A non-local area preserving curve flow

In this paper, we consider a kind of area preserving non-local flow for convex curves in the plane. We show that the flow exists globally, the length of evolving curve is non-increasing, and the curve converges to a circle in \(C^{\infty }\) sense as time goes into infinity.     

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.     

On the mean length of the chords of a closed curve

Let us consider theN-gons with unit length of sides in the plane. What is the maximum of the arithmetical mean of the length of diagonals? We give an elementary solution for this problem and some more general ones. We deal with continuous analogons too.     

On the length preserving approximation of plane curves by circular arcs

A technique for the length preserving approximation of plane curves by two circular arcs is analyzed. The conditions under which this technique can be applied are extended, and certain consequences of the proved results unrelated to the approximation problem are discussed. More precisely, inequalities for the length of a convex spiral arc subject to the given boundary conditions are obtained. Conjectures on curve closeness conditions obtained using computer simulation are discussed.     

On queue length in a queueing system with Erlang incoming flow

A single-channel queueing system with an Erlang incoming flow and random server unavailability intervals as the system releases is considered. The nonstationary and stationary distributions of queue length are obtained.     

On the curve diffusion flow of closed plane curves

In this paper, we consider the steepest descent H ^?1-gradient flow of the length functional for immersed plane curves, known as the curve diffusion flow. It is known that under this flow there exist both initially immersed curves that develop at least one singularity in finite time and initially embedded curves that self-intersect in finite time. We prove that under the flow closed curves with initial data close to a round circle in the sense of normalised L ² oscillation of curvature exist for all time and converge exponentially fast to a round circle. This implies that for a sufficiently large ‘waiting time’, the evolving curves are strictly convex. We provide an optimal estimate for this waiting time, which gives a quantified feeling for the magnitude to which the maximum principle fails. We are also able to control the maximum of the multiplicity of the curve along the evolution. A corollary of this estimate is that initially embedded curves satisfying the hypotheses of the global existence theorem remain embedded. Finally, as an application we obtain a rigidity statement for closed planar curves with winding number one.     

Point-based methods for estimating the length of a parametric curve

This paper studies a general method for estimating the length of a parametric curve using only samples of points. We show that by making a special choice of points, namely the Gauss–Lobatto nodes, we get higher orders of approximation, similar to the behaviour of Gauss quadrature, and we derive some explicit examples.     

On a curve veering aberration

In numerous places in the literature of eigenvalue problems of mathematical physics one finds curves which approach each other and suddenly veer away. The author postulates that this ugly behavior may be the result of approximation in the representation of physical reality. In the present paper such behavior is demonstrated to arise from the application of the well-known Ritz-Galerkin method to the classical eigenvalue problem of the free vibration of a rectangular membrane.     

Linear operators strongly preserving digraphs whose maximum cycle length

We characterize those linear operators L such that for all digraphsD,L(D) has maximal cycle length L if an only if D has maximum cycle length L. We do this for L =0 and L = 1, the cases L ≤2 having been done previously.     

On shape preserving semigroups

Motivated by positivity-, monotonicity-, and convexity preserving differential equations, we introduce a definition of shape preserving operator semigroups and analyze their fundamental properties. In particular, we prove that the class of shape preserving semigroups is preserved by perturbations and taking limits. These results are applied to partial delay differential equations.     

On order preserving contractions

Let (Ω,Σ,μ) be a measure space and letP be an operator onL ₂(Ω,Σ,μ) with ‖P‖≦1,Pf≧0 a.e. wheneverf≧0. If the subspaceK is defined byK={x| ||P ⁿ x||=||P ^*n x||=||x||,n=1,2,...} thenK=L ₂(Ω,Σ₁,μ), where Σ₁ ⊂ Σ and onK the operatorP is “essentially” a measure preserving transformation. Thus the eigenvalues ofP of modulus one, form a group under multiplication. This last result was proved by Rota for finiteμ here finiteness is not assumed) and is a generalization of a theorem of Frobenius and Perron on positive matrices. The research reported in this document has been sponsored in part by Air Force Office of Scientific Research, OAR through the European Office, Aerospace Research, United States Air Force.     

On mappings preserving pentagons

Summary We introduce a new characterization of linear isometries. More precisely, we prove that if a one-to-one mapping f:<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>ℝⁿ→ℝⁿ(2<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>≦n<<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>∞) maps every regular pentagon of side length a> 0 onto a pentagon with side length b> 0, then there exists a linear isometry I :<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>ℝⁿ→ℝⁿup to translation such that f(x) = (b/a) I(x).