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

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77 The a coecient can now be removed. Distribute its inverse and cancel. ([([a]([[x]][]))])([([b][x])])([c]) Distribution ( [a]([[x]][]) )( [b][x] )([c]) Involution ( ([[x]][]) )( [b][x] )([c]) Inversion Now the x 2 has a unary coecient. Dene d and e. d = : ([b]) e = : ([c]) Work them into the equation. (([[x]][]))([x][b][]) ([c]) Inversion (([[x]][]))([x][b][]) ([c]) Collection (([[x]][]))([x][([b])][])([c]) Involution (([[x]][]))([x][d][])e Denition <strong>of</strong> d and e Flatten and complete the square. ([x][x])([x][d][])e Cardinality ([x][x])([x][d][])([d][d])e Inversion Now factor it. ([x][x])([x][d])([x][d])([d][d])e Cardinality ([x][xd])([xd][d])e Distribution (2x) ([xd][xd])e Distribution (2x) Make a dierence <strong>of</strong> squares. ([xd][xd]) Inverse Cancellation ([xd][xd]) Collection ([xd][xd]) Involution ([xd][xd]) Inversion ([xd][xd]) Involution Dene f to clean up the negative square. f : = (([[([d][d])]])) Put it into the equation and atten to a product. ([xd][xd]) ([xd][xd]) Separate into two factors. Denition <strong>of</strong> f Cardinality
78 ([xd][xd])([xd][f]) ([xd][xd])([xd][f])([xd][])([f][]) ([xd][xdf])([xdf][]) ([xdf][xd]) ([x][x]) Inversion Promotion Distribution Distribution Inverse Cancellation This gives two solutions. x = and x = Expand these for the recognized solutions. Recall f. f : = (([[([d][d])]])) First expand the interior <strong>of</strong> f. ([d][d]) = = ([([b])][([b])]) Denition <strong>of</strong> d = ( [b] [b] ) Involution = ([b][b]) Denition <strong>of</strong> e = ([b][b]) Inversion = ([b][b]) Collection = ([([b][b])]) Involution = ([([b][b])])
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A Calculus <strong
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University ofWashi
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4.2 The Two-Boundary Calcul
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A.2 Functions : : : : : : : : : : :
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LIST OF TABLES 1.1 Simple Calculati
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as an undergraduate at the Universi
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Chapter 1 INTRODUCTION Numb
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3 Table 1.1: Simple Calculations in
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5 contents, they are also drawn as
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7 Table 1.4: Axioms of</str
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Chapter 2 PRIOR WORK Boundary mathe
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11 3 2 Given Number</stro
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13 this system uses string rewrite
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15 Table 2.3: Denition of</
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17 Figure 3.1: The Void. Figure 3
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19 connectivities denes the system
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Chapter 4 INSTANCE AND ABSTRACT 4.1
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23 The third rule creates new eleme
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25 Distribution: (A[BC]) : =(A[B])(
Page 40 and 41: 27 Multiplication of</stron
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
Page 92: 79 f = (([[(([][]) [(([[b]][]))])]]
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