Then, a vertical torus over a

closed curve [gamma] in M, [S.sub.[gamma]] = [[pi].sup.-1]([gamma]), is a Willmore surface in N(K, [tau]), if and only if, [S.sub.[gamma]] is an extremal of the following Willmore-like energy:

r(i) represents the number of red

closed curves contained in the ith yellow

closed curve; [r.sub.i] represents the number of red

closed curves, which are crossed when the aircraft is passing through the red

closed curve contained in the ith yellow

closed curve.

Caption: Figure 3: The initial value is (0.9112,0.01228), (a) the bifurcation diagram in [delta]-u plane, it exhibits the dynamical behaviors of model (5), (b) [delta]=1, the equilibrium point is stable, (c) [delta] = 1.5, one invariant

closed curve emerges from the equilibrium point, and (d) the solution curve when [delta] = 1.5, which shows that the invariant

closed curve is not orbit of model (5).

We also arrange that each simple

closed curve on the border of a disk in [f.sup.-1]([F.sub.0]) is mapped to [lambda] by a monotone map determined by the local degree, which we may assume to be nonzero.

The principle of this procedure is that we treat a

closed curve as a closed polygon of N vertices and N sides, with the N vertices corresponding to the N points of the discretized curve.

However, if tumor antigenicity is high, a stable

closed curve exists and a patient will experience tumor regression and relapse.

The difference between the areas of two

closed curves can be calculated from the formula by [pi] x a x b as following Figure 4 in [4].

(a) The UST image slice, (b) the input image after contrast enhancement, (c) the edge image detected by Canny with an adaptive thresholding, (d) the convex polygon vertex image produced by using convex hull searching and postprocessing, (e) the

closed curve image produced by using Hermite cubic curve algorithm, and (f) the resultant image extracted by GrabCut.

Indirect encoding of the genes of a

closed curve for interactively create innovative car silhouettes.

Figure 9 shows the phase portrait of this experiment, and we can find that the phase portrait depicts a

closed curve. The circles of either end of the limit cycle are observed due to the upper body vibration caused by foot-landing impact.

Under the UDG model, the shortest

closed curve enclosing the unit circle centered at node [n.sub.0] is the hexagon also centered at node [n.sub.0], and because any node on the 2-hop contour of node [n.sub.0] must reside outside of the unit circle, and, accordingly, the

closed curve satisfying the CLOSED-CYCLE condition must involve more nodes than the hexagon, we can conclude that [L.sub.th] = 7.

Since [partial derivative][OMEGA] is a simple

closed curve, we have [partial derivative][OMEGA] [subset or equal to] D(p, r).