Article
Mathematics, Applied
Zenggui Wang, Ran Ding
Summary: This paper considers the motion of plane curves under hyperbolic inverse mean curvature with a constant force field c. It proves the preserving convexity of the evolving curve under this flow. Furthermore, an example is given to understand how the forced term c affects the evolution of plane curves. The asymptotic behavior of the flow is discussed in particular.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics
Laiyuan Gao, Shengliang Pan, Dong-Ho Tsai
Summary: This paper investigates the evolution of smooth convex closed plane curves under the 1/k(alpha)-type area-preserving nonlocal flow, showing that the evolving curve will smoothly converge to a circle if the curvature does not blow up in finite time.
JOURNAL OF FUNCTIONAL ANALYSIS
(2021)
Article
Mathematics
Xiao-Liu Wang
Summary: In this paper, we investigate an area-preserving inverse curvature flow and a length-preserving inverse curvature flow for immersed locally convex closed plane curves with rotation number m is an element of N+. The main findings show that the global-in-time flows smoothly converge to m-fold round circles as time goes to infinity, and sufficient conditions on the initial curve are identified to ensure the occurrence of the flow's singularity at a finite time.
JOURNAL OF FUNCTIONAL ANALYSIS
(2023)
Article
Mathematics
Zenggui Wang, Shixia Lv, Bin Zhao
Summary: This paper examines the behavior of one-dimensional dissipative hyperbolic mean curvature flow. We analyze the evolution of a family of circles to understand the asymptotic properties of this flow. Our propositions and proofs demonstrate that if the initial velocity's minimum is nonnegative, the flow converges to either a point or a limit curve with discontinuous curvature in finite time. If the initial velocity's maximum is positive, the flow initially expands and then converges to a point or a limit curve with discontinuous curvature.
JOURNAL OF FUNCTIONAL ANALYSIS
(2023)
Article
Mathematics
Ruixia Hao, Run Zhang
Summary: The aim of this paper is to introduce a convex curve evolution problem based on both local (curvature?) and global (area A) geometric quantities of the evolving curve. The flow will reduce the perimeter and area of the curve, resulting in a progressively more circular shape. Eventually, as t tends to infinity, the limiting curve will converge to a finite circle in the C-8 metric.
INTERNATIONAL JOURNAL OF MATHEMATICS
(2023)
Article
Mathematics, Applied
Leonardo Alese
Summary: The paper investigates the conditions that a pair of real functions must satisfy to be the curvature in the arc-length of a closed planar curve for all real lambda. It points out several equivalent conditions, demonstrates the essentiality of certain periodic behaviors, and explicitly constructs a family of such pairs. Additionally, the discrete counterpart of the problem is also explored.
RESULTS IN MATHEMATICS
(2021)
Article
Mathematics
Yee Meng Teh, R. U. Gobithaasan, Kenjiro T. Miura, Diya' J. Albayari, Wen Eng Ong
Summary: This work introduces a new type of surface called the Log Aesthetic Patch (LAP), which is an extension of the Coons surface patch. The LAP is approximated from a hyperbolic paraboloid using lines of curvature (LoC) information. The study investigates the LAP in terms of curvature, derivative of curvature, torsion, and Logarithmic Curvature Graph (LCG), and confirms its high quality compared to the original hyperbolic paraboloid. Finally, the LAP is projected onto a plane to create strips that simulate hot and cold bending processes in the shipbuilding industry.
Article
Mathematics
Mohammad Ghomi, James Wenk
Summary: In Euclidean 3-space, it is proven that the length of a closed curve outside the unit sphere containing the sphere within its convex hull is at least 4 pi, with equality only achieved when the curve is composed of four semicircles arranged in the shape of a baseball seam, as conjectured by V. A. Zalgaller in 1996.
JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK
(2021)
Article
Mathematics, Applied
Emilio Musso, Alvaro Pampano
Summary: In this study, we investigate critical trajectories in the hyperbolic plane, considering the 1/2-Bernoulli's bending energy with a length constraint. Critical trajectories with periodic curvature are classified into three types based on the causal character of their momentum. We demonstrate that closed trajectories only occur when the momentum is a time-like vector. Moreover, we prove the existence of countably many closed trajectories with time-like momentum, depending on a pair of relatively prime natural numbers.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2023)
Article
Mathematics, Applied
Tim Binz, Balazs Kovacs
Summary: An algorithm is proposed for generalized mean curvature flow of closed two-dimensional surfaces, with error estimates and convergence proof provided. The algorithm combines evolving surface finite elements and linearly implicit backward difference formulae for time integration, with stability analysis independent of geometric arguments.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2022)
Article
Mathematics, Applied
Muhittin Evren Aydin, Adela Mihai, Asif Yokus
Summary: This paper introduces new concepts for plane curves, compares them with traditional concepts, makes some conclusions and classifies curves with constant equiaffine curvature. It also provides examples for illustration.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Review
Engineering, Electrical & Electronic
Satyakam Baraha, Ajith Kumar Sahoo, Sowjanya Modalavalasa
Summary: This paper presents a review of major speckle filtering algorithms for synthetic aperture radar (SAR) images, focusing on nonlocal means (NLMs) and variational models (VMs) methods.
Article
Multidisciplinary Sciences
Luka Mesarec, Wojciech Gozdz, Veronika Kralj-Iglic, Samo Kralj, Ales Iglic
Summary: The impact of the intrinsic curvature of in-plane orientationally ordered curved flexible nematic molecules attached to closed 3D flexible shells was studied numerically. A Helfrich-Landau-de Gennes-type mesoscopic approach was adopted where the flexible shell's curvature field and in-plane nematic field are coupled and concomitantly determined in the process of free energy minimisation. We demonstrate that this coupling has the potential to generate a rich diversity of qualitatively new shapes of closed 3D nematic shells and the corresponding specific in-plane orientational ordering textures, which strongly depend on the shell's volume-to-surface area ratio, so far not predicted in mesoscopic-type numerical studies of 3D shapes of closed flexible nematic shells.
SCIENTIFIC REPORTS
(2023)
Article
Mathematics, Applied
Annalisa Cesaroni, Valerio Pagliari
Summary: Nonlocal curvature functionals with positive interaction kernels are considered, and it is shown that local anisotropic mean curvature functionals can be obtained from them in a blow-up limit. As a result, it is proven that the viscosity solutions to rescaled nonlocal geometric flows locally uniformly converge to the viscosity solution to anisotropic mean curvature motion. This result is achieved by combining compactness argument with a set-theoretic approach related to De Giorgi's barriers for evolution equations.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
(2021)
Article
Chemistry, Applied
Haiping Zhang, Ke Wang, Hui Wang, Hongfei Lin, Ying Zheng
Summary: In this study, a two-phase hydrothermal synthesis approach was proposed to fabricate high-curvature bulk surface with engineered in-plane defects. Experimental results showed that higher curvature degree led to a better catalytic performance. MoS2 synthesized with supercritical solvent exhibited the highest activity among the investigated catalysts. The formation of curved structure facilitated the surface defects generation, promoting the catalyst activity.
