arXiv - MATH - Dynamical Systems最新文献_第8页

Variational construction of singular characteristics and propagation of singularities 奇异特征的变量构造和奇异点的传播

arXiv - MATH - Dynamical Systems Pub Date : 2024-09-02 DOI: arxiv-2409.00961

Piermarco Cannarsa, Wei Cheng, Jiahui Hong, Kaizhi Wang

{"title":"Variational construction of singular characteristics and propagation of singularities","authors":"Piermarco Cannarsa, Wei Cheng, Jiahui Hong, Kaizhi Wang","doi":"arxiv-2409.00961","DOIUrl":"https://doi.org/arxiv-2409.00961","url":null,"abstract":"On a smooth closed manifold $M$, we introduce a novel theory of maximal slope\u0000curves for any pair $(phi,H)$ with $phi$ a semiconcave function and $H$ a\u0000Hamiltonian. By using the notion of maximal slope curve from gradient flow theory, the\u0000intrinsic singular characteristics constructed in [Cannarsa, P.; Cheng, W.,\u0000textit{Generalized characteristics and Lax-Oleinik operators: global theory}.\u0000Calc. Var. Partial Differential Equations 56 (2017), no. 5, 56:12], the smooth\u0000approximation method developed in [Cannarsa, P.; Yu, Y. textit{Singular\u0000dynamics for semiconcave functions}. J. Eur. Math. Soc. 11 (2009), no. 5,\u0000999--1024], and the broken characteristics studied in [Khanin, K.; Sobolevski,\u0000A., textit{On dynamics of Lagrangian trajectories for Hamilton-Jacobi\u0000equations}. Arch. Ration. Mech. Anal. 219 (2016), no. 2, 861--885], we prove\u0000the existence and stability of such maximal slope curves and discuss certain\u0000new weak KAM features. We also prove that maximal slope curves for any pair\u0000$(phi,H)$ are exactly broken characteristics which have right derivatives\u0000everywhere. Applying this theory, we establish a global variational construction of\u0000strict singular characteristics and broken characteristics. Moreover, we prove\u0000a result on the global propagation of cut points along generalized\u0000characteristics, as well as a result on the propagation of singular points\u0000along strict singular characteristics, for weak KAM solutions. We also obtain\u0000the continuity equation along strict singular characteristics which clarifies\u0000the mass transport nature in the problem of propagation of singularities.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195945","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}

引用次数: 0

Finding Large Independent Sets in Networks Using Competitive Dynamics 利用竞争动力学寻找网络中的大型独立集

arXiv - MATH - Dynamical Systems Pub Date : 2024-09-02 DOI: arxiv-2409.01336

Niek Mooij, Ivan Kryven

{"title":"Finding Large Independent Sets in Networks Using Competitive Dynamics","authors":"Niek Mooij, Ivan Kryven","doi":"arxiv-2409.01336","DOIUrl":"https://doi.org/arxiv-2409.01336","url":null,"abstract":"Many decision-making algorithms draw inspiration from the inner workings of\u0000individual biological systems. However, it remains unclear whether collective\u0000behavior among biological species can also lead to solutions for computational\u0000tasks. By studying the coexistence of species that interact through simple\u0000rules on a network, we demonstrate that the underlying dynamical system can\u0000recover near-optimal solutions to the maximum independent set problem -- a\u0000fundamental, computationally hard problem in graph theory. Furthermore, we\u0000observe that the optimality of these solutions is improved when the competitive\u0000pressure in the system is gradually increased. We explain this phenomenon by\u0000showing that the cascade of bifurcation points, which occurs with rising\u0000competitive pressure in our dynamical system, naturally gives rise to Katz\u0000centrality-based node removal in the network. By formalizing this connection,\u0000we propose a biologically inspired discrete algorithm for approximating the\u0000maximum independent set problem on a graph. Our results indicate that complex\u0000systems may collectively possess the capacity to perform non-trivial\u0000computations, with implications spanning biology, economics, and other fields.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"68 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195961","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}

引用次数: 0

Four closed characteristics on compact star-shaped hypersurfaces in $mathbb{R}^{8}$ $mathbb{R}^{8}$中紧凑星形超曲面上的四个封闭特征

arXiv - MATH - Dynamical Systems Pub Date : 2024-09-02 DOI: arxiv-2409.04460

Huagui Duan, Dong Xie

引用次数: 0

Distributions of periodic points for the Dyck shift and the heterochaos baker maps 戴克平移和异相贝克图的周期点分布

arXiv - MATH - Dynamical Systems Pub Date : 2024-09-02 DOI: arxiv-2409.01261

Hiroki Takahasi

引用次数: 0

A geometric approach to Mather quotient problem 解决马瑟商数问题的几何方法

arXiv - MATH - Dynamical Systems Pub Date : 2024-09-02 DOI: arxiv-2409.00958

Wei Cheng, Wenxue Wei

引用次数: 0

Conformal measures of (anti)holomorphic correspondences 反）全态对应的共形测量

arXiv - MATH - Dynamical Systems Pub Date : 2024-09-02 DOI: arxiv-2409.01361

Nils Hemmingsson, Xiaoran Li, Zhiqiang Li

引用次数: 0

Rapidly yawing spheroids in viscous shear flow: Emergent loss of symmetry 粘性剪切流中快速偏航的球体：新出现的对称性丧失

arXiv - MATH - Dynamical Systems Pub Date : 2024-09-02 DOI: arxiv-2409.01273

Mohit P. Dalwadi

引用次数: 0

Latent Space Dynamics Learning for Stiff Collisional-radiative Models 刚性碰撞辐射模型的潜空间动力学学习

arXiv - MATH - Dynamical Systems Pub Date : 2024-09-01 DOI: arxiv-2409.05893

Xuping Xie, Qi Tang, Xianzhu Tang

{"title":"Latent Space Dynamics Learning for Stiff Collisional-radiative Models","authors":"Xuping Xie, Qi Tang, Xianzhu Tang","doi":"arxiv-2409.05893","DOIUrl":"https://doi.org/arxiv-2409.05893","url":null,"abstract":"Collisional-radiative (CR) models describe the atomic processes in a plasma\u0000by tracking the population density in the ground and excited states for each\u0000charge state of the atom or ion. These models predict important plasma\u0000properties such as charge state distributions and radiative emissivity and\u0000opacity. Accurate CR modeling is essential in radiative plasma modeling for\u0000magnetic fusion, especially when significant amount of impurities are\u0000introduced into the plasmas. In radiative plasma simulations, a CR model, which\u0000is a set of high-dimensional stiff ordinary differential equations (ODE), needs\u0000to be solved on each grid point in the configuration space, which can overwhelm\u0000the plasma simulation cost. In this work, we propose a deep learning method\u0000that discovers the latent space and learns its corresponding latent dynamics,\u0000which can capture the essential physics to make accurate predictions at much\u0000lower online computational cost. To facilitate coupling of the latent space CR\u0000dynamics with the plasma simulation model in physical variables, our latent\u0000space in the autoencoder must be a grey box, consisting of a physical latent\u0000space and a data-driven or blackbox latent space. It has been demonstrated that\u0000the proposed architecture can accurately predict both the full-order CR\u0000dynamics and the critical physical quantity of interest, the so-called\u0000radiative power loss rate.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"68 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195950","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}

引用次数: 0

Polynomial shape adic systems are inherently expansive 多项式形状自适应系统本身具有扩展性

arXiv - MATH - Dynamical Systems Pub Date : 2024-09-01 DOI: arxiv-2409.00762

Sarah Frick, Karl Petersen, Sandi Shields

引用次数: 0

Data Assimilation to the Primitive Equations in $H^2$ 对 $H^2$ 原始方程的数据同化

arXiv - MATH - Dynamical Systems Pub Date : 2024-09-01 DOI: arxiv-2409.00579

Ken Furukawa

引用次数: 0