If [gamma] is a

closed curve in M, then its vertical torus [S.sub.[gamma]] = [[pi].sup.-1]([gamma]) is a Willmore-like surface, if and only if, [gamma] is a extremal of the following elastic-like energy [E.sub.[phi]]([gamma]) = [[integral].sub.[gamma]]([[kappa].sup.2] + 4[phi])ds.

The storm area is composed of cumulonimbus clouds and is a

closed curve with inward increasing precipitation grades.

Up to now, the researches on discrete models are still focused on the dynamical behaviors (including stability, periodic solutions, bifurcations, chaos, and chaotic control; see[1-6]), and most of the scholars have studied the Neimark-Sacker bifurcation, a codimension-1 bifurcation, which has shown one invariant

closed curve bifurcating from an equilibrium point.

Consider a handle H (either a 0- or 1-handle), which is bordered by the simple

closed curve [partial derivative]H.

What is the mathematical term for a twodimensional

closed curve having the shape of an elongated circle?

Section 2 reviews how FDs are used for representing a

closed curve on the plane and discuss the issue of parametrization in detail.

In the paper "Dynamics, Circuit Design, and Synchronization of a New Chaotic System with

Closed Curve Equilibrium," X.

For high antigenicity of the tumor, a globally

closed curve exists when the dosage of [CD4.sup.+] T cells is small.

In plane geometry Holditch Theorem states if a chord of fixed length is allowed to rotate inside a convex

closed curve then the locus of a point on the chord a distance p from one end and a distance q from the other is a

closed curve whose area is less than that of the original curve by [pi]pq.

The

closed curve in red is the Hermite curve with the inputting of the 36 points refined and shown in Figure 2(e).

The paper, through the eye-movement experiment, obtains characteristics that influence product style, utilizes shape grammar to extract the contour of feature elements and transforms it to two-dimensional

closed curve, then makes similarity analysis on different feature elements via Fourier decomposition.