Changes

Now, to the Example.

Now, to the Example.

Once again, let us begin with a 1-feature alphabet <math>\mathcal{X} = \{ x_1 \} = \{ x \}.</math>  In the discussion that follows I will consider a class of trajectories that are ruled by the constraint that <math>d^k x = 0\!</math> for all <math>k\!</math> greater than some fixed <math>m,\!</math> and I will indulge in the use of some picturesque speech to describes salient classes of such curves.  Given this finite order condition, there is a highest order non-zero difference <math>d^m x\!</math> that is exhibited at each point in the course of any determinate trajectory.

Once again, let us begin with a 1-feature alphabet <math>\mathcal{X} = \{ x_1 \} = \{ x \}.</math>  In the discussion that follows I will consider a class of trajectories that are ruled by the constraint that <math>d^k x = 0\!</math> for all <math>k\!</math> greater than some fixed <math>m,\!</math> and I will indulge in the use of some picturesque speech to describes salient classes of such curves.  Given this finite order condition, there is a highest order non-zero difference <math>d^m x\!</math> that is exhibited at each point in the course of any determinate trajectory.  Relative to any point of the corresponding orbit or curve, let us call this highest order differential feature <math>d^m x\!</math> the ''drive'' at that point.  Curves of constant drive <math>d^m x\!</math> are then referred to as ''<math>m^\text{th}\!</math> gear curves''.

Relative to any point of the corresponding orbit or curve, let us call this highest order differential feature <math>d^m x\!</math> the ''drive'' at that point.  Curves of constant drive <math>d^m x\!</math> are then referred to as ''<math>m^\text{th}\!</math> gear curves''.

One additional piece of notation will be needed here.  Starting from the base alphabet <math>\mathcal{X} = \{ x \},</math> we define and notate <math>\operatorname{E}^j \mathcal{X} = \{ x, d^1 x, d^2 x, \ldots, d^j x \}</math> as the ''<math>j^\text{th}\!</math> order extended alphabet over <math>\mathcal{X}</math>''.

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Let us now consider the family of 4^th gear curves through

Let us now consider the family of 4^th gear curves through

the extended space E^4.X = <|x, dx, d^2.x, d^3.x, d^4.x|>.

the extended space E^4.X = <|x, dx, d^2.x, d^3.x, d^4.x|>.