary: Rigorous Model Analysis and Robust Simulations","authors":"Shuang Liu, Li-Tien Cheng, Bo Li","doi":"10.1137/22m1506766","DOIUrl":"https://doi.org/10.1137/22m1506766","url":null,"abstract":"SIAM Journal on Applied Mathematics, Ahead of Print. <br/> Abstract. Cell polarity and movement are fundamental to many biological functions. Experimental and theoretical studies have indicated that interactions of certain proteins lead to the cell polarization which plays a key role in controlling the cell movement. We study the cell polarity and movement based on a class of biophysical models that consist of reaction-diffusion equations for different proteins and the dynamics of a moving cell boundary. Such a moving boundary is often simulated by a phase-field model. We first apply the matched asymptotic analysis to give a rigorous derivation of the sharp-interface model of the cell boundary from a phase-field model. We then develop a robust numerical approach that combines the level-set method to track the sharp boundary of a moving cell and accurate discretization techniques for solving the reaction-diffusion equations on the moving cell region. Our extensive numerical simulations predict the cell polarization under various kinds of stimuli and capture both the linear and the circular trajectories of a moving cell for a long period of time. In particular, we have identified some key parameters controlling different cell trajectories that are less accurately predicted by reduced models. Our work has linked different models and also developed tools that can be adapted for the challenging three-dimensional simulations.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2023-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138528016","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Applied Mathematics, Volume 83, Issue 6, Page 2212-2236, December 2023. Abstract. Because of the significance of remediating contaminated ecosystems, many mathematical models have been developed to describe the interactions between populations and toxicants in polluted aquatic environments. These models typically neglect the consequences of toxicant-induced behavioral changes on population dynamics. Taking into account that individuals may flee from areas with high toxicant concentrations to areas with low toxicant concentrations in order to improve their chances of survival, growth, and reproduction, we develop a diffusive population-toxicant model with toxicant-taxis. We establish the global well-posedness of our model and prove the global stability of spatially homogeneous toxicant-only steady states and population-toxicant coexistence steady states under some conditions. We find conditions under which stable spatially inhomogeneous steady states become unstable to trigger spatial pattern formations as the toxicant-taxis is strong. We also identify a narrow parameter regime in which toxicant-only and population-toxicant coexistence steady states are bistable. Numerical simulations are performed to illustrate that spatial aggregation and segregation patterns between the population and the toxicant will typically emerge. Our study highlights the important effects of toxicant-induced movement responses on the spatial distributions of populations in polluted aquatic environments.
{"title":"Global Dynamics and Pattern Formation in a Diffusive Population-Toxicant Model with Negative Toxicant-Taxis","authors":"Xiumei Deng, Qihua Huang, Zhi-An Wang","doi":"10.1137/22m1510881","DOIUrl":"https://doi.org/10.1137/22m1510881","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 83, Issue 6, Page 2212-2236, December 2023. <br/> Abstract. Because of the significance of remediating contaminated ecosystems, many mathematical models have been developed to describe the interactions between populations and toxicants in polluted aquatic environments. These models typically neglect the consequences of toxicant-induced behavioral changes on population dynamics. Taking into account that individuals may flee from areas with high toxicant concentrations to areas with low toxicant concentrations in order to improve their chances of survival, growth, and reproduction, we develop a diffusive population-toxicant model with toxicant-taxis. We establish the global well-posedness of our model and prove the global stability of spatially homogeneous toxicant-only steady states and population-toxicant coexistence steady states under some conditions. We find conditions under which stable spatially inhomogeneous steady states become unstable to trigger spatial pattern formations as the toxicant-taxis is strong. We also identify a narrow parameter regime in which toxicant-only and population-toxicant coexistence steady states are bistable. Numerical simulations are performed to illustrate that spatial aggregation and segregation patterns between the population and the toxicant will typically emerge. Our study highlights the important effects of toxicant-induced movement responses on the spatial distributions of populations in polluted aquatic environments.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":"3 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138528000","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Applied Mathematics, Ahead of Print. Abstract. The existence of HIV post-treatment controllers (PTCs) offers hope for an HIV functional cure, and understanding the critical mechanisms determining PTC represents a key step toward this goal. Here, we have studied these mechanisms by analyzing an established mathematical model for HIV viral dynamics. In mathematical models, critical mechanisms are represented by parameters that affect the tipping points to induce qualitatively different dynamics, and in cases with multiple stability, the initial conditions of the system also play a role in determining the fate of the solution. As such, for the tipping points in parameter space, we developed and implemented a sensitivity analysis of the threshold conditions of the associated bifurcations to identify the critical mechanisms for this model. Our results suggest that the infected cell death rate and the saturation parameter for cytotoxic T lymphocyte proliferation significantly affect post-treatment control. For the case with multiple stability, in state space of initial conditions, we first investigated the saddle-type equilibrium point to identify its stable manifold, which delimits trapping regions associated to the high and low viral set points. The identified stable manifold serves as a guide for the loads of immune cells and HIV virus at the time of therapy termination to achieve post-treatment control.
{"title":"Detecting and Resetting Tipping Points to Create More HIV Post-Treatment Controllers with Bifurcation and Sensitivity Analysis","authors":"Wenjing Zhang, Leif A. Ellingson","doi":"10.1137/22m1485255","DOIUrl":"https://doi.org/10.1137/22m1485255","url":null,"abstract":"SIAM Journal on Applied Mathematics, Ahead of Print. <br/> Abstract. The existence of HIV post-treatment controllers (PTCs) offers hope for an HIV functional cure, and understanding the critical mechanisms determining PTC represents a key step toward this goal. Here, we have studied these mechanisms by analyzing an established mathematical model for HIV viral dynamics. In mathematical models, critical mechanisms are represented by parameters that affect the tipping points to induce qualitatively different dynamics, and in cases with multiple stability, the initial conditions of the system also play a role in determining the fate of the solution. As such, for the tipping points in parameter space, we developed and implemented a sensitivity analysis of the threshold conditions of the associated bifurcations to identify the critical mechanisms for this model. Our results suggest that the infected cell death rate and the saturation parameter for cytotoxic T lymphocyte proliferation significantly affect post-treatment control. For the case with multiple stability, in state space of initial conditions, we first investigated the saddle-type equilibrium point to identify its stable manifold, which delimits trapping regions associated to the high and low viral set points. The identified stable manifold serves as a guide for the loads of immune cells and HIV virus at the time of therapy termination to achieve post-treatment control.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":"15 1 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138527991","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
One of the most important branches of nonlinear control theory of dynamical systems is the so-called sliding mode. Its aim is the design of a (nonlinear) feedback law that brings and maintains the state trajectory of a dynamic system on a given sliding surface. Here, dynamics become completely independent of the model parameters and can be tuned accordingly to the desired target. In this paper we study this problem when the feedback law is subject to strong structural constraints. In particular, we assume that the control input may take values only over two bounded and disjoint sets. Such sets could be also not perfectly known a priori. An example is a control input allowed to switch only between two values. Under these peculiarities, we derive the necessary and sufficient conditions that guarantee sliding-mode control effectiveness for a class of time-varying continuous-time linear systems that includes all the stationary state-space linear models. Our analysis covers several scientific fields. It is only apparently confined to the linear setting and also allows one study an important set of nonlinear models. We describe fundamental examples related to epidemiology where the control input is the level of contact rate among people and the sliding surface permits to control the number of infected. We prove the global convergence of epidemic sliding-mode control schemes applied to two popular dynamical systems used in epidemiology, i.e., SEIR and SAIR, and based on the introduction of severe restrictions like lockdowns. Results obtained in the literature regarding control of many other epidemiological models are also generalized by casting them within a general sliding-mode theory.
{"title":"Sliding-Mode Theory Under Feedback Constraints and the Problem of Epidemic Control","authors":"Mauro Bisiacco, Gianluigi Pillonetto","doi":"10.1137/22m1535309","DOIUrl":"https://doi.org/10.1137/22m1535309","url":null,"abstract":"One of the most important branches of nonlinear control theory of dynamical systems is the so-called sliding mode. Its aim is the design of a (nonlinear) feedback law that brings and maintains the state trajectory of a dynamic system on a given sliding surface. Here, dynamics become completely independent of the model parameters and can be tuned accordingly to the desired target. In this paper we study this problem when the feedback law is subject to strong structural constraints. In particular, we assume that the control input may take values only over two bounded and disjoint sets. Such sets could be also not perfectly known a priori. An example is a control input allowed to switch only between two values. Under these peculiarities, we derive the necessary and sufficient conditions that guarantee sliding-mode control effectiveness for a class of time-varying continuous-time linear systems that includes all the stationary state-space linear models. Our analysis covers several scientific fields. It is only apparently confined to the linear setting and also allows one study an important set of nonlinear models. We describe fundamental examples related to epidemiology where the control input is the level of contact rate among people and the sliding surface permits to control the number of infected. We prove the global convergence of epidemic sliding-mode control schemes applied to two popular dynamical systems used in epidemiology, i.e., SEIR and SAIR, and based on the introduction of severe restrictions like lockdowns. Results obtained in the literature regarding control of many other epidemiological models are also generalized by casting them within a general sliding-mode theory.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":"10 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136227557","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Lachlan Elam, Mónica C. Quiñones-Frías, Ying Zhang, Avital A. Rodal, Thomas G. Fai
. The transport of particles in cells is influenced by the properties of intracellular networks they traverse while searching for localized target regions or reaction partners. Moreover, given the rapid turnover in many intracellular structures, it is crucial to understand how temporal changes in the network structure affect diffusive transport. In this work, we use network theory to characterize complex intracellular biological environments across scales. We develop both a coarse-grained model and an efficient computational method to compute the mean first passage times for simulating a particle diffusing along two-dimensional planar networks extracted from fluorescence microscopy imaging. We first benchmark this methodology in the context of synthetic networks, and subsequently apply it to live-cell data from endoplasmic reticulum tubular networks.
{"title":"Fast Solver for Diffusive Transport Times on Dynamic Intracellular Networks","authors":"Lachlan Elam, Mónica C. Quiñones-Frías, Ying Zhang, Avital A. Rodal, Thomas G. Fai","doi":"10.1137/22m1509308","DOIUrl":"https://doi.org/10.1137/22m1509308","url":null,"abstract":". The transport of particles in cells is influenced by the properties of intracellular networks they traverse while searching for localized target regions or reaction partners. Moreover, given the rapid turnover in many intracellular structures, it is crucial to understand how temporal changes in the network structure affect diffusive transport. In this work, we use network theory to characterize complex intracellular biological environments across scales. We develop both a coarse-grained model and an efficient computational method to compute the mean first passage times for simulating a particle diffusing along two-dimensional planar networks extracted from fluorescence microscopy imaging. We first benchmark this methodology in the context of synthetic networks, and subsequently apply it to live-cell data from endoplasmic reticulum tubular networks.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":"31 35","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134954127","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The study of emergent behavior of swarms is of great interest for applied sciences. One of the most fundamental questions for self-organizing swarms is whether the swarms disperse or remain in a spatially cohesive configuration. In this paper we study dissipativity properties and spatial cohesiveness of the swarm of self-propelled particles governed by the model , where , , and is a symmetric positive semidefinite matrix. The self-propulsion term is assumed to be continuously differentiable and to grow faster than , that is, as . We establish that the velocity and acceleration of the particles are ultimately bounded. We show that when is trivial, the positions of the particles are also ultimately bounded. For systems with , we show that, while the system might infinitely drift away from its initial location, the particles remain within a bounded distance from the generalized center of mass of the system, which geometrically coincides with the weighted average of agent positions. The weights are determined by the coefficients of the projection matrix onto . We also discuss the ultimate boundedness for systems with bounded coupling, including the Morse potential systems, and systems governed by power-law potentials with strong repulsion properties. We show that the former systems are ultimately bounded in the velocity-acceleration domain, whereas the models based on the power-law potentials are not.
{"title":"On Spatial Cohesiveness of Second-Order Self-Propelled Swarming Systems","authors":"Constantine Medynets, Irina Popovici","doi":"10.1137/23m1553388","DOIUrl":"https://doi.org/10.1137/23m1553388","url":null,"abstract":"The study of emergent behavior of swarms is of great interest for applied sciences. One of the most fundamental questions for self-organizing swarms is whether the swarms disperse or remain in a spatially cohesive configuration. In this paper we study dissipativity properties and spatial cohesiveness of the swarm of self-propelled particles governed by the model , where , , and is a symmetric positive semidefinite matrix. The self-propulsion term is assumed to be continuously differentiable and to grow faster than , that is, as . We establish that the velocity and acceleration of the particles are ultimately bounded. We show that when is trivial, the positions of the particles are also ultimately bounded. For systems with , we show that, while the system might infinitely drift away from its initial location, the particles remain within a bounded distance from the generalized center of mass of the system, which geometrically coincides with the weighted average of agent positions. The weights are determined by the coefficients of the projection matrix onto . We also discuss the ultimate boundedness for systems with bounded coupling, including the Morse potential systems, and systems governed by power-law potentials with strong repulsion properties. We show that the former systems are ultimately bounded in the velocity-acceleration domain, whereas the models based on the power-law potentials are not.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":" 48","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135241977","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Wearable Data Assimilation to Estimate the Circadian Phase","authors":"Dae Wook Kim, Minki P. Lee, Daniel B. Forger","doi":"10.1137/22m1509680","DOIUrl":"https://doi.org/10.1137/22m1509680","url":null,"abstract":"","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":" 7","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135242165","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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