Pub Date : 2024-09-18DOI: 10.1017/s0956792524000299
Youshan Tao, Michael Winkler
In a smoothly bounded domain $Omega subset mathbb{R}^n$ , $nge 1$ , this manuscript considers the homogeneous Neumann boundary problem for the chemotaxis system begin{eqnarray*} left { begin{array}{l} u_t = Delta u - nabla cdot (unabla v), [5pt] v_t = Delta v + u - alpha uv, end{array} right . end{eqnarray*} with parameter $alpha gt 0$ and with coincident production and uptake of attractants, as recently emphasized by Dallaston et al. as relevant for the understanding of T-cell dynamics. It is shown that there exists $delta _star =delta _star (n)gt 0$ such that for any given $alpha ge frac{1}{delta _star }$ and for any suitably regular initial data satisfying $v(cdot, 0)le delta _star$ , this problem admits a unique classical solution that stabilizes to the constant equilibrium $(frac{1}{|Omega |}int _Omega u(cdot, 0), , frac{1}{alpha })$ in the large time limit.
{"title":"Stabilization in a chemotaxis system modelling T-cell dynamics with simultaneous production and consumption of signals","authors":"Youshan Tao, Michael Winkler","doi":"10.1017/s0956792524000299","DOIUrl":"https://doi.org/10.1017/s0956792524000299","url":null,"abstract":"In a smoothly bounded domain <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0956792524000299_inline1.png\"/> <jats:tex-math> $Omega subset mathbb{R}^n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0956792524000299_inline2.png\"/> <jats:tex-math> $nge 1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, this manuscript considers the homogeneous Neumann boundary problem for the chemotaxis system<jats:disp-formula> <jats:alternatives> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" position=\"float\" xlink:href=\"S0956792524000299_eqnU1.png\"/> <jats:tex-math> begin{eqnarray*} left { begin{array}{l} u_t = Delta u - nabla cdot (unabla v), [5pt] v_t = Delta v + u - alpha uv, end{array} right . end{eqnarray*} </jats:tex-math> </jats:alternatives> </jats:disp-formula>with parameter <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0956792524000299_inline3.png\"/> <jats:tex-math> $alpha gt 0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and with coincident production and uptake of attractants, as recently emphasized by Dallaston et al. as relevant for the understanding of T-cell dynamics. It is shown that there exists <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0956792524000299_inline4.png\"/> <jats:tex-math> $delta _star =delta _star (n)gt 0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> such that for any given <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0956792524000299_inline5.png\"/> <jats:tex-math> $alpha ge frac{1}{delta _star }$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and for any suitably regular initial data satisfying <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0956792524000299_inline6.png\"/> <jats:tex-math> $v(cdot, 0)le delta _star$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, this problem admits a unique classical solution that stabilizes to the constant equilibrium <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0956792524000299_inline7.png\"/> <jats:tex-math> $(frac{1}{|Omega |}int _Omega u(cdot, 0), , frac{1}{alpha })$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> in the large time limit.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259870","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}
Pub Date : 2024-09-18DOI: 10.1017/s0956792524000317
Arthur Stephanovitch, Anqi Dong, Tryphon T. Georgiou
We address the problem of optimal transport with a quadratic cost functional and a constraint on the flux through a constriction along the path. The constriction, conceptually represented by a toll station, limits the flow rate across. We provide a precise formulation which, in addition, is amenable to generalization in higher dimensions. We work out in detail the case of transport in one dimension by proving existence and uniqueness of solution. Under suitable regularity assumptions, we give an explicit construction of the transport plan. Generalization of flux constraints to higher dimensions and possible extensions of the theory are discussed.
{"title":"Optimal transport through a toll station","authors":"Arthur Stephanovitch, Anqi Dong, Tryphon T. Georgiou","doi":"10.1017/s0956792524000317","DOIUrl":"https://doi.org/10.1017/s0956792524000317","url":null,"abstract":"We address the problem of optimal transport with a quadratic cost functional and a constraint on the flux through a constriction along the path. The constriction, conceptually represented by a toll station, limits the flow rate across. We provide a precise formulation which, in addition, is amenable to generalization in higher dimensions. We work out in detail the case of transport in one dimension by proving existence and uniqueness of solution. Under suitable regularity assumptions, we give an explicit construction of the transport plan. Generalization of flux constraints to higher dimensions and possible extensions of the theory are discussed.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259868","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}
Pub Date : 2024-09-18DOI: 10.1017/s0956792524000263
Zhongyang Li
We study the community detection problem on a Gaussian mixture model, in which vertices are divided into $kgeq 2$ distinct communities. The major difference in our model is that the intensities for Gaussian perturbations are different for different entries in the observation matrix, and we do not assume that every community has the same number of vertices. We explicitly find the necessary and sufficient conditions for the exact recovery of the maximum likelihood estimation, which can give a sharp phase transition for the exact recovery even though the Gaussian perturbations are not identically distributed; see Section 7. Applications include the community detection on hypergraphs.
{"title":"Exact recovery of community detection in k-community Gaussian mixture models","authors":"Zhongyang Li","doi":"10.1017/s0956792524000263","DOIUrl":"https://doi.org/10.1017/s0956792524000263","url":null,"abstract":"We study the community detection problem on a Gaussian mixture model, in which vertices are divided into <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0956792524000263_inline1.png\"/> <jats:tex-math> $kgeq 2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> distinct communities. The major difference in our model is that the intensities for Gaussian perturbations are different for different entries in the observation matrix, and we do not assume that every community has the same number of vertices. We explicitly find the necessary and sufficient conditions for the exact recovery of the maximum likelihood estimation, which can give a sharp phase transition for the exact recovery even though the Gaussian perturbations are not identically distributed; see Section 7. Applications include the community detection on hypergraphs.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259840","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}
Pub Date : 2024-09-18DOI: 10.1017/s0956792524000305
Quentin Griette, Pierre Magal, Min Zhao
This work describes a hyperbolic model for cell-cell repulsion with population dynamics. We consider the pressure produced by a population of cells to describe their motion. We assume that cells try to avoid crowded areas and prefer locally empty spaces far away from the carrying capacity. Here, our main goal is to prove the existence of travelling waves with continuous profiles. This article complements our previous results about sharp travelling waves. We conclude the paper with numerical simulations of the PDE problem, illustrating such a result. An application to wound healing also illustrates the importance of travelling waves with a continuous and discontinuous profile.
{"title":"Travelling waves with continuous profile for hyperbolic Keller-Segel equation","authors":"Quentin Griette, Pierre Magal, Min Zhao","doi":"10.1017/s0956792524000305","DOIUrl":"https://doi.org/10.1017/s0956792524000305","url":null,"abstract":"This work describes a hyperbolic model for cell-cell repulsion with population dynamics. We consider the pressure produced by a population of cells to describe their motion. We assume that cells try to avoid crowded areas and prefer locally empty spaces far away from the carrying capacity. Here, our main goal is to prove the existence of travelling waves with continuous profiles. This article complements our previous results about sharp travelling waves. We conclude the paper with numerical simulations of the PDE problem, illustrating such a result. An application to wound healing also illustrates the importance of travelling waves with a continuous and discontinuous profile.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259871","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}
Pub Date : 2024-09-18DOI: 10.1017/s0956792524000251
Meraj Alam, Adrian Muntean, Raja Sekhar G P
This paper is concerned with the development and analysis of a mathematical model that is motivated by interstitial hydrodynamics and tissue deformation mechanics (poro-elasto-hydrodynamics) within an in-vitro solid tumour. The classical mixture theory is adopted for mass and momentum balance equations for a two-phase system. A main contribution of this study is we treat the physiological transport parameter (i.e., hydraulic resistivity) as anisotropic and heterogeneous, thus the governing system is strongly coupled and non-linear. We derived a weak formulation and then formulated the equivalent fixed-point problem. This enabled us to use the Galerkin method, and the classical results on monotone operators combined with the well-known Schauder and Banach fixed-point theorems to prove the existence and uniqueness of results.
{"title":"Non-linear biphasic mixture model: Existence and uniqueness results","authors":"Meraj Alam, Adrian Muntean, Raja Sekhar G P","doi":"10.1017/s0956792524000251","DOIUrl":"https://doi.org/10.1017/s0956792524000251","url":null,"abstract":"This paper is concerned with the development and analysis of a mathematical model that is motivated by interstitial hydrodynamics and tissue deformation mechanics (poro-elasto-hydrodynamics) within an in-vitro solid tumour. The classical mixture theory is adopted for mass and momentum balance equations for a two-phase system. A main contribution of this study is we treat the physiological transport parameter (i.e., hydraulic resistivity) as anisotropic and heterogeneous, thus the governing system is strongly coupled and non-linear. We derived a weak formulation and then formulated the equivalent fixed-point problem. This enabled us to use the Galerkin method, and the classical results on monotone operators combined with the well-known Schauder and Banach fixed-point theorems to prove the existence and uniqueness of results.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259865","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}
Pub Date : 2024-09-18DOI: 10.1017/s0956792524000287
Huaian Diao, Xiaoxu Fei, Hongyu Liu
The purpose of the paper is twofold. First, we show that partial-data transmission eigenfunctions associated with a conductive boundary condition vanish locally around a polyhedral or conic corner in $mathbb{R}^n$ , $n=2,3$ . Second, we apply the spectral property to the geometrical inverse scattering problem of determining the shape as well as its boundary impedance parameter of a conductive scatterer, independent of its medium content, by a single far-field measurement. We establish several new unique recovery results. The results extend the relevant ones in [26] in two directions: first, we consider a more general geometric setup where both polyhedral and conic corners are investigated, whereas in [26] only polyhedral corners are concerned; second, we significantly relax the regularity assumptions in [26] which is particularly useful for the geometrical inverse problem mentioned above. We develop novel technical strategies to achieve these new results.
{"title":"Local geometric properties of conductive transmission eigenfunctions and applications","authors":"Huaian Diao, Xiaoxu Fei, Hongyu Liu","doi":"10.1017/s0956792524000287","DOIUrl":"https://doi.org/10.1017/s0956792524000287","url":null,"abstract":"The purpose of the paper is twofold. First, we show that partial-data transmission eigenfunctions associated with a conductive boundary condition vanish locally around a polyhedral or conic corner in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0956792524000287_inline1.png\"/> <jats:tex-math> $mathbb{R}^n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0956792524000287_inline2.png\"/> <jats:tex-math> $n=2,3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Second, we apply the spectral property to the geometrical inverse scattering problem of determining the shape as well as its boundary impedance parameter of a conductive scatterer, independent of its medium content, by a single far-field measurement. We establish several new unique recovery results. The results extend the relevant ones in [26] in two directions: first, we consider a more general geometric setup where both polyhedral and conic corners are investigated, whereas in [26] only polyhedral corners are concerned; second, we significantly relax the regularity assumptions in [26] which is particularly useful for the geometrical inverse problem mentioned above. We develop novel technical strategies to achieve these new results.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259864","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}
Pub Date : 2024-05-29DOI: 10.1017/s095679252400024x
Iryna Zabaikina, Pavol Bokes
All vital functions of living cells rely on the production of various functional molecules through gene expression. The production periods are burst-like and stochastic due to the discrete nature of biochemical reactions. In certain contexts, the concentrations of RNA or protein require regulation to maintain a fine internal balance within the cell. Here we consider a motif of two types of RNA molecules – mRNA and an antagonistic microRNA – which are encoded by a shared coding sequence and form a feed forward loop (FFL). This control mechanism is shown to be perfectly adapting in the deterministic context. We demonstrate that the adaptation (of the mean value) becomes imperfect if production occurs in random bursts. The FFL nevertheless outperforms the benchmark feedback loop in terms of counterbalancing variations in the signal. Methodologically, we adapt a hybrid stochastic model, which has widely been used to model a single regulatory molecule, to the current case of a motif involving two species; the use of the Laplace transform thereby circumvents the problem of moment closure that arises owing to the mRNA–microRNA interaction. We expect that the approach can be applicable to other systems with nonlinear kinetics.
{"title":"Maintenance of steady-state mRNA levels by a microRNA-based feed forward loop in the presence of stochastic gene expression noise","authors":"Iryna Zabaikina, Pavol Bokes","doi":"10.1017/s095679252400024x","DOIUrl":"https://doi.org/10.1017/s095679252400024x","url":null,"abstract":"<p>All vital functions of living cells rely on the production of various functional molecules through gene expression. The production periods are burst-like and stochastic due to the discrete nature of biochemical reactions. In certain contexts, the concentrations of RNA or protein require regulation to maintain a fine internal balance within the cell. Here we consider a motif of two types of RNA molecules – mRNA and an antagonistic microRNA – which are encoded by a shared coding sequence and form a feed forward loop (FFL). This control mechanism is shown to be perfectly adapting in the deterministic context. We demonstrate that the adaptation (of the mean value) becomes imperfect if production occurs in random bursts. The FFL nevertheless outperforms the benchmark feedback loop in terms of counterbalancing variations in the signal. Methodologically, we adapt a hybrid stochastic model, which has widely been used to model a single regulatory molecule, to the current case of a motif involving two species; the use of the Laplace transform thereby circumvents the problem of moment closure that arises owing to the mRNA–microRNA interaction. We expect that the approach can be applicable to other systems with nonlinear kinetics.</p>","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141168459","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}
Pub Date : 2024-05-21DOI: 10.1017/s0956792524000093
Simone Fagioli, Gianluca Favre
We study a system of nonlocal aggregation cross-diffusion PDEs that describe the evolution of opinion densities on a network. The PDEs are coupled with a system of ODEs that describe the time evolution of the agents on the network. Firstly, we apply the Deterministic Particle Approximation (DPA) method to the aforementioned system in order to prove the existence of solutions under suitable assumptions on the interactions between agents. Later on, we present an explicit model for opinion formation on an evolving network. The opinions evolve based on both the distance between the agents on the network and the ’attitude areas’, which depend on the distance between the agents’ opinions. The position of the agents on the network evolves based on the distance between the agents’ opinions. The goal is to study radicalisation, polarisation and fragmentation of the population while changing its open-mindedness and the radius of interaction.
{"title":"Opinion formation on evolving network: the DPA method applied to a nonlocal cross-diffusion PDE-ODE system","authors":"Simone Fagioli, Gianluca Favre","doi":"10.1017/s0956792524000093","DOIUrl":"https://doi.org/10.1017/s0956792524000093","url":null,"abstract":"\u0000 We study a system of nonlocal aggregation cross-diffusion PDEs that describe the evolution of opinion densities on a network. The PDEs are coupled with a system of ODEs that describe the time evolution of the agents on the network. Firstly, we apply the Deterministic Particle Approximation (DPA) method to the aforementioned system in order to prove the existence of solutions under suitable assumptions on the interactions between agents. Later on, we present an explicit model for opinion formation on an evolving network. The opinions evolve based on both the distance between the agents on the network and the ’attitude areas’, which depend on the distance between the agents’ opinions. The position of the agents on the network evolves based on the distance between the agents’ opinions. The goal is to study radicalisation, polarisation and fragmentation of the population while changing its open-mindedness and the radius of interaction.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141117525","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}
Pub Date : 2024-05-16DOI: 10.1017/s0956792524000226
Tim Laux, Kerrek Stinson, Clemens Ullrich
The purpose of this paper is to derive anisotropic mean curvature flow as the limit of the anisotropic Allen–Cahn equation. We rely on distributional solution concepts for both the diffuse and sharp interface models and prove convergence using relative entropy methods, which have recently proven to be a powerful tool in interface evolution problems. With the same relative entropy, we prove a weak–strong uniqueness result, which relies on the construction of gradient flow calibrations for our anisotropic energy functionals.
{"title":"Diffuse-interface approximation and weak–strong uniqueness of anisotropic mean curvature flow","authors":"Tim Laux, Kerrek Stinson, Clemens Ullrich","doi":"10.1017/s0956792524000226","DOIUrl":"https://doi.org/10.1017/s0956792524000226","url":null,"abstract":"The purpose of this paper is to derive anisotropic mean curvature flow as the limit of the anisotropic Allen–Cahn equation. We rely on distributional solution concepts for both the diffuse and sharp interface models and prove convergence using relative entropy methods, which have recently proven to be a powerful tool in interface evolution problems. With the same relative entropy, we prove a weak–strong uniqueness result, which relies on the construction of gradient flow calibrations for our anisotropic energy functionals.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141062608","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}
Pub Date : 2024-05-16DOI: 10.1017/s0956792524000238
Xiaoxue Zhao, Zhuchun Li
Given a set of standard binary patterns and a defective pattern, the pattern retrieval task is to find the closest pattern to the defective one among these standard patterns. The Hebbian network of Kuramoto oscillators with second-order coupling provides a dynamical model for this task, and the mutual orthogonality in memorised patterns enables us to distinguish these memorised patterns from most others in terms of stability. For the sake of error-free retrieval for general problems lacking orthogonality, a unified approach was proposed which transforms the problem into a series of subproblems with orthogonality using the orthogonal lift for two patterns. In this work, we propose the least orthogonal lift for three patterns, which evidently reduces the time of solving subproblems and even the dimensions of subproblems. Furthermore, we provide an estimate for the critical strength for stability/instability of binary patterns, which is convenient in practical use. Simulation results are presented to illustrate the effectiveness of the proposed approach.
{"title":"Binary pattern retrieval with Kuramoto-type oscillators via a least orthogonal lift of three patterns","authors":"Xiaoxue Zhao, Zhuchun Li","doi":"10.1017/s0956792524000238","DOIUrl":"https://doi.org/10.1017/s0956792524000238","url":null,"abstract":"\u0000 Given a set of standard binary patterns and a defective pattern, the pattern retrieval task is to find the closest pattern to the defective one among these standard patterns. The Hebbian network of Kuramoto oscillators with second-order coupling provides a dynamical model for this task, and the mutual orthogonality in memorised patterns enables us to distinguish these memorised patterns from most others in terms of stability. For the sake of error-free retrieval for general problems lacking orthogonality, a unified approach was proposed which transforms the problem into a series of subproblems with orthogonality using the orthogonal lift for two patterns. In this work, we propose the least orthogonal lift for three patterns, which evidently reduces the time of solving subproblems and even the dimensions of subproblems. Furthermore, we provide an estimate for the critical strength for stability/instability of binary patterns, which is convenient in practical use. Simulation results are presented to illustrate the effectiveness of the proposed approach.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140970801","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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