一种保持周长的平面曲线流

The purpose of this paper is to investigate a new type of evolution problem for closedconvex plane curves which will preserves the perimeter of the curve but expands the enclosedarea and the final limiting curve is a circle in the Hausdorff metric in the plane.     

Expanding convex immersed closed plane curves

We study the evolution driven by curvature of a given convex immersed closed plane curve. We show that it will converge to a self-similar solution eventually. This self-similar solution may or may not contain singularities. In case it does, we also have estimate on the curvature blow-up rate.     

On a non-local perimeter-preserving curve evolution problem for convex plane curves

This paper deals with a non-local evolution problem for closed convex plane curves which preserves the perimeter of the evolving curve but enlarges the area it bounds and makes the evolving curve more and more circular during the evolution process. And the final shape of the evolving curve will be a circle in the C ^∞ metric as the time t goes to infinity. The first author is supported in part by the National Science Foundation of China (No.10671066) and the Shanghai Leading Academic Discipline Project (No. B407).     

Central affine curve flow on the plane

We give the following for Pinkall’s central affine curve flow on the plane: (i) a systematic and simple way to construct the known higher commuting curve flows, conservation laws, and a bi-Hamiltonian structure, (ii) Bäcklund transformations and a permutability formula, (iii) infinitely many families of explicit solutions. We also solve the Cauchy problem for periodic initial data.     

On maximally looped closed curves

In this paper, the maximally looped closed curves are discussed and the relation between the number of triple points and the number of loops is presented. Also, the rotation number of maximally looped curves is calculated.     

Bonnesen's inequality for the isoperimetric deficiency of closed curves in the plane

An inequality of T. Bonnesen for the isoperimetric deficiency of a convex closed curve in the plane is extended to arbitrary simple closed curves. As a primary tool it is shown that, for any such curve, there exist two concentric circles such that the curve is between these and passes at least four times between them. Next, it is shown that Bonnesen's inequality holds for any such pair of concentric circles.This work was supported by The Danish Natural Science Council and carried out during the author's stay 1989–90 as a Visiting Member at the Institute for Advanced Study, Princeton, N.J., U.S.A.     

Some global properties and constructions for closed curves in the plane

This is an extended comment on the paper [1] by W. Cielak and W. Mozgawa. It is shown that several theorems presented there for rosettes can be extended to a larger class of closed curves by using different methods of proof. Furthermore, the notion of rosettes of constant width is discussed in more detail and some classical constructions for convex curves of constant width are generalized to this class of curves.     

On multi-Frobenius non-classical plane curves

In this paper, we characterize the plane curves over \mathbb F_q{\mathbb {F}_q} which are Frobenius non-classical for different powers of q.     

On the equivalence of plane curve singularities

A systematic method to calculate cleft extensions for pointed Hopf algebras is developed and applied to U_q(sl ₂) and the Frobenius-Lusztig kernel U_q(sl ₂)'.     

On a property of plane curves

Let γ:[0,1]→²[0,1] be a continuous curve such that γ(0)=(0,0), γ(1)=(1,1), and γ(t)∈²(0,1) for all t∈(0,1). We prove that, for each n∈N, there exists a sequence of points Ai, 0?i?n+1, on γ such that A₀=(0,0), An+1=(1,1), and the sequences and , 0?i?n, are positive and the same up to order, where π₁, π₂ are projections on the axes.     

On minimal non-potentially closed subsets of the plane

We study the Borel subsets of the plane that can be made closed by refining the Polish topology on the real line. These sets are called potentially closed. We first compare Borel subsets of the plane using products of continuous functions. We show the existence of a perfect antichain made of minimal sets among non-potentially closed sets. We apply this result to graphs, quasi-orders and partial orders. We also give a non-potentially closed set minimum for another notion of comparison. Finally, we show that we cannot have injectivity in the Kechris-Solecki-Todor?evi? dichotomy about analytic graphs.     

On the intersection number of two plane curves

The intersection number of two algebraic curves in a plane over any infinite field is shown to be uniquely defined by a fairly weak set of postulates. These postulates also imply some of the basic properties of that number. The treatment is completely elementary.     

On the renewal of a Wiener field on a plane with the use of its values on closed curves

Forw(u, v), (u, v)∉ γ (here,w(x, y), x≥0, y≥0, is a Wiener field and γ is a certain closed curve on a plane), we construct the best mean-square estimate on the basis of the values ofw(x, y) for (x, y)∈ γ. We also calculate the error of this estimate. Donetsk University, Donetsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 6, pp. 744–752, June, 1999.