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27 Multiplication <strong>of</strong> natural numbers requires some calculation to return values to the canonical form <strong>of</strong> collections <strong>of</strong> units. This reduction uses distribution to strip away units one at a time, along with involution to remove unit multiplications. 3 2 Given 2 <strong>Number</strong> Rewrite ([][]) Multiplication Rewrite ([][])([][]) Distribution ([] )([] ) Involution Involution 4.4 Cardinality In the boundary calculus, repeated references to the same element can be rewritten as a cardinality <strong>of</strong> that element. Theorem 1 (Cardinality) Multiple references in the same context may be reduced to a single reference with multiple units signifying its quantity, as A:::A=([A][:::]): Cardinality diers from multiplication in that this transformation is not semantically tied to its context, as it is in multiplication. Cardinality applies at any depth. Everything has a count <strong>of</strong> one, shown directly by involution: A=([A])=([A][]): Here, the lone denotes the single cardinality <strong>of</strong>a: Greater cardinalities can be formed by collecting many units in this space using distribution <strong>of</strong> separate single cardinalities. For example, two references can be distributed into a cardinality <strong>of</strong> two: AA=([A][])([A][])=([A][]): The induction step to all natural numbers is, A:::AA=([A][:::])([A][])=([A][:::]):
28 4.4.1 Dominion A zero cardinality, ([A]2), reduces to the void as a void multiplication. In this form, the empty abstract, 2, dominates its context. This element is called a black hole because <strong>of</strong> this collapsing property. A general form <strong>of</strong> this collapsing derives from void multiplication. Theorem 2 (Dominion) Elements collected with a black hole are irrelevant, as 2A=2: Dominion is proven from void multiplication with an involution: 2A=[(2A)]=2: 4.4.2 Generalized Cardinality The cardinality form is independent <strong>of</strong> its context and can be applied to repetition in both addition and multiplication. Cardinality applies to all repetition in the same context, regardless <strong>of</strong> the form repeated or the depth <strong>of</strong> this context. Any collection <strong>of</strong> identical elements can be counted this way. Cardinality applies to addition directly. It applies to multiplication by counting abstracted elements inside the context <strong>of</strong> a surrounding instance boundary. For example, ([a][a])=(([[a]][])). Cardinality <strong>of</strong>multiplication translates to exponentiation. Cardinality appears in standard and boundary numbers as: x + :::+x= nx versus x:::x=([x][:::]) x :::x= x n versus ([x]:::[x])=(([[x]][:::])) 4.5 Conclusions Though a single boundary represents natural numbers, a second boundary provides convenient forms for algebraic multiplication and exponentiation. The involution and distribution axioms completely dene computation on the natural numbers for these functions. The two-boundary calculus is summarized in Table 4.1. The two-boundary calculus constructs a generalized form <strong>of</strong> cardinality. This form applies to all situations <strong>of</strong> repeated reference in space, including addition and
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Page 5 and 6: 4.2 The Two-Boundary Calcul
Page 7 and 8: A.2 Functions : : : : : : : : : : :
Page 9 and 10: LIST OF TABLES 1.1 Simple Calculati
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Page 14 and 15: Chapter 1 INTRODUCTION Numb
Page 16 and 17: 3 Table 1.1: Simple Calculations in
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Page 22 and 23: Chapter 2 PRIOR WORK Boundary mathe
Page 24 and 25: 11 3 2 Given Number</stro
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Page 28 and 29: 15 Table 2.3: Denition of</
Page 30 and 31: 17 Figure 3.1: The Void. Figure 3
Page 32 and 33: 19 connectivities denes the system
Page 34 and 35: Chapter 4 INSTANCE AND ABSTRACT 4.1
Page 36 and 37: 23 The third rule creates new eleme
Page 38 and 39: 25 Distribution: (A[BC]) : =(A[B])(
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 56 and 57: Chapter 6 PHASE 6.1 Introduction Th
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
Page 66 and 67: 53 ematics problems could be phrase
Page 68 and 69: 55 boundary form, in search <strong
Page 70 and 71: 57 Technology interface sof
Page 72 and 73: 59 such asintegrals and Fourier tra
Page 74 and 75: 61 mulas specify how they transform
Page 76 and 77: 63 dominion A theorem stating that
Page 78 and 79: BIBLIOGRAPHY [1] B. Banaschewski. O
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
Page 88 and 89: 75 B.2 Square Root The following sq
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77 The a coecient can now be remove
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79 f = (([[(([][]) [(([[b]][]))])]]
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