By G. Viennot
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Homogeneous areas of linear algebraic teams lie on the crossroads of algebraic geometry, thought of algebraic teams, classical projective and enumerative geometry, harmonic research, and illustration concept. through general purposes of algebraic geometry, for you to remedy a variety of difficulties on a homogeneous area, it's common and priceless to compactify it whereas keeping an eye on the crowd motion, i.
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Additional info for Algèbres de Lie libres et monoïdes libres : Bases des algèbres de Lie libres et factorisations des monoïdes libres
To determine graphically which form an attractor assumes, we can draw its projection on one face of the box. In the former case we shall see curves, but in the latter the separate surfaces will project on top of each other and completely fill an area on the face. In the present instance, we can find the projection of the cross section by simply repeating the computational procedure that we used in producing the attractor in Figure 12, ignoring the fact that 54 A JOURNEY INTO CHAOS u actually varies on the attractor.
We next allow each sled to descend 5 meters, so that x equals 2. 5 again. In the lower left panel we have plotted the five thousand points representing the newly acquired cross-slope positions and velocities. We find that the points have gathered themselves into a more or less elliptical region with two thin arms extending from it. There are large empty areas, representing states that cannot occur except transiently. Points on the left and right edges are confined to narrow bands near the midpoints of these edges, where V is close to zero; this implies that the sleds represented by these points are moving from nearly due north.
In the present example, the velocity remains constant. If two nearby objects have identical velocities—speeds and directions—and differ in their initial states only by virtue of differing in their positions, they will not move apart. Let us therefore amend our definition of chaos. First, for this particular purpose, let us refer to any quantity whose value remains unaltered when a system evolves without our interference, but may be altered if we introduce new initial conditions, as a virtual constant.
Algèbres de Lie libres et monoïdes libres : Bases des algèbres de Lie libres et factorisations des monoïdes libres by G. Viennot