AIMS Mathematics最新文献

Multi-scale Jones polynomial and persistent Jones polynomial for knot data analysis. 多尺度琼斯多项式和持久琼斯多项式的结数据分析。

IF 1.8 3区数学

AIMS Mathematics Pub Date : 2025-01-01 Epub Date: 2025-01-22 DOI: 10.3934/math.2025068

Ruzhi Song, Fengling Li, Jie Wu, Fengchun Lei, Guo-Wei Wei

引用次数: 0

Persistent de Rham-Hodge Laplacians in Eulerian representation for manifold topological learning. 流形拓扑学习的欧拉表示中的持久de Rham-Hodge拉普拉斯算子。

IF 1.8 3区数学

AIMS Mathematics Pub Date : 2024-01-01 Epub Date: 2024-09-23 DOI: 10.3934/math.20241333

Zhe Su, Yiying Tong, Guo-Wei Wei

{"title":"Persistent de Rham-Hodge Laplacians in Eulerian representation for manifold topological learning.","authors":"Zhe Su, Yiying Tong, Guo-Wei Wei","doi":"10.3934/math.20241333","DOIUrl":"10.3934/math.20241333","url":null,"abstract":"Recently, topological data analysis has become a trending topic in data science and engineering. However, the key technique of topological data analysis, i.e., persistent homology, is defined on point cloud data, which does not work directly for data on manifolds. Although earlier evolutionary de Rham-Hodge theory deals with data on manifolds, it is inconvenient for machine learning applications because of the numerical inconsistency caused by remeshing the involving manifolds in the Lagrangian representation. In this work, we introduced persistent de Rham-Hodge Laplacian, or persistent Hodge Laplacian (PHL), as an abbreviation for manifold topological learning. Our PHLs were constructed in the Eulerian representation via structure-persevering Cartesian grids, avoiding the numerical inconsistency over the multi-scale manifolds. To facilitate the manifold topological learning, we proposed a persistent Hodge Laplacian learning algorithm for data on manifolds or volumetric data. As a proof-of-principle application of the proposed manifold topological learning model, we considered the prediction of protein-ligand binding affinities with two benchmark datasets. Our numerical experiments highlighted the power and promise of the proposed method.","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":"9 10","pages":"27438-27470"},"PeriodicalIF":1.8,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12462892/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145187160","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Evolutionary Khovanov homology. 进化Khovanov同源性。

IF 1.8 3区数学

AIMS Mathematics Pub Date : 2024-01-01 Epub Date: 2024-09-10 DOI: 10.3934/math.20241277

Li Shen, Jian Liu, Guo-Wei Wei

引用次数: 0

Efficient numerical approaches with accelerated graphics processing unit (GPU) computations for Poisson problems and Cahn-Hilliard equations. 利用图形处理器（GPU）加速计算泊松问题和卡恩-希利亚德方程的高效数值方法。

IF 1.8 3区数学

AIMS Mathematics Pub Date : 2024-01-01 Epub Date: 2024-09-23 DOI: 10.3934/math.20241334

Saulo Orizaga, Maurice Fabien, Michael Millard

{"title":"Efficient numerical approaches with accelerated graphics processing unit (GPU) computations for Poisson problems and Cahn-Hilliard equations.","authors":"Saulo Orizaga, Maurice Fabien, Michael Millard","doi":"10.3934/math.20241334","DOIUrl":"10.3934/math.20241334","url":null,"abstract":"In this computational paper, we focused on the efficient numerical implementation of semi-implicit methods for models in materials science. In particular, we were interested in a class of nonlinear higher-order parabolic partial differential equations. The Cahn-Hilliard (CH) equation was chosen as a benchmark problem for our proposed methods. We first considered the Cahn-Hilliard equation with a convexity-splitting (CS) approach coupled with a backward Euler approximation of the time derivative and tested the performance against the bi-harmonic-modified (BHM) approach in terms of accuracy, order of convergence, and computation time. Higher-order time-stepping techniques that allow for the methods to increase their accuracy and order of convergence were then introduced. The proposed schemes in this paper were found to be very efficient for 2D computations. Computed dynamics in 2D and 3D are presented to demonstrate the energy-decreasing property and overall performance of the methods for longer simulation runs with a variety of initial conditions. In addition, we also present a simple yet powerful way to accelerate the computations by using MATLAB built-in commands to perform GPU implementations of the schemes. We show that it is possible to accelerate computations for the CH equation in 3D by a factor of 80, provided the hardware is capable enough.","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":"9 10","pages":"27471-27496"},"PeriodicalIF":1.8,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11466300/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142401649","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Fejér type inequalities for harmonically convex functions 调和凸函数的fejsamr型不等式

IF 2.2 3区数学

AIMS Mathematics Pub Date : 2023-08-18 DOI: 10.3934/math.2022835

Muhammad Amer Latif