The Experts below are selected from a list of 22692 Experts worldwide ranked by ideXlab platform
Ljubica S. Velimirović - One of the best experts on this subject based on the ideXlab platform.
-
The Total Curvature of knots under second-order infinitesimal bending
Journal of Knot Theory and Its Ramifications, 2019Co-Authors: Marija S. Najdanović, Svetozar R. Rančić, Louis H. Kauffman, Ljubica S. VelimirovićAbstract:In this paper, we consider infinitesimal bending of the second-order of curves and knots. The Total Curvature of the knot during the second-order infinitesimal bending is discussed and expressions for the first and the second variation of the Total Curvature are given. Some examples aimed to illustrate infinitesimal bending of knots are shown using figures. Colors are used to illustrate Curvature values at different points of bent knots and the Total Curvature is numerically calculated.
-
The Total Curvature of knots under second-order infinitesimal bending
Journal of Knot Theory and Its Ramifications, 2019Co-Authors: Marija S. Najdanović, Svetozar R. Rančić, Louis H. Kauffman, Ljubica S. VelimirovićAbstract:In this paper, we consider infinitesimal bending of the second-order of curves and knots. The Total Curvature of the knot during the second-order infinitesimal bending is discussed and expressions ...
Francisco J. López - One of the best experts on this subject based on the ideXlab platform.
-
interpolation and optimal hitting for complete minimal surfaces with finite Total Curvature
Calculus of Variations and Partial Differential Equations, 2019Co-Authors: Antonio Alarcon, Ildefonso Castroinfantes, Francisco J. LópezAbstract:We prove that, given a compact Riemann surface \(\Sigma \) and disjoint finite sets \(\varnothing \ne E\subset \Sigma \) and \(\Lambda \subset \Sigma \), every map \(\Lambda \rightarrow \mathbb {R}^3\) extends to a complete conformal minimal immersion \(\Sigma \setminus E\rightarrow \mathbb {R}^3\) with finite Total Curvature. This result opens the door to study optimal hitting problems in the framework of complete minimal surfaces in \(\mathbb {R}^3\) with finite Total Curvature. To this respect we provide, for each integer \(r\ge 1\), a set \(A\subset \mathbb {R}^3\) consisting of \(12r+3\) points in an affine plane such that if A is contained in a complete nonflat orientable immersed minimal surface \(X:M\rightarrow \mathbb {R}^3\), then the absolute value of the Total Curvature of X is greater than \(4\pi r\). In order to prove this result we obtain an upper bound for the number of intersections of a complete immersed minimal surface of finite Total Curvature in \(\mathbb {R}^3\) with a straight line not contained in it, in terms of the Total Curvature and the Euler characteristic of the surface.
-
interpolation and optimal hitting for complete minimal surfaces with finite Total Curvature
arXiv: Differential Geometry, 2017Co-Authors: Antonio Alarcon, Ildefonso Castroinfantes, Francisco J. LópezAbstract:We prove that, given a compact Riemann surface $\Sigma$ and disjoint finite sets $\varnothing\neq E\subset\Sigma$ and $\Lambda\subset\Sigma$, every map $\Lambda \to \mathbb{R}^3$ extends to a complete conformal minimal immersion $\Sigma\setminus E\to \mathbb{R}^3$ with finite Total Curvature. This result opens the door to study optimal hitting problems in the framework of complete minimal surfaces in $\mathbb{R}^3$ with finite Total Curvature. To this respect we provide, for each integer $r\ge 1$, a set $A\subset\mathbb{R}^3$ consisting of $12r+3$ points in an affine plane such that if $A$ is contained in a complete nonflat orientable immersed minimal surface $X\colon M\to\mathbb{R}^3$, then the absolute value of the Total Curvature of $X$ is greater than $4\pi r$.
-
Uniform Approximation by Complete Minimal Surfaces of Finite Total Curvature in $\mathbb{R}^3$
arXiv: Differential Geometry, 2009Co-Authors: Francisco J. LópezAbstract:An approximation theorem for minimal surfaces by complete minimal surfaces of finite Total Curvature in $\mathbb{R}^3$ is obtained. This Mergelyan type result can be extended to the family of complete minimal surfaces of weak finite Total Curvature, that is to say, having finite Total Curvature on proper regions of finite conformal type. We deal only with the orientable case.
-
uniform approximation by complete minimal surfaces of finite Total Curvature in mathbb r 3
arXiv: Differential Geometry, 2009Co-Authors: Francisco J. LópezAbstract:An approximation theorem for minimal surfaces by complete minimal surfaces of finite Total Curvature in $\mathbb{R}^3$ is obtained. This Mergelyan type result can be extended to the family of complete minimal surfaces of weak finite Total Curvature, that is to say, having finite Total Curvature on proper regions of finite conformal type. We deal only with the orientable case.
-
The classification of complete minimal surfaces with Total Curvature greater than -12
Transactions of the American Mathematical Society, 1992Co-Authors: Francisco J. LópezAbstract:We classify complete orientable minimal surfaces with finite Total Curvature -87r . INTRODUCTION The classification of complete minimal surfaces with finite Total Curvature in R3 has been an important problem in the classical differential geometry. Some basic properties of these surfaces were studied by R. Osserman (see [8, 9]), who showed the first nontrivial result about this subject. Concretely, he characterized the catenoid and Enneper surface as the unique complete orientable minimal surfaces of Total Curvature -47 . However until recent years no more relevant results have been obtained. W. H. Meeks [6] gave the classification of nonorientable complete minimal surfaces with Total Curvature greater than -87 . This paper is concerned with the Total classification of orientable complete minimal surfaces with Total Curvature -87. Chen and Gackstatter [1] discovered the first example of a complete minimal surface properly of genus 1 (see Theorem 1). The picture of Chen-Gackstatter surface is obtained by joining a handle on Enneper's surface. This genus one minimal surface has Total Curvature -87, and no other examples of such surfaces were found. So, it is expected that no other genus one orientable minimal surface of Total Curvature -87 does exist. In this paper we give a proof of this fact. More precisely, we prove that "Chen-Gackstatter surface is the only genus one orientable complete minimal surface with finite Total Curvature -87 ." Of course, it is not difficult to find genus zero minimal surfaces with Total Curvature -8i . A geometrically interesting example, described by Jorge and Meeks [5], is the trinoid. This surface has three embedded catenoid ends. Moreover its normal vectors at these ends are placed symmetrically in an equator of S2. Received by the editors April 27, 1990 and, in revised form, July 20, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 53A10. Research partially supported by DGICYT Grant PS87-01 15-C03-02. (
Pedro Solórzano - One of the best experts on this subject based on the ideXlab platform.
-
Riemannian rigidity of the parallel postulate in Total Curvature
Mathematische Annalen, 2019Co-Authors: Luis Guijarro, Pedro SolórzanoAbstract:Without any symmetry assumptions and under natural integrability conditions on the Gaussian Curvature, we show that the Euclidean plane is unique among all the complete 2-dimensional Riemannian manifolds that satisfy the Euclid’s fifth postulate. Namely, we prove that the Euclidean plane is the only Riemannian surface free of conjugate points and admitting Total Curvature that satisfies Playfair’s version of the parallel postulate.
-
Riemannian Rigidity of the Parallel Postulate in Total Curvature
arXiv: Differential Geometry, 2016Co-Authors: Luis Guijarro, Pedro SolórzanoAbstract:We prove that the Euclidean plane is the only Riemannian plane with Total Curvature and free of conjugate points that satisfies Playfair's version of the parallel postulate.
Hong Lin - One of the best experts on this subject based on the ideXlab platform.
-
Identification of Spinal Deformity Classification With Total Curvature Analysis and Artificial Neural Network
IEEE transactions on bio-medical engineering, 2008Co-Authors: Hong LinAbstract:In this paper, a multilayer feed-forward, back-propagation (MLFF/BP) artificial neural network (ANN) was implemented to identify the classification patterns of the scoliosis spinal deformity. At the first step, the simplified 3D spine model was constructed based on the coronal and sagittal X-ray images. The features of the central axis curve of the spinal deformity patterns in 3D space were extracted by the Total Curvature analysis. The discrete form of the Total Curvature, including the Curvature and the torsion of the central axis of the simplified 3D spine model was derived from the difference quotients. The Total Curvature values of 17 vertebrae from the first thoracic to the fifth lumbar spine formed a Euclidean space of 17 dimensions. The King classification model was tested on this MLFF/BP ANN identification system. The 17 Total Curvature values were presented to the input layer of MLFF/BP ANN. In the output layer there were five neurons representing five King classification types. A Total of 37 spinal deformity patterns from scoliosis patients were selected. These 37 patterns were divided into two groups. The training group had 25 patterns and testing group had 12 patterns. The 25-pattern training group was further divided into five subsets. Based on the definition of King classification system, each subset contained all five King types. The network training was conducted on these five subsets by the hold-out method, one of cross-validation variants, and the early stop method. In each one of the five cross-validation sessions, four subsets were alternatively used for estimation learning and one subset left was used for validation learning. Final network testing was conducted with remaining 12 patterns in testing group after the MLFF/BP ANN was trained by all five subsets in training group. The performance of the neural network was evaluated by comparing between two network topologies, one with one hidden layer and another with two hidden layers. The results are shown in three tables. The first table shows network errors in estimation learning and the second table shows identification rates in validation learning. The network errors and identification rates in the last round of network training and testing are shown in the third table. Each table has a comparison for both one hidden layer and two hidden layer networks.
-
Identification of Spinal Deformity Classification with Total Curvature Analysis and Artificial Neural Network
2005 IEEE Engineering in Medicine and Biology 27th Annual Conference, 2005Co-Authors: Hong LinAbstract:In this study, a multilayer feedforward, back-propagation artificial neural network is implemented to identify the classification patterns of the scoliotic spinal deformity. At first step the simplified three-dimensional spine model is constructed from coronal and sagittal X-ray images. The features of the central axis curve of the spinal deformity patterns in 3D space are extracted by the Total Curvature analysis. The discrete form of the Total Curvature, including the Curvature and the torsion of the central axis of the simplified 3D spine model is derived from the difference quotients. The values of Total Curvature of 17 vertebrae from first thoracic to the fifth lumbar spine formed a Euclidean space of 17 dimensions. Either the Curvature or the torsion of the three-dimensional curve of the central axis of the spine model could provide the input of the artificial neural network. The King classification model is tested on the neural network. Five sets of King spinal deformity patterns are randomly selected by the definition of King classification. The output layer of the artificial neural network has five neurons representing the five King classification types. The network validation was conducted by the hold-out method, one of cross-validation variant. The performance of the neural network is compared between two network topologies, one with one hidden layer and another with two hidden layers. The results are shown in a table with each of five datasets leave-out and all five datasets participating the training, with either one hidden layer or two hidden layer network
Peng Zhu - One of the best experts on this subject based on the ideXlab platform.
-
Hypersurfaces in Spheres with Finite Total Curvature
Results in Mathematics, 2019Co-Authors: Peng ZhuAbstract:Suppose that M is a complete noncompact hypersurface in a sphere $$\mathbb {S}^{n+1}$$ $$(n\ge 3)$$ with finite Total Curvature. We show that each p-th space of reduced $$L^2$$-cohomology on M has finite dimension, for $$0\le p\le n$$. This result solves the conjecture posed in Zhu (Ann Braz Acad Sci 88:2053–2065, 2016).
-
On reduced L2 cohomology of hypersurfaces in spheres with finite Total Curvature.
Anais da Academia Brasileira de Ciencias, 2016Co-Authors: Peng ZhuAbstract:In this paper, we prove that the dimension of the second space of reduced L2 cohomology of M is finite if is a complete noncompact hypersurface in a sphere
-
A gap theorem on submanifolds with finite Total Curvature in spheres
Journal of Mathematical Analysis and Applications, 2014Co-Authors: Peng Zhu, Shouwen FangAbstract:Abstract We study a complete noncompact submanifold M n in a sphere S n + p . We prove that there admit no nontrivial L 2 -harmonic 1-forms on M if the Total Curvature is bounded from above by a constant depending only on n. The gap theorem is a generalized version of Carronʼs, Yunʼs, Cavalcanteʼs and the first authorʼs results on submanifolds in Euclidean spaces and Seoʼs result on submanifolds in hyperbolic space without the condition of minimality.
