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Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0 (view source)
Revision as of 11:34, 7 August 2008
, 11:34, 7 August 2008→Example 1. A Square Rigging: HTML → TeX
By way of example, suppose that we are given the initial condition <math>A = \operatorname{d}A</math> and the law <math>\operatorname{d}^2 A = (A).</math> Since the equation <math>A = \operatorname{d}A</math> is logically equivalent to the disjunction <math>A\ \operatorname{d}A\ \operatorname{or}\ (A)(\operatorname{d}A),</math> we may infer two possible trajectories, as displayed in Table 11. In either case the state <math>A\ (\operatorname{d}A)(\operatorname{d}^2 A)</math> is a stable attractor or a terminal condition for both starting points.
By way of example, suppose that we are given the initial condition <math>A = \operatorname{d}A</math> and the law <math>\operatorname{d}^2 A = (A).</math> Since the equation <math>A = \operatorname{d}A</math> is logically equivalent to the disjunction <math>A\ \operatorname{d}A\ \operatorname{or}\ (A)(\operatorname{d}A),</math> we may infer two possible trajectories, as displayed in Table 11. In either case the state <math>A\ (\operatorname{d}A)(\operatorname{d}^2 A)</math> is a stable attractor or a terminal condition for both starting points.
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:96%"
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:96%"
|+ '''Table 11. A Pair of Commodious Trajectories'''
|+ '''Table 11. A Pair of Commodious Trajectories'''
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|- style="background:ghostwhite"
| <math>\operatorname{Time}</math>
| <math>\operatorname{Trajectory}\ 1</math>
| <math>\operatorname{Trajectory}\ 2</math>
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Because the initial space ''X'' = 〈''A''〉 is one-dimensional, we can easily fit the second order extension E<sup>2</sup>''X'' = 〈''A'', d''A'', d<sup>2</sup>''A''〉 within the compass of a single venn diagram, charting the couple of converging trajectories as shown in Figure 12.
Because the initial space ''X'' = 〈''A''〉 is one-dimensional, we can easily fit the second order extension E<sup>2</sup>''X'' = 〈''A'', d''A'', d<sup>2</sup>''A''〉 within the compass of a single venn diagram, charting the couple of converging trajectories as shown in Figure 12.