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New Trends in Algebraic Geometry and Its Applications, 2nd Edition
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New Trends in Algebraic Geometry and Its Applications, 2nd Edition

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Algebra, Geometry and Topology".

Deadline for manuscript submissions: 31 May 2024 | Viewed by 5237

Special Issue Editor

Prof. Dr. Sonia Pérez-Díaz

E-Mail Website
Guest Editor
Department of Physic and Mathematics, University of Alcalá, Madrid, Spain
Interests: study and analysis of symbolic; numeric and approximate algorithms; applications to the theory of curves and surfaces
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Unirational algebraic varieties, in particular curves and surfaces, play an important role in many fields, such as algebraic geometry, computational geometry, computer-aided design, computer graphics, and geometric modeling. Consequently, algorithms and results dealing with different algebraic and geometric problems involving these geometric objects are very important.

The Special Issue on “New Trends in Algebraic Geometry and Its Applications” is directed at mathematicians and computer scientists who have a particular interest in algebraic geometry and its applications. In particular, this Special Issue is intended to facilitate communication between researchers who are addressing fundamental algorithmic issues in the treatment of curves and surfaces (from the symbolic and also numeric points of view). 

We welcome papers that explore the interplay between geometry, algebra, and numerical computation when designing algorithms for varieties, or provide complexity analysis on the running time of such algorithms.

We are soliciting high-quality original research articles or review articles focused on recent problems concerning algebraic geometry and its applications. Potential topics include but are not limited to:

  • Algorithms for computing the topology of algebraic varieties;
  • Algorithms for visualizing algebraic varieties;
  • Applications in scientific computing;
  • Complexity analysis of algorithms for algebraic varieties;
  • Meshing and refinement;
  • Computational algebra and applications;
  • Computational and algebraic geometry and applications;
  • Computer algebra problems solved by means of symbolic–numeric techniques.

Prof. Dr. Sonia Pérez-Díaz
Guest Editor

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. Mathematics is an international peer-reviewed open access semimonthly 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 2600 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

  • algebraic geometry
  • computer science
  • symbolic computation
  • geometric computation
  • geometry
  • algebra

Published Papers (3 papers)

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19 pages, 404 KiB  
Article
Asymptotic Chow Semistability Implies Ding Polystability for Gorenstein Toric Fano Varieties
by Naoto Yotsutani
Mathematics 2023, 11(19), 4114; https://doi.org/10.3390/math11194114 - 28 Sep 2023
Viewed by 593
Abstract
In this paper, we prove that if a Gorenstein toric Fano variety (X,−KX) is asymptotically Chow semistable, then it is Ding polystable with respect to toric test configurations (Theorem 3). This extends the known result obtained by [...] Read more.
In this paper, we prove that if a Gorenstein toric Fano variety (X,−KX) is asymptotically Chow semistable, then it is Ding polystable with respect to toric test configurations (Theorem 3). This extends the known result obtained by others (Theorem 2) to the case where X admits Gorenstein singularity. We also show the additivity of the Mabuchi constant for the product toric Fano varieties in Proposition 2 based on the author’s recent work (Ono, Sano and Yotsutani in arxiv:2305.05924). Applying this formula to certain toric Fano varieties, we construct infinitely many examples that clarify the difference between relative K-stability and relative Ding stability in a systematic way (Proposition 1). Finally, we verify the relative Chow stability for Gorenstein toric del Pezzo surfaces using the combinatorial criterion developed in (Yotsutani and Zhou in Tohoku Math. J.71 (2019), 495–524.) and specifying the symmetry of the associated polytopes as well. Full article
(This article belongs to the Special Issue New Trends in Algebraic Geometry and Its Applications, 2nd Edition)
24 pages, 1304 KiB  
Article
Challenges and Opportunities in Machine Learning for Geometry
by Rafael Magdalena-Benedicto, Sonia Pérez-Díaz and Adrià Costa-Roig
Mathematics 2023, 11(11), 2576; https://doi.org/10.3390/math11112576 - 04 Jun 2023
Cited by 1 | Viewed by 2329
Abstract
Over the past few decades, the mathematical community has accumulated a significant amount of pure mathematical data, which has been analyzed through supervised, semi-supervised, and unsupervised machine learning techniques with remarkable results, e.g., artificial neural networks, support vector machines, and principal component analysis. [...] Read more.
Over the past few decades, the mathematical community has accumulated a significant amount of pure mathematical data, which has been analyzed through supervised, semi-supervised, and unsupervised machine learning techniques with remarkable results, e.g., artificial neural networks, support vector machines, and principal component analysis. Therefore, we consider as disruptive the use of machine learning algorithms to study mathematical structures, enabling the formulation of conjectures via numerical algorithms. In this paper, we review the latest applications of machine learning in the field of geometry. Artificial intelligence can help in mathematical problem solving, and we predict a blossoming of machine learning applications during the next years in the field of geometry. As a contribution, we propose a new method for extracting geometric information from the point cloud and reconstruct a 2D or a 3D model, based on the novel concept of generalized asymptotes. Full article
(This article belongs to the Special Issue New Trends in Algebraic Geometry and Its Applications, 2nd Edition)
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12 pages, 2186 KiB  
Article
Rib Reinforcement Bionic Topology Optimization under Multi-Scale Cyclic Excitation
by Zhongmin Xiao, Longfei Wu, Dachang Zhu, Wenqiang Wu, Chunliang Zhang and Fangyi Li
Mathematics 2023, 11(11), 2478; https://doi.org/10.3390/math11112478 - 28 May 2023
Viewed by 1107
Abstract
Thin-walled structures have problems such as low stiffness, large deflection, and vibration. The layout of rib reinforcement in thin-walled structures plays a vital role in providing structural strength and rigidity and reducing structural weight. A multi-scale bionic topology optimization method with a cyclic [...] Read more.
Thin-walled structures have problems such as low stiffness, large deflection, and vibration. The layout of rib reinforcement in thin-walled structures plays a vital role in providing structural strength and rigidity and reducing structural weight. A multi-scale bionic topology optimization method with a cyclic variable load is proposed in this paper to optimize dynamic flexibility by simulating the growth law of leaf vein formation and distribution. A material interpolation method is adopted to penalize the material attributes of rib reinforcement according to their thickness, based on polynomial interpolation. Combined with the layout of rib reinforcement and SIMP, the mathematical model of rib reinforcement layout optimization with cyclic variable loading is proposed, and the sensitivity of thin-walled dynamic flexibility to the rib reinforcement thickness is analyzed. Two typical examples of thin-walled structures are presented to validate the proposed method. Considering the impact effect of multi-scale cyclic loads such as wind speed, pressure, and raindrops acting on the leaf vein, the natural frequencies of bionic topological structures of heart-shaped and elliptical leaf veins are increased by 63.44% and 47.2%, respectively. Considering the change in radial thickness, the mass of the automotive door inner panel with a bionic topological structure increased by 3.2%, the maximum stress value was reduced by 1.4% and 36.8%, and deformation was reduced by 37.6% and 27.1% under the anti-concave and sinking conditions, respectively. Moreover, the first-order natural frequency of the automotive door’s inner panel with a bionic topological structure increased to 30.45%, 3.7% higher than the original. Full article
(This article belongs to the Special Issue New Trends in Algebraic Geometry and Its Applications, 2nd Edition)
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