Article
Computer Science, Software Engineering
Kestutis Karciauskas, Jorg Peters
Summary: This study focuses on minimal single-valence quad meshes, which have irregular vertices with the same valence and closest pairs separated by a regular, 4-valent vertex. Smooth skins of high quality can be generated using bi-cubic patches in many practical configurations.
COMPUTER-AIDED DESIGN
(2022)
Article
Mathematics
Graziano Gentili, Anna Gori, Giulia Sarfatti
Summary: This paper focuses on the study of affine quaternionic manifolds and a potential classification of all compact affine quaternionic curves and surfaces. It is established that there is only one affine quaternionic structure on an affine quaternionic manifold. The classification of compact affine quaternionic curves is limited to quaternionic tori and the primary Hopf surface S3xS1, while for compact affine quaternionic surfaces, a path towards their classification is identified through the study of fundamental groups and inspection of all nilpotent hypercomplex simply connected 8-dimensional Lie Groups.
JOURNAL OF GEOMETRIC ANALYSIS
(2021)
Article
Mathematics
Amit Ghosh, Peter Sarnak
Summary: This paragraph discusses the properties of the affine cubic surface V-k and proves the validity of the Hasse Principle for most values of k. It also introduces conjectures concerning class numbers and Hasse failures, along with numerical analysis pointing towards these conjectures. Furthermore, it mentions the extension of some analysis to less special affine cubic surfaces.
INVENTIONES MATHEMATICAE
(2022)
Article
Mathematics, Applied
Alexander Perepechko
Summary: For a smooth del Pezzo surface of degree 3 polarized by a very ample divisor not proportional to the anticanonical one, the affine cone over it is flexible in codimension one, with an open subset having an infinitely transitive action of the special automorphism group on it.
FORUM MATHEMATICUM
(2021)
Article
Mathematics, Applied
Yuchao Wang, Weili Yao
Summary: This paper investigates the distribution of rational points on singular cubic surfaces, with coordinates having few prime factors. The key tools employed are universal torsors, the circle method, and results on linear equations in primes.
SCIENCE CHINA-MATHEMATICS
(2023)
Article
Mathematics
Muhammad Ammad, Md Yushalify Misro, Muhammad Abbas, Abdul Majeed
Summary: This paper introduces a new approach for the fabrication of generalized developable cubic trigonometric Bezier surfaces with shape parameters and discusses the influence of shape parameters on the surfaces.
Article
Mathematics, Applied
Ce Ce Li, Cheng Xing, Hui Yang Xu
Summary: In this paper, the authors study affine hypersurfaces with parallel cubic form relative to the affine a-connection by analyzing the affine a-connection of statistical manifolds. The main results include the classification of such hypersurfaces under specific affine metrics and a new characterization of the generalized Cayley hypersurfaces.
ACTA MATHEMATICA SINICA-ENGLISH SERIES
(2023)
Article
Green & Sustainable Science & Technology
Saeed Aligholi, Manoj Khandelwal
Summary: According to chaos theory, certain underlying patterns can reveal the order of disordered systems. This study discusses the intermittency of rough rock fractured surfaces as an orderable disorder at intermediate length scales, which is more complex than simple fractal or multi-scaling behaviors. The introduced parameters effectively capture the systematic behavior and quantify the intermittency of the surfaces, providing a framework for quantifying and modeling the roughness of fractured surfaces and analyzing fluid flow and shear strength in rock media. This framework can also be used in analyzing the intermittency of time series and developing new models for predicting seismic or flood events with higher accuracy in a short time.
Article
Mathematics
Zhibek Kadyrsizova, Jennifer Kenkel, Janet Page, Jyoti Singh, Karen E. Smith, Adela Vraciu, Emily E. Witt
Summary: Cubic surfaces in characteristic two were studied from the perspective of prime characteristic commutative algebra. It was proved that non-Frobenius split cubic surfaces form a linear subspace of codimension four in the 19-dimensional space of all cubics, and there are finitely many non-Frobenius split cubic surfaces up to projective equivalence. The defining equations for each were explicitly described, and they were characterized as extremal based on configurations of lines on them, with a cubic surface in characteristic two failing to be Frobenius split if and only if no three lines on it form a triangle.
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
(2021)
Article
Mathematics, Applied
Juan Gerardo Alczar, Georg Muntingh
Summary: We introduce a characterization for affine equivalence of two surfaces of translation defined by either rational or meromorphic generators, and this characterization also applies to minimal surfaces. In the rational case, our results provide algorithms for detecting affine equivalence of these surfaces as well as the symmetries of translation surfaces and minimal surfaces of the considered types. Additionally, we apply our results to design surfaces of translation and minimal surfaces with symmetries and to compute the symmetries of higher-order Enneper surfaces.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Boris Bilich
Summary: In this paper, we extend the classification of two-dimensional normal affine commutative algebraic monoids to noncommutative monoid structures on normal affine surfaces. We prove that two-dimensional algebraic monoids are toric and show how to find all monoid structures on a normal toric surface. These structures are induced by a comultiplication formula involving Demazure roots. Descriptions of opposite monoids, quotient monoids, and boundary divisors are also given.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2022)
Article
Mathematics, Applied
Alexander Kolpakov, Alexey Talambutsa
Summary: In this paper, we extend some of Klarner's work on free semigroups of affine maps acting on the real line by using a classical approach from geometric group theory (the Ping-Pong lemma). We also investigate the boundaries where Klarner's necessary condition is applicable.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2022)
Article
Mathematics
Cece Li, Cheng Xing, Huiyang Xu
Summary: In this paper, we study locally strongly convex affine hypersurfaces with semi-parallel cubic form relative to the Levi-Civita connection of affine metric. We provide two results on such hypersurfaces, including the classification of non-affine hyperspheres and the classification of affine hyperspheres with constant scalar curvature. For the latter, we translate the classification into that of affine hypersurfaces with parallel cubic form, which has been previously completed by Hu-Li-Vrancken (J Differ Geom 87:239-307, 2011) by proving the parallelism of their cubic forms.
JOURNAL OF GEOMETRIC ANALYSIS
(2023)
Article
Computer Science, Software Engineering
K. Karciauskas, J. Peters
Summary: Point-Augmented Subdivision (PAS) is a subdivision method that replaces complex geometry-dependent guided subdivision with explicit subdivision formulas. It can generate high-quality surfaces and is suitable for high-end geometric design, eliminating shape artifacts near extraordinary points.
COMPUTER GRAPHICS FORUM
(2022)
Article
Mathematics, Applied
Huiyang Xu, Cece Li
Summary: In this paper, we classify conformally flat, locally strongly convex affine hypersurfaces with pseudo-parallel cubic form relative to the Levi-Civita connection of affine metric. The classification includes two natural classes: all affine surfaces; all affine hypersurfaces with constant sectional curvature. Specifically, we establish the classification for 3-dimensional locally strongly convex affine hypersurfaces with pseudo-parallel cubic form.
JOURNAL OF GEOMETRY AND PHYSICS
(2023)