A Calculus of Number Based on Spatial Forms - University of ...

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Chapter 6 PHASE 6.1 Introduction This chapter extends the boundary calculus to complex numbers and basic transcendentals by interpreting boundaries functionally and by adding some manipulative constraints. The inverse boundary can be put into an arithmetic form to which cardinality can be applied (the boundary itself cannot be counted). This form is called the phase element, orJ, and it translates to the radian value i in standard numbers. Fractional cardinalities <strong>of</strong> J build into the complex roots <strong>of</strong> one, just as fractions <strong>of</strong> i determine a radian angle in the complex plane. This property <strong>of</strong> J undermines additive space because it has indeterminate sign and ambiguous magnitude, a complication similar to using radian numbers to represent complex numbers. This issue is addressed in terms <strong>of</strong> cardinality and inverse. The instance and abstract boundaries act as exponential and logarithmic functions, respectively. Interpreting these as the natural exponent, e x ! (x); and the natural logarithm, ln x ! [x]; introduces these transcendental functions into the calculus. With these functions, basic transcendentals such aseand can be constructed. 6.2 Phase When manipulating boundary expressions, the inverse boundary falls into an arithmetical form that cardinality can apply to. Recall the concept <strong>of</strong> negative cardinality, =([a][]); written as ,a = a ,1 in standard notation. In the boundary form, the inverse boundary has been separated from what it previously contained, into the encapsulated form []: This form shall be called J: J =[]: :
44 J has a stability unparalleled in the calculus: it uses all three boundaries exactly once. All other legal congurations <strong>of</strong> three dierent boundaries reduce to void because the abstract and instance boundaries cancel out. (The sixth case, (); is undened.) [()];([]);;: 6.2.1 Phase Independence The conguration <strong>of</strong> three boundaries also serves as phase operator, given as []: This operator possesses a curious property <strong>of</strong> independence from its contents. Theorem 1 (Phase Independence) In a nesting <strong>of</strong> abstract, inverse, and instance boundaries, the contents <strong>of</strong> the instance can be moved to the context <strong>of</strong> the conguration. Pro<strong>of</strong>. For all A 2 B, [] Given [ ( [ ])] Involution [ (A [ ])] Dominion [ (A [() ])] Inversion [ (A [()])(A [])] Distribution [ (A )(A [])] Involution [ (A [])] Inversion A [] Involution Phase independence simplies manipulations with the inverse. In particular, it simplies the derivation <strong>of</strong> negative cardinality, as =([])=([A][]): Phase independence also gives a dierent pro<strong>of</strong> <strong>of</strong> inverse promotion. Rather than generate, distribute, and cancel (as in the pro<strong>of</strong> in Section 5.3.1), with phase independence, the inverse is wrapped into the phase operator and moved. =([])=(A[])=(A[]):
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Page 38 and 39: 25 Distribution: (A[BC]) : =(A[B])(
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Page 42 and 43: 29 Table 4.1: The Two-Boundary <str
Page 44 and 45: 31 transformations. 5.2.1 Elements
Page 46 and 47: 33 Inversion: A = : Equivalent Expr
Page 48 and 49: 35 Boundary integers add by collect
Page 50 and 51: 37 Multiplication requires manageme
Page 52 and 53: 39 Proof. Rational
Page 54 and 55: 41 For example, a half-cardinality
Page 58 and 59: 45 Using J, two concise and useful
Page 60 and 61: 47 The ambiguous sign makes the car
Page 62 and 63: 49 Table 6.1: Transcendentals in th
Page 64 and 65: Chapter 7 FUTURE WORK 7.1 Introduct
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Page 80 and 81: 67 [28] L. Lamport. LaTeX: A Docume
Page 82 and 83: 69 A.2 Functions The standard algeb
Page 84 and 85: 71 Distribution of
Page 86 and 87: 73 Radians 1+e i =0 ([]) = Euler's
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Page 90 and 91: 77 The a coecient can now be remove
Page 92: 79 f = (([[(([][]) [(([[b]][]))])]]
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