Duality properties of indicatrices of knots

The bridge index and superbridge index of a knot are important invariants in knot theory. We define the bridge map of a knot conformation, which is closely related to these two invariants, and interpret it in terms of the tangent indicatrix of the knot conformation. Using the concepts of dual and derivative curves of spherical curves as introduced by Arnold, we show that the graph of the bridge map is the union of the binormal indicatrix, its antipodal curve, and some number of great circles. Similarly, we define the inflection map of a knot conformation, interpret it in terms of the binormal indicatrix, and express its graph in terms of the tangent indicatrix. This duality relationship is also studied for another dual pair of curves, the normal and Darboux indicatrices of a knot conformation. The analogous concepts are defined and results are derived for stick knots.     

柱面的由一些重要曲线刻画的特征

通过对曲线副法线曲面的研究,得到了柱面的由一些重要曲线刻画的特征.     

Integration of the equations of gas dynamics for 2.5-dimensional solutions

We integrate the equations of gas dynamics in finite form for the solutions in which the thermodynamic parameters depend only on one spatial variable. The corresponding motion of gas represents the nonlinear superposition of the one-dimensional gas motion corresponding to the invariant system and the two-dimensional motion determined by noninvariant functions. These motions are called 2.5-dimensional. We reduce the invariant system to a first-order implicit ordinary differential equation. We study various solutions of the latter. We construct some continuous and discontinuous solutions to the equations of gas dynamics and give their physical interpretation.     

Topological recursion relations on M_(3,2)

We give some new genus-3 universal equations for Gromov-Witten invariants of compact symplectic manifolds. These equations were obtained by studying relations in the tautological ring of the moduli space of2-pointed genus-3 stable curves. A byproduct of our search for genus-3 equations is a new genus-2 universal equation for Gromov-Witten invariants.     

Integrable Equations Arising from Motions of Plane Curves. II

Integrable equations satisfied by the curvature of plane curves or curves on the real line under inextensible motions in some Klein geometries are identified. These include the Euclidean, similarity, and projective geometries on the real line, and restricted conformal, conformal, and projective geometries in the plane. Together with Chou and Qu [Physica D 162 (2002), 9–33], we determine inextensible motions and their associated integrable equations in all Klein geometries in the plane. The relations between several pairs of these geometries provide a natural geometric explanation of the existence of transformations of Miura and Cole-Hopf type.     

Functional relations in lattice statistical mechanics, enumerative combinatorics, and discrete dynamical systems

We recall some non-trivial, non-linear functional relations appearing in various domains of mathematics and physics, such as lattice statistical mechanics, quantum mechanics, or enumerative combinatorics. We focus, more particularly, on the analyticity properties of the solutions of these functional relations. We then consider discrete dynamical systems corresponding to birational transformations. The rational expressions for dynamical zeta functions obtained for a particular two-dimensional birational mapping, depending on two parameters, are recalled, as well as some non-trivial functional relations satisfied by these dynamical zeta functions. We finally give some functional equations corresponding to some singled out orbits of this two-dimensional birational mapping for particular values of the two parameters. This example shows that functional equations associated with curves, for real values of the variables, are actually compatible with a chaotic dynamical system.     

Nonlinear Waves in a Rod: Results for Incompressible Elastic Materials

A nonlinear intrinsic theory is used to describe the motions of a straight round elastic rod including the influence of radial shear and inertia. Consideration of steady wave motions reduces the two coupled partial differential equations to ordinary differential equations for which two integrals of the motion may be found. For incompressible elastic materials with the restriction of small strain gradients, but arbitrary finite strains, a large variety of exact solutions may be found by quadrature. These include large amplitude periodic waves (which may contain shocks), solitary waves, and in some cases waves that are transitional from one stress level to another. Such solutions may be found for uniform stress strain curves that are concave up or down or that contain inflections, and even for nonmontonic curves, which have been used to represent phase transitions.     

极分解与广义*-Aluthge变换

首先给出了Hilbert空间上有界线性算子极分解的的若干性质.其次指出广义的*-Aluthge变换与*-Aluthge变换具有许多相似性质;例如,T_(α,β)^((*))=U|T_(α,β)((*))=U|T_(α,β)^((*))|当且仅当T是双正规的,即[|T|,|T*|]=0,其中对任意两个算子A和B,[A,B]=AB-BA.     

Lightlike surfaces of Darboux-like indicatrixes and binormal indicatrixes of spacelike curves in lightcone 3-space

In this paper, by introducing a new frame on spacelike curves lying in lightcone 3-space, we investigate the geometric properties of the lightlike surface of the Darboux-like indicatrix and the lightlike surface of the binormal indicatrix generated by spacelike curves in lightcone 3-space. As an extension of our previous work and an application of the singularity theory, the singularities of the lightlike surfaces of the Darboux-like indicatrix and the lightlike surface of the binormal indicatrix are classified, several new invariants of spacelike curves are discovered to be useful for characterizing these singularities, meanwhile, it is found that the new invariants also measure the order of contact between spacelike curves or principal normal indicatrixes of spacelike curves located in lightcone 3-space and two-dimensional lightcone whose vertices are at the singularities of lightlike surfaces. One concrete example is provided to illustrate our results.     

Integrable Systems in Three-Dimensional Riemannian Geometry

Summary. In this paper we introduce a new infinite-dimensional pencil of Hamiltonian structures. These Poisson tensors appear naturally as the ones governing the evolution of the curvatures of certain flows of curves in 3-dimensional Riemannian manifolds with constant curvature. The curves themselves are evolving following arclength-preserving geometric evolutions for which the variation of the curve is an invariant combination of the tangent, normal, and binormal vectors. Under very natural conditions, the evolution of the curvatures will be Hamiltonian and, in some instances, bi-Hamiltonian and completely integrable. Received May 31, 2001; accepted January 2, 2002 Online publication March 11, 2002 Communicated by A. Bloch Communicated by A. Bloch rid="     

On the topological classification of rarefaction curves in systems of three conservation laws

In this paper we study the topological properties of integral curves of a system of implicit differential equations associated to rarefaction curves of a system of three conservation laws. This system of equations becomes singular at the points of eigenvalue of multiplicity greater or equal to two. We focus our attention to the generic case of multiplicity two and three. We give local weak topological models for these equations.     

An analytical method for computing Hopf bifurcation curves in neural field networks with space-dependent delays

We give an analytical parametrization of the curves of purely imaginary eigenvalues in the delay-parameter plane of the linearized neural field network equations with space-dependent delays. In order to determine if the rightmost eigenvalue is purely imaginary, we have to compute a finite number of such curves; the number of curves is bounded by a constant for which we give an expression. The Hopf bifurcation curve lies on these curves.     

On the Geometry of Complex‐Lamellar Magnetohydrodynamics: Universal Motions

A geometric formulation is adopted for a nonlinear magnetohydrodynamic system wherein the magnetic field is aligned with the direction of the binormal to the streamlines. It is established that, for complex‐lamellar motion, if the divergence of the binormal field vanishes then the fluid streamlines are geodesics on generalized helicoids. The latter constitute the Maxwellian surfaces and the magnetic lines are helices thereon. The key geometric and physical parameters of the magneto‐hydrodynamic motion are all determined in terms of the torsion τ of the streamlines. A superposition principle is presented whereby a more general class of magnetohydrodynamic motions may be isolated with streamlines and magnetic lines no longer restricted to be geodesics or parallels on the Maxwellian surfaces. Moreover, the class so generated is not subject to the complex‐lamellar constraint. In an appendix, a particular reduction is obtained to the integrable Da Rios system.     

Secant varieties and birational geometry

We show how to use information about the equations defining secant varieties to smooth projective varieties in order to construct a natural collection of birational transformations. These were first constructed as flips in the case of curves by M. Thaddeus via Geometric Invariant Theory, and the first flip in the sequence was constructed by the author for varieties of arbitrary dimension in an earlier paper. We expose the finer structure of a second flip; again for varieties of arbitrary dimension. We also prove a result on the cubic generation of the secant variety and give some conjectures on the behavior of equations defining the higher secant varieties. Received: 29 November 1999; in final form: 4 September 2000 / Published online: 23 July 2001     

Critical curves for the total normal curvature in surfaces of 3-dimensional space forms

A variational problem closely related to the bending energy of curves contained in surfaces of real 3-dimensional space forms is considered. We seek curves in a surface which are critical for the total normal curvature energy (and its generalizations). We start by deriving the first variation formula and the corresponding Euler–Lagrange equations of these energies and apply them to study critical special curves (geodesics, asymptotic lines, lines of curvature) on surfaces. Then, we show that a rotation surface in a real space form for which every parallel is a critical curve must be a special type of a linear Weingarten surface. Finally, we give some classification and existence results for this family of rotation surfaces.     

Heun equations coming from geometry

We give a list of Heun equations which are Picard-Fuchs associated to families of algebraic varieties. Our list is based on the classification of families of elliptic curves with four singular fibers done by Herfurtner. We also show that pull-backs of hypergeometric functions by rational Belyi functions with restricted ramification data give rise to Heun equations.     

On self-similar Ovsyannikov’s vortex

We study a multidimensional self-similar solution of the dynamic equations of an ideal compressible fluid. The solution describes swirling motions of a gas and is partially invariant with respect to the rotation group extended by dilations. The analysis of the solution reduces to the analysis of the singular points and manifolds of a system of fourth-order ordinary differential equations. We also give an example of a solution that describes the expansion of a swirling gas cloud into vacuum.     

Sanya Lectures: Geometry of Spherical Varieties

These are expanded notes from lectures on the geometry of spherical varieties given in Sanya. We review some aspects of the geometry of spherical varieties. We first describe the structure of B-orbits. Using the local structure theorems, we describe the Picard group and the group of Weyl divisors and give some necessary conditions for smoothness. We later on consider B-stable curves and describe in details the structure of the Chow group of curves as well as the pairing between curves and divisors. Building on these results we give an explicit B-stable canonical divisor on any spherical variety.     

Spacelike helices in Minkowski 4-space $${E_1^4 }$$

In this paper, we give some characterizations for spacelike helices in Minkowski space–time E₁⁴{E_1^4}. We find the differential equations characterizing the spacelike helices and also give the integral characterizations for these curves in Minkowski space–time E₁⁴{E_1^4}.