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Axioms
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Discrete Curvatures and Laplacians
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Discrete Curvatures and Laplacians

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Geometry and Topology".

Deadline for manuscript submissions: 31 May 2024 | Viewed by 6368

Special Issue Editors

Dr. Emil Saucan

E-Mail Website
Guest Editor
Department of Applied Mathematics, Braude College, Karmiel 2161002, Israel
Interests: discrete differential geometry and its applications; mathematical imaging and vision; complex networks; quasi-conformal mappings and applications; topology and its applications
Special Issues, Collections and Topics in MDPI journals
Dr. David Xianfeng Gu

E-Mail Website
Guest Editor
Department of Computer Science, State University of New York at Stony Brook, New York, NY 11794, USA
Interests: computer graphics; computer vision; geometric modeling; networking; medical imaging

Special Issue Information

Dear Colleagues,

The artificial separation and delimitation of mathematics into “continuous” and “discrete” that sadly still permeates many curricula, thus forever skewing students’ perception, is nowhere less true than in the context of curvature and Laplacians, where the boundary is fluid and where ideas and methods from the classical setting not only influence the discrete setting, but where the latter becomes largely the mainstream, influential setting.

We therefore invite you to submit papers appertaining to the whole spectrum spanned by these notions, being they theoretical or applied, and in particular to the discretizations of curvature and Laplacians and their manifold uses in complex networks, graphics, imaging and deep learning.

Dr. Emil Saucan
Dr. David Xianfeng Gu
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Axioms is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • discrete curvature
  • Discrete Laplace operators
  • Ollivier Ricci curvature
  • Forman Ricci curvature
  • geometric flow and applications (Ricci curvature flow, mean curvature flow, etc.)
  • Combinatorial Hodge theory
  • geometric deep learning
  • digital geometry processing
  • geometric modelling
  • information geometry

Published Papers (6 papers)

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Research

15 pages, 261 KiB  
Article
On the Noteworthy Properties of Tangentials in Cubic Structures
by Vladimir Volenec and Ružica Kolar-Šuper
Axioms 2024, 13(2), 122; https://doi.org/10.3390/axioms13020122 - 16 Feb 2024
Viewed by 787
Abstract
The cubic structure, a captivating geometric structure, finds applications across various areas of geometry through different models. In this paper, we explore the significant characteristics of tangentials in cubic structures of ranks 0, 1, and 2. Specifically, in the cubic structure of rank [...] Read more.
The cubic structure, a captivating geometric structure, finds applications across various areas of geometry through different models. In this paper, we explore the significant characteristics of tangentials in cubic structures of ranks 0, 1, and 2. Specifically, in the cubic structure of rank 2, we derive the Hessian configuration (123,164) of points and lines. Finally, we introduce and investigate the de Vries configuration of points and lines in a cubic structure. Full article
(This article belongs to the Special Issue Discrete Curvatures and Laplacians)
14 pages, 8246 KiB  
Article
A Novel Robust Topological Denoising Method Based on Homotopy Theory for Virtual Colonoscopy
by Ming Ma, Wei Chen, Na Lei and Xianfeng Gu
Axioms 2023, 12(10), 942; https://doi.org/10.3390/axioms12100942 - 30 Sep 2023
Viewed by 863
Abstract
Virtual colonoscopy plays an important role in polyp detection of colorectal cancer. Noise in the colon data acquisition process can result in topological errors during surface reconstruction. Topological denoising can be employed to remove these errors on surfaces for subsequent geometry processing, such [...] Read more.
Virtual colonoscopy plays an important role in polyp detection of colorectal cancer. Noise in the colon data acquisition process can result in topological errors during surface reconstruction. Topological denoising can be employed to remove these errors on surfaces for subsequent geometry processing, such as surface simplification and parameterization. Many methods have been proposed for this task. However, many existing methods suffer from failure in computation of all the non-trivial loops, due to high genus or complex topological structures. In this paper, we propose a novel robust topological denoising method for surfaces based on homotopy theory. The proposed method was evaluated on two datasets of colon meshes. We compared our method with the State-of-the-Art persistent-homology-based method. Our method can successfully compute the loops on all colon data for topological denoising, whereas the persistent homology method fails on some colon data. Moreover, our method detects all loops with shorter lengths than those detected by the persistent homology method. Our experimental results show that the proposed method is effective and robust in topological denoising, and that it has the potential for practical application to virtual colonoscopy. Full article
(This article belongs to the Special Issue Discrete Curvatures and Laplacians)
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10 pages, 294 KiB  
Article
The Recursive Structures of Manin Symbols over Q, Cusps and Elliptic Points on X0 (N)
by Sanmin Wang
Axioms 2023, 12(6), 597; https://doi.org/10.3390/axioms12060597 - 16 Jun 2023
Viewed by 683
Abstract
Firstly, we present a more explicit formulation of the complete system D(N) of representatives of Manin’s symbols over Q, which was initially given by Shimura. Then, we establish a bijection between [...] Read more.
Firstly, we present a more explicit formulation of the complete system D(N) of representatives of Manin’s symbols over Q, which was initially given by Shimura. Then, we establish a bijection between D(M)×D(N) and D(MN) for (M,N)=1, which reveals a recursive structure between Manin’s symbols of different levels. Based on Manin’s complete system Π(N) of representatives of cusps on X0(N) and Cremona’s characterization of the equivalence between cusps, we establish a bijection between a subset C(N) of D(N) and Π(N), and then establish a bijection between C(M)×C(N) and C(MN) for (M,N)=1. We also provide a recursive structure for elliptical points on X0(N). Based on these recursive structures, we obtain recursive algorithms for constructing Manin symbols over Q, cusps, and elliptical points on X0(N). This may give rise to more efficient algorithms for modular elliptic curves. As direct corollaries of these recursive structures, we present a recursive version of the genus formula and prove constructively formulas of the numbers of D(N), cusps, and elliptic points on X0(N). Full article
(This article belongs to the Special Issue Discrete Curvatures and Laplacians)
34 pages, 6306 KiB  
Article
Bakry–Émery Curvature Sharpness and Curvature Flow in Finite Weighted Graphs: Implementation
by David Cushing, Supanat Kamtue, Shiping Liu, Florentin Münch, Norbert Peyerimhoff and Ben Snodgrass
Axioms 2023, 12(6), 577; https://doi.org/10.3390/axioms12060577 - 11 Jun 2023
Viewed by 880
Abstract
In this paper, we discuss the implementation of a curvature flow on weighted graphs based on the Bakry–Émery calculus. This flow can be adapted to preserve the Markovian property and its limits as time goes to infinity turn out to be curvature sharp [...] Read more.
In this paper, we discuss the implementation of a curvature flow on weighted graphs based on the Bakry–Émery calculus. This flow can be adapted to preserve the Markovian property and its limits as time goes to infinity turn out to be curvature sharp weighted graphs. After reviewing some of the main results of the corresponding paper concerned with the theoretical aspects, we present various examples (random graphs, paths, cycles, complete graphs, wedge sums and Cartesian products of complete graphs, and hypercubes) and exhibit various properties of this flow. One particular aspect of our investigations is asymptotic stability and instability of curvature flow equilibria. The paper ends with a description of the Python functions and routines freely available in an ancillary file on arXiv or via github. We hope that the explanations of the Python implementation via examples will help users to carry out their own curvature flow experiments. Full article
(This article belongs to the Special Issue Discrete Curvatures and Laplacians)
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31 pages, 397 KiB  
Article
Ricci Curvature on Birth-Death Processes
by Bobo Hua and Florentin Münch
Axioms 2023, 12(5), 428; https://doi.org/10.3390/axioms12050428 - 26 Apr 2023
Viewed by 715
Abstract
In this paper, we study curvature dimension conditions on birth-death processes which correspond to linear graphs, i.e., weighted graphs supported on the infinite line or the half line. We give a combinatorial characterization of Bakry and Émery’s CD(K,n) [...] Read more.
In this paper, we study curvature dimension conditions on birth-death processes which correspond to linear graphs, i.e., weighted graphs supported on the infinite line or the half line. We give a combinatorial characterization of Bakry and Émery’s CD(K,n) condition for linear graphs and prove the triviality of edge weights for every linear graph supported on the infinite line Z with non-negative curvature. Moreover, we show that linear graphs with curvature decaying not faster than −R2 are stochastically complete. We deduce a type of Bishop-Gromov comparison theorem for normalized linear graphs. For normalized linear graphs with non-negative curvature, we obtain the volume doubling property and the Poincaré inequality, which yield Gaussian heat kernel estimates and parabolic Harnack inequality by Delmotte’s result. As applications, we generalize the volume growth and stochastic completeness properties to weakly spherically symmetric graphs. Furthermore, we give examples of infinite graphs with a positive lower curvature bound. Full article
(This article belongs to the Special Issue Discrete Curvatures and Laplacians)
22 pages, 5497 KiB  
Article
Object-Based Dynamics: Applying Forman–Ricci Flow on a Multigraph to Assess the Impact of an Object on The Network Structure
by Haim Cohen, Yinon Nachshon, Anat Maril, Paz M. Naim, Jürgen Jost and Emil Saucan
Axioms 2022, 11(9), 486; https://doi.org/10.3390/axioms11090486 - 19 Sep 2022
Cited by 2 | Viewed by 1436
Abstract
Temporal information plays a central role in shaping the structure of a network. In this paper, we consider the impact of an object on network structure over time. More specifically, we use a novel object-based dynamic measure to reflect the extent to which [...] Read more.
Temporal information plays a central role in shaping the structure of a network. In this paper, we consider the impact of an object on network structure over time. More specifically, we use a novel object-based dynamic measure to reflect the extent to which an object that is represented in the network by a vertex affects the topology of the network over time. By way of multigraph and Forman–Ricci flow, we assess the object’s impact on graph weights by comparing two graphs, one in which the object is present and one in which the object is absent. After using a case study to demonstrate the impact of Forman–Ricci flow on the network structure, we apply our measure in a semantic network to assess the effects of a word on the interactions between other words that follow it. In addition, we compare our novel measure to centrality and curvature measures so that we can ascertain the advantages of our measure over ones that already exist. Full article
(This article belongs to the Special Issue Discrete Curvatures and Laplacians)
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