Simple closed curve

For example, consider the embedded graph G illustrated in Figure 14. The total collection of arcs and simple closed curves that we get by performing this operation at the vertices is illustrated in Figure 15, and the elements of the set T(G) are illustrated in Figure 16. 

Figure 15. The arcs and simple closed curves that we get from G.

FIGURE 7 Typical shapes of subharmonic trajectories. A subharmonic period 4 within the 4/3 resonance horn is a three-peaked oscillation in time (a), has three loops in its phase plane projection (b), and four loops in its x-cos 0 projection (c) (Brusselator, a = 0.0072, o = 4/3). The subharmonic period 4 within the 4/1 resonance horn has one loop in its phase plane projection (e), four loops in the x-cos projection (f) and is a one-peaked oscillation in time (d). Stroboscopic points are denoted by O. Try to imagine them winding around the doughnut in three-dimensional space An interesting shape shows up at the period 2 resonance in the 2/3 resonance horn (surface model >/aio = 2/3, alao = 0.1, o0 = 0.001) (g, h). These shapes are comparatively simple because of the shape of the unperturbed limit cycle which for all cases was a simple closed curve. 

Unlike other closed surfaces the Mobius strip is bounded. The boundary is a simple closed curve, but unlike an opening in the surface of a sphere it cannot be physically shrunk away in three-dimensional space. When the boundary is shrunk away the resulting closed surface is topologically a real projective plane. In other words, the Mobius strip is a real projective plane with a hole cut out of it. 

Definition A.12 (Simply Connected Domain) A domain is simply connected if every simple closed curve C in the domain encloses only points of the domain. 

Consider the family of linear systems x = xcosa-ysina, y = xsma + y cos a, where a is a parameter that runs over the range 0 < a < r. Let C be a simple closed curve that does not pass through the origin. 

Next considering the case of reactive motion at the same energy E, we realize that the situation at hand is not very different. The phase space of qi is still elliptical. The phase space of is not elliptical, but it is a simple closed curve, and it still therefore has the same topology as a one-dimensional sphere (every point on a closed curve can be uniquely mapped onto a sphere). Thus, the phase space of reactive motion consists of foliated tori that span both sides of the potential barrier. These reactive tori will be skinny when sliced along the ( 2 Pz) compared to the trapped tori, because they have less energy in the vibrational coordinate and more in the reaction coordinate. In Figure 8 these are labeled Qab j... 

The first to discuss the question of constructing nontrivial surfaces with closed geodesics was evidently Darboux [189]. We will say that a Riemannian manifold satisfies the 5C-property if there exists a number / > 0 such that any geodesic on Af is a simple closed curve of length / ( or its multiplicities). 

The corresponding manifold is the SC-manifold aU of whose geodesics are simple closed curves of equal length. 

At this point we have a mathematical structure that describes fields in bulk matter. A complementary description of conditions prevailing at interfaces between two materials can be derived by applying the same balance equations to a disc-shaped volume of area wa and height h in each material and to a simple closed curve of height h in each material and side S. Using the divergence theorem and the disc-shaped volume with equation 2.1, we have in the limit as A -+ 0... 

In order to apply the Euler-Lagrange formalism to the problem of determining the minimum area contained by a simple closed curve it is necessary to generalize the result of Appendix I to the case of two independent variables. 

A simple closed curve T in space, with a projection on the z = 0 plane that forms a simple closed curve, has surfaces bounded by F that can be written in the form... 

---