Article
Mathematics
Maria Colombo, Nick Edelen, Luca Spolaor
Summary: In this article, we adapt Simon's method to prove a C-1, C-alpha regularity theorem for minimal varifolds resembling a cone structure over an equiangular geodesic net. Additionally, for varifold classes satisfying a no-hole condition on the singular set, we establish C-1, C-alpha regularity near the cone structure. These results are significant for the structure of minimizing clusters and size-minimizing currents.
JOURNAL OF DIFFERENTIAL GEOMETRY
(2022)
Article
Mathematics
M. B. Alves, J. B. Gomes, K. M. Pedroso
Summary: This paper introduces the conic dimension and conic basis for polyhedral cones in Euclidean space, establishes a conic version of the rank-nullity theorem, and discusses the decomposition and union of conic basis involving the lineality space of the cone. It also introduces the signature of a polyhedral cone and shows results on the injectivity of a linear map and the preservation of the signature of a polyhedral cone under linear maps.
LINEAR & MULTILINEAR ALGEBRA
(2022)
Article
Mathematics, Applied
Nicholas Edelen, Chao Li
Summary: This paper proves an Allard-type regularity theorem for free-boundary minimal surfaces in Lipschitz domains locally modeled on convex polyhedra. It shows that if such a minimal surface is sufficiently close to an appropriate free-boundary plane, then the surface is C1,alpha graphical over this plane. The theorem is applied to prove partial regularity results for free-boundary minimizing hypersurfaces and relative isoperimetric regions.
COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Muhittin Evren Aydin, Ayla Erdur Kara
Summary: In this paper, we study surfaces with minimal potential energy under gravitational forces, which are called singular minimal surfaces. We prove that a ruled singular minimal surface in a Euclidean 3-space is cylindrical, specifically an alpha-catenary cylinder, based on a result by Lopez. This result is also extended to Lorentz-Minkowski 3-space.
JOURNAL OF GEOMETRY AND PHYSICS
(2024)
Article
Mathematics, Applied
Luiz C. B. da Silva, Rafael Lopez
Summary: This paper investigates the hanging chain problem in the simply isotropic plane and its 2-dimensional analog in the simply isotropic space. The hanging chain and surface problems are well-posed in the degenerate metric if we employ the relative arc length and relative area to measure the weight. Furthermore, the simply isotropic catenary is proven to be the generating curve of a minimal surface of revolution in the simply isotropic space, and the shape of a hanging surface of revolution is determined.
RESULTS IN MATHEMATICS
(2023)
Article
Mathematics
Leon Simon
Summary: With respect to a C infinity metric close to the standard Euclidean metric, a class of embedded (N + )-dimensional pound hypersurfaces without boundary is constructed, which are minimal and strictly stable, and have a singular set equal to an arbitrary preassigned closed subset K C {0} x Re. This settles the question of whether there can be gaps or fractional dimensional parts in the singular set with a strong affirmative. The construction involves the analysis of solutions u of the symmetric minimal surface equation on domains 52 C Rn whose symmetric graphs lie on one side of a cylindrical minimal cone.
ANNALS OF MATHEMATICS
(2023)
Article
Mathematics, Applied
Rafael Lopez
Summary: We investigate the instability of singular minimal surfaces and provide explicit bounds, stability conditions, and numerical evidences for various cases.
COMMUNICATIONS ON PURE AND APPLIED ANALYSIS
(2022)
Article
Computer Science, Software Engineering
Mark Gillespie, Boris Springborn, Keenan Crane
Summary: This paper introduces a numerical method for surface parameterization that produces locally injective and discretely conformal maps on any manifold triangle mesh. The method is extremely robust in practice and provides high-quality interpolation even on meshes with poor elements.
ACM TRANSACTIONS ON GRAPHICS
(2021)
Article
Mathematics, Applied
Antonio Martinez, A. L. Martinez-Trivino
Summary: In this paper, we provide a Weierstrass representation formula for translating solitons and singular minimal surfaces in R-3. As an application, we study the case when the Euclidean Gauss map has a harmonic argument and solve a general Cauchy problem in this class of surfaces.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2022)
Article
Mathematics, Applied
Te Ba, Shengyu Li, Yaping Xu
Summary: This paper investigates the rigidity of bordered polyhedral surfaces and proves that they are determined by boundary value and discrete curvatures on the interior edges using the variational principle. As a corollary, the classical result that congruent Euclidean cyclic polygons (or hyperbolic cyclic polygons) have equal side lengths is reproved.
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Federico Franceschini, Joaquim Serra
Summary: For the thin obstacle problem in Rn, we prove that except for a (n-3)(n-3)-dimensional set, the solution at all free boundary points differs from its blow-up by higher order corrections. This expansion leads to a C1,1-type free boundary regularity result, up to a codimension 3 set.
COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS
(2023)
Article
Mathematics
Scott Mullane
Summary: The study demonstrates that the pseudoeffective cone is not polyhedral under certain conditions, as the class of the fibre of the morphism forgetting one point forms an extremal ray of the dual nef cone.
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
(2021)
Article
Operations Research & Management Science
A. Farajzadeh
Summary: This study presents an existence theorem for the maximum points of a set preordered by a convex cone in a real linear space. The proof of the theorem differs from the usual technique of using the separation theorem. The main result of this study provides a positive answer to an open problem and can be seen as a new version of the main theorem appeared in previous papers with mild assumptions and without using the separation theory and related concepts.
OPTIMIZATION LETTERS
(2023)
Article
Mathematics, Applied
Ludvig Of Klinteberg, Chiara Sorgentone, Anna-Karin Tornberg
Summary: This paper presents a method for estimating quadrature errors in the evaluation of layer potentials defined over smooth curved surfaces. The results are highly accurate, with low computational cost, and can be practically applied.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Chemistry, Physical
Benjamin Heuser, Kurt V. Mikkelsen, James E. Avery
Summary: In their study, researchers investigated the synthesis path of C-60-Buckyball fullerene using density functional theory methods, finding that the cardboard with hinges model shows promise in approximating reaction paths for molecules of this type.
PHYSICAL CHEMISTRY CHEMICAL PHYSICS
(2021)