Space curves of constant breadth in Minkowski 3-space

In this paper, we study spacelike and timelike curves of constant breadth in Minkowski 3-space. We show that in Minkowski 3-space spacelike and timelike curves of constant breadth are normal, helices, and spherical curves in some special cases. Furthermore, we give that the total torsion of a closed spacelike curve of constant breadth is zero while the total torsion of a simple closed timelike curve is equal to ${2\pi n, (n \in Z)}$ .     

The total mixed curvature of open curves in $$E^3$$

The total mixed curvature of a curve in \(E^3\) is defined as the integral of \(\sqrt{\kappa ^2+\tau ^2}\), where \(\kappa \) is the curvature and \(\tau \) is the torsion. The total mixed curvature is the length of the spherical curve defined by the principal normal vector field. We study the infimum of the total mixed curvature in a family of open curves whose endpoints and principal normal vectors at the endpoints are prescribed. In our previous works, we studied similar problems for the total absolute curvature, which is the length of the spherical curve defined by the unit tangent vector, and for the total absolute torsion, which is the length of the spherical curve defined by the binormal vector.     

On curves of constant torsion I

We give an explicit construction of a closed curve with constant torsion and everywhere positive curvature. We also discuss the restrictions on closed curves of constant torsion when they are constrained to lie on convex surfaces.     

常挠率运动曲线生成曲面上的贝克隆变换

该文给出由常挠率运动曲线生成曲面上的贝克隆变换,其中运动曲线的曲率满足修正KdV方程,从而得到著名的对于修正KdV方程贝克隆变换的一个几何实现.作为应用,取圆柱面作为种子曲面,构造了一些由周期运动曲线生成的新曲面,其中周期运动曲线在xy平面上的投影是闭曲线.     

Total torsion of curves in three-dimensional manifolds

A classical result in differential geometry assures that the total torsion of a closed spherical curve in the three-dimensional space vanishes. Besides, if a surface is such that the total torsion vanishes for all closed curves, it is part of a sphere or a plane. Here we extend these results to closed curves in three dimensional Riemannian manifolds with constant curvature. We also extend an interesting companion for the total torsion theorem, which was proved for surfaces in by L. A. Santaló, and some results involving the total torsion of lines of curvature. Dedicated to Professor Manfredo P. do Carmo on his 80th birthday.     

Nonlinear partial differential equations associated with binormal motions of constant torsion curves in Minkowski 3-space

We give three nonlinear partial differential equations which are associated with binormal motions of constant torsion curves in Minkowski 3-space. We also give B?cklund transformations for these equations, as well as for surfaces swept out by related moving curves. As applications, from some trivial binormal motions we construct some new binormal motions.     

Explicit Elastic Curves

The equilibria of thin rods are given by curves which are critical points of the modified total squared curvature. The critical curves are known as elastic curves. It is shown how all the elastic curves are given explicitly in terms of elliptic functions as soon as a certain set of three parameters is known. Every regular curve can be parametrized to have a constant speed but the parametrization is rarely known explicitly. Remarkably, all the elastic curves are here explicitly parametrized to have a constant speed. Curves with fixed distinct endpoints as well as closed curves are admitted. The tangent direction may be constrained at one, both, or neither of the endpoints. There are three major strands of formulas corresponding to: fixed length L, variable length without tension, and variable length with tension (let > 0 and add a term L to the total squared curvature). In the most complicated cases the three parameters are given as solutions to a non-linear system of three equations. In the least complicated case everything is given explicitly in terms of elliptic functions. If the length is variable and there is no tension, at least one of the parameters is completely determined (the elliptic modulus m = 1/2). Using the same set of parameters explicit formulas are given for: the length when it is variable, the total squared curvature, and the tangent angle along the elastic curve. A number of examples are presented which illustrate the full range of constraints.     

High rank elliptic curves with prescribed torsion group over quadratic fields

There are 26 possibilities for the torsion groups of elliptic curves defined over quadratic number fields. We present examples of high rank elliptic curves with a given torsion group which set the current rank records for most of the torsion groups. In particular, we show that for each possible torsion group, except maybe for \(\mathbb {Z}/15\mathbb {Z}\) , there exists an elliptic curve over some quadratic field with this torsion group and with rank \(\ge 2\) .     

Maximally movable Riemannian spaces with torsion

We prove that, among all Riemannian spaces of constant curvature, only three-dimensional spaces have torsion which is invariant under the group of motions. The torsion tensor in these spaces is covariantly constant and determines the torsion form. The ratio of the integral of this form over a bounded domain to its volume is a constant determining the torsion of the space. We introduce the notions of volume torsion and scalar torsion.     

Self-linking of spatial curves without inflections and its applications

The self-linking number of generic smooth closed curves in Euclidean 3-space is studied. A formula expressing the self-linking number via the signs of the double points of a generic projection of the curve on a plane and the signs of the torsion at the points that are projected into inflection points is obtained. Every local invariant of generic curves is proved to be equal, up to an additive constant, to a linear combination of two basic local invariants: the number of flattening points and the self-linking number. Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italy. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 34, No. 2, pp. 1–8, April–June, 2000. Translated by F. Aicardi     

Sequences of rational torsions on abelian varieties

Summary We address the question of how fast the available rational torsion on abelian varieties over increases with dimension. The emphasis will be on the derivation of sequences of torsion divisors on hyperelliptic curves. Work of Hellegouarch and Lozach (and Klein) may be made explicit to provide sequences of curves with rational torsion divisors of orders increasing linearly with respect to genus. The main results in §2) are applications of a new technique which provide sequences of hyperelliptic curves for all torsions in an interval [a _g ,a _g +b _g ]] wherea _g is quadratic ing andb _g is linear ing. As well as providing an improvement from linear to quadratic, these results provide a wide selection of torsion orders for potential use by those involved in computer integration. We conclude by considering possible techniques for divisors of non-hyperelliptic curves, and for general abelian varieties.     

A procedure to calculate torsion¶of elliptic curves over ℚ

We present an algorithm which uses the analytic parameterization of elliptic curves to rapidly calculate torsion subgroups, and calculate its running time. This algorithm is much faster than the “traditional” Lutz–Nagell algorithm used by most computer algebra systems to calculate torsion subgroups. Received: 7 August 1997 / Revised version: 28 November 1997     

$ L^2 $-torsion of Hyperbolic Manifolds of Finite Volume

Suppose is a compact connected odd-dimensional manifold with boundary, whose interior M comes with a complete hyperbolic metric of finite volume. We will show that the -topological torsion of and the -analytic torsion of the Riemannian manifold M are equal. In particular, the -topological torsion of is proportional to the hyperbolic volume of M, with a constant of proportionality which depends only on the dimension and which is known to be nonzero in odd dimensions [HS]. In dimension 3 this proves the conjecture [Lü2, Conjecture 2.3] or [LLü, Conjecture 7.7] which gives a complete calculation of the -topological torsion of compact -acyclic 3-manifolds which admit a geometric JSJT-decomposition.?In an appendix we give a counterexample to an extension of the Cheeger-Müller theorem to manifolds with boundary: if the metric is not a product near the boundary, in general analytic and topological torsion are not equal, even if the Euler characteristic of the boundary vanishes. Submitted: March 1998, revised: July 1998.     

The Total Absolute Curvature of Nonclosed Curves in S2

The total absolute curvature of nonclosed curves in S² is studied. We look at the set of curves with fixed endpoints and end-directions and see how the infimum of the total absolute curvature in this set depends on the endpoints and the end-directions. We consider both the case when the length of curves is fixed and the case when the length is free, and see the difference of results between them.     

Countably valued lattice-ordered groups

A countably valued lattice-ordered group is a lattice-ordered group in which every element has only countably many values. Such lattice-ordered groups are proven to be normal-valued and, though not necessarily special-valued, every element in a countably valuedl-group must have a special value. The class of countably valuedl-groups forms a torsion class, and the torsion radical determined by this class is anl-ideal that is the intersection of all maximal countably valued subgroups.Countably valuedl-groups are shown to be closed with respect toeventually constant sequence extensions, and it is shown that many properties of anl-group pass naturally to its eventually constant sequence extension.Presented by M. Henriksen.     

任意多连通截面扭转应力函数的有限元解法

杆件扭转问题的求解,主要有基于扭转理论翘曲函数的边界元法和有限元法、基于薄壁杆件理论的数值解法和基于扭转理论应力函数的有限元法．根据任意多连通截面直杆扭转问题的应力函数理论,讨论并改进了与微分方程及定解条件等效的泛函,在此基础上推导了求解多连通截面扭转应力函数的有限元列式,将扭转问题的翘曲位移单值条件转化为边界节点上的集中荷载．采用主从节点法满足孔洞边界上应力函数的同值条件,实现了任意多连通复杂截面扭转应力函数的有限元直接求解,通过应力函数积分获得截面的扭转常数．算例验证了方法的可行性和有效性．     

Stress relaxation in physically nonlinear polymer materials under simultaneous tension and torsion

Conclusions 1. Application of our method of determining the parameters of physically nonlinear polymer materials from tests in uniaxial tension or in torsion ensures a satisfactory agreement between theoretical and experimental stress-relaxation curves corresponding to other modes of loading.2. Theoretical stress-relaxation curves for shear (tension) calculated on the basis of parameters which have been determined from approximations of stress-relaxation curves for tension (torsion) fall closely within the confidence interval for an individual measurement (with =95%).3. The agreement between theoretical and experimental stress-relaxation curves for a plane state of stress is 2–3 times worse, in terms of the mean-square relative difference, than the agreement between the respective curves for uniaxial tension.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 797–803, September–October, 1977.