A book concerning symbolic sequence processing

A book concerning symbolic sequence processing b

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by Daniel J. Greenhoe

⚅ ⚅ ⚅⚅ ⚅ ⚅ ⚅ ⚅ ⚄⚄⚄ ⚄ ⚄ ⚄⚄ ⚄⚄ ⚄ ⚄ ⚄ ⚄ ⚃ ⚃ ⚃ ⚃ ⚃ ⚃ ⚃ ⚃⚃ ⚂ ⚂ ⚂ ⚂ ⚁ ⚁ ⚁⚁ ⚁ ⚁ ⚁ ⚁ ⚁ ⚁ ⚁ ⚁ ⚁⚁ ⋯ ⚀ ⚀ ⚀ ⚀ ⚀ ⚀ ⚀⚀⚀ 0 10 20 30 40 5 15 25 35 45 50

A book concerning symbolic sequence processing

by Daniel J. Greenhoe

Copyright © 2016 Daniel J. Greenhoe All rights reserved

This text was typeset using XƎLATEX, which is part of the TEXfamily of typesetting engines, which is arguably the greatest development since the Gutenberg Press. Graphics were rendered using the pstricks and related packages, and LATEX graphics support. The main roman, italic, and bold font typefaces used are all from the Heuristica family of typefaces (based on the Utopia typeface, released by Adobe Systems Incorporated). The math font is XITS from the XITS font project. The font used in quotation boxes is adapted from Zapf Chancery Medium Italic, originally from URW++ Design and Development Incorporated. The font used for the text in the title is Adventor (similar to Avant-Garde) from the TEX-Gyre Project. The font used for the version in the footer of individual pages is Liquid Crystal (Liquid Crystal) from FontLab Studio. The Latin handwriting font is Lavi from the Free Software Foundation. Chinese glyphs appearing in the text are from the font 王漢宗中明體繁.1 The ship appearing throughout this text is loosely based on the Golden Hind, a sixteenth century English galleon famous for circumnavigating the globe.

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pinyin: Wáng Hàn Zōng Zhōng Mińg Tǐ Fán; translation: Hàn Zōng Wáng's Medium-weight Mińg-style Traditional Characters; literal: 王漢宗∼font designer's name; 中∼medium; 明∼Mińg (a dynasty); 體∼style; 繁∼traditional

page iii

Abstract A real-valued random variable 𝖷 is a measurable function that maps from a probability space (𝛺, 𝔼, 𝖯) to (ℝ, ≤, 𝖽, +, ⋅, ℬ) where ℬ is the usual Borel σ-algebra on the “real line” (ℝ, ≤, 𝖽, +, ⋅) and where ℝ is the set of real numbers, ≤ is the standard linear order relation on ℝ, 𝖽(𝑥, 𝑦) ≜ |𝑥 − 𝑦| is the usual metric on ℝ, and (ℝ, +, ⋅, 0, 1) is the standard field on ℝ. This text demonstrates that this definition of random variable is often a poor choice for computing statistics when the probability space that 𝖷 maps from has structure that is dissimilar to that of the real line. This text proposes two alternative statistical systems that, unlike the traditional method of a random variable 𝖷 mapping exclusively to the real line, 𝖷 instead maps more generally to (1) a weighted graph (2) an ordered distance linear space ℝ𝘕 . In each mapping method, the structure that 𝖷 maps to is preferably one that has order and metric geometry structures similar to that of the underlying stochastic process. And ideally the structure 𝖷 maps from and the structure 𝖷 maps to are, with respect to each other, both isomorphic and isometric.

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❝Here, on the level sand,

Between the sea and land, What shall I build or write Against the fall of night? ❞

❝Tell me of runes to grave

That hold the bursting wave, Or bastions to design For longer date than mine. ❞

Alfred Edward Housman, English poet (1859–1936)

2

❝The uninitiated imagine that one must await inspiration in order to

create. That is a mistake. I am far from saying that there is no such thing as inspiration; quite the opposite. It is found as a driving force in every kind of human activity, and is in no wise peculiar to artists. But that force is brought into action by an effort, and that effort is work. Just as appetite comes by eating so work brings inspiration, if inspiration is not discernible at the beginning. ❞ Igor Fyodorovich Stravinsky (1882–1971), Russian-born composer

3

❝As

I think about acts of integrity and grace, I realise that there is nothing in my knowledge to compare with Frege's dedication to truth. His entire life's work was on the verge of completion, much of his work had been ignored to the benefit of men infinitely less capable, his second volume was about to be published, and upon finding that his fundamental assumption was in error, he responded with intellectual pleasure clearly submerging any feelings of personal disappointment. It was almost superhuman and a telling indication of that of which men are capable if their dedication is to creative work and knowledge instead of cruder efforts to dominate and be known.❞ Bertrand Russell (1872–1970), British mathematician, in a 1962 November 23 letter to Dr. van Heijenoort.

4

2 3 4

quote: image: quote: image: quote: image:

📘 Housman (1936), page 64 ⟨“Smooth Between Sea and Land”⟩, 📘 Hardy (1940) ⟨section 7⟩ http://en.wikipedia.org/wiki/Image:Housman.jpg 📘 Ewen (1961), page 408, 📘 Ewen (1950) http://en.wikipedia.org/wiki/Image:Igor_Stravinsky.jpg 📘 Heijenoort (1967), page 127 http://www-history.mcs.st-andrews.ac.uk/PictDisplay/Russell.html

ACKNOWLEDGEMENTS

It is not far from the truth to say that much of the research presented in this text started with this email from Professor Po-Ning Chen of National Chiao-Tung University, Taiwan:5 Sat, 25 Jan 2014 03:45:51 -0800 (PST) Dear Dan: So far people are mostly (and are used to) dealing with “frequency-transform” of a numerical sequences. However, sometimes, we need to find out the ”occurrence frequency” of a symbolic sequence. I am wondering “Can we design a wavelet transform for a symbolic sequence like DNA?” What do you think? See the attached paper as an example. Po-Ning So I would like to say one more time to Professor Po-Ning Chen, “Thank you for introducing me to the topic”(!) which has to a large extent resulted in the research described in this text.6

5

Po-Ning Chen: 陳伯寧 (pinyin: Chén Bó Nińg) National Chiao-Tung University: 國立交通大學 (Gúo Lì Jiāo Tōng Dà Xúe). Taiwan: 台灣 (pinyin: Tái Wān). 6 The “attached paper” referred to by Professor Chen's email is 📃 Galleani and Garello (2010).

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SYMBOLS

Symbols ❝Rule XVI. As for things which do not

❝rugula

XVI. Quae vero praesentem mentis attentionem non requirunt, etiamsi ad conclusionem necessaria sint, illa melius est per brevissimas notas designare quam per integras figuras: ita enim memoria non poterit falli, nec tamen interim cogitatio distrahetur ad haec retinenda, dum aliis deducendis incumbit.❞

require the immediate attention of the mind, however necessary they may be for the conclusion, it is better to represent them by very concise symbols rather than by complete figures. It will thus be impossible for our memory to go wrong, and our mind will not be distracted by having to retain these while it is taken up with deducing other matters.❞

René Descartes (1596–1650), French philosopher and mathematician

7

❝In

signs one observes an advantage in discovery which is greatest when they express the exact nature of a thing briefly and, as it were, picture it; then indeed the labor of thought is wonderfully diminished.❞ Gottfried Leibniz (1646–1716), German mathematician,

symbol

8

description

reference

empty set integers whole numbers natural numbers extended set of integers real numbers non-negative real numbers positive real numbers extended real numbers complex numbers

Definition 1.1 (page 5) Definition 1.2 (page 5) Definition 1.2 (page 5) Definition 1.2 (page 5) Definition 1.2 (page 5) Definition 1.2 (page 5) Definition 1.2 (page 5) Definition 1.2 (page 5) Definition 1.2 (page 5) Definition 1.10 (page 6)

pi positive infinity negative infinity

3.14159265 …

relation ordered pair Cartesian product of 𝑋 and 𝑌 ordered pair inverse of relation Ⓡ

Definition 1.5 (page 6) Definition 1.3 (page 5) Definition 1.4 (page 6) Definition 1.3 (page 5) Definition 1.7 (page 6) Definition 1.9 (page 6)

sets: ∅ ℤ 𝕎 ℕ ℤ∗ ℝ ℝ⊢ ℝ+ ℝ∗ ℂ values: 𝜋 ∞ −∞ relations: Ⓡ (△, ▽) 𝑋×𝑌 (𝑎, 𝑏) Ⓡ−1 𝑋𝘕

…continued on next page… 7

quote: 📘 Descartes (1684a) ⟨rugula XVI⟩, translation: 📘 Descartes (1684b) ⟨rule XVI⟩, image: Frans Hals (circa 1650), http://en.wikipedia.org/wiki/Descartes, public domain 8 quote: 📘 Cajori (1993) ⟨paragraph 540⟩, image: http://en.wikipedia.org/wiki/File:Gottfried_Wilhelm_von_Leibniz.jpg, public domain

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symbol (𝑎, 𝑏, 𝑐) |𝑧| = ≜ → ∈ ∉ 𝓓(Ⓡ) 𝓘(Ⓡ) 𝓡(Ⓡ) 𝓝 (Ⓡ) relations on sets: ⊆ ⊂ ⊇ ⊃ ⊈ ⊄ sets of relations: 𝟚𝑋𝑌 𝑌𝑋 operations on sets: 𝐴∪𝐵 𝐴∩𝐵 𝐴△𝐵 𝐴⧵𝐵 𝐴𝖼 |⋅| logic: ¬ ∧ ∨ ⊕ ⟹ ⟸ ⟺ ∀ ∃ order: ∨ ∧ ≤ ≥ < > ( 𝑋, ≤) [𝑎 ∶ 𝑏] (𝑎 ∶ 𝑏] [𝑎 ∶ 𝑏)

description ordered triple absolute value of a complex number 𝑧 equality relation equality by definition maps to is an element of is not an element of domain of a relation Ⓡ image of a relation Ⓡ range of a relation Ⓡ null space of a relation Ⓡ

reference Definition 1.11 (page 6) Definition 1.24 (page 9)

Definition 1.12 (page 6) Definition 1.12 (page 6) Definition 1.12 (page 6) Definition 1.12 (page 6)

subset proper subset super set proper superset is not a subset of is not a proper subset of set of all relations in 𝑋 × 𝑌 set of all functions in 𝑋 × 𝑌

Definition 1.5 (page 6) Definition 1.6 (page 6)

set union set intersection set symmetric difference set difference set complement set order

Definition 1.13 (page 7)

logical NOT operation logical AND operation logical inclusive OR operation logical exclusive OR operation “implies”; “implied by”; “if and only if”; universal quantifier: existential quantifier:

“only if” “if” “implies and is implied by” “for each” “there exists”

join or least upper bound meet or greatest lower bound reflexive ordering relation reflexive ordering relation irreflexive ordering relation irreflexive ordering relation ordered set closed interval from 𝑎 to 𝑏 half-open interval from 𝑎 to 𝑏 half-open interval from 𝑎 to 𝑏

Definition 1.26 (page 10) Definition 1.27 (page 10) Definition 1.20 (page 8) Definition 1.22 (page 9) Definition 1.22 (page 9) Definition 1.22 (page 9) Definition 1.20 (page 8) Definition 1.23 (page 9) Definition 1.23 (page 9) Definition 1.23 (page 9)

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page x symbol (𝑎 ∶ 𝑏) |𝑥| ⌈𝑥⌉ ⌊𝑥⌋ ring operations: ∑𝑛∈𝔻 𝑥𝑛 𝖬𝜙 (⦅𝑥𝑛 ⦆) 𝖬𝜙(𝑥;𝑝) (⦇𝑥𝑛 ⦈) set structures: 𝑻 𝑹 𝑨 𝟚𝑋 topology: (𝑋, 𝑻 ) 𝐴− 𝐴∘ lim𝑛→∞ ⦅𝑥𝑛 ⦆ ⦅𝑥𝑛 ⦆ → 𝑥 distance spaces: 𝖽(𝑝, 𝑞) (𝑋, 𝖽) 𝖡(𝑥, 𝑟) 𝖡 (𝑥, 𝑟) diam 𝐴 𝜏 ▵ (𝑝, 𝜎; 𝖽) ○ probability: (𝛺, 𝔼, 𝖯) 𝛺 𝔼 𝖯 𝖷 𝖤(𝖷) 𝖵𝖺𝗋(𝖷) graphs: (𝑋, 𝐸) (𝑋, 𝐸, 𝖽, 𝗐) ∁(𝙂) outcome subspaces: (𝛺, ≤, 𝖽, 𝔼, 𝖯) 𝑅𝗑𝗒 (𝑛) 𝑅𝗑𝗑 (𝑛) 𝗆𝑛 (𝑥, 𝑦) 𝗆(𝑥, 𝑦) ̇ ∁(𝙂) ∁𝖺̇ (𝙂) ∁𝗀̇ (𝙂) ∁𝗁̇ (𝙂) version 0.50

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description open interval from 𝑎 to 𝑏 absolute value of 𝑥 ceiling of 𝑥 floor of 𝑥

reference Definition 1.23 (page 9) Definition 1.24 (page 9) Definition 1.28 (page 10) Definition 1.28 (page 10)

sum over 𝔻 of ⦅𝑥𝑛 ⦆ weighted 𝜙-mean power mean with parameter 𝑝

Definition D.12 (page 164) Definition D.14 (page 165) Definition D.15 (page 166)

a topology of sets a ring of sets an algebra of sets power set on a set 𝑋

Definition D.16 (page 170)

Definition 1.8 (page 6)

topological space on a set 𝑋 closure of a set 𝐴 interior of a set 𝐴 limit of ⦅𝑥𝑛 ⦆ limit of ⦅𝑥𝑛 ⦆

Definition D.16 (page 170) Definition D.18 (page 171) Definition D.18 (page 171) Definition D.20 (page 172) Definition D.20 (page 172)

distance function distance space open ball centered at 𝑥 with radius 𝑟 closed ball centered at 𝑥 with radius 𝑟 diameter of a set 𝐴 power triangle function power triangle inequality

Definition B.1 (page 127) Definition B.1 (page 127) Definition B.4 (page 128) Definition B.4 (page 128) Definition B.2 (page 127) Definition C.1 (page 137) Definition C.2 (page 138)

probability space set of outcomes, σ-algebra on 𝛺 probability function on 𝔼 random variable traditional expected value of 𝖷 traditional variance of 𝖷

Definition 1.38 (page 22) Definition 1.38 (page 22) Definition 1.38 (page 22) Definition 1.38 (page 22) Definition 1.39 (page 22) Definition 1.40 (page 22) Definition 1.40 (page 22)

graph weighted graph center of a graph 𝙂

Definition 1.17 (page 8) Definition 1.18 (page 8) Definition 1.19 (page 8)

extended probability space cross-correlation of ⦅𝑥𝑛 ⦆ and ⦅𝑦𝑛 ⦆ auto-correlation of ⦅𝑥𝑛 ⦆ 𝑛th-moment from 𝑥 to 𝑦 moment from 𝑥 to 𝑦 outcome center of 𝙂 arithmetic center of 𝙂 geometric center of 𝙂 harmonic center of 𝙂

Definition 2.1 (page 27) Definition 3.2 (page 78) Definition 3.2 (page 78) Definition 2.2 (page 27) Definition 2.2 (page 27) Definition 2.3 (page 28) Definition 2.4 (page 28) Definition 2.4 (page 28) Definition 2.4 (page 28)

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symbol description ̇∁ (𝙂) minimal center of 𝙂 𝗆 ̇∁ (𝙂) maxmin center of 𝙂 𝖬 ̈ 𝖤(𝖷) outcome expected value ̈ 𝖵𝖺𝗋(𝖷; 𝖤𝑥 ) outcome variance ̈ 𝖵𝖺𝗋(𝖷) outcome variance (𝑋, ≤, 𝖽) ordered distance space (ℤ, ≤, |⋅|) integer line (ℝ, |⋅|, ≤) real line ({⚀, ⚁, ⚂, ⚃, ⚄, ⚅}, 𝖽,̇ ≤,̇ 𝖯̇ ) ({➀, ➁, ➂, ➃, ➄, ➅}, 𝖽,̇ ∅, 𝖯̇ ) 𝘛 ◻, 𝘊 ◻}, 𝘎 𝘈 ◻, 𝖽,̇ ∅, 𝖯̇ ) ({◻, 𝘕 𝘛 ◻, 𝘊 ◻, 𝘎 ◻}, 𝘈 ◻, 𝖽,̇ ∅, 𝖯̇ ) ({◻, sequences: sequence over domain 𝔻 ⦅𝑥𝑛 ⦆𝑛∈𝔻 sequence ⦅𝑥𝑛 ⦆ ⦅𝑧𝑛 ⦆𝔻 ≜ ⦅𝑥𝑛 ⦆𝔻1 ⋆ ⦅𝑦𝑛 ⦆𝔻2 convolution of ⦅𝑥𝑛 ⦆ and ⦅𝑦𝑛 ⦆ 𝖣𝖥𝖳⦅𝑥𝑛 ⦆ Discrete Fourier Transform on ⦅𝑥𝑛 ⦆ 𝛼 + ⦅𝑥𝑛 ⦆ constant-sequence addition constant-sequence addition ⦅𝑥𝑛 ⦆ + 𝛼 𝛼 ⦅𝑥𝑛 ⦆ constant-sequence multiplication constant-sequence multiplication ⦅𝑥𝑛 ⦆ 𝛼 linear spaces: linear space over 𝑋 (𝑋, +, ⋅, (𝔽 , ∔, ⨰)) 𝑋, +, ⋅, (𝔽 , ∔, ⨰), 𝖽 metric linear space ( ) 𝘕 𝘕 ℝ ordered distance linear space (ℝ , ≤, 𝖽) vector norm ‖⋅‖ ring operations: ⊕ addition operator ⊗ multiplication operator

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reference Definition 2.4 (page 28) Definition 2.4 (page 28) Definition 2.14 (page 46) Definition 2.14 (page 46) Definition 2.14 (page 46) Definition 1.34 (page 20) Definition 1.36 (page 21) Definition 1.35 (page 21) Definition 2.7 (page 28) Definition 2.10 (page 29) Definition 2.11 (page 29) Definition 2.12 (page 29) Definition 1.41 (page 23) Definition 1.41 (page 23) Definition 1.43 (page 24) Definition 1.52 (page 26) Definition 1.44 (page 25) Definition 1.44 (page 25) Definition 1.44 (page 25) Definition 1.44 (page 25) Definition D.1 (page 149) Definition D.3 (page 150) Definition D.4 (page 151) Definition 1.15 (page 7) Definition 1.16 (page 7)

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CONTENTS

Front matter Title page . . . . . . . . Typesetting information Abstract . . . . . . . . . Quotes . . . . . . . . . Acknowledgements . . Symbols . . . . . . . . . Contents . . . . . . . . Preface . . . . . . . . .

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i . i . ii . iii . v . vii . viii . xiii . xvi

1 Foundations 1.1 Introduction . . . . . . . . . . . . . . . . . . 1.2 Further mathematical support . . . . . . . . 1.3 Standard definitions . . . . . . . . . . . . . 1.3.1 Sets . . . . . . . . . . . . . . . . . . 1.3.2 Relations . . . . . . . . . . . . . . . 1.3.3 Field operator pairs . . . . . . . . . 1.3.4 Graph Theory . . . . . . . . . . . . . 1.4 Order space concepts . . . . . . . . . . . . 1.4.1 Order . . . . . . . . . . . . . . . . . 1.4.2 Lattices . . . . . . . . . . . . . . . . 1.4.3 Isomorphic spaces . . . . . . . . . . 1.4.4 Monotone functions on ordered sets 1.5 Metric space concepts . . . . . . . . . . . . 1.5.1 Motivation . . . . . . . . . . . . . . . 1.5.2 Isometric spaces . . . . . . . . . . . 1.6 Ordered metric spaces . . . . . . . . . . . . 1.6.1 Definitions . . . . . . . . . . . . . . . 1.6.2 Examples . . . . . . . . . . . . . . . 1.7 Traditional probability . . . . . . . . . . . . . 1.8 Sequences . . . . . . . . . . . . . . . . . . 1.8.1 Sequences . . . . . . . . . . . . . . 1.8.2 Filtering . . . . . . . . . . . . . . . . 1.8.3 Discrete Fourier Transform . . . . .

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1 1 4 5 5 5 7 8 8 8 10 12 15 19 19 19 20 20 21 22 23 23 25 26

2 Stochastic processing on weighted graphs 2.1 Outcome subspaces . . . . . . . . . . . . 2.1.1 Definitions . . . . . . . . . . . . . . 2.1.2 Specific outcome subspaces . . . 2.1.3 Example calculations . . . . . . . 2.2 Random variables on outcome subspaces 2.2.1 Definitions . . . . . . . . . . . . . . 2.2.2 Properties . . . . . . . . . . . . . . 2.2.3 Problem statement . . . . . . . . . 2.2.4 Examples . . . . . . . . . . . . . .

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CONTENTS

2.3 Operations on outcome subspaces 2.3.1 Summation . . . . . . . . . 2.3.2 Multiplication . . . . . . . . 2.3.3 Metric transformation . . .

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3 Symbolic sequence processing on RN 3.1 Outcome subspace sequences . . . . . . . . . . 3.1.1 Definitions . . . . . . . . . . . . . . . . . . 3.1.2 Examples of symbolic sequence statistics 3.2 Extending to distance linear spaces . . . . . . . 3.2.1 Motivation . . . . . . . . . . . . . . . . . . 3.2.2 Some random variables . . . . . . . . . . 3.2.3 Some ordered distance linear spaces . . 3.3 Symbolic sequence processing applications . . . 3.3.1 Low pass filtering/Smoothing . . . . . . . 3.3.2 High pass filtering . . . . . . . . . . . . . 3.3.3 Fourier Analysis . . . . . . . . . . . . . . 3.3.4 Wavelet Analysis . . . . . . . . . . . . . .

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A Lagrange arc distance A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.1 The spherical metric . . . . . . . . . . . . . . . . . A.1.2 Linear interpolation . . . . . . . . . . . . . . . . . . A.1.3 Polar linear interpolation . . . . . . . . . . . . . . . A.1.4 Distance in terms of polar linear interpolation arcs A.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4.1 Arc function R(p,q) properties . . . . . . . . . . . . A.4.2 Distance function d(p,q) properties . . . . . . . . . A.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . .

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B Distance spaces B.1 Introduction and summary . . . . . . . . . B.2 Fundamental structure of distance spaces B.2.1 Definitions . . . . . . . . . . . . . . B.2.2 Properties . . . . . . . . . . . . . . B.3 Open sets in distance spaces . . . . . . . B.3.1 Definitions . . . . . . . . . . . . . . B.3.2 Properties . . . . . . . . . . . . . . B.4 Sequences in distance spaces . . . . . . B.4.1 Definitions . . . . . . . . . . . . . . B.4.2 Properties . . . . . . . . . . . . . . B.5 Examples . . . . . . . . . . . . . . . . . .

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C Power distance spaces C.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . C.2 Properties . . . . . . . . . . . . . . . . . . . . . . . C.2.1 Relationships of the power triangle function C.2.2 Properties of power distance spaces . . . . C.3 Examples . . . . . . . . . . . . . . . . . . . . . . .

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137 137 138 138 139 145

D Some mathematical tools D.1 Linear spaces . . . . . . . . . . . . . . . . . . . . D.1.1 Structure . . . . . . . . . . . . . . . . . . D.1.2 Metric Linear Spaces . . . . . . . . . . . D.1.3 Normed Linear Spaces . . . . . . . . . . D.1.4 Relationship between metrics and norms D.2 Metric spaces . . . . . . . . . . . . . . . . . . . . D.2.1 Algebraic structure . . . . . . . . . . . . . D.2.2 Metric preserving functions . . . . . . . . D.2.3 Product metrics . . . . . . . . . . . . . . .

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CONTENTS D.3 Sums . . . . . . . . . . . . . D.3.1 Summation . . . . . . D.3.2 Convexity . . . . . . . D.3.3 Power means . . . . . D.3.4 Inequalities . . . . . . D.4 Topological Spaces . . . . . . D.5 Polynomial interpolation . . . D.5.1 Lagrange interpolation D.5.2 Newton interpolation .

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E C++ source code support E.1 Symbolic sequence routines . . E.2 Die routines . . . . . . . . . . . E.3 Real die routines . . . . . . . . E.4 Spinner routines . . . . . . . . E.5 DNA routines . . . . . . . . . . E.6 Legrange arc distance routines

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Back Matter References . . . Reference Index Subject Index . . End of document

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PREFACE

Preface ❝You

know that I write slowly. This is chiefly because I am never satisfied until I have said as much as possible in a few words, and writing briefly takes far more time than writing at length.❞ Karl Friedrich Gauss (1777–1855), German mathematician

9

This book is largely based on four papers written by the same author as this text: 1.

📃

2.

📃

3.

📃

4.

Greenhoe (2016b): Properties of distance spaces with power triangle inequalities. http://www.journals.pu.if.ua/index.php/cmp/article/view/483 https://peerj.com/preprints/2055 https://www.researchgate.net/publication/281831459 This paper is the basis for APPENDIX B and APPENDIX C. It has been published in the 2016 volume 8 number 1 edition of the journal Carpathian Mathematical Publications.

9

Greenhoe (2015): Order and metric geometry compatible stochastic processing. https://peerj.com/preprints/844/ https://www.researchgate.net/publication/303057738 This paper is the basis for CHAPTER 2. Greenhoe (2016c): Order and Metric Compatible Symbolic Sequence Processing. https://peerj.com/preprints/2052/ https://www.researchgate.net/publication/302947227 This paper is the basis for CHAPTER 3. The C++ source code used to generate the 128 or so TeX files for the data plots in the paper can be downloaded from here: https://www.researchgate.net/publication/302953844 Greenhoe (2016a): An extension to the spherical metric using polar linear interpolation. https://www.researchgate.net/publication/303984226 This paper is the basis for APPENDIX A. 📃

quote: 📘 Simmons (2007), page 177 image: http://en.wikipedia.org/wiki/Karl_Friedrich_Gauss

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CHAPTER 1 FOUNDATIONS

❝I

was especially delighted with the mathematics, on account of the certitude and evidence of their reasonings; but I had not as yet a precise knowledge of their true use; and thinking that they but contributed to the advancement of the mechanical arts, I was astonished that foundations, so strong and solid, should have had no loftier superstructure reared on them.1 ❞

❝Je

me plaisois surtout aux mathématiques, à cause de la certitude et de l'évidence de leurs raisons: mais je ne remarquois point encore leur vrai usage; et, pensant qu'elles ne servoient qu'aux arts mécaniques, je m'étonnois de ce que leurs fondements étant si fermes et si solides, on n'avoit rien bâti dessus de plus relevé:❞

René Descartes, philosopher and mathematician (1596–1650)

1.1 Introduction In traditional stochastic processing , a real-valued random variable 𝖷 (or we might even say “a traditional random variable 𝖷) first maps the underlying probability space (𝛺, 𝔼, 𝖯) to 2 (ℝ, ≤, 𝖽, +, ⋅, ℬ) where ℬ is the usual Borel σ-algebra on the “real line” (ℝ, ≤, 𝖽, +, ⋅) (see Fig- 10 ure 1.1 page 2), and then operations (such as the expected value operation 𝖤(𝑋)) is performed −1 −2 on 𝖷. Here, real line refers to the structure (ℝ, |⋅|, ≤), where ℝ is the set of real numbers, ≤ is the standard linear order relation on ℝ, and 𝖽(𝑥, 𝑦) ≜ |𝑥 − 𝑦| is the usual metric on ℝ. This is all well and good when the physical process being analyzed (in the case of statistical estimation or system analysis) or being processed (as in the case of signal processing, including digital signal processing) is also linearly ordered and has a metric geometry similar to the one induced by the usual metric on ℝ. Be that as it may, in several real world applications, this is simply not the case. Take these processes for example: 1

quote: translation: image:

📘 Descartes (1637a) 📘 Descartes (1637b) ⟨part I, paragraph 10⟩ http://en.wikipedia.org/wiki/Image:Descartes_Discourse_on_Method.png

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1.1. INTRODUCTION 1 6 1 6

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Figure 1.1: traditional random variable 𝖷 mapping a stochastic process to the real line The values of a fair die {⚀, ⚁, ⚂, ⚃, ⚄, ⚅} have absolutely no order structure, and have no metric except the discrete metric. On a fair die, ⚁ is not greater or less than ⚀; rather ⚁ and ⚀ are simply symbols without order. Moreover, ⚀ is not “closer” to ⚁ than it is to ⚂; rather, ⚀, ⚁, and ⚂ are simply symbols without any inherit order or metric geometry. Genomic Signal Processing (GSP) analyzes biological sequences called genomes. These sequences are constructed over a set of 4 sym𝘛 ◻, 𝘊 and ◻, 𝘎 each of which cor𝘈 ◻, bols that are commonly referred to as ◻, responds to a nucleobase (adenine, thymine, cytosine, and guanine, respectively).2 A typical genome sequence contains a large number of symbols (about 3 billion for humans, 29751 for the SARS virus).3

1 6 1 6

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A linear congruential pseudo-random number generator induced by the equation 𝑦𝑛+1 = (𝑦𝑛 + 2) mod 5 with 𝑦0 = 1. The sequential nature of the structure induces both a natural order and distance.

0 1 2 4

In all three processes, the symbols in general have an order structure and a metric geometry that is fundamentally dissimilar from that of the real line. Therefore, statistical inferences based on the real line will likely lead to results that arguably have little relationship with intuition or reality. So we can observe the following: 1. A traditional random variable 𝖷 maps to the real line. 2. The structure of the underlying stochastic process (the domain of 𝖷) may be very dissimilar to that of the real line. 3. To fix the problem, we need random variables that map to alternative structures that are more similar to the underlying stochastic processes. 4. Such a structure should have both an order relation and a distance function defined on it that are similar to the stochastic process. In particular, such a structure should be an 1

📃 Mendel (1853) ⟨Mendel (1853): gene coding uses discrete symbols⟩, 📃 Watson and Crick (1953a) page 737 ⟨Watson and Crick (1953): gene coding symbols are adenine, thymine, cytosine, and guanine⟩, 📃 Watson and Crick (1953b) page 965, 📘 Pommerville (2013) page 52 2 💻 GenBank (2014) ⟨http://www.ncbi.nlm.nih.gov/genome/guide/human/⟩⟨Homo sapiens, NC_000001– NC_000022 (22 chromosome pairs), NC_000023 (X chromosome), NC_000024 (Y chromosome), NC_012920 (mitochondria)⟩, 💻 GenBank (2014) ⟨http://www.ncbi.nlm.nih.gov/nuccore/30271926⟩⟨SARS coronavirus, NC_004718.3⟩ 📃 S. G. Gregory (2006) ⟨homo sapien chromosome 1⟩, 📃 Runtao He (2004) ⟨SARS coronavirus⟩

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CHAPTER 1. FOUNDATIONS

ordered distance space. And ideally, the stochastic process and the ordered distance space should be both isomorphic and isometric with respect to each other. This text proposes two ordered distance spaces for stochastic processing: CHAPTER 2 proposes mapping to directed weighted graphs. In such a structure, order is represented by direction of it's edges, and distance by the lengths of it's edges. Furthermore, probability can be represented by weights assigned to it's vertices.

1 50

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CHAPTER 3 proposes mapping to an ordered distance linear space 𝑛 (ℝ , ≤, 𝖽, +, ⋅, ℝ, ∔, ⨰), where (ℝ, ∔, ⨰, 0, 1) is a field, + is the vector addition operator on ℝ𝑛 × ℝ𝑛 , and ⋅ is the scalar-vector multiplication operator on ℝ × ℝ𝑛 . Probability has no natural representation here, and must be assigned through a separate probability function 𝖯.

⚂ ⚄ ⚀ 𝑥

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b

⚃

Mapping to a weighted graph as in CHAPTER 2 is useful for analysis of a single random variable 𝖷. But the traditional expectation value 𝖤(𝖷) of 𝖷 is often a poor choice of a statistic. For example, the traditional expected value of a fair die is 𝖤(𝑋) = 16 (1 + 2 + ⋯ + 6) = 3.5. But this result has no relationship with reality or with intuition because the result implies that we expect the value of ⚂ or ⚃ more than we expect the outcome of say ⚀ or ⚁. The fact is, that for a fair die, we would expect any pair of values equally. The reason for this is that the values of the face of a fair die are merely symbols with no order, and with no metric geometry other than the discrete metric geometry. Weighted graphs on the other hand offer structures more similar to the underlying stochastic process. And the expectation 𝖤𝖷 of 𝖷 can be defined simply as the center of the weighted graph (as illustrated above for a weighted die with 𝖤𝖷 shaded in blue). However, the mapping has limitations with regards to a sequence of random variables in performing sequence analysis (using for example Fourier analysis or wavelet analysis), in performing sequence processing (using for example FIR filtering or IIR filtering ), in making diagnostic measurements (using a post-transform metric space), or in making “optimal” decisions (based on “distance” measurements in a metric space or more generally a distance space). ⚅ ⚅ ⚅ ⚅ ⚅⚅ ⚄ ⚄ ⚄ ⚄ ⚄⚄ ⚄⚄⚄⚄⚄ ⚄ ⚄⚄ ⚃ ⚃ ⚃⚃ ⚃ ⚃ ⚃⚃ ⚃ ⚂ ⚂ ⚂ ⚂ ⚂ ⚂ ⚂ ⚂ ⋯ ⚁ ⚁ ⚁ ⚁ ⚁ ⚁ ⚁ ⚁ ⚀⚀ ⚀ ⚀ ⚀ ⚀⚀ ⚀ ⚀ ⚀ ⚀⚀ 0 10 20 30 40 5 15 25 35 45 50

Mapping to an ordered distance linear space 𝙔 as in CHAPTER 3 provides the linear space component of 𝙔 , which in turn provides a much more convenient framework for sequence analysis and processing (as compared to the weighted graph). The ordered set and distance space components of 𝙔 allow one to preserve the order structure and distance geometry inherent in the underlying stochastic process, which can provide a less distorted (as compared to the real line) framework for analysis, diagnostics, and optimal decision making. For any given stochastic process, there are an infinite number of possible random variable mappings, with some being “better”—with respect to some criteria (probably involving isomorphic and isometric properties)—than others. Multiple mappings using random variables 𝖶, 𝖷, 𝖸, and 𝖹 for the “real die” stochastic process 𝙂 are illustrated in Figure 1.2 (page 4).

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1.2. FURTHER MATHEMATICAL SUPPORT 1 6

1 6

1 6

1 6

1 6

1 6

1

2

3

4

5

6

(ℤ, |⋅|, ≤)

𝖷(⋅) 1 6 1 6

1 6

6 3

5

1 6

4

2 1

⚅ ⚄

⚂

𝖸(⋅)

𝖧 1 6

1 6

1 6 1 6

1 6

1 6

⚃

⚁ ⚀

1 6

3

1 6

0 0/6 1 6

1 6

𝖪

5

𝖹(⋅)

𝙂

1 6

6

4

2 1

1 6

1 6

1 6

𝖶(⋅) 1 6

1 6

1 6

1 6

1 6

1 6

1

2

3

4

5

6

(ℝ, |⋅|, ≤)

Figure 1.2: several random variable mappings for the real die

1.2 Further mathematical support

Further mathematical support for these two methods is provided in appendices. Here is partial summary of what may be found there: APPENDIX A introduces what is herein called the Lagrange arc distance function. It is important in this text because it is used in CHAPTER 3 in the processing of real die sequences in ℝ3 and spinner sequences in ℝ2 . The function is an extension to all of ℝ𝘕 of the spherical metric, which has as domain only a “spherical” surface in ℝ𝘕 . The Lagrange arc distance 𝖽(𝑝, 𝑞) draws arcs between certain pairs of points (𝑝, 𝑞) using Lagrange interpolation. APPENDIX B presents distance spaces. A distance space is a metric space in which the triangle inequality does not necessarily hold. Distance spaces are important in this text because the Lagrange arc distance is a distance function, and not a metric in that 𝖽(𝑝, 𝑟) ≤ 𝖽(𝑝, 𝑞)+𝖽(𝑞, 𝑟) does not necessarily hold for all triples (𝑝, 𝑞, 𝑟) in ℝ𝘕 .

2 1 b

b

b

−2

−1 −1 −2

b

b

1

2

b

b

1 𝑞 b

b

−1

𝑟

b

𝑝

1

APPENDIX C introduces what is herein called the power distance space. This space is a generalization of the metric space, and is a distance space that satisfies what is herein called the power triangle inequality: 1 𝖽(𝑥, 𝑦) ≤ 2𝜎 [ 21 𝖽𝑝 (𝑥, 𝑧) + 12 𝖽𝑝 (𝑧, 𝑦)] 𝑝 ∀(𝑝,𝜎)∈ℝ∗ ×ℝ⊢ , 𝑥,𝑦,𝑧∈𝑋 1 It turns out that the inequality 2𝜎 ≤ 2 𝑝 has special significance with regards to these spaces and appears repeatedly in APPENDIX C. It is plotted in Figure 1.3 (page 5). Power distance spaces are not explicitly used in this text. However, they may prove useful to future research in symbolic sequence processing space in which the triangle inequality fails to hold, but in which some of the key properties of a metric space are still required. version 0.50


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CHAPTER 1. FOUNDATIONS 𝜎 geometric inequality at (𝑝, 𝜎) = (0, 12 )

4

2nd order inflection point at (− 12 ln 2, 12 𝑒−2 )

3

triangle inequality at (𝑝, 𝜎) = (1, 1)

2

quadratic mean inequality at 2, √2 ( 2 )

harmonic inequality at (𝑝, 𝜎) = (−1, 14 )

square mean root (SMR) inequality at ( 12 , 2)

1 to minimal inequality at (𝑝, 𝜎) = (∞, 12 )

to inframetric inequality at (𝑝, 𝜎) = (∞, 12 ) 𝑝

1 2

−3

−2

−1

0

1

1

1

2

3

ln 2 Figure 1.3: 𝜎 = 12 (2 𝑝 ) = 2 𝑝 −1 or 𝑝 = ln(2𝜎)

1.3 Standard definitions 1.3.1 Sets Definition 1.1.

3

Let 𝑋 be a SET.4 The empty set ∅ is defined as

∅ ≜ {𝑥 ∈ 𝑋|𝑥 ≠ 𝑥}.

Definition 1.2. 5 Let ℝ be the set of real numbers. Let ℝ⊢ ≜ {𝑥 ∈ ℝ|𝑥 ≥ 0} be the set of non-negative real numbers. Let ℝ+ ≜ {𝑥 ∈ ℝ|𝑥 > 0} be the set of postive real numbers. Let ℝ∗ ≜ ℝ ∪ {−∞, ∞} be the set of extended real numbers. 6 Let ℤ be the set of integers. Let 𝕎 ≜ {𝑛 ∈ ℤ|𝑛 ≥ 0} be the set of whole numbers. Let ℕ ≜ {𝑛 ∈ ℤ|𝑛 ≥ 1} be the set of natural numbers. Let ℤ∗ ≜ ℤ ∪ {−∞, ∞} be the EXTENDED SET OF INTEGERS.

1.3.2 Relations One of the most fundamental structures in mathematics is the ordered pair, and one of the most common definitions of ordered pair is due to Kuratowski (1921) and is presented next:7 Definition 1.3.

8

The ordered pair (𝑎, 𝑏) is defined as

(𝑎, 𝑏) ≜ {{𝑎}, {𝑎, 𝑏}}.

Proposition 1.1 (next) and Corollary 1.1 demonstrate that the the definition of ordered pair given by Definition 1.3 allows 𝑎 and 𝑏 to be unambiguously extracted from (𝑎, 𝑏) and that (𝑎, 𝑏) is well defined. Proposition 1.1. {𝑎} = ⋂ (𝑎, 𝑏) = ⋂ {{𝑎}, {𝑎, 𝑏}} = {𝑎} ∩ {𝑎, 𝑏} {𝑏} = △ (𝑎, 𝑏) = △{{𝑎}, {𝑎, 𝑏}} = {𝑎}△{𝑎, 𝑏} 3

📘 Halmos (1960) page 8, 📘 Kelley (1955) page 3, 📘 Kuratowski (1961), page 26 The mathematical structure called set is left undefined in this paper. For more information on sets, see for example 📃 Zermelo (1908a) pages 263–267 (7 axioms), 📘 Zermelo (1908b) ⟨(English translation of previous reference)⟩, 📃 Fraenkel (1922), 📘 Halmos (1960) pages 1–6 ⟨Naive set theory⟩, 📘 Wolf (1998), page 139 5 Notation ℝ, 𝕎, ℕ, etc.: Bourbaki notation. References: 📘 Davis (2005) page 9, 📘 Cohn (2012) page 3 6 📘 Rana (2002) pages 385–388 ⟨Appendix A⟩ 7 As an alternative to the Kuratowski definition, the ordered pair can also be taken as an axiom. References: 📘 Bourbaki (1968), page 72, 📘 Munkres (2000), page 13 8 📘 Suppes (1972) page 32, 📘 Halmos (1960) page 23, 📘 Kuratowski (1961), page 39, 📃 Kuratowski (1921) ⟨Def. V, page 171⟩, 📃 Wiener (1914) 4

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1.3. STANDARD DEFINITIONS

Corollary 1.1. ✎PROOF:

9

(𝑎, 𝑏) = (𝑐, 𝑑)

⟺

{𝑎 = 𝑐 and 𝑏 = 𝑑 }

{𝑎} = ⋂ (𝑎, 𝑏) = ⋂(𝑐, 𝑑) = {𝑐} by Proposition 1.1 and left hypothesis {𝑏} = △ (𝑎, 𝑏) = △(𝑐, 𝑑) = {𝑑} by Proposition 1.1 and left hypothesis (𝑎, 𝑏) = (𝑐, 𝑑) by right hypothesis

✏

Definition 1.4. 10 Let 𝑋 and 𝑌 be sets. The Cartesian product 𝑋 × 𝑌 is defined as 𝑋 × 𝑌 ≜ {(𝑥, 𝑦) |(𝑥 ∈ 𝑋) and (𝑦 ∈ 𝑌 )} Definition 1.5. 11 Let 𝑋 and 𝑌 be SETS. A relation Ⓡ on 𝑋 and 𝑌 is any subset of 𝑋 × 𝑌 such that Ⓡ ⊆ 𝑋 × 𝑌 . The set 𝟚𝑋𝑌 is the set of all relations in 𝑋 × 𝑌 . Definition 1.6. 12 Let 𝑋 and 𝑌 be SETS. A RELATION 𝖿 ∈ 𝟚𝑋𝑌 is a function if ⟹ {(𝑥, 𝑦1 ) ∈ 𝖿 and (𝑥, 𝑦2 ) ∈ 𝖿 } {𝑦1 = 𝑦2 }. The set 𝑌 𝑋 is the set of all functions in 𝟚𝑋𝑌 . A function does not always have an inverse that is also a function. But unlike functions, every relation has an inverse that is also a relation. Note that since all functions are relations, every function does have an inverse that is at least a relation, and in some cases this inverse is also a function. Definition 1.7. 13 Let Ⓡ be a relation in 𝟚𝑋𝑌 . Ⓡ−1 is the inverse of relation Ⓡ if Ⓡ−1 ≜ {(𝑦, 𝑥) ∈ 𝑌 × 𝑋| (𝑥, 𝑦) ∈ Ⓡ} The inverse relation Ⓡ−1 is also called the converse of Ⓡ. Definition 1.8. Let 𝑋 be a set. The quantity 𝟚𝑋 is the POWER SET OF 𝑋 such that 𝟚𝑋 ≜ {𝐴 ⊆ 𝑋} (the set of all subsets of 𝑋). Definition 1.9. Let 𝑌 be a SET. The structure 𝑌 𝑛 for 𝑛 ∈ ℕ is a SET defined as 𝑌1 ≜ 𝑌 and 𝑌 𝑛 ≜ 𝑌 × 𝑌 𝑛−1 for 𝑛 = 2, 3, 4, … Definition 1.10. The set of complex numbers ℂ is defined as ℂ ≜ ℝ2 . Definition 1.11. Let 𝑌1 , 𝑌2 , …𝑌𝘕 be SETS. The structure (𝑥1 , 𝑥2 , ⋯ , 𝑥𝑛 ) is an n-tuple on 𝑌1 ×𝑌2 ×⋯×𝑌𝘕 if (𝑥1 , 𝑥2 , ⋯ , 𝑥𝑛 ) is an element in the set 𝑌1 × 𝑌2 × ⋯ × 𝑌𝘕 . A 3-TUPLE is also called an ordered triple or simply a triple. Definition 1.12. 14 Let Ⓡ ∈ 𝟚𝑋𝑌 be a RELATION (Definition 1.5 page 6). The domain of Ⓡ is 𝓓(Ⓡ) ≜ {𝑥 ∈ 𝑋|∃𝑦 such that (𝑥, 𝑦) ∈ Ⓡ} . The image set of Ⓡ is 𝓘(Ⓡ) ≜ {𝑦 ∈ 𝑌 |∃𝑥 such that (𝑥, 𝑦) ∈ Ⓡ} . The null space of Ⓡ is 𝓝 (Ⓡ) ≜ {𝑥 ∈ 𝑋| (𝑥, 0) ∈ Ⓡ} . The range of Ⓡ is any set 𝓡(Ⓡ) such that 𝓘(Ⓡ) ⊆ 𝓡(Ⓡ). 9

📘 Apostol (1975) page 33, 📘 Hausdorff (1937) page 15 📘 Halmos (1960) page 24, G. Frege, 2007 August 25, http://groups.google.com/group/sci.logic/msg/ 3b3294f5ac3a76f0 11 📘 Maddux (2006) page 4, 📘 Halmos (1960) pages 26–30, 📘 Suppes (1972) page 86, 📘 Kelley (1955) page 10, 📘 Bourbaki (1939), 📘 Bottazzini (1986) page 7, 📘 Comtet (1974) page 4 ⟨| 𝑌 𝑋 |⟩; The notation 𝟚𝑋𝑌 is motivated by the fact that for finite 𝑋 and 𝑌 , | 𝟚𝑋𝑌 | = 2| 𝑋 |⋅| 𝑌 | . 12 📘 Maddux (2006) page 4, 📘 Halmos (1960) pages 26–30, 📘 Suppes (1972) page 86, 📘 Kelley (1955) page 10, 📘 Bourbaki (1939), 📘 Bottazzini (1986) page 7, 📘 Comtet (1974) page 4 ⟨| 𝑌 𝑋 |⟩; The notation 𝑌 𝑋 is motivated by the fact that for finite 𝑋 and 𝑌 , | 𝑌 𝑋 | = | 𝑌 || 𝑋 | . 13 📘 Suppes (1972) page 61 ⟨Defintion 6, inverse=“converse”⟩ 📘 Kelley (1955) page 7 📘 Peirce (1883) page 188 ⟨inverse=“converse”⟩ 14 📘 Munkres (2000), page 16, 📘 Kelley (1955) page 7 10

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CHAPTER 1. FOUNDATIONS 𝑋

Definition 1.13. The SET FUNCTION15 | 𝐴 | ∈ ℤ∗𝟚 is the CARDINALITY OF 𝐴 such that the number of elements in 𝐴 for FINITE 𝐴 ∀𝐴 ∈ 𝟚𝑋 |𝐴| ≜ { ∞ } otherwise

𝑋

𝑌

𝑋 𝑦4 𝑦3 𝑦2 𝑦1

𝑥4 𝑥3 𝑥2 𝑥1

𝑥4 𝑥3 𝑥2 𝑥1

“into” (arbitrary function in 𝑌 𝑋 )

𝑌

𝑋 𝑦3 𝑦2 𝑦1

“onto” surjective in 𝑌 𝑋

𝑌

𝑋 𝑦4 𝑦3 𝑦2 𝑦1

𝑥3 𝑥2 𝑥1

𝑥4 𝑥3 𝑥2 𝑥1

“one-to-one” injective in 𝑌 𝑋

𝑌 𝑦4 𝑦3 𝑦2 𝑦1

“one-to-one and onto” bijective in 𝑌 𝑋

Figure 1.4: types of functions (Definition 1.14 page 7) Definition 1.14. 16 𝖿 is surjective 𝖿 is injective 𝖿 is bijective

Let 𝖿 be a FUNCTION in 𝑌 𝑋 (Definition 1.6 page 6). (also called onto) if 𝖿(𝑋) = 𝑌 . (also called one-to-one) if 𝖿(𝑥) = 𝖿(𝑦) ⟹ 𝑥 = 𝑦. (also called one-to-one and onto) if 𝖿 is both SURJECTIVE and INJECTIVE.

1.3.3 Field operator pairs Definition 1.15. Let 𝑋, 𝑌 , and 𝑍 be SETs. Let (ℝ, +, ⋅, 0, 1) be the standard FIELD OF REAL NUMBERS. The addition operator ⊕ ∶ 𝑋 × 𝑌 → 𝑍 is defined as shown in Table 1.1 (page 7). Moreover, for some sequence ⦅𝑥𝑛 ⦆, 𝘕

⨁

𝑥𝑛 ≜ 𝑥 1 ⊕ 𝑥2 ⊕ ⋯ ⊕ 𝑥 𝘕

1. 2. 3. 4. 5. 6.

If If If If If If

𝑋×𝑌 𝑋×𝑌 𝑋×𝑌 𝑋×𝑌 𝑋×𝑌 𝑋×𝑌

and

⨁

𝑥𝑛 ≜ 𝑥𝛼∈𝔻 ⊕ 𝑥𝛽∈𝔻 ⊕ ⋯ ⊕ 𝑥𝛾∈𝔻

𝑛∈𝔻

𝑛=1

≜ℝ×ℝ ≜ℂ×ℂ ≜ ℝ 𝑛 × ℝ𝑛 ≜ ℂ𝑛 × ℂ𝑛 ≜ℝ×ℂ ≜ℂ×ℝ

then then then then then then

𝑥⊕𝑦 (𝑎, 𝑏) ⊕ (𝑐, 𝑑) (𝑥1 , 𝑥2 , ⋯ , 𝑥𝑛 ) ⊕ (𝑦1 , 𝑦2 , ⋯ , 𝑦𝑛 ) (𝑥1 , 𝑥2 , ⋯ , 𝑥𝑛 ) ⊕ (𝑦1 , 𝑦2 , ⋯ , 𝑦𝑛 ) 𝑥 ⊕ (𝑎, 𝑏) 𝑥⊕𝑦

≜ ≜ ≜ ≜ ≜ ≜

𝑥+𝑦 (𝑎 + 𝑐, 𝑏 + 𝑑) (𝑥1 + 𝑦1 , ⋯ , 𝑥𝑛 + 𝑦𝑛 ) (𝑥1 ⊕ 𝑦1 , ⋯ , 𝑥𝑛 ⊕ 𝑦𝑛 ) (𝑥 + 𝑎, 𝑥 + 𝑏) 𝑦⊕𝑥

∈ℝ ∈ℂ ∈ ℝ𝑛 ∈ ℂ𝑛 ∈ℂ ∈ℂ

. . . . . .

Table 1.1: Definition of the addition operator ⊕ ∶ 𝑋 × 𝑌 → 𝑍 (see Definition 1.15 page 7) Definition 1.16. Let 𝑋, 𝑌 , and 𝑍 be SETs. Let (ℝ, +, ⋅, 0, 1) be the standard FIELD OF REAL NUMBERS. Let the JUXTAPOSITION operator 𝑥𝑦 on 𝑥 and 𝑦 be equivalent to the real field operator 𝑥 ⋅ 𝑦 on 𝑥 and 𝑦 such that 𝑥𝑦 ≜ 𝑥 ⋅ 𝑦. The multiplication operator ⊗ ∶ 𝑋 × 𝑌 → 𝑍 is defined as shown in Table 1.2 (page 8). 15

set function: 📘 Pap (1995) page 8 ⟨Definition 2.3: extended real-valued set function⟩, 📘 Halmos (1950) page 30 ⟨§7. MEASURE ON RINGS⟩ 16 📘 Michel and Herget (1993) pages 14–15, 📘 Fuhrmann (2012) page 2, 📘 Comtet (1974) page 5, 📘 Durbin (2000) pages 16–17 2016 Jul 4

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1.4. ORDER SPACE CONCEPTS

If If If If If If If If If If

𝑋×𝑌 𝑋×𝑌 𝑋×𝑌 𝑋×𝑌 𝑋×𝑌 𝑋×𝑌 𝑋×𝑌 𝑋×𝑌 𝑋×𝑌 𝑋×𝑌

≜ℝ×ℝ ≜ℝ×ℂ ≜ℂ×ℂ ≜ ℝ × ℝ𝑛 ≜ ℂ × ℝ𝑛 ≜ ℂ × ℂ𝑛 ≜ℂ×ℝ ≜ ℝ𝑛 × ℝ ≜ ℝ𝑛 × ℂ ≜ ℂ𝑛 × ℂ

then then then then then then then then then then

𝑥⊗𝑦 𝑥 ⊗ (𝑎, 𝑏) (𝑎, 𝑏) ⊗ (𝑐, 𝑑) 𝑥 ⊗ ( 𝑦1 , 𝑦 2 , ⋯ , 𝑦 𝑛 ) 𝑥 ⊗ ( 𝑦1 , 𝑦 2 , ⋯ , 𝑦 𝑛 ) 𝑥 ⊗ ( 𝑦1 , 𝑦 2 , ⋯ , 𝑦 𝑛 ) 𝑥⊗𝑦 𝑥⊗𝑦 𝑥⊗𝑦 𝑥⊗𝑦

≜ ≜ ≜ ≜ ≜ ≜ ≜ ≜ ≜ ≜

𝑥𝑦 (𝑥𝑎, 𝑥𝑏) (𝑎𝑐 − 𝑏𝑑, 𝑎𝑑 + 𝑏𝑐) (𝑥𝑦1 , 𝑥𝑦2 , ⋯ , 𝑥𝑦𝑛 ) (𝑦1 ⊗ 𝑥, 𝑦2 ⊗ 𝑥, ⋯ , 𝑦𝑛 ⊗ 𝑥) ( 𝑥 ⊗ 𝑦1 , 𝑥 ⊗ 𝑦 2 , ⋯ , 𝑥 ⊗ 𝑦 𝑛 ) 𝑦⊗𝑥 𝑦⊗𝑥 𝑦⊗𝑥 𝑦⊗𝑥

∈ℝ ∈ℂ ∈ℂ ∈ ℝ𝑛 ∈ ℂ𝑛 ∈ ℂ𝑛 ∈ℂ ∈ ℝ𝑛 ∈ ℂ𝑛 ∈ ℂ𝑛

. . . . . . . . . .

Table 1.2: Definition of the multiplication operation ⊗ ∶ 𝑋 × 𝑌 → 𝑍 (see Definition 1.16 page 7)

1.3.4 Graph Theory Definition 1.17. 17 Let 𝟚𝑋𝑋 be the set of all RELATIONs (Definition 1.5 page 6) on a set 𝑋. The pair (𝑋, 𝐸) is a graph if 𝐸 ⊆ 𝟚𝑋𝑋 . A graph (𝑋, 𝐸) is undirected if (𝑥, 𝑦) ∈ 𝐸 ⟹ (𝑦, 𝑥) ∈ 𝐸. A graph (𝑋, 𝐸) is directed if it is NOT undirected. A GRAPH that is DIRECTED is a directed graph. A GRAPH that is UNDIRECTED is an undirected graph. The elements of 𝑋 are the vertices and the ordered pairs of 𝐸 are the arcs of a GRAPH (𝑋, 𝐸). The element 𝑥 is the tail and 𝑦 is the head of an arc (𝑥, 𝑦). Definition 1.18. The tuple (𝑋, 𝐸, 𝖽, 𝗐) is a weighted graph if (𝑋, 𝐸) is a GRAPH (Definition 1.17 page 8), 𝖽(𝑥, 𝑦) is a FUNCTION in (ℝ⊢ )𝑋×𝑋 (Definition 1.6 page 6), and 𝗐(𝑥) is a function in (ℝ⊢ )𝑋 . The function 𝖽 is called the edge weight, and the function 𝗐 is called the vertice weight. Definition 1.19. Let 𝙂 ≜ (𝑋, 𝐸, 𝖽, 𝗐) be a WEIGHTED GRAPH (Definition 1.18 page 8). The center ∁(𝙂) of 𝙂 is ∁(𝙂) ≜ arg min max 𝖽(𝑥, 𝑦) 𝗐(𝑦). 𝑥∈𝑋

𝑦∈𝑋

1.4 Order space concepts 1.4.1 Order Definition 1.20. 18 Let 𝑋 be a set. A relation ≤ is an order relation in 𝟚𝑋𝑋 (Definition 1.5 page 6) if 1. 𝑥 ≤ 𝑥 ∀𝑥∈𝑋 ( REFLEXIVE) and PREORDER 2. 𝑥 ≤ 𝑦 and 𝑦 ≤ 𝑧 ⟹ 𝑥 ≤ 𝑧 ∀𝑥,𝑦,𝑧∈𝑋 ( TRANSITIVE) and 3. 𝑥 ≤ 𝑦 and 𝑦 ≤ 𝑥 ⟹ 𝑥 = 𝑦 ∀𝑥,𝑦∈𝑋 ( ANTI-SYMMETRIC). An ordered set is the pair (𝑋, ≤). If ≤= ∅ (Definition 1.1 page 5), then (𝑋, ≤) is an unordered set. If 𝑥 ≤ 𝑦 or 𝑦 ≤ 𝑥, then elements 𝑥 and 𝑦 are said to be comparable; otherwise they are incomparable. Definition 1.21. 19 A relation ≤ is a linear order relation on 𝑋 if 1. ≤ is an ORDER RELATION (Definition 1.20 page 8) and 2. 𝑥 ≤ 𝑦 or 𝑦 ≤ 𝑥 ∀𝑥,𝑦∈𝑋 ( COMPARABLE). A linearly ordered set is the pair (𝑋, ≤). A linearly ordered set is also called a totally ordered set, a fully ordered set, and a chain. 17

📘 orgen Bang-Jensen and Gutin (2007), page 2 ⟨§1.2⟩, 📘 Haray (1969), page 9 📘 MacLane and Birkhoff (1999) page 470, 📘 Beran (1985) page 1, 📃 Korselt (1894) page 156 ⟨I, II, (1)⟩, 📃 Dedekind (1900) page 373 ⟨I–III⟩. An order relation is also called a partial order relation. An ordered set is also called a partially ordered set or poset. 19 📘 MacLane and Birkhoff (1999) page 470, 📃 Ore (1935) page 410 18

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The familiar relations ≥, (next) can be defined in terms of the order relation ≤ (Definition 1.20—previous). Definition 1.22. 𝑥≥𝑦

20

def

𝑦≤𝑥

∀𝑥,𝑦∈𝑋

𝑥 ≤ 𝑦 and 𝑥 ≠ 𝑦

∀𝑥,𝑦∈𝑋

𝑥 ⪈ 𝑦 ⟺ 𝑥 ≥ 𝑦 and 𝑥 ≠ 𝑦 The relation ≥ is called the dual of ≤.

∀𝑥,𝑦∈𝑋

𝑥⪇𝑦

⟺

Let (𝑋, ≤) be an ordered set. The relations ≥, ∈ 𝟚𝑋𝑋 are defined as follows:

def

⟺ def

Example 1.1 (Coordinatewise order relation). 21 Let (𝑋, ≤) be an ordered set. Let 𝒙 ≜ (𝑥1 , 𝑥2 , … , 𝑥𝑛 ) and 𝒚 ≜ (𝑦1 , 𝑦2 , … , 𝑦𝑛 ). The coordinatewise order relation ⋜ on the Cartesian product 𝑋 𝑛 is defined for all 𝒙, 𝒚 ∈ 𝑋 𝑛 as 𝒙⋜𝒚

def

{𝑥1 ≤ 𝑦1 and 𝑥2 ≤ 𝑦2 and … and 𝑥𝑛 ≤ 𝑦𝑛 } Example 1.2 (Lexicographical order relation). 22 Let (𝑋, ≤) be an ordered set. Let 𝒙 ≜ (𝑥1 , 𝑥2 , … , 𝑥𝑛 ) and 𝒚 ≜ (𝑦1 , 𝑦2 , … , 𝑦𝑛 ). The lexicographical order relation ⋜ on the Cartesian product 𝑋 𝑛 is defined forall 𝒙, 𝒚 ∈ 𝑋 𝑛 as ( 𝑥1 < 𝑦1 ) or ⎫ ⎧ ⎪ ⎪ 𝑥 < 𝑦 and 𝑥 = 𝑦 ) or ⎪ 2 1 1 ⎪ ( 2 def ⎪ 𝑥 < 𝑦3 and (𝑥1 , 𝑥2 ) = (𝑦1 , 𝑦2 ) ) or ⎪ 𝒙⋜𝒚 ⟺ ⎨ ( 3 ⎬ … … ) or ⎪ ⎪ ( … ⎪ ( 𝑥𝑛−1 < 𝑦𝑛−1 and (𝑥1 , 𝑥2 , … , 𝑥𝑛−2 ) = (𝑦1 , 𝑦2 , … , 𝑦𝑛−2 ) ) or ⎪ ⎪ ⎪ and (𝑥1 , 𝑥2 , … , 𝑥𝑛−1 ) = (𝑦1 , 𝑦2 , … , 𝑦𝑛−1 ) ) ⎩ ( 𝑥𝑛 ≤ 𝑦𝑛 ⎭ The lexicographical order relation is also called the dictionary order relation or alphabetic order relation. Definition 1.23. 23 the SET [𝑥 ∶ 𝑦] the SET (𝑥 ∶ 𝑦] the SET [𝑥 ∶ 𝑦) the SET (𝑥 ∶ 𝑦)

⟺

In an ORDERED SET ( 𝑋, ≜ {𝑧 ∈ 𝑋|𝑥 ≤ 𝑧 ≤ 𝑦} ≜ {𝑧 ∈ 𝑋|𝑥 < 𝑧 ≤ 𝑦} ≜ {𝑧 ∈ 𝑋|𝑥 ≤ 𝑧 < 𝑦} ≜ {𝑧 ∈ 𝑋|𝑥 < 𝑧 < 𝑦}

≤), is a closed interval on ( 𝑋, ≤) is a half-open interval on ( 𝑋, ≤) is a half-open interval on ( 𝑋, ≤) is an open interval. on ( 𝑋, ≤)

and and and

Definition 1.24. Let ( ℝ, ≤) be the ORDERED SET OF REAL NUMBERS (Definition 1.20 page 8). −𝑥 for 𝑥 ≤ 0 The absolute value |⋅| ∈ ℝℝ is defined as24 . |𝑥| ≜ { 𝑥 otherwise } Definition 1.25. 25 Let ( 𝑋, ≤) be an ORDERED SET. A subset 𝐷 ⊆ 𝑋 is convex in 𝑋 if 𝑥, 𝑦 ∈ 𝐷 ⟹ (𝑥 ∶ 𝑦) ⊆ 𝐷. 20

Peirce (1880) page 2 📘 Shen and Vereshchagin (2002) page 43 22 📘 Shen and Vereshchagin (2002) page 44, 📘 Halmos (1960) page 58, 📘 Hausdorff (1937) page 54 23 📘 Apostol (1975) page 4, 📃 Ore (1935) page 409, 📃 Duthie (1942) page 2, 📃 Ore (1935) page 425 ⟨quotient structures⟩ 24 A more general definition for absolute value is available for any commutative ring : Let 𝑅 be a commutative ring . A function |⋅| in 𝑅𝑅 is an absolute value, or modulus, on 𝑅 if 1. 𝑥∈ℝ (non-negative) and |𝑥| ≥ 0 2. |𝑥| = 0 ⟺ 𝑥 = 0 𝑥∈ℝ (nondegenerate) and 3. 𝑥,𝑦∈ℝ (homogeneous / submultiplicative) and |𝑥𝑦| = |𝑥| ⋅ |𝑦| 4. |𝑥 + 𝑦| ≤ |𝑥| + |𝑦| 𝑥,𝑦∈ℝ (subadditive / triangle inequality) Reference: 📘 Cohn (2002) page 312 25 📘 Barvinok (2002) page 5 📃

21

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Example 1.3. Convex subsets of ℤ under the usual integer ordering relation include ∅, ℤ, 𝕎, ℕ, {0, 1, 2, 3, 4, 5}, and {−2, −1, 0, 1, 2, 3}. Definition 1.26. Let (𝑋, ≤) be an ordered set. For any set 𝐴 ∈ 𝟚𝑋 , 𝑐 is an upper bound of 𝐴 in (𝑋, ≤) if 𝑥 ≤ 𝑐 ∀𝑥 ∈ 𝐴. An element 𝑏 is the least upper bound, or lub, of 𝐴 in (𝑋, ≤) if 𝑏 and 𝑐 are UPPER BOUNDs of 𝐴 ⟹ 𝑏 ≤ 𝑐. The least upper bound of the set 𝐴 is denoted ⋁ 𝐴. The join 𝑥 ∨ 𝑦 of 𝑥 and 𝑦 is defined as 𝑥 ∨ 𝑦 ≜ ⋁ {𝑥, 𝑦}. Definition 1.27. Let (𝑋, ≤) be an ordered set. For any set 𝐴 ∈ 𝟚𝑋 , 𝑝 is a lower bound of 𝐴 in (𝑋, ≤) if 𝑝 ≤ 𝑥 ∀𝑥 ∈ 𝐴. An element 𝑎 is the greatest lower bound, or glb, of 𝐴 in (𝑋, ≤) if 𝑎 and 𝑝 are LOWER BOUNDs of 𝐴 ⟹ 𝑝 ≤ 𝑎. The greatest lower bound of the set 𝐴 is denoted ⋀ 𝐴. The meet 𝑥 ∧ 𝑦 of 𝑥 and 𝑦 is defined as 𝑥 ∧ 𝑦 ≜ ⋀ {𝑥, 𝑦}. Lemma 1.1. Let ( 𝑋, ≤) be an ORDERED SET (Definition 1.20 page 8). Let ⋁ 𝐴 be the LEAST UPPER BOUND ∈ 𝟚𝑋 (Definition 1.8 page 6). Let ⋀ 𝐴 be the GREATEST LOWER BOUND (Definition 1.27 𝑋 page 10) of a set 𝐴 ∈ 𝟚 . 1. ⋁ 𝐴 = {𝑎 ∈ 𝑋|𝑥 ≤ 𝑎, ∀𝑥, 𝑎 ∈ 𝑋} and {𝐴 = 𝑋} ⟹ { 2. ⋀ 𝐴 = {𝑎 ∈ 𝑋|𝑎 ≤ 𝑥, ∀𝑥, 𝑎 ∈ 𝑋} . }

(Definition 1.26 page 10) of a set 𝐴

Definition 1.28. Let (ℝ, ≤) be the STANDARD ORDERED SET OF REAL NUMBERS. The floor function ⌊𝑥⌋ ∈ ℤℝ and the ceiling function ⌈𝑥⌉ ∈ ℤℝ are defined as ⌊𝑥⌋ ≜ {𝑛 ∈ ℤ|𝑛 ≤ 𝑥} and ⌈𝑥⌉ ≜ {𝑛 ∈ ℤ|𝑛 ≥ 𝑥}. ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⋁ ⋀ FLOOR FUNCTION

CEILING FUNCTION

1.4.2 Lattices The structure available in an ordered set (Definition 1.20 page 8) tends to be insufficient to ensure “wellbehaved” mathematical systems. This situation is greatly remedied if every pair of elements in the ordered set has both a least upper bound and a greatest lower bound (Definition 1.27 page 10) in the set; in this case, that ordered set is a lattice (next definition).26 Definition 1.29. 27 An algebraic structure 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) is a lattice if 1. (𝑋, ≤) is an ORDERED SET (Definition 1.20 page 8) and 2. 𝑥, 𝑦 ∈ 𝑋 ⟹ 𝑥 ∨ 𝑦 ∈ 𝑋 (Definition 1.26 page 10) and 3. 𝑥, 𝑦 ∈ 𝑋 ⟹ 𝑥 ∧ 𝑦 ∈ 𝑋 (Definition 1.27 page 10). The LATTICE 𝙇 is linear if (𝑋, ≤) is a LINEARLY ORDERED SET (Definition 1.21 page 8). Theorem 1.1. 28 ( 𝑋, 𝑥∨𝑥 = ⎧ ⎪ 𝑥∨𝑦 = ⎨ (𝑥 ∨ 𝑦) ∨ 𝑧 = ⎪ ⎩ 𝑥 ∨ (𝑥 ∧ 𝑦) =

∨, ∧ ; ≤) is a LATTICE (Definition 1.29 page 10) 𝑥 𝑥∧𝑥 = 𝑥 𝑦∨𝑥 𝑥∧𝑦 = 𝑦∧𝑥 𝑥 ∨ (𝑦 ∨ 𝑧) (𝑥 ∧ 𝑦) ∧ 𝑧 = 𝑥 ∧ (𝑦 ∧ 𝑧) 𝑥 𝑥 ∧ (𝑥 ∨ 𝑦) = 𝑥

⟺ ∀𝑥∈𝑋

( IDEMPOTENT)

and

∀𝑥,𝑦∈𝑋

( COMMUTATIVE)

and

∀𝑥,𝑦,𝑧∈𝑋

( ASSOCIATIVE)

and

∀𝑥,𝑦∈𝑋

( ABSORPTIVE).

⎫ ⎪ ⎬ ⎪ ⎭

26

Gian-Carlo Rota (1932–1999) has illustrated the advantage of lattices over simple ordered sets by pointing out that the ordered set of partitions of an integer “is fraught with pathological properties”, while the lattice of partitions of a set “remains to this day rich in pleasant surprises”. 📃 Rota (1997) page 1440 ⟨(illustration)⟩, 📃 Rota (1964) page 498 ⟨partitions of a set⟩ 27 📘 MacLane and Birkhoff (1999) page 473, 📘 Birkhoff (1948) page 16, 📃 Ore (1935), 📃 Birkhoff (1933) page 442, 📘 Maeda and Maeda (1970), page 1 28 📘 MacLane and Birkhoff (1999) pages 473–475 ⟨LEMMA 1, THEOREM 4⟩, 📘 Burris and Sankappanavar (1981) pages 4–7, 📘 Birkhoff (1938), pages 795–796, 📃 Ore (1935) page 409 ⟨(𝛼)⟩, 📃 Birkhoff (1933) page 442, 📃 Dedekind (1900) pages 371–372 ⟨(1)–(4)⟩

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Minimax inequality. Suppose we arrange a finite sequence of values into 𝑚 groups of 𝑛 elements per group. This could be represented as an 𝑚 × 𝑛 matrix. Suppose now we find the minimum value in each row, and the maximum value in each column. We can call the maximum of all the minimum row values the maximin, and the minimum of all the maximum column values the minimax. Now, which is greater, the maximin or the minimax? The minimax inequality demonstrates that the maximin is always less than or equal to the minimax. The minimax inequality is illustrated below and stated formerly in Theorem 1.2 (page 11). 𝑛

𝑚

⎧ ⎫ ⎪ ⋀ { 𝑥11 𝑥12 ⋯ 𝑥1𝑛 } ⎪ ⎪ 1 ⎪ ⎪ 𝑛 ⎪ ⎪ 𝑥 𝑥 ⋯ 𝑥 21 22 2𝑛 𝑚 { } ⎪ ⎪ ⋀ ⎪ 1 ⎨ ⎬ 𝑛 ⋁ 1 ⎪ ⎪ ⋮ ⋱ ⋱ ⋮ } ⎪ ⎪ ⋀{ ⎪ 1𝑛 ⎪ ⎪ ⎪ ⎪ ⋀ { 𝑥𝑚1 𝑥𝑚2 ⋯ 𝑥𝑚𝑛 } ⎪ ⎩ 1 ⎭ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

≤

Theorem 1.2 (minimax inequality). 𝑛

𝑛

𝑚

𝑚

minimax

maximin

𝑚

𝑚

⎫ ⎧ ⎪ ⋁ ⋁ ⋁ ⋁ ⎪ ⎪ 1 1 1 1 ⎪ ⎪ ⎪ 𝑥 𝑥 ⋯ 𝑥 ⎪ ⎪ 11 12 1𝑛 𝑛 ⎪ ⎪ ⎨ ⋀ 𝑥21 𝑥22 ⋯ 𝑥2𝑛 ⎬ 1 ⎪ ⎪ ⎪ ⎪ ⋱ ⋱ ⋮ ⎪ ⎪ ⋮ ⎪ ⎪ ⎪ ⎪ 𝑥 𝑥 ⋯ 𝑥 𝑚𝑛 ⎭ ⎩ 𝑚1 𝑚2 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

29

Let ( 𝑋, ∨, ∧ ; ≤) be a LATTICE (Definition 1.29 page 10).

𝑚

𝑥 𝑥 ≤ ⋀ ⋁ 𝑖𝑗 ⋁ ⋀ 𝑖𝑗 𝑖=1 𝑗=1 𝑗=1 𝑖=1 ⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟

∀𝑥𝑖𝑗 ∈ 𝑋

minimax: smallest of the largest

maxmini: largest of the smallest

✎PROOF: 𝑛

𝑥𝑖𝑘 (⋀ ) 𝑘=1 ⏟⏟⏟⏟⏟⏟⏟

𝑛

≤ 𝑥𝑖𝑗 ≤

smallest for any given 𝑖 𝑛 𝑚

⟹

𝑥 ⋁ (⋀ 𝑖𝑘 ) 𝑖=1 𝑘=1 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ largest amoung all 𝑖s of the smallest values 𝑚 𝑛

⟹

𝑛

≤

𝑥𝑘𝑗 (⋁ ) 𝑘=1 ⏟⏟⏟⏟⏟⏟⏟

∀𝑖, 𝑗

largest for any given 𝑗 𝑚

𝑥 ⋀ (⋁ 𝑘𝑗 ) 𝑘=1 𝑗=1 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ smallest amoung all 𝑗s of the largest values 𝑛 𝑚

𝑥 ≤ 𝑥 ⋁ (⋀ 𝑖𝑗 ) ⋀ (⋁ 𝑖𝑗 ) 𝑖=1 𝑗=1 𝑗=1 𝑖=1 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ maxmini

(change of variables)

minimax

✏ Special cases of the minimax inequality include three distributive inequalities (next theorem). If for some lattice any one of these inequalities is an equality, then all three are equalities; and in this case, the lattice is a called a distributive lattice. Theorem 1.3 (distributive inequalities). 30 ( 𝑋, ∨, ∧ ; ≤) is a LATTICE (Definition 1.29 page 10) 𝑥 ∧ (𝑦 ∨ 𝑧) ≥ (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧) ∀𝑥,𝑦,𝑧∈𝑋 ( JOIN SUPER-DISTRIBUTIVE) and ⎧ ⎪ 𝑥 ∨ (𝑦 ∧ 𝑧) ≤ (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧) ∀𝑥,𝑦,𝑧∈𝑋 ( MEET SUB-DISTRIBUTIVE) and ⎨ ⎪ ⎩ (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧) ∨ (𝑦 ∧ 𝑧) ≤ (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧) ∧ (𝑦 ∨ 𝑧) ∀𝑥,𝑦,𝑧∈𝑋 ( MEDIAN INEQUALITY).

⟹ ⎫ ⎪ ⎬ ⎪ ⎭

29

📘 Birkhoff (1948) pages 19–20 📘 Davey and Priestley (2002) page 85, 📘 Grätzer (2003) page 38, page 157, 📘 Müller-Olm (1997) page 13 ⟨terminology⟩ 30

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Birkhoff (1933) page 444,

Daniel J. Greenhoe

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Korselt (1894)

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✎PROOF: 1. Proof that ∧ sub-distributes over ∨: (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧) ≤ (𝑥 ∨ 𝑥) ∧ (𝑦 ∨ 𝑧) = 𝑥 ∧ (𝑦 ∨ 𝑧)

by minimax inequality (Theorem 1.2 page 11) by idempotent property of lattices (Theorem 1.1 page 10)

⎧ ⎪ ⎪ ⋀{ 𝑥 𝑦 } ⎫ ⎬ ⋁⎨ ⋀ ⎪ { 𝑥 𝑧 } ⎪ ⎩ ⎭

≤

⋁ ⋁ ⎫ ⎧ ⎪ ⎪ 𝑥 𝑦 ⎬ ⋀⎨ ⎪ ⎩ 𝑥 𝑧 ⎪ ⎭

2. Proof that ∨ super-distributes over ∧: 𝑥 ∨ (𝑦 ∧ 𝑧) = (𝑥 ∧ 𝑥) ∨ (𝑦 ∧ 𝑧) ≤ (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧)

by idempotent property of lattices (Theorem 1.1 page 10) by minimax inequality (Theorem 1.2 page 11)

⎧ ⎪ ⋀{ 𝑥 𝑥 } ⎫ ⎪ ⎬ ⋁⎨ ⋀ ⎪ { 𝑦 𝑧 } ⎪ ⎩ ⎭

≤

⋁ ⋁ ⎫ ⎧ ⎪ ⎪ 𝑥 𝑥 ⎬ ⋀⎨ ⎪ ⎩ 𝑦 𝑧 ⎪ ⎭

3. Proof that of median inequality: by minimax inequality (Theorem 1.2 page 11)

✏ Besides the distributive property, another consequence of the minimax inequality is the modularity inequality (next theorem). A lattice in which this inequality becomes equality is said to be modular. Theorem 1.4 (Modular inequality). 31 Let ( 𝑋, ∨, ∧ ; ≤) be a LATTICE (Definition 1.29 page 10). 𝑥≤𝑦 ⟹ 𝑥 ∨ (𝑦 ∧ 𝑧) ≤ 𝑦 ∧ (𝑥 ∨ 𝑧) ✎PROOF: 𝑥 ∨ (𝑦 ∧ 𝑧) = (𝑥 ∧ 𝑥) ∨ (𝑦 ∧ 𝑧)

by absorptive property (Theorem 1.1 page 10)

≤ (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧)

by the minimax inequality (Theorem 1.2 page 11)

= 𝑦 ∧ (𝑥 ∨ 𝑧)

by left hypothesis

⎧ ⎪ ⋀{ 𝑥 𝑥 } ⎫ ⎪ ⎬ ⋁⎨ ⋀ ⎪ { 𝑦 𝑧 } ⎪ ⎩ ⎭

≤

⋁ ⋁ ⎫ ⎧ ⎪ ⎪ 𝑥 𝑥 ⎬ ⋀⎨ ⎪ ⎩ 𝑦 𝑧 ⎪ ⎭

✏

1.4.3 Isomorphic spaces Definition 1.30. 32 Let 𝙓 ≜ (𝑋, ≤) and 𝙔 ≜ (𝑌 , ⋜) be ordered sets. (𝑥 ≤ 𝑦) ⟹ (𝜃(𝑥) ⋜ 𝜃(𝑦)) A function 𝜃 ∈ 𝑌 𝑋 is order preserving in (𝙓 , 𝙔 ) if 31 32

∀𝑥,𝑦∈𝑋 .

📘 Birkhoff (1948) page 19, 📘 Burris and Sankappanavar (1981) page 11, 📃 Dedekind (1900) page 374 📘 Burris and Sankappanavar (2000), page 10

version 0.50


Daniel J. Greenhoe

2016 Jul 4

10:12am UTC

page 13


Definition 1.31. Let 𝙇1 ≜ ( 𝑋, ∨, ∧ ; ≤) and 𝙇2 ≜ ( 𝑌 , >, ? ; ⋜) be LATTICEs. 𝙇1 and 𝙇2 are isomorphic on (𝙓 , 𝙔 ) if there exists a function 𝜃 ∈ 𝑌 𝑋 such that 1. 𝜃(𝑥 ∨ 𝑦) = 𝜃(𝑥) > 𝜃(𝑦) x,y X ( PRESERVES JOINS) and 2. 𝜃(𝑥 ∧ 𝑦) = 𝜃(𝑥) ? 𝜃(𝑦) x,y X ( PRESERVES MEETS). In this case, the function 𝜃 is said to be an isomorphism from 𝙇1 to 𝙇2 , and the isomorphic relationship between 𝙇1 and 𝙇2 is denoted as 𝙇1 ≡ 𝙇2 . Theorem 1.5. 33 Let ( 𝑋, ∨, ∧ ; ≤) and ( 𝑌 , >, ? ; ⋜) be lattices and 𝜃 ∈ 𝑌 𝑋 be a BIJECTIVE function with inverse 𝜃 −1 ∈ 𝑋 𝑌 . 𝑥1 ≤ 𝑥2 ⟹ 𝜃(𝑥1 ) ⋜ 𝜃(𝑥2 ) ∀𝑥1 ,𝑥2 ∈𝑋 and ⟺ (⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑋, ∨, ∧ ; ≤) ≡ ( 𝑌 , >, ? ; ⋜) −1 −1 } 𝑦 ⋜ 𝑦 ⟹ 𝜃 (𝑦 ) ⋜ 𝜃 (𝑦 ) ∀𝑦 1 ,𝑦2 ∈𝑌 1 2 1 2 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ISOMORPHIC

𝜃 and 𝜃 −1 are ORDER PRESERVING

✎PROOF:

Let 𝜃 ∈ 𝑌 𝑋 be the isomorphism between lattices ( 𝑋, ∨, ∧ ; ≤) and ( 𝑌 , >, ? ; ⋜).

1. Proof that order preserving ⟹ preserves joins: (a) Proof that 𝜃(𝑥1 ∨ 𝑥2 ) ⧁ 𝜃(𝑥1 ) > 𝜃(𝑥2 ): i. Note that 𝑥1 ≤ 𝑥 1 ∨ 𝑥 2 𝑥2 ≤ 𝑥 1 ∨ 𝑥 2 . ii. Because 𝜃 is order preserving 𝜃(𝑥1 ) ⋜ 𝜃(𝑥1 ∨ 𝑥2 ) 𝜃(𝑥2 ) ⋜ 𝜃(𝑥1 ∨ 𝑥2 ). iii. We can then finish the proof of item (1a): 𝜃(𝑥1 ) > 𝜃(𝑥2 ) ⋜ 𝜃(𝑥 ⏟⏟⏟⏟⏟ 1 ∨ 𝑥2 ) > 𝜃(𝑥 ⏟⏟⏟⏟⏟ 1 ∨ 𝑥2 ) 𝑥1 ≤ 𝑥 1 ∨ 𝑥2

by order preserving hypothesis

𝑥2 ≤ 𝑥1 ∨ 𝑥2

by idempotent property page 10

= 𝜃(𝑥1 ∨ 𝑥2 ) (b) Proof that 𝜃(𝑥1 ∨ 𝑥2 ) ⋜ 𝜃(𝑥1 ) > 𝜃(𝑥2 ):

i. Just as in item (1a), note that 𝜃 −1 (𝑦1 ) ∨ 𝜃 −1 (𝑦2 ) ≤ 𝜃 −1 (𝑦1 > 𝑦2 ): −1 −1 𝜃 −1 (𝑦1 ) ∨ 𝜃 −1 (𝑦2 ) ≤ 𝜃⏟⏟⏟⏟⏟⏟⏟⏟⏟ (𝑦1 > 𝑦2 ) ∨ 𝜃⏟⏟⏟⏟⏟⏟⏟⏟⏟ (𝑦1 > 𝑦2 ) 𝑦1 ⋜ 𝑦1 > 𝑦2


𝑦2 ⋜ 𝑦1 > 𝑦2

= 𝜃 −1 (𝑦1 > 𝑦2 )


ii. Because 𝜃 is order preserving 𝜃 [𝜃 −1 (𝑦1 ) ∨ 𝜃 −1 (𝑦2 )] ⋜ 𝜃𝜃 −1 (𝑦1 > 𝑦2 ) = 𝑦1 > 𝑦2

by item (1(b)i) page 13 by definition of inverse function 𝜃 −1

iii. Let 𝑢1 ≜ 𝜃(𝑥1 ) and 𝑢2 ≜ 𝜃(𝑥2 ). iv. We can then finish the proof of item (1b): 𝜃(𝑥1 ∨ 𝑥2 ) = 𝜃 [𝜃 −1 𝜃(𝑥1 ) ∨ 𝜃 −1 𝜃(𝑥2 )]

33

by definition of inverse function 𝜃 −1

= 𝜃 [𝜃 −1 (𝑢1 ) ∨ 𝜃 −1 (𝑢2 )]

by definition of 𝑢1 , 𝑢2 , item (1(b)iii)

⋜ 𝑢1 > 𝑢2

by item (1(b)ii)

= 𝜃(𝑥1 ) > 𝜃(𝑥2 )

by definition of 𝑢1 , 𝑢2 , item (1(b)iii)

📘 Burris and Sankappanavar (2000), page 10

2016 Jul 4

10:12am UTC


Daniel J. Greenhoe

version 0.50

page 14


(c) And so, combining item (1a) and item (1b), we have 𝜃(𝑥1 ∨ 𝑥2 ) ⧁ 𝜃(𝑥1 ) > 𝜃(𝑥2 ) 𝜃(𝑥1 ∨ 𝑥2 ) ⋜ 𝜃(𝑥1 ) > 𝜃(𝑥2 )

(item (1a) page 13)

and

𝜃(𝑥1 ∨ 𝑥2 ) = 𝜃(𝑥1 ) > 𝜃(𝑥2 )

⟹

}

(item (1b) page 13)

2. Proof that order preserving ⟹ preserves meets: (a) Proof that 𝜃(𝑥1 ∧ 𝑥2 ) ⋜ 𝜃(𝑥1 ) ? 𝜃(𝑥2 ): 𝜃(𝑥1 ) ? 𝜃(𝑥2 ) ⧁ 𝜃(𝑥 ⏟⏟⏟⏟⏟ 1 ∧ 𝑥2 ) ? 𝜃(𝑥 ⏟⏟⏟⏟⏟ 1 ∧ 𝑥2 )


𝑥2 ≥ 𝑥 1 ∧ 𝑥2

𝑥1 ≥ 𝑥1 ∧ 𝑥2

= 𝜃(𝑥1 ∧ 𝑥2 )


(b) Proof that 𝜃(𝑥1 ∧ 𝑥2 ) ⧁ 𝜃(𝑥1 ) ? 𝜃(𝑥2 ): i. Just as in item (2a), note that 𝜃 −1 (𝑦1 ) ∧ 𝜃 −1 (𝑦2 ) ≥ 𝜃 −1 (𝑦1 ? 𝑦2 ): −1 −1 𝜃 −1 (𝑦1 ) ∧ 𝜃 −1 (𝑦2 ) ≥ 𝜃⏟⏟⏟⏟⏟⏟⏟⏟⏟ (𝑦1 ? 𝑦2 ) ? 𝜃⏟⏟⏟⏟⏟⏟⏟⏟⏟ (𝑦1 ? 𝑦2 )


𝑦2 ⧁ 𝑦1 ? 𝑦2

𝑦1 ⧁ 𝑦1 ? 𝑦2

= 𝜃 (𝑦1 ? 𝑦2 ) −1


ii. Because 𝜃 is order preserving 𝜃 [𝜃 −1 (𝑦1 ) ∧ 𝜃 −1 (𝑦2 )] ⧁ 𝜃𝜃 −1 (𝑦1 ? 𝑦2 )

by item (2(b)i)

= 𝑦1 ? 𝑦2 iii. Let 𝑣1 ≜ 𝜃(𝑥1 ) and 𝑣2 ≜ 𝜃(𝑥2 ). iv. We can then finish the proof of item (2a): 𝜃(𝑥1 ∧ 𝑥2 ) = 𝜃 [𝜃 −1 𝜃(𝑥1 ) ∧ 𝜃 −1 𝜃(𝑥2 )] = 𝜃 [𝜃 −1 (𝑣1 ) ∧ 𝜃 −1 (𝑣2 )]

by item (2(b)iii)

⧁ 𝑣 1 ? 𝑣2

by item (2(b)ii)

= 𝜃(𝑥1 ) ? 𝜃(𝑥2 )

by item (2(b)iii)

(c) And so, combining item (2a) and item (2b), we have 𝜃(𝑥1 ∧ 𝑥2 ) ⋜ 𝜃(𝑥1 ) ? 𝜃(𝑥2 ) 𝜃(𝑥1 ∧ 𝑥2 ) ⧁ 𝜃(𝑥1 ) ? 𝜃(𝑥2 )

(item (2a) page 14)

and

(item (2b) page 14)

}

⟹

𝜃(𝑥1 ∧ 𝑥2 ) = 𝜃(𝑥1 ) ? 𝜃(𝑥2 )

3. Proof that order preserving ⟸ isomorphic: 𝑥 ≤ 𝑦 ⟹ 𝜃(𝑦) = 𝜃(𝑥 ∨ 𝑦) = 𝜃(𝑥) > 𝜃(𝑦)

by right hypothesis

⟹ 𝜃(𝑥) ⋜ 𝜃(𝑦) 𝑥 ≤ 𝑦 ⟹ 𝜃(𝑥) = 𝜃(𝑥 ∧ 𝑦) = 𝜃(𝑥) ? 𝜃(𝑦)

by right hypothesis

⟹ 𝜃(𝑥) ⋜ 𝜃(𝑦)

✏ Example 1.4. 34 In the diagram to the right, the function 𝜃 ∈ 𝑌 𝑋 is order preserving with respect to ≤ and ⋜. Note that 𝜃 −1 is not order preserving and that the ordered sets (𝑋, ≤) and (𝑌 , ⋜) are not isomorphic—as already demonstrated by Theorem 1.5 that they cannot be. 34

(𝑋, ≤)

(𝑌 , ⋜)

𝜃 ∈ 𝑌𝑋

📘 Burris and Sankappanavar (2000), page 10

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Daniel J. Greenhoe

2016 Jul 4

10:12am UTC

page 15


Example 1.5. In the diagram to the right, the function 𝜃 ∈ 𝑌 𝑋 is order preserving with respect to ≤ and ⋜. Note that 𝜃 −1 is not order preserving. Like Example 1.4 (page 14), this example also illustrates the fact that that order preserving does not imply isomorphic.

(𝑋, ≤)

Example 1.6. In the diagram to the right, the function 𝜃 ∈ 𝑌 𝑋 is order preserving with respect to ≤ and ⋜. Note that 𝜃 −1 is also order preserving and that the ordered sets (𝑋, ≤) and (𝑌 , ⋜) are isomorphic—as already demonstrated by Theorem 1.5 that they must be.

(𝑋, ≤)

(𝑌 , ⋜)

𝜃∈𝑌

𝑋

(𝑌 , ⋜)

𝜃 ∈ 𝑌𝑋

1.4.4 Monotone functions on ordered sets Definition 1.32. 35 Let ( 𝑋, ≤) and ( 𝑌 , ⊑) be ORDERED SETs (Definition 1.20 page 8). Let 𝜙 be a function in 𝑌 𝑋 (Definition 1.6 page 6). 𝜙 is isotone in ( 𝑌 , ⊑)( 𝑋, ≤) if 𝑥 ≤ 𝑦 ⟹ 𝜙(𝑥) ⊑ 𝜙(𝑦) ∀𝑥,𝑦∈𝑋. 𝜙 is strictly isotone in ( 𝑌 , ⊑)( 𝑋, ≤) if 𝑥 < 𝑦 ⟹ 𝜙(𝑥) ⊏ 𝜙(𝑦) ∀𝑥,𝑦∈𝑋. 𝜓 is antitone in ( 𝑌 , ⊑)( 𝑋, ≤) if 𝑥 ≤ 𝑦 ⟹ 𝜓(𝑦) ⊑ 𝜓(𝑥) ∀𝑥,𝑦∈𝑋. 𝜓 is strictly antitone in ( 𝑌 , ⊑)( 𝑋, ≤) if 𝑥 < 𝑦 ⟹ 𝜓(𝑦) ⊏ 𝜓(𝑥) ∀𝑥,𝑦∈𝑋. A FUNCTION is monotone if it is ISOTONE or ANTITONE and strictly monotone if it is STRICTLY ISOTONE or STRICTLY ANTITONE. An ISOTONE function in ( 𝑌 , ⊑)( 𝑋, ≤) is also said to be order preserving in ( 𝑌 , ⊑)( 𝑋, ≤) . Lemma 1.2. Let ( 𝑋, ∨, ∧ ; ≤) and ( 𝑌 , ⊔, ⊓ ; ⊑) be LATTICEs (Definition 1.29 page 10). Let 𝖿 be a FUNCTION in 𝑋 𝑋 . Let 𝜙 be a FUNCTION in 𝑌 𝑋 (Definition 1.6 page 6). ⎧ 1. arg ⋁ 𝖿(𝑥) ⊆ arg ⨆ 𝜙[𝖿(𝑥)] 𝑎𝑛𝑑 ⎪ 𝜙 is ISOTONE 𝑥∈𝑋 𝑥∈𝑋 ⟹ ⎨ { (Definition 1.32 page 15) } ⎪ 2. arg ⋀ 𝖿 (𝑥) ⊆ arg ⨅ 𝜙[𝖿(𝑥)] . 𝑥∈𝑋 𝑥∈𝑋 ⎩

⎫ ⎪ ⎬ ⎪ ⎭

✎PROOF: arg

𝖿(𝑥) = arg𝑥 {𝖿(𝑥)|𝖿 (𝑦) ≤ 𝖿 (𝑥)

⋁

∀𝑥, 𝑦 ∈ 𝑋}

because 𝖿 ∈ 𝑋 𝑋 and by Lemma 1.1 page 10

𝑥∈𝑋

⊆ arg𝑥 {𝖿(𝑥)|𝜙[𝖿(𝑦)] ≤ 𝜙[𝖿(𝑥)] ∀𝑥, 𝑦 ∈ 𝑋}

by isotone hypothesis (Definition 1.32 page 15)

= arg𝑥 {𝜙[𝖿 (𝑥)]|𝜙[𝖿(𝑦)] ≤ 𝜙[𝖿(𝑥)] ∀𝑥, 𝑦 ∈ 𝑋}

because arg𝑥 {𝖿(𝑥)|𝑃 (𝑥)} = arg𝑥 {𝗀[𝖿(𝑥)]|𝑃 (𝑥)}

≜ arg


⨆

𝜙[𝖿 (𝑥)]

𝑥∈𝑋

arg

⋀

𝖿(𝑥) = arg𝑥 {𝖿(𝑥)|𝖿 (𝑥) ≤ 𝖿 (𝑦)

∀𝑥, 𝑦 ∈ 𝑋}


𝑥∈𝑋

⊆ arg𝑥 {𝖿(𝑥)|𝜙[𝖿(𝑥)] ≤ 𝜙[𝖿 (𝑦)] ∀𝑥, 𝑦 ∈ 𝑋}


= arg𝑥 {𝖿(𝑥)|𝜙[𝖿(𝑥)] ≤ 𝜙[𝖿 (𝑦)] ∀𝑥, 𝑦 ∈ 𝑋}


= arg𝑥 {𝜙[𝖿 (𝑥)]|𝖿(𝑥) ≤ 𝖿 (𝑦)

because arg𝑥 {𝖿(𝑥)|𝑃 (𝑥)} = arg𝑥 {𝗀[𝖿(𝑥)]|𝑃 (𝑥)}

≜ arg

⨅

∀𝑥, 𝑦 ∈ 𝑋}


𝜙[𝖿 (𝑥)]

𝑥∈𝑋

✏ 35

📘 Rudeanu (2001) page 182 ⟨Definition 2.1⟩, 📘 Burris and Sankappanavar (2000), page 10, page 308 2016 Jul 4

10:12am UTC


Daniel J. Greenhoe

📃

Topkis (1978)

version 0.50

page 16


Remark 1.1. 36 Let ( 𝑋, ≤) and ( 𝑌 , ⊑) be ordered sets (Definition 1.20 page 8). Let 𝜙 be a function in 𝑌 𝑋 (Definition 1.6 page 6). Note that even if 𝜙 is bijective (Definition 1.14 page 7) and strictly isotone (Definition 1.32 page 15), 𝑥 1 4. Let 𝑥′ represent the die face which, when its numeric value is summed “in the usual way” with the numeric value of the die face 𝑥, equals 7. Then at each point 𝑥 in 𝙃 , three open balls are possible: {𝑥} for 0 < 𝑟 ≤ 1 ′ . 𝖡(𝑥, 𝑟) = {𝑥, 𝑥 } for 1 < 𝑟 ≤ 2 } { 𝛺 for 𝑟 > 2 version 0.50


Daniel J. Greenhoe

2016 Jul 4

10:12am UTC

page 33

CHAPTER 2. STOCHASTIC PROCESSING ON WEIGHTED GRAPHS

5. The open balls of (𝛺, 𝖽) or (𝛺, 𝗉) in turn induce a base for a topology 𝑻 , such that 𝑻 = {𝑈 ∈ 𝟚𝛺 |𝑈 is a union of open balls}. The topology induced by 𝙂 is the discrete topology 𝟚𝛺 (Defini𝛺 tion 1.8 page 6). The topology induced by 𝙃 is also the discrete topology 𝟚 . 6. So the metrics of 𝙂 and 𝙃 are different. And the balls induced by 𝙂 and those induced by 𝙃 are different. However, the topologies induced by 𝙂 and 𝙃 are the same.

✏ Example 2.3. The weighted real die outcome subspace (Definition 2.6 page 28) illustrated in Figure 2.1 page 29 (C) has the following geometric characteristics: 101 ≈ 0.33 ̇ ̇ ∁(𝙂) = ∁𝖺̇ (𝙂) = {⚃} 𝖵𝖺𝗋(𝙂) = 300 ̇ ≈ 0.847 ∁𝗀̇ (𝙂) = {⚁} 𝖵𝖺𝗋(𝙂; ∁𝗀̇ ) = 127 150 145 ̇∁ (𝙂) = {⚄} ̇ ̇ 𝖵𝖺𝗋(𝙂; ∁𝗁 ) = 150 ≈ 0.967 . 𝗁 ̇∁ (𝙂) = {⚀, ⚂, ⚃, ⚅} ̇ 𝖵𝖺𝗋(𝙂; ∁𝗆̇ ) = 11 = 0.22 𝗆 50 11 ̇∁ (𝙂) = {⚁, ⚄} ̇ ̇ 𝖵𝖺𝗋(𝙂; ∁𝖬 ) = 50 ≈ 0.767 𝖬 ̇ Note that the outcome center ∁(𝙂) and arithmetic center ∁𝖺̇ (𝙂) again yield identical results. Also note ̇ ̇ ̇ ̇ that of the four center measures of cardinality 1 (| ∁(𝙂) | = | ∁𝖺 (𝙂) | = | ∁𝗀 (𝙂) | = | ∁𝗁 (𝙂) | = 1 Definition 1.13 page 7), ∁̇ and ∁𝖺̇ yield by far the lowest variance measures. ✎PROOF: ̇ ∁(𝙂) ≜ arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑥∈𝙂

≜ arg min max 𝑥∈𝙂

by definition of ∁̇ (Definition 2.3 page 28)

𝑦∈𝙂

𝑦∈𝙂

1 𝖽(𝑥, 𝑦) 𝖯(𝑦)300 300

≜ arg min max 𝖽(𝑥, 𝑦) 300𝖯(𝑦) 𝑥∈𝙂

by Lemma 1.7 (page 17)

𝑦∈𝙂

𝖽(⚀, ⚀)𝖯(⚀) 𝖽(⚀, ⚁)𝖯(⚁) ⎧ ⎪ 𝖽(⚁, ⚀)𝖯(⚀) 𝖽(⚁, ⚁)𝖯(⚁) = arg min max 300⎨ ⋮ ⋮ 𝑦∈𝙂 𝑥∈𝙂 ⎪ ⎩ 𝖽(⚅, ⚀)𝖯(⚀) 𝖽(⚅, ⚁)𝖯(⚁) 0 × 30 1 × 15 1 × 10 1 × 180 ⎧ 1 × 30 0 × 15 1 × 10 1 × 180 ⎪ ⎪ 1 × 30 1 × 15 0 × 10 2 × 180 = arg min max ⎨ 1 × 30 1 × 15 2 × 10 0 × 180 𝑦∈𝙂 𝑥∈𝙂 ⎪ 1 × 30 2 × 15 1 × 10 1 × 180 ⎪ ⎩ 2 × 30 1 × 15 1 × 10 1 × 180 ∁𝖺̇ (𝙂) ≜ arg min 𝑥∈𝙂

= arg min 𝑥∈𝙂


∑

𝑥∈𝙂

= arg min 𝑥∈𝙂 2016 Jul 4

10:12am UTC

𝖺

𝑦∈𝙂

1 𝖽(𝑥, 𝑦) 𝖯(𝑦)300 ∑ 300 𝑦∈𝙂 ∑

by definition of 𝙂 by Lemma 1.2 (page 15)

𝖽(𝑥, 𝑦) 𝖯(𝑦)300

𝑦∈𝙂

15 0 15 15 30 15

+ + + + + +

𝖽(𝑥, 𝑦) ∏ [

𝖯(𝑦)

⎧ ⎪ ⎪ = arg min ⎨ 𝑥∈𝙂 ⎪ ⎪ ⎩ ∁𝗀̇ (𝙂) ≜ arg min

𝖽(𝑥, 𝑦) 𝖯(𝑦)

𝖽(⚀, ⚂)𝖯(⚂) ⋯ 𝖽(⚀, ⚅)𝖯(⚅) ⎪ 𝖽(⚁, ⚂)𝖯(⚂) ⋯ 𝖽(⚁, ⚅)𝖯(⚅) ⎫ ⎬ ⋮ ⋱ ⋮ 𝖽(⚅, ⚂)𝖯(⚂) ⋯ 𝖽(⚅, ⚅)𝖯(⚅) ⎪ ⎭ 1 × 6 2 × 10 180 ⎧ ⎫ 2 × 6 1 × 10 ⎫ ⎪ ⎪ 180 ⎫ ⎪ ⎧ ⎪ ⎪ ⎪ 360 ⎪ ⎪ ⎪ 1 × 6 1 × 10 ⎪ min ⎨ 30 ⎬ = ⎨ ⚃ ⎬ 1 × 6 1 × 10 ⎬ = arg 𝑥∈𝙂 ⎪ ⎪ 180 ⎪ ⎪ 0 × 6 1 × 10 ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ 1 × 6 0 × 10 ⎭ ⎩ 180 ⎪ ⎭ ⎩ by definition of ∁̇ (Definition 2.4 page 28)

0 30 30 30 30 60

+ + + + + +

10 10 0 20 10 10

+ + + + + +

180 180 360 0 180 180

+ 6 + 20 ⎧ + 12 + 10 ⎫ ⎪ ⎪ ⎪ + 6 + 10 ⎪ min ⎨ + 6 + 10 ⎬ = arg 𝑥∈𝙂 ⎪ + 0 + 10 ⎪ ⎪ ⎪ + 6 + 0 ⎭ ⎩

]

231 242 421 81 260 271

⎫ ⎫ ⎪ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬=⎨ ⚃ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎭ ⎩ ̇ by definition of ∁ (Definition 2.4 page 28) 𝗀

𝑦∈𝙂⧵{𝑥}

1

𝖽(𝑥, 𝑦)300𝖯(𝑦) 300 ] ∏ [



Daniel J. Greenhoe

version 0.50

page 34

2.1. OUTCOME SUBSPACES 1

⎛ ⎞ 300 300𝖯(𝑦) = arg min ⎜ 𝖽(𝑥, 𝑦) ]⎟⎟ ∏ [ 𝑥∈𝙂 ⎜𝑦∈𝙂⧵ {𝑥} ⎝ ⎠ 300𝖯(𝑦) = arg min 𝖽(𝑥, 𝑦) ] ∏ [ 𝑥∈𝙂



⎧ ⎪ ⎪ = arg min ⎨ 𝑥∈𝙂 ⎪ ⎪ ⎩

130 130 130 130 230

× × × × × ×

115 × × 115 × 115 × 215 × 115 ×

110 × 1180 110 × 1180 × 2180 10 2 × 110 × 1180 110 × 1180

⎛ ⎞ 1 ∁𝗁̇ (𝙂) ≜ arg min ⎜ 𝖯(𝑦)⎟ ∑ 𝖽(𝑥, 𝑦) ⎟ 𝑥∈𝙂 ⎜𝑦∈𝙂⧵ {𝑥} ⎝ ⎠ 1 = arg max 𝖯(𝑦) ∑ 𝖽(𝑥, 𝑦) 𝑥∈𝙂 𝑦∈𝙂⧵ {𝑥} 𝑥∈𝙂

𝑥∈𝙂

16

× × × × ×

210 110 110 110 110

30 1 30 1 30 1 30 1 30 2

15 1

+ + + + + +

by Lemma 1.4 page 16

by Lemma 1.2 (page 15) + + + + + +

15 1 15 1 15 2 15 1

10 1 10 1 10 2 10 1 10 1

+ + + + + +

180 1 180 1 180 2 180 1 180 1

+ + + + + +

6 1 6 2 6 1 6 1 6 1

+ + + + + +

10 2 10 1 10 1 10 1 10 1

⎫ ⎪ ⎧ ⎪ ⎪ ⎪ ⎪ ⎬ = arg max ⎨ 𝑥∈𝙂 ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎭

0 30 30 30 30 60

15 0 15 15 30 15

10 10 0 20 10 10

180 6 20 ⎧ 180 12 10 ⎫ ⎪ ⎪ ⎪ 360 6 10 ⎪ min ⎨ 0 6 10 ⎬ = arg 𝑥∈𝙂 ⎪ 180 0 10 ⎪ ⎪ ⎪ 180 6 0 ⎭ ⎩

∁𝖬̇ (𝙂) ≜ arg max min 𝖽(𝑥, 𝑦) 𝖯(𝑦)

⎫ ⎫ ⎪ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬=⎨ ⎬ ⎪ ⎪ ⚄ ⎪ ⎪ ⎪ ⎪ ⎭ ⎭ ⎩

6 10 6 6 10 6

⚀ ⎫ ⎫ ⎪ ⎧ ⎪ ⎪ ⎪ ⎪ ⚂ ⎪ ⎬=⎨ ⚃ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ ⚅ ⎭

by definition of ∁𝖬̇ (Definition 2.4 page 28)

𝑦∈𝙂

by ∁𝗆̇ (𝙂) result

= arg max {6, 10, 6, 6, 10, 6} 𝑥∈𝙂

= {⚁, ⚄} ∑

216.0 233.0 151.0 66.0 237.5 226.0

by definition of ∁𝗆̇ (Definition 2.4 page 28)

𝑦∈𝙂

⎧ ⎪ ⎪ = arg min min ⎨ 𝑥∈𝙂 𝑦∈𝙂⧵{𝑥} ⎪ ⎪ ⎩

̇ 𝖵𝖺𝗋(𝙂) ≜

⎫ ⎪ ⎧ ⎪ ⚁ ⎫ ⎪ ⎪ ⎪ ⎪ = ⎬ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎭

by definition of ∁𝗁̇ (Definition 2.4 page 28)

∁𝗆̇ (𝙂) ≜ arg min min 𝖽(𝑥, 𝑦) 𝖯(𝑦)

𝑥∈𝙂

210 26 2180 210 215 230

−1

300𝖯(𝑦) ∑ 𝖽(𝑥, 𝑦) 𝑦∈𝙂⧵{𝑥}

⎧ ⎪ ⎪ ⎪ = arg max ⎨ 𝑥∈𝙂 ⎪ ⎪ ⎪ ⎩ 𝑥∈𝙂

⎫ ⎧ ⎪ ⎪ ⎪ ⎪ ⎬ = arg min ⎨ 𝑥∈𝙂 ⎪ ⎪ ⎪ ⎪ ⎩ ⎭

1 1 300𝖯(𝑦) ∑ 𝖽(𝑥, 𝑦) 300 𝑦∈𝙂⧵ {𝑥}

= arg max = arg max

16 26 16 16

× × × × × ×

by definition of 𝙂

̇ 𝑥) 𝖯(𝑥) 𝖽2 (∁(𝙂),

̇ (Definition 2.5 page 28) by definition of 𝖵𝖺𝗋

𝖽2 (⚃, 𝑥) 𝖯(𝑥)

̇ by ∁(𝙂) result

𝑥∈𝙂

=

∑

𝑥∈𝙂

1 1 1 3 1 1 + 12 × + 22 × + 0 2 × + 12 × + 12 × 10 20 30 5 50 30 1 101 = (30 + 15 + 40 + 0 + 6 + 10) = ≈ 0.337 300 300 ̇ 𝖵𝖺𝗋(𝙂; ∁𝗀̇ ) ≜ 𝖽2 ∁̇ (𝙂), 𝑥) 𝖯(𝑥) ∑ (𝗀 = 12 ×


𝑥∈𝙂

=

∑

by ∁𝗀̇ (𝙂) result

𝖽2 (⚁, 𝑥) 𝖯(𝑥)

𝑥∈𝙂

version 0.50


Daniel J. Greenhoe

2016 Jul 4

10:12am UTC

page 35


1 1 1 3 1 1 + 02 × + 12 × + 12 × + 2 2 × + 12 × 10 20 30 5 50 30 1 254 127 = (30 + 0 + 10 + 180 + 24 + 10) = = ≈ 0.847 300 300 150 ̇ 𝖵𝖺𝗋(𝙂; ∁𝗁̇ ) ≜ 𝖽2 ∁̇ (𝙂), 𝑥) 𝖯(𝑥) ∑ (𝗁 = 12 ×


𝑥∈𝙂

=

∑

by ∁𝗁̇ (𝙂) result

𝖽2 (⚄, 𝑥) 𝖯(𝑥)

𝑥∈𝙂

1 1 1 3 1 1 + 22 × + 12 × + 12 × + 0 2 × + 12 × 10 20 30 5 50 30 1 290 145 = (30 + 60 + 10 + 180 + 0 + 10) = = ≈ 0.967 300 300 150 ̇ 𝖵𝖺𝗋(𝙂; ∁𝗆̇ ) ≜ 𝖽2 ∁̇ (𝙂), 𝑥) 𝖯(𝑥) ∑ (𝗆 = 12 ×


𝑥∈𝙂

=

∑


𝖽2 (⚀, ⚂, ⚃, ⚅, 𝑥) 𝖯(𝑥)

𝑥∈𝙂

1 1 1 3 1 1 + 12 × + 02 × + 02 × + 1 2 × + 02 × 10 20 30 5 50 30 1 66 11 = (0 + 60 + 0 + 0 + 6 + 0) = = = 0.22 300 300 50 ̇ 𝖵𝖺𝗋(𝙂; ∁𝖬̇ ) ≜ 𝖽2 ∁̇ (𝙂), 𝑥) 𝖯(𝑥) ∑ (𝖬 = 02 ×


𝑥∈𝙂

=

∑

by ∁𝖬̇ (𝙂) result

𝖽2 (⚁, ⚄, 𝑥) 𝖯(𝑥)

𝑥∈𝙂

1 1 1 3 1 1 + 02 × + 12 × + 12 × + 0 2 × + 12 × 10 20 30 5 50 30 1 230 11 = (30 + 0 + 10 + 180 + 0 + 10) = = ≈ 0.767 300 300 50 = 12 ×

✏ Example 2.4 (board game spinner outcome subspace). The six value spinner outcome subspace (Definition 2.10 page 29) has the following geometric values: ̇ ∁(𝙂) = ∁𝖺̇ (𝙂) = ∁𝗀̇ (𝙂) = ∁𝗁̇ (𝙂) = ∁𝗆̇ (𝙂) = ∁𝖬̇ (𝙂) = {1, 2, 3, 4, 5, 6} ̇ ̇ ̇ ̇ ̇ ̇ 𝖵𝖺𝗋(𝙂) = 𝖵𝖺𝗋(𝙂; ∁𝖺̇ ) = 𝖵𝖺𝗋(𝙂; ∁𝗀̇ ) = 𝖵𝖺𝗋(𝙂; ∁𝗁̇ ) = 𝖵𝖺𝗋(𝙂; ∁𝗆̇ ) = 𝖵𝖺𝗋(𝙂; ∁𝖬̇ ) = 0 ✎PROOF: ̇ ∁(𝙂) ≜ arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑥∈𝙂

= arg min max 𝖽(𝑥, 𝑦) 𝑥∈𝙂


𝑦∈𝙂

𝑦∈𝙂

1 6



𝑦∈𝙂

𝖽(1, 1) ⎧ ⎪ 𝖽(2, 1) ⎪ 𝖽(3, 1) = arg min max ⎨ 𝖽(4, 1) 𝑦∈𝙂 𝑥∈𝙂 ⎪ 𝖽(5, 1) ⎪ ⎩ 𝖽(6, 1) = {⚀, ⚁, ⚂, ⚃, ⚄, ⚅} ∁𝖺̇ (𝙂) ≜ arg min 𝑥∈𝙂

= arg min 𝑥∈𝙂 2016 Jul 4

10:12am UTC

∑


⋯ ⋯ ⋯ ⋯ ⋯ ⋯

1 because 𝖿 (𝑥) = 𝑥 is strictly isotone and by Lemma 1.7 (page 17) 6 𝖽(1, 6) 3 0 1 2 3 2 1 ⎧ ⎧ 𝖽(2, 6) ⎫ ⎪ ⎪ ⎪ 3 ⎫ ⎪ ⎪ 1 0 1 2 3 2 ⎫ ⎪ 3 ⎪ ⎪ 2 1 0 1 2 3 ⎪ 𝖽(3, 6) ⎪ min max ⎨ 3 2 1 0 1 2 ⎬ = arg min ⎨ 3 ⎬ 𝖽(4, 6) ⎬ = arg 𝑦∈𝙂 𝑥∈𝙂 ⎪ 𝑥∈𝙂 ⎪ 2 3 2 1 0 1 ⎪ ⎪ 3 ⎪ 𝖽(5, 6) ⎪ ⎪ ⎪ ⎪ 𝖽(6, 6) ⎭ ⎩ 3 ⎪ ⎩ 1 2 3 2 1 0 ⎭ ⎭ by definition of 𝙂 by definition of ∁̇ (Definition 2.4 page 28) 𝖺

𝑦∈𝙂

∑

𝑦∈𝙂

𝖽(𝑥, 𝑦)

1 6

by definition of 𝙂 A book concerning symbolic sequence processing

Daniel J. Greenhoe

version 0.50

page 36

2.1. OUTCOME SUBSPACES


∑

𝑦∈𝙂

⎧ ⎪ ⎪ = arg min ⎨ 𝑥∈𝙂 ⎪ ⎪ ⎩ ∁𝗀̇ (𝙂) ≜ arg min 𝑥∈𝙂


1 because 𝖿 (𝑥) = 𝑥 is strictly isotone and by Lemma 1.2 (page 15) 6

𝖽(𝑥, 𝑦)

0 1 2 3 2 1

+ + + + + +

1 0 1 2 3 2

+ + + + + +

2 1 0 1 2 3

+ + + + + +

𝖽(𝑥, 𝑦)𝖯(𝑦) ] ∏ [

3 2 1 0 1 2

+ + + + + +

2 3 2 1 0 1

+ + + + + +

1 2 3 2 1 0

9 ⎫ ⎧ ⎪ ⎪ 9 ⎫ ⎪ ⎧ ⎪ ⎪ ⎪ 9 ⎪ ⎪ ⎬ = arg min ⎨ 9 ⎬ = ⎨ 𝑥∈𝙂 ⎪ ⎪ 9 ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ 9 ⎪ ⎭ ⎭ ⎪ by definition of ∁̇ (Definition 2.4 page 28)

⚀ ⚁ ⚂ ⚃ ⚄ ⚅

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

⚀ ⚁ ⚂ ⚃ ⚄ ⚅

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

𝗀


1

𝖽(𝑥, 𝑦) 6 ] ∏ [



1

⎛ ⎞6 = arg min ⎜ 𝖽(𝑥, 𝑦)⎟ ∏ ⎟ 𝑥∈𝙂 ⎜𝑦∈𝙂⧵ {𝑥} ⎝ ⎠ = arg min 𝖽(𝑥, 𝑦) ∏ 𝑥∈𝙂



⎧ ⎪ ⎪ = arg min ⎨ 𝑥∈𝙂 ⎪ ⎪ ⎩

1 2 3 2 1

× × × × × ×

1 × × 1 × 2 × 3 × 2 ×

2 × 3 1 × 2 × 1 1 × 2 × 1 3 × 2

⎞ ⎛ 1 𝖯(𝑦)⎟ ∁𝗁̇ (𝙂) ≜ arg min ⎜ ∑ 𝖽(𝑥, 𝑦) ⎟ 𝑥∈𝙂 ⎜𝑦∈𝙂⧵ {𝑥} ⎠ ⎝ 1 = arg max 𝖯(𝑦) ∑ 𝖽(𝑥, 𝑦) 𝑥∈𝙂 𝑦∈𝙂⧵ {𝑥} = arg max 𝑥∈𝙂

= arg max 𝑥∈𝙂

1 1 1 2 1 3 1 2 1 1

+ + + + + +

1 1

1 2 3 2 1

⎫ ⎧ ⎪ ⎪ ⎪ ⎪ ⎬ = arg min ⎨ 𝑥∈𝙂 ⎪ ⎪ ⎪ ⎪ ⎩ ⎭

12 12 12 12 12 12

⎫ ⎪ ⎧ ⎪ ⎪ ⎪ ⎬=⎨ ⎪ ⎪ ⎪ ⎩ ⎭ ⎪

by definition of ∁𝗁̇ (Definition 2.4 page 28) because 𝜙(𝑥) ≜ 𝑥−1 is strictly antitone and by Lemma 1.5 page 16

by Lemma 1.2 (page 15) + + + + + +

1 1 1 2 1 3 1 2


= arg min min 𝖽(𝑥, 𝑦) 𝑦∈𝙂⧵{𝑥}

1 2 1 1 1 1 1 2 1 3

+ + + + + +

1 3 1 2 1 1 1 1 1 2

+ + + + + +

1 2 1 3 1 2 1 1 1 1

+ + + + + +

1 1 1 2 1 3 1 2 1 1

⎫ ⎪ ⎧ ⎪ ⎪ ⎪ 1⎪ ⎬ = arg max 6 ⎨ 𝑥∈𝙂 ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎭

20 20 20 20 20 20

⎫ ⎪ ⎧ ⎪ ⎪ ⎪ ⎬=⎨ ⎪ ⎪ ⎪ ⎩ ⎭ ⎪

⚀ ⚁ ⚂ ⚃ ⚄ ⚅

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭


1 6

= arg min min 𝖽(𝑥, 𝑦) 𝑥∈𝙂

× × × × × 1 ×

−1

1 ∑ 𝖽(𝑥, 𝑦) 𝑦∈𝙂⧵{𝑥}

∁𝗆̇ (𝙂) ≜ arg min min 𝖽(𝑥, 𝑦) 𝖯(𝑦)

𝑥∈𝙂

2 3 2 1

1 1 ∑ 𝖽(𝑥, 𝑦) 6 𝑦∈𝙂⧵{𝑥}

⎧ ⎪ ⎪ ⎪ = arg max ⎨ 𝑥∈𝙂 ⎪ ⎪ ⎪ ⎩ 𝑥∈𝙂

× × × × × ×



= arg min {1, 1, 1, 1, 1, 1} 𝑥∈𝙂

= {⚀, ⚁, ⚂, ⚃, ⚄, ⚅} ∁𝖬̇ (𝙂) ≜ arg max min 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑥∈𝙂


= arg max {1, 1, 1, 1, 1, 1} 𝑥∈𝙂

version 0.50

by definition of 𝙂 by definition of ∁𝖬̇ (Definition 2.4 page 28) by ∁𝗆̇ (𝙂) result


Daniel J. Greenhoe

2016 Jul 4

10:12am UTC

page 37


= {⚀, ⚁, ⚂, ⚃, ⚄, ⚅}


̇ ̇ ̇ ̇ ̇ ̇ 𝖵𝖺𝗋(𝙂; ∁𝖺̇ ) = 𝖵𝖺𝗋(𝙂; ∁𝗀̇ ) = 𝖵𝖺𝗋(𝙂; ∁𝗁̇ ) = 𝖵𝖺𝗋(𝙂; ∁𝗆̇ ) = 𝖵𝖺𝗋(𝙂; ∁𝖬̇ ) = 𝖵𝖺𝗋(𝙂) ≜

2

̇ 𝖽 ∁(𝙂), 𝑥)] 𝖯(𝑥) ∑[ (


1 02 ∑ ( )6

̇ because ∁(𝙂) =𝙂

𝑥∈𝙂

=

𝑥∈𝙂

=0

by field property of additive identity element 0

✏ Example 2.5 (weighted spinner outcome subspace). The six value weighted spinner outcome subspace 𝙂 (Definition 2.1 page 27) illustrated to the right has the following geometric values: ̇ ̇ ∁(𝙂) = ∁𝖺̇ (𝙂) = ∁𝗀̇ (𝙂) = {1, 6} 𝖵𝖺𝗋(𝙂) = 53 ≈ 1.667 ̇ ∁𝗁̇ (𝙂) = {2, 5} 𝖵𝖺𝗋(𝙂; ∁𝗁̇ ) = 43 ≈ 1.333 ̇ ∁𝗆̇ (𝙂) = ∁𝖬̇ (𝙂) = {1, 2, 3, 4, 5, 6} 𝖵𝖺𝗋(𝙂; ∁𝗆̇ ) = 0 = 0

1 6

3 6

➃

1 6

➄

➂

➅

➁ ➀

1 6

1 6

3 6

The outcome center result is used later in Example 2.16 (page 57). Note that, unlike the weighted real die outcome subspace (Example 2.3 page 33), of the center measures of cardinality 2 or less, the harmonic center ∁𝗁̇ (𝙂) yields the lowest outcome variance (Definition 2.5 page 28). This is surprising since it suggests that ∁𝗁̇ (𝙂) is superior to all the other center measures (Definition 2.3 page 28, Definition 2.4 page 28), but yet unlike the other center measures, it yields center values that are not maximally likely. ✎PROOF: ̇ ∁(𝙂) ≜ arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑥∈𝙂

⎧ ⎪ 1⎪ = arg min max ⎨ 𝑦∈𝙂 10 𝑥∈𝙂 ⎪ ⎪ ⎩ ∁𝖺̇ (𝙂) ≜ arg min 𝑥∈𝙂


𝑦∈𝙂

∑

0×3 1×3 2×3 3×3 2×3 1×3

1×1 0×1 1×1 2×1 3×1 2×1


2×1 1×1 0×1 1×1 2×1 3×1

3×1 2×1 1×1 0×1 1×1 2×1

2×1 1×3 ⎧ 3×1 2×3 ⎫ ⎪ ⎪ 1⎪ 2×1 3×3 ⎪ min ⎨ 1 × 1 2 × 3 ⎬ = arg 10 ⎪ 𝑥∈𝙂 0×1 1×3 ⎪ ⎪ ⎪ 1×1 0×3 ⎭ ⎩ by definition of ∁̇ (Definition 2.4 page 28)

3 6 9 9 6 3

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

1 ⎧ ⎪ ⎪ =⎨ ⎪ ⎪ ⎩ 6

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

𝖺

𝑦∈𝙂

0 × 3+1 × 1+2 × 1+3 × 1+2 × 1+1 × 3 ⎧ 1 × 3+0 × 1+1 × 1+2 × 1+3 × 1+2 × 3⎫ ⎪ ⎪ 1 ⎪ ⎪ 2 × 3+1 × 1+0 × 1+1 × 1+2 × 1+3 × 3 min ⎨ 3 × 3 + 2 × 1 + 1 × 1 + 0 × 1 + 1 × 1 + 2 × 3 ⎬ = arg 10 ⎪ 𝑥∈𝙂 2 × 3+3 × 1+2 × 1+1 × 1+0 × 1+1 × 3⎪ ⎪ ⎪ 1 × 3+2 × 1+3 × 1+2 × 1+1 × 1+0 × 3⎭ ⎩ 𝖯(𝑦) ̇ ̇∁ (𝙂) ≜ arg min by definition of ∁𝗀 (Definition 2.4 page 28) 𝖽(𝑥, 𝑦) ] 𝗀 ∏ [ 𝑥∈𝙂 ⎧ ⎪ 1⎪ = arg min ⎨ 10 ⎪ 𝑥∈𝙂 ⎪ ⎩

11 15 19 19 15 11

1 ⎫ ⎪ ⎧ ⎪ ⎪ ⎪ ⎬=⎨ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 6

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭



𝖽(𝑥, 𝑦) ∏ [


6𝖯(𝑦) 16

] 1

⎞6 ⎛ 6𝖯(𝑦) ⎟ = arg min ⎜ 𝖽(𝑥, 𝑦) ]⎟ ∏ [ 𝑥∈𝙂 ⎜𝑦∈𝙂⧵ {𝑥} ⎠ ⎝ 6𝖯(𝑦) = arg min 𝖽(𝑥, 𝑦) ] ∏ [ 𝑥∈𝙂



2016 Jul 4

10:12am UTC


Daniel J. Greenhoe

version 0.50

page 38


⎧ ⎪ ⎪ = arg min ⎨ 𝑥∈𝙂 ⎪ ⎪ ⎩

3

1 23 33 23 13

11 × × 1 1 × 21 × 31 × 21 ×

× × × × × ×

21 × 31 11 × 21 × 11 1 1 × 21 × 11 31 × 21

⎛ ⎞ 1 ∁𝗁̇ (𝙂) ≜ arg min ⎜ 𝖯(𝑦)⎟ ∑ 𝖽(𝑥, 𝑦) ⎟ 𝑥∈𝙂 ⎜𝑦∈𝙂⧵ {𝑥} ⎝ ⎠ 1 = arg max 𝖯(𝑦) ∑ 𝖽(𝑥, 𝑦) 𝑥∈𝙂 𝑦∈𝙂⧵ {𝑥} = arg max 𝑥∈𝙂

= arg max 𝑥∈𝙂

13 23 33 23 13

⎫ ⎧ ⎪ ⎪ ⎪ ⎪ ⎬ = arg min ⎨ 𝑥∈𝙂 ⎪ ⎪ ⎪ ⎪ ⎩ ⎭

22 × 31 24 × 31 24 × 33 24 × 33 24 × 31 22 × 31

⎫ 1 ⎪ ⎧ ⎪ ⎫ ⎪ ⎪ ⎪ ⎪ ⎬=⎨ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 6 ⎪ ⎭ ⎭

−1


6𝖯(𝑦) ∑ 𝖽(𝑥, 𝑦) 𝑦∈𝙂⧵{𝑥} 3 1 3 2 3 3 3 2 3 1

1 1

+ + + + + +

by Lemma 1.2 (page 15) + + + + + +

1 1 1 2 1 3 1 2

∁𝗆̇ (𝙂) ≜ arg min min 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑦∈𝙂

⎧ ⎪ ⎪ = arg min min ⎨ 𝑦∈𝙂⧵ {𝑥} 𝑥∈𝙂 ⎪ ⎪ ⎩

1 2 1 1 2 1 3 2 2 3

3 6 9 6 3

∁𝖬̇ (𝙂) ≜ arg max min 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑥∈𝙂

11

× × × × ×

1 1 6𝖯(𝑦) ∑ 𝖽(𝑥, 𝑦) 6 𝑦∈𝙂⧵{𝑥}

⎧ ⎪ ⎪ ⎪ = arg max ⎨ 𝑥∈𝙂 ⎪ ⎪ ⎪ ⎩ 𝑥∈𝙂

21 31 21 11

× × × × × ×

𝑦∈𝙂

= arg max {1, 1, 1, 1, 1, 1} 𝑥∈𝙂

= {1, 2, 3, 4, 5, 6} ̇ ̇ ̇ 𝖵𝖺𝗋(𝙂; ∁𝖺̇ ) = 𝖵𝖺𝗋(𝙂; ∁𝗀̇ ) = 𝖵𝖺𝗋(𝙂) ̇ ≜ 𝖽2 ∁(𝙂), 𝑥) 𝖯(𝑥) ∑ (

1 2 1 1 1 1 1 2 1 3

+ + + + + +

1 3 1 2 1 1 1 1 1 2

+ + + + + +

1 2 1 3 1 2 1 1 1 1

+ + + + + +

3 1 3 2 3 3 3 2 3 1

⎫ ⎪ ⎧ ⎪ ⎪ ⎪ 1⎪ ⎬ = arg max 6 ⎨ 𝑥∈𝙂 ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎭

32 38 30 30 38 32

⎫ ⎪ ⎧ ⎪ 2 ⎫ ⎪ ⎪ ⎪ ⎪ ⎬=⎨ ⎬ ⎪ ⎪ 5 ⎪ ⎪ ⎩ ⎪ ⎭ ⎭ ⎪

by definition of ∁𝗆̇ (Definition 2.4 page 28) 3 2 1 ⎧ 2 3 2 ⎫ ⎪ ⎪ ⎪ 1 2 3 ⎪ = arg min ⎨ 1 2 ⎬ 𝑥∈𝙂 ⎪ 1 1 ⎪ ⎪ ⎪ 2 1 ⎩ ⎭

1 1 1 1 1 1

⎫ ⎪ ⎪ ⎧ ⎪ ⎪ = ⎬ ⎨ ⎪ ⎪ ⎪ ⎭ ⎪ ⎩

1 2 3 4 5 6

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

by definition of ∁𝖬̇ (Definition 2.4 page 28) by ∁𝗆̇ (𝙂) result


𝑥∈𝙂

=

∑

𝖽2 ({1, 6}, 𝑥) 𝖯(𝑥)


𝑥∈𝙂

3 1 1 1 1 3 = (0)2 + (1)2 + (2)2 + (2)2 + (1)2 + (0)2 6 6 6 6 6 6 2 10 5 = = 1 ≈ 1.667 = 6 3 3 2 ̇ ̇ ̇ (Definition 2.5 page 28) ̇ by definition of 𝖵𝖺𝗋 𝖵𝖺𝗋(𝙂; ∁𝗁 ) ≜ 𝖽 ∁ (𝙂), 𝑥) 𝖯(𝑥) ∑ (𝗁 𝑥∈𝙂

=

∑

𝖽2 ({2, 5}, 𝑥) 𝖯(𝑥)


𝑥∈𝙂

1 1 1 1 3 3 = (1)2 + (0)2 + (1)2 + (1)2 + (0)2 + (1)2 6 6 6 6 6 6 8 4 = = ≈ 1.333 6 3 ̇ ̇ ̇ 𝖵𝖺𝗋(𝙂; ∁𝖬 ) = 𝖵𝖺𝗋(𝙂; ∁𝗆̇ ) version 0.50


Daniel J. Greenhoe

2016 Jul 4

10:12am UTC

page 39


≜

∑

𝖽2 (∁𝗆̇ (𝙂), 𝑥) 𝖯(𝑥)


𝖽2 ({1, 2, 3, 4, 5, 6}, 𝑥) 𝖯(𝑥)


𝑥∈𝙂

=

∑

𝑥∈𝙂

=

∑

02 𝖯(𝑥) = 0

𝑥∈𝙂

✏ Example 2.6 (weighted ring). The weighted five element ring illustrated to the right has the geometric values below. The outcome center result is used later in Example 2.17 (page 59). ̇ ̇ 1 ∁(𝙂) = {4} 𝖵𝖺𝗋(𝙂) = 119 ≈ 1.222 9 1 1 ̇ 9 ∁𝖺̇ (𝙂) = {3, 4} 𝖵𝖺𝗋(𝙂; ∁𝖺̇ ) = 79 ≈ 0.778 2 ̇ ∁𝗀̇ (𝙂) = {3} 𝖵𝖺𝗋(𝙂; ∁𝗀̇ ) = 169 ≈ 1.778 0 29 𝙂 . ̇ ∁𝗁̇ (𝙂) = {1, 2, 3} 𝖵𝖺𝗋(𝙂; ∁𝗁̇ ) = 59 ≈ 0.556 3 2 ̇ ∁𝗆̇ (𝙂) = {0, 3, 4} 𝖵𝖺𝗋(𝙂; ∁𝗆̇ ) = 29 ≈ 0.222 9 4 3 9 ̇ ∁𝖬̇ (𝙂) = {1, 2} 𝖵𝖺𝗋(𝙂; ∁𝖬̇ ) = 169 ≈ 1.778 ✎PROOF: ̇ ∁(𝙂) ≜ arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑥∈𝙂


𝑦∈𝙂

1 = arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦)9 𝑦∈𝙂 9 𝑥∈𝙂 = arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦)9 𝑥∈𝙂

𝑦∈𝙂

𝖽(0, 0)𝖯(0)9 ⎧ ⎪ 𝖽(1, 0)𝖯(0)9 = arg min max ⎨ ⋮ 𝑦∈𝙂 𝑥∈𝙂 ⎪ ⎩ 𝖽(4, 0)𝖯(0)9 0×2 1×1 ⎧ 1×2 0×1 ⎪ = arg min max ⎨ 2 × 2 1 × 1 𝑦∈𝙂 𝑥∈𝙂 ⎪ 2×2 2×1 ⎩ 1×2 2×1 ∁𝖺̇ (𝙂) ≜ arg min 𝑥∈𝙂



∑

𝑥∈𝙂


𝖺

𝑦∈𝙂

1 𝖽(𝑥, 𝑦) 𝖯(𝑦)9 ∑9 𝑦∈𝙂 ∑

1 because 𝖿(𝑥) = 𝑥 is strictly isotone and by Lemma 1.2 (page 15) 9

𝖽(𝑥, 𝑦) 𝖯(𝑦)9

𝑦∈𝙂

⎧ ⎪ = arg min ⎨ 𝑥∈𝙂 ⎪ ⎩ ∁𝗀̇ (𝙂) ≜ arg min


1 because 𝖿(𝑥) = 𝑥 is strictly isotone and by Lemma 1.7 (page 17) 9 𝖽(0, 1)𝖯(1)9 𝖽(0, 2)𝖯(2)9 𝖽(0, 3)𝖯(3)9 𝖽(0, 4)𝖯(4)9 ⎪ 𝖽(1, 1)𝖯(1)9 𝖽(1, 2)𝖯(2)9 𝖽(1, 3)𝖯(3)9 𝖽(1, 4)𝖯(4)9 ⎫ ⎬ ⋮ ⋮ ⋮ ⋮ 𝖽(4, 1)𝖯(1)9 𝖽(4, 2)𝖯(2)9 𝖽(4, 3)𝖯(3)9 𝖽(4, 4)𝖯(4)9 ⎪ ⎭ 4 2×1 2×2 1×3 ⎧ 6 ⎫ ⎧ ⎫ 1×1 2×2 2×3 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ 0 × 1 1 × 2 2 × 3 ⎬ = arg min ⎨ 6 ⎬ = ⎨ ⎬ 𝑥∈𝙂 ⎪ 4 ⎪ 1×1 0×2 1×3 ⎪ ⎪ ⎪ 2×1 1×2 0×3 ⎭ ⎩ 4 ⎭ ⎩ 2 ⎭ by definition of ∁̇ (Definition 2.4 page 28)

0×2 2×2 1×2 1×2 2×2

∏

+ + + + +

2×1 0×1 2×1 1×1 1×1

+ + + + +

1×1 2×1 0×1 2×1 1×1

+ + + + +

1×2 1×2 2×2 0×2 1×2

+ + + + +

2×3 1×3 1×3 1×3 0×3

⎫ ⎪ ⎬ ⎪ ⎭

⎧ ⎪ = arg min ⎨ 𝑥∈𝙂 ⎪ ⎩

11 11 11 8 8

⎫ ⎪ ⎬ ⎪ ⎭

⎧ ⎫ ⎪ ⎪ = arg min ⎨ ⎬ 𝑥∈𝙂 ⎪ 3 ⎪ ⎩ 4 ⎭

by definition of ∁𝗀̇ (Definition 2.4 page 28)

𝖽(𝑥, 𝑦)𝖯(𝑦)

𝑦∈𝙂⧵{𝑥} 1

∏

𝖽(𝑥, 𝑦)9𝖯(𝑦) 9

𝑦∈𝙂⧵{𝑥} 1

⎡ ⎤9 = arg min ⎢ 𝖽(𝑥, 𝑦)9𝖯(𝑦) ⎥ ∏ ⎥ 𝑥∈𝙂 ⎢𝑦∈𝙂⧵ {𝑥} ⎣ ⎦ = arg min 𝑥∈𝙂 2016 Jul 4

10:12am UTC

∏

𝖽(𝑥, 𝑦)9𝖯(𝑦)

1

because 𝖿(𝑥) ≜ 𝑥 9 is strictly isotone and by Lemma 1.2 (page 15)

𝑦∈𝙂⧵{𝑥} A book concerning symbolic sequence processing

Daniel J. Greenhoe

version 0.50

page 40


⎧ ⎪ ⎪ = arg min ⎨ 𝑥∈𝙂 ⎪ ⎪ ⎩

2

2 12 12 22

21 × 11 21 21 × 11 × 21 11 × 11

× × × ×

1 𝖯(𝑦) ∑ (𝑦∈𝛺 𝖽(𝑥, 𝑦) )

= arg max

1 𝖯(𝑦) ∑ (𝑦∈𝛺 𝖽(𝑥, 𝑦) )

𝑥∈𝛺

= arg max 𝑥∈𝛺


16 8 8 2 4

⎫ ⎪ ⎬ ⎪ ⎭

⎧ ⎫ ⎪ ⎪ = arg min ⎨ ⎬ 𝑥∈𝙂 ⎪ 3 ⎪ ⎩ ⎭


1 1 𝖯(𝑦)9 ∑ ( 9 𝑦∈𝛺 𝖽(𝑥, 𝑦) ) 9𝖯(𝑦) ∑ 𝖽(𝑥, 𝑦) 𝑦∈𝛺


1 1 2 3 ⎧ 02 + 2 + 11 + 12 + 32 ⎪ + 0 + 2 + 1 + 1 ⎪ 2 = arg max ⎨ 21 + 21 + 0 + 22 + 31 𝑥∈𝙂 ⎪ 2 + 1 + 1 + 0 + 3 ⎪ 12 + 11 + 12 + 2 + 01 ⎩ 2 1 1 1

∁𝗆̇ (𝙂) ≜ arg min min 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑥∈𝙂

⎧ ⎪ = arg min ⎨ 𝑥∈𝙂 ⎪ ⎩

−1

∁𝗁̇ (𝙂) ≜ arg min 𝑥∈𝛺

× 12 × 23 ⎫ × 12 × 13 ⎪ ⎪ 2 2 × 13 ⎬ × 13 ⎪ ⎪ 2 × 1 ⎭


⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

⎧ 1⎪ = arg max ⎨ 2⎪ 𝑥∈𝙂 ⎩

10 13 13 13 11

⎫ ⎪ ⎬ ⎪ ⎭

⎧ 1 ⎫ ⎪ ⎪ = arg min ⎨ 2 ⎬ 𝑥∈𝙂 ⎪ 3 ⎪ ⎩ ⎭


1 𝖽(𝑥, 𝑦) 𝖯(𝑦)9 𝑦∈𝙂⧵{𝑥} 9

= arg min min 𝑥∈𝙂

= arg min min 𝖽(𝑥, 𝑦) 𝖯(𝑦)9 𝑥∈𝙂


⎧ ⎪ = arg min min ⎨ 𝑦∈𝙂⧵ {𝑥} 𝑥∈𝙂 ⎪ ⎩

2×1 2×2 1×2 2×1 1×2 1×1 2×2 1×1



= arg max {1, 2, 2, 1, 1} 𝑥∈𝙂

1 because 𝖿(𝑥) = 𝑥 is strictly isotone and by Lemma 1.7 (page 17) 9 1×1 1×2 2×3 1 0 ⎧ 2 ⎫ ⎧ ⎫ 2×1 1×2 1×3 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ 2 × 2 1 × 3 ⎬ = arg min ⎨ 2 ⎬ = arg min ⎨ ⎬ 𝑥∈𝙂 ⎪ 1 ⎪ 𝑥∈𝙂 ⎪ 3 ⎪ 2×1 1×3 ⎪ 1×1 1×2 ⎩ 4 ⎭ ⎩ 1 ⎭ ⎭ ̇ by definition of ∁ (Definition 2.4 page 28) 𝖬


= {1, 2} ̇ 𝖵𝖺𝗋(𝙂) ≜

∑

̇ 𝖽2 (∁(𝙂), 𝑥) 𝖯(𝑥)


𝖽2 ({4}, 𝑥) 𝖯(𝑥)


𝑥∈𝙂

=

∑

𝑥∈𝙂

2 1 1 2 3 11 = (1)2 + (2)2 + (2)2 + (1)2 + (0)2 = ≈ 1.222 9 9 9 9 9 9 ̇ (Definition 2.5 page 28) ̇ by definition of 𝖵𝖺𝗋 𝖵𝖺𝗋(𝙂; ∁𝖺̇ ) ≜ 𝖽2 ∁̇ (𝙂), 𝑥) 𝖯(𝑥) ∑ (𝖺 𝑥∈𝙂

=

∑

𝖽2 ({3, 4}, 𝑥) 𝖯(𝑥)

by ∁𝖺̇ (𝙂) result

𝑥∈𝙂

2 1 1 2 3 7 = (1)2 + (2)2 + (1)2 + (0)2 + (0)2 = ≈ 0.778 9 9 9 9 9 9 2 ̇ ̇ ̇ (Definition 2.5 page 28) ̇ by definition of 𝖵𝖺𝗋 𝖵𝖺𝗋(𝙂; ∁𝗀 ) ≜ 𝖽 ∁ (𝙂), 𝑥) 𝖯(𝑥) ∑ (𝗀 𝑥∈𝙂

=

∑

𝖽2 ({3}, 𝑥) 𝖯(𝑥)


𝑥∈𝙂

version 0.50


Daniel J. Greenhoe

2016 Jul 4

10:12am UTC

page 41


2 1 1 2 3 16 = (2)2 + (2)2 + (1)2 + (0)2 + (1)2 = ≈ 1.778 9 9 9 9 9 9 ̇ ̇ (Definition 2.5 page 28) 𝖵𝖺𝗋(𝙂; ∁𝗁̇ ) ≜ 𝖽2 ∁̇ (𝙂), 𝑥) 𝖯(𝑥) by definition of 𝖵𝖺𝗋 ∑ (𝗁 𝑥∈𝙂

=

∑


𝖽2 ({1, 2, 3}, 𝑥) 𝖯(𝑥)

𝑥∈𝙂

2 1 1 2 3 5 = (1)2 + (0)2 + (0)2 + (0)2 + (1)2 = ≈ 0.556 9 9 9 9 9 9 ̇ ̇ (Definition 2.5 page 28) 𝖵𝖺𝗋(𝙂; ∁𝗆̇ ) ≜ 𝖽2 ∁̇ (𝙂), 𝑥) 𝖯(𝑥) by definition of 𝖵𝖺𝗋 ∑ (𝗆 𝑥∈𝙂

=

∑


𝖽2 ({0, 3, 4}, 𝑥) 𝖯(𝑥)

𝑥∈𝙂

2 1 1 2 3 2 = (0)2 + (1)2 + (1)2 + (0)2 + (0)2 = ≈ 0.222 9 9 9 9 9 9 ̇ ̇ (Definition 2.5 page 28) 𝖵𝖺𝗋(𝙂; ∁𝖬̇ ) ≜ 𝖽2 ∁̇ (𝙂), 𝑥) 𝖯(𝑥) by definition of 𝖵𝖺𝗋 ∑ (𝖬 𝑥∈𝙂

=

∑


𝖽2 ({1, 2}, 𝑥) 𝖯(𝑥)

𝑥∈𝙂

1 1 2 3 16 2 = (1)2 + (0)2 + (0)2 + (1)2 + (2)2 = ≈ 1.778 9 9 9 9 9 9

✏ Example 2.7. The weighted five element structure illustrated to 2 9 the right has the following geometric values: 2 3 9 10 ̇∁(𝙂) = ∁̇ (𝙂) = {3} ̇ 0 𝖵𝖺𝗋(𝙂) = 9 ≈ 1.111 𝗀 4 ̇∁ (𝙂) = {3, 4} ̇ ̇ 𝙂 1 19 𝖵𝖺𝗋(𝙂; ∁𝖺 ) = 9 ≈ 0.444 𝖺 ̇ ∁𝗁̇ (𝙂) = {1, 2, 3} 𝖵𝖺𝗋(𝙂; ∁𝗁̇ ) = 95 ≈ 0.555 . 2 1 4 3 9 2 ̇∁ (𝙂) = {0, 3, 4} ̇ ̇ 𝖵𝖺𝗋(𝙂; ∁𝗆 ) = 9 ≈ 0.222 𝗆 9 ̇ ∁𝖬̇ (𝙂) = {1, 2} 𝖵𝖺𝗋(𝙂; ∁𝖬̇ ) = 97 ≈ 0.778 The outcome center result is used later in Example 2.18 (page 61). Note that only the operators ∁̇ ̇ ̇ and ∁𝗀̇ were able to successfully isolate a single center point (| ∁(𝙂) | = | ∁𝗀 (𝙂) | = | {3} | = 1). ✎PROOF: ̇ ∁(𝙂) ≜ arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑥∈𝙂


𝑦∈𝙂

1 = arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦)9 𝑦∈𝙂 9 𝑥∈𝙂 = arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦)9 𝑥∈𝙂

𝑦∈𝙂

𝖽(0, 0)𝖯(0)9 ⎧ ⎪ 𝖽(1, 0)𝖯(0)9 = arg min max ⎨ ⋮ 𝑦∈𝙂 𝑥∈𝙂 ⎪ ⎩ 𝖽(4, 0)𝖯(0)9 0×2 2×1 ⎧ 2×2 0×1 ⎪ = arg min max ⎨ 1 × 2 2 × 1 𝑦∈𝙂 𝑥∈𝙂 ⎪ 1×2 1×1 ⎩ 2×2 1×1 ∁𝖺̇ (𝙂) ≜ arg min 𝑥∈𝙂

∑

10:12am UTC

𝖺

𝑦∈𝙂

⎧ ⎪ = arg min ⎨ 𝑥∈𝙂 ⎪ ⎩ 2016 Jul 4


1 because 𝖿(𝑥) = 𝑥 is strictly isotone and by Lemma 1.7 (page 17) 9 𝖽(0, 1)𝖯(1)9 𝖽(0, 2)𝖯(2)9 𝖽(0, 3)𝖯(3)9 𝖽(0, 4)𝖯(4)9 ⎪ 𝖽(1, 1)𝖯(1)9 𝖽(1, 2)𝖯(2)9 𝖽(1, 3)𝖯(3)9 𝖽(1, 4)𝖯(4)9 ⎫ ⎬ ⋮ ⋮ ⋮ ⋮ 𝖽(4, 1)𝖯(1)9 𝖽(4, 2)𝖯(2)9 𝖽(4, 3)𝖯(3)9 𝖽(4, 4)𝖯(4)9 ⎪ ⎭ 1×1 1×2 2×3 6 ⎧ 4 ⎫ ⎧ ⎫ 2×1 1×2 1×3 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ 0 × 1 2 × 2 1 × 3 ⎬ = arg min ⎨ 4 ⎬ = arg min ⎨ ⎬ 𝑥∈𝙂 ⎪ 3 ⎪ 𝑥∈𝙂 ⎪ 3 ⎪ 2×1 0×2 1×3 ⎪ 1×1 1×2 0×3 ⎭ ⎩ ⎭ ⎩ 4 ⎭ by definition of ∁̇ (Definition 2.4 page 28)

0×2 2×2 1×2 1×2 2×2

+ + + + +

2×1 0×1 2×1 1×1 1×1

+ + + + +

1×1 2×1 0×1 2×1 1×1

+ + + + +

1×2 1×2 2×2 0×2 1×2


+ + + + +

2×3 1×3 1×3 1×3 0×3

⎫ ⎪ ⎬ ⎪ ⎭

⎧ ⎪ = arg min ⎨ 𝑥∈𝙂 ⎪ ⎩

Daniel J. Greenhoe

11 11 11 8 8

⎫ ⎪ ⎬ ⎪ ⎭

⎧ ⎫ ⎪ ⎪ = arg min ⎨ ⎬ 𝑥∈𝙂 ⎪ 3 ⎪ ⎩ 4 ⎭ version 0.50

page 42


∁𝗀̇ (𝙂) ≜ arg min 𝑥∈𝙂

∏

𝑦∈𝙂⧵{𝑥} 1




∏

𝖽(𝑥, 𝑦)9𝖯(𝑦) 9

𝑦∈𝙂⧵{𝑥} 1

⎡ ⎤9 9𝖯(𝑦) ⎥ ⎢ = arg min 𝖽(𝑥, 𝑦) ∏ ⎥ 𝑥∈𝙂 ⎢𝑦∈𝙂⧵ {𝑥} ⎣ ⎦ = arg min 𝑥∈𝙂

∏

1




⎧ ⎪ ⎪ = arg min ⎨ 𝑥∈𝙂 ⎪ ⎪ ⎩

2

2 12 12 22

21 × 11 21 1 2 × 11 × 21 11 × 11

× × × ×

1 𝖯(𝑦) ∑ (𝑦∈𝛺 𝖽(𝑥, 𝑦) )

= arg max

1 𝖯(𝑦) ∑ (𝑦∈𝛺 𝖽(𝑥, 𝑦) )

𝑥∈𝛺



24 23 23 21 22

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

⎧ ⎫ ⎪ ⎪ = arg min ⎨ ⎬ 𝑥∈𝙂 ⎪ 3 ⎪ ⎩ ⎭


1 1 𝖯(𝑦)9 ∑ 9 𝖽(𝑥, 𝑦) ) ( 𝑦∈𝛺 9𝖯(𝑦) ∑ 𝖽(𝑥, 𝑦) 𝑦∈𝛺


1 1 2 3 ⎧ 02 + 2 + 11 + 12 + 32 ⎪ + 0 + 2 + 1 + 1 ⎪ 2 = arg max ⎨ 21 + 21 + 0 + 22 + 31 𝑥∈𝙂 ⎪ 2 + 1 + 1 + 0 + 3 ⎪ 12 + 11 + 12 + 2 + 01 ⎩ 2 1 1 1


⎧ ⎪ ⎪ = arg min ⎨ 𝑥∈𝙂 ⎪ ⎪ ⎩

−1


× 12 × 23 ⎫ × 12 × 13 ⎪ ⎪ 2 2 × 13 ⎬ × 13 ⎪ ⎪ 2 × 1 ⎭


⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

⎧ 1⎪ = arg max ⎨ 2⎪ 𝑥∈𝙂 ⎩

10 13 13 13 10

⎫ ⎪ ⎬ ⎪ ⎭

⎧ 1 ⎫ ⎪ ⎪ =⎨ 2 ⎬ ⎪ 3 ⎪ ⎩ ⎭


1 𝖽(𝑥, 𝑦) 𝖯(𝑦)9 𝑦∈𝙂⧵{𝑥} 9




⎧ ⎪ = arg min min ⎨ 𝑦∈𝙂⧵ {𝑥} 𝑥∈𝙂 ⎪ ⎩

2×1 2×2 1×2 2×1 1×2 1×1 2×2 1×1



= arg max {1, 2, 2, 1, 1} 𝑥∈𝙂

= {1, 2} ̇ ̇ 𝖵𝖺𝗋(𝙂) = 𝖵𝖺𝗋(𝙂; ∁𝗀̇ ) ≜

∑

1 because 𝖿(𝑥) = 𝑥 is strictly isotone and by Lemma 1.7 (page 17) 9 1 1×1 1×2 2×3 0 ⎧ 2 ⎫ ⎧ ⎫ 2×1 1×2 1×3 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ 2 × 2 1 × 3 ⎬ = arg min ⎨ 2 ⎬ = ⎨ ⎬ 𝑥∈𝙂 ⎪ 1 ⎪ 2×1 1×3 ⎪ ⎪ 3 ⎪ 1×1 1×2 ⎩ 4 ⎭ ⎩ 1 ⎭ ⎭ by definition of ∁̇ (Definition 2.4 page 28) 𝖬


̇ by ∁(𝙂) and ∁𝗀̇ (𝙂) results

𝖽2 (∁𝗀̇ (𝙂), 𝑥) 𝖯(𝑥)


𝖽2 ({3}, 𝑥) 𝖯(𝑥)


𝑥∈𝙂

=

∑

𝑥∈𝙂

2 1 1 2 3 10 = (1)2 + (1)2 + (2)2 + (0)2 + (1)2 = ≈ 1.111 9 9 9 9 9 9 A book concerning version 0.50 symbolic sequence processing

Daniel J. Greenhoe

2016 Jul 4

10:12am UTC

page 43


̇ 𝖵𝖺𝗋(𝙂; ∁𝖺̇ ) ≜

∑

𝖽2 (∁𝖺̇ (𝙂), 𝑥) 𝖯(𝑥)


𝖽2 ({3, 4}, 𝑥) 𝖯(𝑥)

by ∁𝖺̇ (𝙂) result

𝑥∈𝙂

=

∑

𝑥∈𝙂

2 1 1 2 3 4 = (1)2 + (1)2 + (1)2 + (0)2 + (0)2 = ≈ 0.444 9 9 9 9 9 9 ̇ ̇ (Definition 2.5 page 28) 𝖵𝖺𝗋(𝙂; ∁𝗁̇ ) ≜ 𝖽2 ∁̇ (𝙂), 𝑥) 𝖯(𝑥) by definition of 𝖵𝖺𝗋 ∑ (𝗁 𝑥∈𝙂

=

∑

𝖽2 ({1, 2, 3}, 𝑥) 𝖯(𝑥)


𝑥∈𝙂

2 1 1 2 3 5 = (1)2 + (0)2 + (0)2 + (0)2 + (1)2 = ≈ 0.555 9 9 9 9 9 9 ̇ ̇ (Definition 2.5 page 28) 𝖵𝖺𝗋(𝙂; ∁𝗆̇ ) ≜ 𝖽2 ∁̇ (𝙂), 𝑥) 𝖯(𝑥) by definition of 𝖵𝖺𝗋 ∑ (𝗆 𝑥∈𝙂

=

∑

𝖽2 ({0, 3, 4}, 𝑥) 𝖯(𝑥)


𝑥∈𝙂

2 1 1 2 3 2 = (0)2 + (1)2 + (1)2 + (0)2 + (0)2 = ≈ 0.222 9 9 9 9 9 9 ̇ ̇ (Definition 2.5 page 28) 𝖵𝖺𝗋(𝙂; ∁𝖬̇ ) ≜ 𝖽2 ∁̇ (𝙂), 𝑥) 𝖯(𝑥) by definition of 𝖵𝖺𝗋 ∑ (𝖬 𝑥∈𝙂

=

∑

𝖽2 ({1, 2}, 𝑥) 𝖯(𝑥)

by ∁𝖬̇ (𝙂) result

𝑥∈𝙂

1 1 2 3 7 2 = (1)2 + (0)2 + (0)2 + (1)2 + (1)2 = ≈ 0.778 9 9 9 9 9 9

✏ 2 Example 2.8. The outcome subspace (Definition 2.1 page 27) illustrated 9 2 3 9 to the right, with a quasi-metric (Definition D.6 page 155) has the 0 following geometric values: 𝙂 1 19 ̇ ̇ ∁(𝙂) = ∁𝖺̇ (𝙂) = ∁𝗀̇ (𝙂) = ∁𝗁̇ (𝙂) = {3} 𝖵𝖺𝗋(𝙂) = 12 ≈ 1.333 9 2 ̇ 1 ∁𝗆̇ (𝙂) = {0, 4} 𝖵𝖺𝗋(𝙂; ∁𝗆̇ ) = 10 ≈ 1.111 4 3 9 9 ̇ 9 ∁𝖬̇ (𝙂) = {1, 2, 3} 𝖵𝖺𝗋(𝙂; ∁𝖬̇ ) = 95 ≈ 0.555 This is the first example in this section to use a directed graph (rather than an undirected graph Definition 1.17 page 8) and to require the use of a quasi-metric (Definition D.6 page 155) that is not a metric. Unlike Example 2.7 (page 41), which had neither of these restrictions, twice as many center operators (4 rather than 2) were able to successfully isolate a single center point. The outcome center result is used later in Example 2.19 (page 62). ✎PROOF:

̇ ∁(𝙂) ≜ arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑥∈𝙂

𝑦∈𝙂


1 ≜ arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦)9 𝑦∈𝙂 9 𝑥∈𝙂 ≜ arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦)9 𝑥∈𝙂

𝑦∈𝙂

𝖽(0, 0)𝖯(0)9 ⎧ ⎪ 𝖽(1, 0)𝖯(0)9 = arg min max ⎨ ⋮ 𝑦∈𝙂 𝑥∈𝙂 ⎪ ⎩ 𝖽(4, 0)𝖯(0)9 0×2 3×1 ⎧ 2×2 0×1 ⎪ = arg min max ⎨ 4 × 2 2 × 1 𝑦∈𝙂 𝑥∈𝙂 ⎪ 1×2 2×1 ⎩ 3×2 1×1 2016 Jul 4

10:12am UTC

1 because 𝖿(𝑥) = 𝑥 is strictly isotone and by Lemma 1.7 (page 17) 9 𝖽(0, 1)𝖯(1)9 𝖽(0, 2)𝖯(2)9 𝖽(0, 3)𝖯(3)9 𝖽(0, 4)𝖯(4)9 ⎪ 𝖽(1, 1)𝖯(1)9 𝖽(1, 2)𝖯(2)9 𝖽(1, 3)𝖯(3)9 𝖽(1, 4)𝖯(4)9 ⎫ ⎬ ⋮ ⋮ ⋮ ⋮ 𝖽(4, 1)𝖯(1)9 𝖽(4, 2)𝖯(2)9 𝖽(4, 3)𝖯(3)9 𝖽(4, 4)𝖯(4)9 ⎪ ⎭ 8 1×1 4×2 2×3 ⎧ 6 ⎫ ⎧ ⎫ 3×1 1×2 2×3 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ 0 × 1 3 × 2 1 × 3 ⎬ = arg min ⎨ 8 ⎬ = ⎨ ⎬ 𝑥∈𝙂 ⎪ 3 ⎪ 2×1 0×2 1×3 ⎪ ⎪ 3 ⎪ 4×1 2×2 0×3 ⎭ ⎩ ⎭ ⎩ 6 ⎭


Daniel J. Greenhoe

version 0.50

page 44


∁𝖺̇ (𝙂) ≜ arg min 𝑥∈𝙂



∑

𝑦∈𝙂

1 𝖽(𝑥, 𝑦) 𝖯(𝑦)9 ∑9 𝑦∈𝙂 ∑

𝑥∈𝙂



𝖽(𝑥, 𝑦) 𝖯(𝑦)9

𝑦∈𝙂

⎧ ⎪ = arg min max ⎨ 𝑦∈𝙂 𝑥∈𝙂 ⎪ ⎩ ∁𝗀̇ (𝙂) ≜ arg min

by definition of ∁𝖺̇ (Definition 2.4 page 28)


∏

0×2 2×2 4×2 1×2 3×2

+ + + + +

3×1 0×1 2×1 2×1 1×1

+ + + + +

1×1 3×1 0×1 2×1 4×1

+ + + + +

4×2 1×2 3×2 0×2 2×2

+ + + + +

2×3 2×3 1×3 1×3 0×3

⎧ ⎪ = arg min ⎨ 𝑥∈𝙂 ⎪ ⎩

⎫ ⎪ ⎬ ⎪ ⎭

18 15 19 9 15

⎫ ⎪ ⎬ ⎪ ⎭

⎧ ⎫ ⎪ ⎪ =⎨ ⎬ ⎪ 3 ⎪ ⎩ ⎭



𝑦∈𝙂⧵{𝑥} 1

∏

𝖽(𝑥, 𝑦)9𝖯(𝑦) 9

𝑦∈𝙂⧵{𝑥} 1

⎡ ⎤9 9𝖯(𝑦) ⎥ ⎢ = arg min 𝖽(𝑥, 𝑦) ∏ ⎥ 𝑥∈𝙂 ⎢𝑦∈𝙂⧵ {𝑥} ⎣ ⎦ = arg min 𝑥∈𝙂

∏

1




⎧ ⎪ ⎪ = arg min ⎨ 𝑥∈𝙂 ⎪ ⎪ ⎩

2

2 42 12 32

× × × ×

31 × 11 31 1 2 × 21 × 21 11 × 41

1 𝖯(𝑦) ∑ 𝖽(𝑥, 𝑦) ) (𝑦∈𝛺

= arg max

1 𝖯(𝑦) ∑ (𝑦∈𝛺 𝖽(𝑥, 𝑦) )

𝑥∈𝛺



⎧ ⎪ = arg min ⎨ 𝑥∈𝙂 ⎪ ⎩

384 192 432 24 144

⎫ ⎪ ⎬ ⎪ ⎭

⎧ ⎫ ⎪ ⎪ =⎨ ⎬ ⎪ 3 ⎪ ⎩ ⎭

−1


× 42 × 23 ⎫ × 12 × 23 ⎪ ⎪ 3 2 × 13 ⎬ × 13 ⎪ ⎪ 2 × 2 ⎭


1 1 𝖯(𝑦)9 ∑ ( 9 𝑦∈𝛺 𝖽(𝑥, 𝑦) ) 9𝖯(𝑦) ∑ 𝖽(𝑥, 𝑦) 𝑦∈𝛺


1 1 2 3 ⎧ 0 + 3 + 1 + 4 + 2 ⎫ ⎪ 2 + 0 + 1 + 2 + 3 ⎪ 3 1 2 ⎪ ⎪ 2 = arg max ⎨ 24 + 21 + 0 + 23 + 31 ⎬ 𝑥∈𝙂 ⎪ 2 + 1 + 1 + 0 + 3 ⎪ ⎪ 12 + 21 + 12 + 2 + 01 ⎪ ⎩ 3 ⎭ 1 4 2

⎧ 1⎪ = arg max ⎨ 2⎪ 𝑥∈𝙂 ⎩

20 27 28 36 35

⎫ ⎪ ⎬ ⎪ ⎭

⎧ ⎫ ⎪ ⎪ =⎨ ⎬ ⎪ 3 ⎪ ⎩ ⎭




1 𝖽(𝑥, 𝑦) 𝖯(𝑦)9 𝑦∈𝙂⧵{𝑥} 9



version 0.50




Daniel J. Greenhoe

2016 Jul 4

10:12am UTC

page 45


⎧ ⎪ = arg min min ⎨ 𝑥∈𝙂 𝑦∈𝙂⧵{𝑥} ⎪ ⎩

3×1 2×2 4×2 2×1 1×2 2×1 3×2 1×1



≜ arg max {1, 2, 2, 2, 1} 𝑥∈𝙂

1×1 4×2 3×1 1×2 3×2 2×1 4×1 2×2

2×3 2×3 1×3 1×3

⎫ ⎪ ⎬ ⎪ ⎭

⎧ ⎪ = arg min ⎨ 𝑥∈𝙂 ⎪ ⎩

1 2 2 2 1

⎫ ⎪ ⎬ ⎪ ⎭

0 ⎧ ⎪ =⎨ ⎪ ⎩ 4

⎫ ⎪ ⎬ ⎪ ⎭

by definition of ∁𝖬̇ (Definition 2.4 page 28) by ∁𝗆̇ (𝙂) result

= {1, 2, 3} ̇ ̇ ̇ ̇ ̇ 𝖵𝖺𝗋(𝙂) = 𝖵𝖺𝗋(𝙂; ∁𝖺̇ ) = 𝖵𝖺𝗋(𝙂; ∁𝗀̇ ) = 𝖵𝖺𝗋(𝙂; ∁𝗁̇ )by ∁(𝙂), ∁𝖺̇ (𝙂), ∁𝗀̇ (𝙂), and ∁𝗁̇ (𝙂) results ≜

∑

𝖽2 (∁𝗁̇ (𝙂), 𝑥) 𝖯(𝑥)


𝖽2 ({3}, 𝑥) 𝖯(𝑥)


𝑥∈𝙂

=

∑

𝑥∈𝙂

2 1 1 2 3 12 4 = (1)2 + (2)2 + (2)2 + (0)2 + (1)2 = = ≈ 1.333 9 9 9 9 9 9 3 2 ̇ ̇ ̇ ̇ (Definition 2.5 page 28) 𝖵𝖺𝗋(𝙂; ∁𝗆 ) ≜ 𝖽 ∁ (𝙂), 𝑥) 𝖯(𝑥) by definition of 𝖵𝖺𝗋 ∑ (𝗆 𝑥∈𝙂

=

∑

𝖽2 ({0, 4}, 𝑥) 𝖯(𝑥)


𝑥∈𝙂

1 1 2 3 10 2 ≈ 1.111 = (0)2 + (1)2 + (1)2 + (2)2 + (0)2 = 9 9 9 9 9 9 ̇ ̇ (Definition 2.5 page 28) 𝖵𝖺𝗋(𝙂; ∁𝖬̇ ) ≜ 𝖽2 ∁̇ (𝙂), 𝑥) 𝖯(𝑥) by definition of 𝖵𝖺𝗋 ∑ (𝖬 𝑥∈𝙂

=

∑

𝖽2 ({1, 2, 3}, 𝑥) 𝖯(𝑥)


𝑥∈𝙂

2 1 1 2 3 5 = (1)2 + (0)2 + (0)2 + (0)2 + (1)2 = ≈ 0.555 9 9 9 9 9 9

✏ Example 2.9 (DNA). Genomic Signal Processing (GSP) analyzes biological sequences called genomes. 𝘛 ◻, 𝘊 𝘈 ◻, These sequences are constructed over a set of 4 symbols that are commonly referred to as ◻, 𝘎 each of which corresponds to a nucleobase (adenine, thymine, cytosine, and guanine, reand ◻, spectively).3 A typical genome sequence contains a large number of symbols (about 3 billion for humans, 29751 for the SARS virus).4 1 1 𝘛 ◻, 𝘊 ◻}, 𝘎 𝘈 ◻, Let 𝙂 ≜ ({◻, 𝖽, ≤, 𝖯) be the outcome subspace (Definition 2.1 page 27) gener4 4 𝘛 𝘈 ated by a genome where 𝖽 is the discrete metric (Definition D.8 page 157), ≤≜ ∅ (com𝘛 = 𝖯(◻) 𝘊 = 𝖯(◻) 𝘎 = 1 . This space is illus𝘈 = 𝖯(◻) pletely unordered set), and 𝖯(◻) 4 𝘊 𝘎 trated by the graph (Definition 1.17 page 8) to the right with shaded center (Definition 2.3 1 1 4 4 page 28). The graph has the following geometric values: ̇ 𝘛 ◻, 𝘊 ◻} 𝘎 𝘈 ◻, (shaded in illustration) ∁(𝙂) = {◻, (Definition 2.3 page 28) ̇∁ (𝙂) = {◻, 𝘛 𝘊 𝘎 𝘈 (shaded in illustration) ◻, ◻, ◻} (Definition 2.4 page 28) 𝖺 ̇ 𝖵𝖺𝗋(𝙂) = 0 (Definition 2.5 page 28) 3


2016 Jul 4

10:12am UTC


Daniel J. Greenhoe

version 0.50

page 46

2.2. RANDOM VARIABLES ON OUTCOME SUBSPACES

✎PROOF: ̇ ∁(𝙂) ≜ arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑦∈𝙂

𝑥∈𝙂

1 4

= arg min max 𝖽(𝑥, 𝑦) 𝑦∈𝙂

𝑥∈𝙂

by definition of 𝙂 1 because 𝖿 (𝑥) = 𝑥 is strictly isotone and by Lemma 1.7 (page 17) 4

= arg min max 𝖽(𝑥, 𝑦) 𝑦∈𝙂

𝑥∈𝙂


= arg min {1, 1, 1, 1}

because for the discrete metric (Definition D.8 page 157), max 𝖽 = 1

𝑥∈𝙂 𝘛 ◻, 𝘊 ◻} 𝘎 𝘈 ◻, = {◻,

∁𝖺̇ (𝙂) ≜ arg min 𝑥∈𝙂

∑

𝖺

𝑦∈𝙂


∑

𝖽(𝑥, 𝑦)

𝑦∈𝙂



by definition of 𝙂 by definition of ∁̇ (Definition 2.4 page 28)

∑

𝖽(𝑥, 𝑦)

𝑦∈𝙂

1 6

by definition of 𝙂 1 because 𝖿 (𝑥) = 𝑥 is strictly isotone and by Lemma 1.7 (page 17) 4

3 0+1+1+1 ⎧ ⎧ ⎫ ⎪ 3 ⎫ ⎪ ⎪ ⎪ 1 + 0 + 1 + 1 𝘛 ◻, 𝘊 ◻} 𝘎 𝘈 ◻, = arg min ⎨ 1 + 1 + 0 + 1 ⎬ = arg min ⎨ 3 ⎬ = {◻, 𝑥∈𝙂 ⎪ 𝑥∈𝙂 ⎪ ⎪ ⎪ 3 ⎩ ⎭ ⎩1+1+1+0⎭ 2 ̇ ̇ ̇ (Definition 2.5 page 28) 𝖵𝖺𝗋(𝙂) ≜ 𝖽 ∁(𝙂), 𝑥)] 𝖯(𝑥) by definition of 𝖵𝖺𝗋 ∑[ ( 𝑥∈𝙂

=

∑

𝑥∈𝙂

(0)

1 6

=0

̇ because ∁(𝙂) =𝙂 by field property of additive identity element 0

✏

2.2 Random variables on outcome subspaces 2.2.1 Definitions The traditional random variable (Definition 1.39 page 22) is a mapping from a probability space (Definition 1.38 page 22) to the real line (Definition 1.35 page 21). This paper extends this definition to include functions with additional structure in the domain and expanded structure in the range (next definition). Definition 2.13. A function 𝖷 ∈ 𝖧𝙂 (Definition 1.6 page 6) is an outcome random variable if 𝙂 is an OUTCOME SUBSPACE (Definition 2.1 page 27) and 𝖧 is an ORDERED QUASI-METRIC SPACE (Definition 1.34 page 20). The definitions of outcome expected value and outcome variance (next definition) of an outcome random variable are, in essence, identical to the outcome center (Definition 2.14 page 46) and outcome variance (Definition 2.14 page 46) of outcome subspaces (Definition 2.1 page 27) that outcome random variables map from and by induction, to. Definition 2.14. Let 𝙂 be an OUTCOME SUBSPACE (Definition 2.1 page 27), 𝖧 an ORDERED QUASI-METRIC SPACE (Definition 1.34 page 20), and 𝖷 be an OUTCOME RANDOM VARIABLE (Definition 2.13 page 46) in ∈ 𝖧𝙂 . Let 𝙃 ≜ (𝛺, ≤,̇ 𝖽,̇ 𝖯̇ ) be the OUTCOME SUBSPACE induced by 𝖧, 𝙂, and 𝖷. Let 𝖤̈ 𝑥 be a function from 𝛺 to version 0.50


Daniel J. Greenhoe

2016 Jul 4

10:12am UTC

page 47


the power set 𝟚𝛺 . ̈ The outcome expected value 𝖤(𝖷)

̈ of 𝖷 is 𝖤(𝖷) ≜ arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦). 𝑦∈𝛺

𝑥∈𝛺

̈ ̈ 𝖵𝖺𝗋(𝖷; 𝖤𝑥 ) of 𝖷 is 𝖵𝖺𝗋(𝖷) ≜

The outcome variance

2

∑

𝖽 (𝖤𝑥 (𝖷), 𝑥) 𝖯(𝑥).

𝑥∈𝛺

̈ ̈ ̈ where 𝖤̈ is the OUTCOME EXPECTED VALUE function. Moreover, 𝖵𝖺𝗋(𝖷) ≜ 𝖵𝖺𝗋(𝖷; 𝖤),

2.2.2 Properties Theorem 2.1. Let 𝖷 ∈ 𝖧𝙂 be a RANDOM VARIABLE (Definition 2.13 page 46) on an ORDERED QUASI-METRIC SPACE (Definition 1.34 page 20) 𝖧. Let 𝙃 ≜ (𝛺, ≤,̇ 𝖽,̇ 𝖯̇ ) be the OUTCOME SUBSPACE (Definition 2.1 page 27) induced ̈ by 𝖧, 𝙂, and 𝖷. Let 𝖵𝖺𝗋(𝖷) be the TRADITIONAL VARIANCE (Definition 1.40 page 22) of 𝖷. Let 𝖵𝖺𝗋(𝖷) be the OUTCOME SUBSPACE VARIANCE of 𝖷 (Definition 2.14 page 46). ̈ 𝖤) = 𝖵𝖺𝗋(𝖷) } ⟹ { 𝖧 ≜ (ℝ, |⋅|, ≤) is the REAL LINE (Definition 1.35 page 21) } { 𝖵𝖺𝗋(𝖷; ✎PROOF: ̈ 𝖵𝖺𝗋(𝖷; 𝖤) ≜

𝖽2 (𝖤(𝖷), 𝑥) 𝖯(𝑥)

̈ (Definition 2.14 page 46) by definition of 𝖵𝖺𝗋

|𝖤(𝖷) − 𝑥|2 𝖯(𝑥)

by definition of real line 𝖧 (Definition 1.35 page 21)

(𝑥 − 𝖤(𝖷))2 𝖯(𝑥) 𝖽𝑥

by definition of Lebesgue integration on ℝ

∑

𝑥∈𝖧

=

∑

𝑥∈ℝ

=

∫ ℝ

= 𝖵𝖺𝗋(𝖷)

by definition of 𝖵𝖺𝗋 (Definition 1.40 page 22)

✏ Remark 2.3. Despite the correspondence of traditional variance and outcome variance on the real line as demonstrated in Theorem 2.1 (page 47), the situation is different for expected values. Even when both are calculated on the same real line, the traditional expected value 𝖤(𝖷) (Definition 1.40 page 22) ̈ and the outcome expected value 𝖤(𝖷) (Definition 2.14 page 46) don't always yield the same value. Demonstrations of this include Example 2.14 (page 54) and Example 2.17 (page 59). However, there is one common situation in which the two statistics do correspond (next theorem). Theorem 2.2. Let 𝖷, 𝖧, 𝙂 be defined as in Theorem 2.1 (page 47). Let 𝖤(𝖷) be the TRADITIONAL ̈ EXPECTED VALUE (Definition 1.40 page 22) and 𝖤(𝖷) the OUTCOME EXPECTED VALUE of 𝖷 (Definition 2.14 page 46). 1. 𝖧 ≜ (ℝ, |⋅|, ≤) ( REAL LINE Definition 1.35 page 21) and ̈ = 𝑎 = 𝖤(𝖷) } ⟹ { 𝖤(𝖷) { 2. 𝖯(𝑎 − 𝑥) = 𝖯(𝑎 + 𝑥) ∀𝑥∈ℝ ( SYMMETRIC about 𝑎) } ✎PROOF: ̈ 𝖤(𝖷) ≜ arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑥∈ℝ

𝑦∈ℝ

= arg min max |𝑥 − 𝑦|𝖯(𝑦) 𝑥∈ℝ

𝑦∈ℝ

by definition of 𝖤̈ (Definition 2.14 page 46) by definition of real line (Definition 1.35 page 21) 𝖿(𝑦) ≜ |𝑥 − 𝑦| 𝖯(𝑦)

= 𝑎

𝑥

𝑦

𝑎

because 𝗁(𝑥) ≜ max |𝑥 − 𝑦|𝖯(𝑦) is minimized when 𝑥 = 𝑎 𝑦∈ℝ

= 𝖤(𝖷)

by Proposition 1.3 (page 23)

✏ 2016 Jul 4

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version 0.50

page 48


Theorem 2.3. Let 𝙂 ≜ (𝛺𝙂 , 𝖽𝙂 , ≤𝙂 , 𝖯𝙂 ) be an OUTCOME SUBSPACE (Definition 2.1 page 27). Let 𝖧 ≜ (𝛺𝖧 , 𝖽𝖧 , ≤𝖧 ) be an ORDERED QUASI-METRIC SPACE (Definition 1.34 page 20). Let 𝖪 ≜ (𝛺𝖪 , 𝖽𝖪 , ≤𝖪 ) be an ORDERED QUASI-METRIC SPACE (Definition 1.34 page 20). 𝙂 Let 𝖷 ∈ 𝖧 be a RANDOM VARIABLE from 𝙂 onto 𝖧 (Definition 2.13 page 46). Let 𝖿 ∈ 𝛺𝖪 𝛺𝖧 be a function from 𝛺𝖧 onto 𝛺𝖪 ( PULLBACK) (Theorem 1.7 page 19). ℝ Let 𝜙 ∈ ℝ be a function from ℝ into ℝ ( PUSHFORWARD) (Definition D.11 page 158). Let 𝙃 ≜ (𝛺𝖧 , 𝖽𝖧 , ≤𝖧 , 𝖯𝖧 ) be an OUTCOME SUBSPACE induced by 𝙂, 𝖧, and 𝖷. Let 𝙆 ≜ (𝛺𝖪 , 𝖽𝖪 , ≤𝖪 , 𝖯𝖪 ) be an OUTCOME SUBSPACE induced by 𝖪, 𝙃 and 𝖿. 1. 𝖿 is INJECTIVE and ⎫ ⎧ ⎪ ⎪ ̈ ̈ ]} and ⎬ ⟹ = 𝖿 [𝖤(𝖷) {𝖤[𝖿(𝖷)] ⎨ 2. 𝜙 is STRICTLY ISOTONE ⎪ ⎪ ⎩ 3. 𝖽𝖧 (𝖿(𝑥), 𝖿 (𝑦))𝖯(𝑦) = 𝜙[𝖽𝖧 (𝑥, 𝑦)𝖯(𝑦)] ⎭ ✎PROOF: ̈ 𝖤[𝖿(𝖷)] = arg min max 𝖽𝖪 (𝑥, 𝑦)𝖯𝖪 (𝑦) 𝑥∈𝛺𝖪

𝑦∈𝛺𝖪

by definition of 𝖤̈ (Definition 2.14 page 46) and 𝖪

= 𝖿 arg min max 𝖽𝖪 (𝖿 (𝑥), 𝖿 (𝑦))𝖯𝖪 (𝖿(𝑦)) [ 𝑥∈𝛺𝖧 𝑦∈𝛺𝖧 ]

by 𝖿 bijection hypothesis

= 𝖿 arg min max 𝖽𝖧 (𝖿 (𝑥), 𝖿 (𝑦))𝖯𝖧 (𝑦) [ 𝑥∈𝛺𝖧 𝑦∈𝛺𝖧 ]

by 𝖿 bijection hypothesis

= 𝖿 arg min max 𝜙[𝖽𝖧 (𝑥, 𝑦)𝖯𝖧 (𝑦)] [ 𝑥∈𝛺𝖧 𝑦∈𝛺𝖧 ]

by 𝖽𝖧 hypothesis

= 𝖿 arg min max 𝖽𝖧 (𝑥, 𝑦)𝖯𝖧 (𝑦) [ 𝑥∈𝛺𝖧 𝑦∈𝛺𝖧 ]

by 𝜙 is strictly isotone hypothesis and Lemma 1.7 page 17

̈ = 𝖿 𝖤(𝖷)

by definition of 𝖤̈ (Definition 2.14 page 46) and 𝖷

✏ Corollary 2.1. Let 𝖧 be an ORDERED METRIC SPACE (Definition 1.34 page 20) and 𝖷 ∈ 𝖧𝙂 a RANDOM VARIABLE (Definition 2.13 page 46) onto 𝖧. Let (ℝ, |⋅|, ≤) be the REAL LINE ORDERED METRIC SPACE (Definition 1.35 page 21). ̈ ̈ = 𝑎𝖤(𝖷) ∀𝑎∈ℝ⊢ } 𝖧 = (ℝ, |⋅|, ≤) ⟹ { 𝖤(𝑎𝖷) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ (real line)

✎PROOF: 1. Proof for 𝑎 = 0 case: ̈ ⋅ 𝖷) = arg min max 𝖽(𝑥, 𝑦)𝖯(𝑦) 𝖤(0 𝑥∈0⋅𝖧 𝑦∈0⋅𝖧

by definition of 𝖤̈ (Definition 2.14 page 46)

= arg min max 𝖽(𝑥, 𝑦)𝖯(𝑦) 𝑥∈{0} 𝑦∈{0}

= arg min max 𝖽(0, 0)𝖯(𝑦) 𝑥∈{0} 𝑦∈{0}

= arg min max 0𝖯(𝑦) 𝑥∈{0} 𝑦∈{0}

by nondegenerate property of 𝖽 (Definition D.7 page 155)

=0 ̈ = 0 ⋅ 𝖤(𝖷) 2. Proof for 𝑎 > 0 case: 𝖽(𝖿 (𝑥), 𝖿 (𝑦))𝖯(𝑦) ≜ |𝑎𝑥 − 𝑎𝑦|𝖯(𝑦) = |𝑎||𝑥 − 𝑦|𝖯(𝑦) ≜ |𝑎||𝖽(𝑥, 𝑦)𝖯(𝑦)| ̈ ̈ 𝖤(𝑎𝖷) = 𝑎𝖤(𝖷) version 0.50

because 𝖿 (𝑥) = 𝑎𝑥 is strictly isotone on the real line and by Theorem 2.3 (page 48) A book concerning symbolic sequence processing

Daniel J. Greenhoe

2016 Jul 4

10:12am UTC

page 49


✏

2.2.3 Problem statement The traditional random variable 𝖷 (Definition 1.39 page 22) is a function that maps from a stochastic process to the real line (Definition 1.35 page 21). The traditional expectation value 𝖤(𝖷) of 𝖷 is then often a poor choice of a statistic when the stochastic process that 𝖷 maps from is a structure other than the real line or some substructure of the real line. There are two fundamental problems: 1. A traditional random variable 𝖷 maps to the linearly ordered real line. However, 𝖷 often maps from a random process that is non-linearly ordered (or even unordered Definition 1.20 page 8, Definition 1.21 page 8). 2. A traditional random variable 𝖷 maps to the real line with a metric geometry (Remark 2.2 page 32) induced by the usual metric (Definition D.9 page 157). But many random processes have a fundamentally different metric geometry, a common one being that induced by the discrete metric (Definition D.8 page 157). Thus, the order structure of the domain and range of 𝖷 are often fundamentally dissimilar, leading to statistics, such as 𝖤(𝖷), that are of poor quality with regards to qualitative intuition and quantitative variance (expected error) measurements, and of dubious suitability for tasks such as decision making, prediction, and hypothesis testing. ̈ + 𝖸) = 𝖤(𝖷) ̈ Remark 2.4. Unlike in traditional statistical processing, it in general not true that 𝖤(𝖷 + ̈ 𝖤(𝖸). See Example 2.33 (page 76) for a counter example. Remark 2.5. A possible solution to the traditional random variable order and metric geometry problem is to allow the random variable to map into the complex plane (Example 1.10 page 21) with the usual metric, rather than into the real line only. However, this is a poor solution, as demonstrated in Example 2.21 (page 64).

2.2.4 Examples Fair die examples 𝙂

(ℝ, |⋅|, ≤) 6 5 4 3 2 1

𝖷(⋅)

1 6 1 6

(ℤ, |⋅|, ≤)

⚅

⚄

⚂

1 6

3.5 1 6

⚃

⚁ 1 6

⚀

1 6

𝖸(⋅)

6 5 4 3 2 1

Figure 2.2: random variable mappings from the fair die to the real line and integer line Example 2.10 (fair die mappings to real line and integer line). Let 𝙂 be the fair die outcome subspace 𝙂 (Example 2.1 page 29). Let 𝖷 ∈ (ℝ, |⋅|, ≤) be a random variable (Definition 2.13 page 46) mapping from 𝙂 to the real line (Definition 1.35 page 21), and 𝖸 ∈ (ℤ, ≤, |⋅|)𝙂 be a random variable (Definition 2.13 page 46) mapping from 𝙂 to the integer line (Definition 1.36 page 21), as illustrated in Figure 2.2 (page 49). Let 𝖤 be the traditional expected value function (Definition 1.40 page 22), 𝖵𝖺𝗋 the traditional variance function (Definition 1.40 page 22), ̈ the outcome variance function 𝖤̈ the outcome expected value function (Definition 2.14 page 46), and 𝖵𝖺𝗋 2016 Jul 4

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Daniel J. Greenhoe

version 0.50

page 50


(Definition 2.14 page 46).

This yields the following statistics: ̇ geometry of 𝙂: ∁(𝙂) traditional statistics on real line: 𝖤(𝖷) ̈ outcome subspace statistics on real line: 𝖤(𝖷) ̈ outcome subspace statistics on integer line: 𝖤(𝖸)

= {⚀, ⚁, ⚂, ⚃, ⚄, ⚅} ̈ = 3.5 𝖵𝖺𝗋(𝖷; 𝖤) = 35 ≈ 2.917 12 35 ≈ 2.917 ̈ = {3.5} 𝖵𝖺𝗋(𝖷; 𝖤)̈ = 12 20 ≈ 1.667 ̈ = {3, 4} 𝖵𝖺𝗋(𝖸; 𝖤)̈ = 12

✎PROOF: ̇ ∁(𝙂) = {⚀, ⚁, ⚂, ⚃, ⚄, ⚅}

by Example 2.1 page 29

𝖤(𝖷) ≜

by definition of 𝖤 (Definition 1.40 page 22)

∑

𝑥𝖯(𝑥)

𝑥∈ℝ

=

1 1 + 2 + 3 + 4 + 5 + 6 21 7 = = = 3.5 𝑥 = ∑ 6 6 6 2 𝑥∈ℝ

̈ 𝖵𝖺𝗋(𝖷; 𝖤) = 𝖵𝖺𝗋(𝖷) ≜

∑

by Theorem 2.1 page 47 2

[𝑥 − 𝖤(𝖷)] 𝖯(𝑥)


𝑥∈ℝ

=

7 21 𝑥− ) ∑( 2 6 𝑥∈ℝ

by 𝖤(𝖷) result

7 2 7 2 7 2 7 2 7 2 7 2 1 = (1 − ) + (2 − ) + (3 − ) + (4 − ) + (5 − ) + (6 − ) [ 2 2 2 2 2 2 ]6 1 = [(2 − 7)2 + (4 − 7)2 + (6 − 7)2 + (8 − 7)2 + (10 − 7)2 + (12 − 7)2 ] 2 2 ×6 25 + 9 + 1 + 1 + 9 + 25 70 35 = = = ≈ 2.917 24 24 12 ̈ 𝖤(𝖷) = 𝖤(𝖷) by Theorem 2.2 (page 47) 7 = { } = {3.5} by 𝖤(𝖷) result 2 ̈ ̈ ̈ (Definition 2.14 page 46) 𝖵𝖺𝗋(𝖷; 𝖤)̈ ≜ 𝖽2 𝖤(𝖷), 𝑥) 𝖯(𝑥) by definition of 𝖵𝖺𝗋 ∑ ( 𝑥∈ℝ

=

∑

̈ by 𝖤(𝖷) and 𝖤(𝖷) results

𝖽2 (𝖤(𝖷), 𝑥) 𝖯(𝑥)

𝑥∈ℝ

̈ 𝖵𝖺𝗋(𝖷; 𝖤) 35 = ≈ 2.917 12 ̈ 𝖤(𝖸) ≜ arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑥∈ℤ

by 𝖵𝖺𝗋(𝖷) result by definition of 𝖤̈ (Definition 2.14 page 46)

𝑦∈𝖧

= arg min max |𝑥 − 𝑦| 𝑥∈ℤ


𝑦∈𝖧

1 6

by definition of integer line (Definition 1.36 page 21) and 𝙂

= arg min max |𝑥 − 𝑦| 𝑥∈ℤ

𝑦∈𝖧

0 1 ⎧ ⎪ 1 0 ⎪ 2 1 = arg min max ⎨ 3 2 𝑦∈𝖧 𝑥∈ℤ ⎪ 4 3 ⎪ ⎩ 5 4 2 ̈ ̈ 𝖵𝖺𝗋(𝖸; 𝖤)̈ ≜ 𝖽 𝖤(𝖸), 𝑥) 𝖯(𝑥) ∑ (

2 1 0 1 2 3

3 2 1 0 1 2

4 3 2 1 0 1

5 4 3 2 1 0

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

1 because 𝜙(𝑥) = 𝑥 is strictly isotone and by Lemma 1.7 page 17 6 5 ⎧ ⎧ ⎪ 4 ⎫ ⎪ ⎪ ⎫ ⎪ ⎪ 3 ⎪ ⎪ 3 ⎪ = arg min ⎨ 3 ⎬ = arg min ⎨ 4 ⎬ 𝑥∈ℤ ⎪ 𝑥∈ℤ ⎪ ⎪ 4 ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ ⎩ 5 ⎭ ̈ (Definition 2.14 page 46) by definition of 𝖵𝖺𝗋

𝑥∈ℤ

=

∑

𝑥∈ℤ

𝖽2 ({3, 4}, 𝑥)

1 6

̈ by 𝖤(𝖸) result and definition of 𝙂

1 = (|3 − 1|2 + |3 − 2|2 + |3 − 3|2 + |4 − 4|2 + |4 − 5|2 + |4 − 6|2 ) 6 A book concerning version 0.50 Daniel J. Greenhoe symbolic sequence processing

2016 Jul 4

10:12am UTC

page 51


1 10 5 = (4 + 1 + 0 + 0 + 1 + 4) = = 6 6 3

✏ The random variable mappings in Example 2.10 (page 49) have two fundamental problems: 1. The order structure of the fair die and the order structure of real line are inherently dissimilar in that while the bijective (Definition 1.14 page 7) mapping 𝖷 is trivially order preserving (Definition 1.32 page 15), its inverse is not order preserving . And this is a problem. In the linearly ordered (Definition 1.21 page 8) range of 𝖷, it is true that 𝖷(⚀) = 1 < 2 = 𝖷(⚁). But in the unordered domain of 𝖷 {⚀, ⚁, … , ⚅}, it is not true that ⚀ < ⚁; rather ⚀ and ⚁ are simply symbols without order. This causes problems when we attempt to use the random variable to make statistical inferences involving moments (Definition 2.2 page 27). The traditional expected value (Definition 1.40 page 22) of a fair die (Example 2.1 page 29) is 𝖤(𝑋) = 1 (1 + 2 + ⋯ + 6) = 3.5. This implies that we expect 6 the outcome of ⚂ or ⚃ more than we expect the outcome of say ⚀ or ⚁. But these results have no relationship with reality or with intuition because the values of a fair die are merely symbols. For a fair die, we would expect any pair of values equally. We would not expect the outcome [⚂ or ⚃] more than we would expect the outcome [⚀ or ⚁], or more than we would expect any other outcome pair. 2. The metric geometry (Remark 2.2 page 32) of the fair die outcome subspace is very dissimilar to the metric geometry of the real line (Definition 1.35 page 21) that it is mapped to by the random variable 𝖷. And this is a problem. In the metric geometry of the fair die induced by the discrete metric (Definition D.8 page 157), ⚀ is no closer to ⚁ than it is to ⚂ (𝖽(⚀, ⚁) = 1 = 𝖽(⚀, ⚂)). However in the metric geometry of the real line induced by the usual metric 𝖽(𝑥, 𝑦) ≜ |𝑥 − 𝑦| (Definition D.9 page 157), 𝖷(⚀) = 1 is closer to 𝖷(⚁) = 2 than it is to 𝖷(⚂) = 3 (|1 − 2| = 1 ≠ 2 = |1 − 3|).

𝙂 1 6

1 6

⚅

⚄

⚂

6

1 6

1 6

1 6

𝙂

𝖧 3

5

1 6

1 6

1 6

⚅

⚄

⚂

1 6

𝖷(⋅)

[h] 1 6

⚃

⚁ 1 6

⚀

1 6

6

1 6

𝖷(⋅) 1 6

4

2 1

1 6

1 6

(A) mapping to isomorphic structure

1 6

⚃

⚁ 1 6

⚀

1 2

5

1 6

1 6

4

1 6

0 0/6

1 2 1 2

1 2

1

𝖧

1 2 1 2

3

2

1 6

1 6

1 6

(B) mapping to extended structure

Figure 2.3: order preserving random variable mappings from fair die

Example 2.11 (fair die mapping to isomorphic structure). Let 𝙂 ≜ ({⚀, ⚁, ⚂, ⚃, ⚄, ⚅}, 𝖽, ∅, 𝖯) be a fair die outcome subspace (Example 2.1 page 29), and 𝖧 ≜ ({1, 2, 3, 4, 5, 6}, 𝖽, ∅) be an unordered metric space (Definition 1.34 page 20). Example 2.10 (page 49) presented mappings from 𝙂 to structures with structures dissimilar to 𝙂. Figure 2.3 page 51 (A) illustrates a mapping to the isomorphic structure 𝙃 ≜ ({1, 2, 3, 4, 5, 6}, 𝖽, ∅, 𝖷(𝖯)), yielding the following statistics: ̈ ̈ 𝖤(𝖷) = {1, 2, 3, 4, 5, 6} 𝖵𝖺𝗋(𝖷) =0 ̈ Here, 𝖤(𝖷) equals the entire base set of 𝖧, indicating a statistic carrying no information about an expected outcome. That is, there is no best guess concerning outcome. This is much different than the traditional probability of 3.5 (Example 2.10 page 49) which deceptively suggests a likely outcome of ⚂ or ⚃. And one could easily argue that no information is much better than misleading information. 2016 Jul 4

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Daniel J. Greenhoe

version 0.50

page 52


✎PROOF: ̈ 𝖤(𝖷) ≜ arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑥∈𝖧

𝑦∈𝖧

= arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑥∈𝙂

𝑦∈𝙂

by definition of 𝖤̈ (Definition 2.14 page 46) because 𝙂 and 𝖧 are isomorphic

̇ = 𝖷[∁(𝙂) ]


= 𝖷[{⚀, ⚁, ⚂, ⚃, ⚄, ⚅}]

by Example 2.10 (page 49)

= {1, 2, 3, 4, 5, 6}

by definition of 𝖷

̈ 𝖵𝖺𝗋(𝖷) ≜

∑

̈ 𝖽 (𝖤(𝖷), 𝑥) 𝖯(𝑥)


∑

̇ 𝖽2 (𝖷(∁(𝙂), 𝑥) 𝖯(𝑥)

because 𝙂 and 𝖧 are isomorphic

2

𝑥∈𝖧

=

𝑥∈𝙂

̈ ≜ 𝖵𝖺𝗋(𝙂)


=0


✏ Although all the coefficients of the polynomial equation 𝑥2 −2𝑥+2 = 0 are in the set of real numbers ℝ, the solutions of the equation (𝑥 = 1 + 𝑖 and 𝑥 = 1 − 𝑖) are not. Rather, the two solutions are in the complex plane ℝ2 (Example 1.10 page 21), of which ℝ is a substructure. This is an example of extending a structure (from ℝ to ℝ2 ) to achieve more useful results. The same idea can be applied to a random variable 𝖷 ∈ 𝖧𝙂 . The definition of an outcome random variable (Definition 2.13 page 46) does not require a bijection between 𝙂 and 𝖧; rather, it only requires that the mapping be “into” the base set of 𝖧 (Definition 1.14 page 7). In Example 2.11 (page 51) in which 𝙂 is isomorphic to 𝙃 , the expected value of 𝖷 is a set with six values. However, we could extend 𝖧, while still preserving the order and metric geometry of 𝙂, to produce a random variable with a simpler expected value (next example). Example 2.12 (fair die mapping with extended range). Let 𝙂 ≜ ({⚀, ⚁, ⚂, ⚃, ⚄, ⚅}, 𝖽, ∅, 𝖯) be a fair die outcome subspace (Example 2.1 page 29), and 𝖧 ≜ ({1, 2, 3, 4, 5, 6, 0}, 𝗉, ∅) be an unordered metric space (Definition 1.34 page 20). Figure 2.3 page 51 (B) illustrates a random variable mapping 𝖷 from 𝙂 to the extended structure 𝖧, yielding the following statistics: ̈ ̈ 𝖤(𝖷) = {0} 𝖵𝖺𝗋(𝖷) = 41 As in Example 2.11, order and metric geometry are still preserved. Here, an expected value of {0} simply means that no real physical value is expected more or less than any other real physical value. Note also that the variance (expected error) is more than 11 times smaller than that of the corresponding statistical estimates on the real line (3/12 versus 35/12 Example 2.10 page 49). ✎PROOF: ̈ 𝖤(𝖷) ≜ arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑥∈𝖧


𝑦∈𝖧

= arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑥∈𝖧

𝑦∈𝖧⧵{0}

1 6

= arg min max 𝖽(𝑥, 𝑦) 𝑥∈𝖧

𝑦∈𝖧⧵{0}


𝑦∈𝖧⧵{0}

⎧ ⎪ ⎪ = arg min max ⎨ 𝑦∈𝖧 𝑥∈𝖧 ⎪ ⎪ ⎩ version 0.50

because 𝖯(0) = 0

𝖽(1, 1) 𝖽(2, 1) 𝖽(3, 1) 𝖽(4, 1) 𝖽(5, 1) 𝖽(6, 1) 𝖽(0, 1)

1 because 𝖿(𝑥) = 𝑥 is strictly isotone and by Lemma 1.7 (page 17) 6 𝖽(1, 2) ⋯ 𝖽(1, 6) 2 011112 ⎧ 2 ⎫ ⎧101121 ⎫ 𝖽(2, 2) ⋯ 𝖽(2, 6) ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ 𝖽(3, 2) ⋯ 𝖽(3, 6) ⎪ ⎪ 2 ⎪ ⎪110211 ⎪ 𝖽(4, 2) ⋯ 𝖽(4, 6) ⎬ = arg min max ⎨ 1 1 2 0 1 1 ⎬ = arg min ⎨ 2 ⎬ 𝑦∈𝖧 𝑥∈𝖧 ⎪ 2 ⎪ 𝑥∈𝖧 𝖽(5, 2) ⋯ 𝖽(5, 6) ⎪ ⎪121101 ⎪ 2 1 1 1 1 0 𝖽(6, 2) ⋯ 𝖽(6, 6) ⎪ ⎪ ⎪ 2 ⎪ ⎪ 𝖽(0, 2) ⋯ 𝖽(0, 6) ⎭ ⎩ 1 ⎭ ⎩111111 ⎭


Daniel J. Greenhoe

2016 Jul 4

10:12am UTC

page 53


= {0} ̈ 𝖵𝖺𝗋(𝖷) ≜

∑

̈ 𝖽2 (𝖤(𝖷), 𝑥) 𝖯(𝑥)


𝖽2 ({0}, 𝑥) 𝖯(𝑥)

̈ by 𝖤(𝖷) result

𝑥∈𝖧

=

∑

𝑥∈𝖧

=

∑

𝖽2 ({0}, 𝑥)

𝑥∈𝖧⧵{0}

1 6


1 21 1 = 6( ) = 2 6 4

by definition of 𝖧

✏

Real die examples

1 6

1 6

1 6

1 6

1 6

1 6

1

2

3

4

5

6

(ℤ, |⋅|, ≤)

𝖷(⋅) 1 6 1 6

1 6

6 3

5

4

2 1

⚅ ⚄

⚂

𝖸(⋅)

𝖧 1 6

1 6

1 6

1 6 1 6

1 6

1 6

⚃

⚁ ⚀

1 6

1 6

1 6

0 0/6 1 6

1 6

𝖪 3

5

𝖹(⋅)

𝙂

1 6

6

4

2 1

1 6

1 6

𝖶(⋅) 1 6

1 6

1 6

1 6

1 6

1 6

1

2

3

4

5

6

(ℝ, |⋅|, ≤)

Figure 2.4: random variable mappings from the real die outcome subspace to several ordered metric spaces (Example 2.13 page 53)

Example 2.13 (real die mappings). Let 𝙂 be the real die outcome subspace (Example 2.2 page 31). Let 𝖶, 𝖷, 𝖸 and 𝖹 be random variable (Definition 2.13 page 46) mappings as illustrated in Figure 2.4 (page 53). ̈ be defined as in Example 2.10 (page 49). This yields the following statistics: Let 𝖤, 𝖵𝖺𝗋, 𝖤,̈ and 𝖵𝖺𝗋 ̇ geometry of 𝙂: ∁(𝙂) = {⚀, ⚁, ⚂, ⚃, ⚄, ⚅} ̈ ≈ 2.917 traditional statistics on real line: 𝖤(𝖶) = 3.5 𝖵𝖺𝗋(𝖶; 𝖤) = 35 12 35 ̈ ̈ ̈ outcome subspace statistics on real line: 𝖤(𝖶) = {3.5} 𝖵𝖺𝗋(𝖶; 𝖤) = 12 ≈ 2.917 ̈ ̈ ≈ 1.667 outcome subspace statistics on integer line: 𝖤(𝖷) = {3, 4} 𝖵𝖺𝗋(𝖷; 𝖤)̈ = 20 12 ̈ ̈ ̈ outcome subspace statistics on isomorphic structure: 𝖤(𝖸) = {1, 2, … , 6} 𝖵𝖺𝗋(𝖸; 𝖤) = 0 ̈ ̈ outcome subspace statistics on extended structure: 𝖤(𝖹) = {0} 𝖵𝖺𝗋(𝖹; 𝖤)̈ = 1 ̈ Similar to Example 2.11 (page 51), the statistic 𝖤(𝖹) = {0} indicates a statistic carrying no information about an expected outcome. Again, one could easily argue that no information is much better than misleading information. 2016 Jul 4

10:12am UTC


Daniel J. Greenhoe

version 0.50

page 54


✎PROOF: ̇ ∁(𝙂) = {⚀, ⚁, ⚂, ⚃, ⚄, ⚅} 7 𝖤(𝖶) = = 3.5 2 35 ̈ 𝖵𝖺𝗋(𝖶; 𝖤) = ≈ 2.917 12 ̈ 𝖤(𝖶) = {3.5} 35 ̈ 𝖵𝖺𝗋(𝖶; 𝖤)̈ = ≈ 2.917 12 ̈ 𝖤(𝖷) = {3, 4} 5 ̈ 𝖵𝖺𝗋(𝖷; 𝖤)̈ = ≈ 1.667 3 ̈ 𝖤(𝖸) = 𝖸({⚀, ⚁, ⚂, ⚃, ⚄, ⚅}) = {1, 2, 3, 4, 5, 6} ̈ 𝖵𝖺𝗋(𝖸; 𝖤)̈ ≜

∑

by Example 2.2 (page 31) by 𝖤(𝖷) result of Example 2.10 page 49 by 𝖵𝖺𝗋(𝖷) result of Example 2.10 page 49 ̈ by 𝖤(𝖷) result of Example 2.10 page 49 ̈ by 𝖵𝖺𝗋(𝖷; 𝖤)̈ result of Example 2.10 page 49 ̈ by 𝖤(𝖸) result of Example 2.10 page 49 ̈ by 𝖵𝖺𝗋(𝖸; 𝖤)̈ result of Example 2.10 page 49 ̇ by ∁(𝙂) result of Example 2.2 page 31 by definition of 𝖸

̈ 𝖽2 (𝖤(𝖸), 𝑥) 𝖯(𝑥)


𝖽2 (𝖧, 𝑥) 𝖯(𝑥)

̈ by 𝖤(𝖸) result

02 𝑥𝖯(𝑥)

by nondegenerate property of quasi-metrics (Definition D.6 page 155)

𝑥∈𝖧

=

∑

𝑥∈𝖧

=

∑

𝑥∈𝖧

=0 ̈ 𝖤(𝖹) ≜ arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑦∈𝖪

𝑥∈𝖪

= arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑥∈𝖪

𝑦∈𝖪⧵{0}

= arg min max 𝖽(𝑥, 𝑦) 𝑥∈𝖪

𝑦∈𝖪⧵{0}

1 6

by definition of 𝖤̈ (Definition 2.14 page 46) because 𝖯(0) = 0 by definition of 𝙂 and 𝖹

1 because 𝖿 (𝑥) = 𝑥 is strictly isotone and by Lemma 1.7 (page 17) 6 𝑥∈𝖪 𝑦∈𝖪⧵{0} 𝖽(1, 1) 𝖽(1, 2) ⋯ 𝖽(1, 6) 011112 2 ⎧ 𝖽(2, 1) 𝖽(2, 2) ⋯ 𝖽(2, 6) ⎫ ⎧101121 ⎫ ⎧ 2 ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 𝖽(3, 1) 𝖽(3, 2) ⋯ 𝖽(3, 6) ⎪ ⎪110211 ⎪ ⎪ 2 ⎪ ⎪ ⎪ = arg min max ⎨ 𝖽(4, 1) 𝖽(4, 2) ⋯ 𝖽(4, 6) ⎬ = arg min max ⎨ 1 1 2 0 1 1 ⎬ = arg min ⎨ 2 ⎬ = ⎨ ⎬ 𝑦∈𝖪 𝑦∈𝖪 𝑥∈𝖪 𝑥∈𝖪 ⎪ 2 ⎪ 𝑥∈𝖪 ⎪ ⎪ ⎪121101 ⎪ ⎪ 𝖽(5, 1) 𝖽(5, 2) ⋯ 𝖽(5, 6) ⎪ ⎪211110 ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ 𝖽(6, 1) 𝖽(6, 2) ⋯ 𝖽(6, 6) ⎪ ⎩111111 ⎭ ⎩ 1 ⎭ ⎩ 0 ⎭ ⎩ 𝖽(0, 1) 𝖽(0, 2) ⋯ 𝖽(0, 6) ⎭ ̈ ̈ ̈ (Definition 2.14 page 46) 𝖵𝖺𝗋(𝖹) ≜ 𝖽2 𝖤(𝖹), 𝑥) 𝖯(𝑥) by definition of 𝖵𝖺𝗋 ∑ ( = arg min max 𝖽(𝑥, 𝑦)

𝑥∈𝖪

=

∑

𝖽2 ({0}, 𝑥) 𝖯(𝑥)

̈ by 𝖤(𝖹) result

𝑥∈𝖪⧵{0}

= 6 × 12 ×

1 =1 6


✏ Example 2.14 (weighted die mappings). Let 𝙂 be weighted die outcome subspace (Example 2.3 page 33), ̈ be deand 𝖷 and 𝖸 be random variables, as illustrated in Figure 2.5 (page 55). Let 𝖤, 𝖵𝖺𝗋, 𝖤,̈ and 𝖵𝖺𝗋 fined as in Example 2.10 (page 49). This yields the following statistics: ̇ geometry of 𝙂: ∁(𝙂) = {⚃} ̈ = 1.43 traditional statistics on real line: 𝖤(𝖷) = 3 𝖵𝖺𝗋(𝖷; 𝖤) = 143 100 25 3361 ̈ ̈ ̈ outcome subspace statistics on real line: 𝖤(𝖷) = { 7 } ≈ {3.57} 𝖵𝖺𝗋(𝖷; 𝖤) = 2940 ≈ 1.143 101 ̈ ̈ outcome subspace stats. on isomorphic structure 𝖤(𝖸) = {4} 𝖵𝖺𝗋(𝖷; 𝖤)̈ = 300 ≈ 0.337 The statistic 𝖤(𝖷) = 3 evaluated on the real line is arguably very poor because it suggests that we “expect” the event ⚂ rather than ⚃, even though 𝖯(⚃) is very large, 𝖯(⚂) is very small, and the version 0.50


Daniel J. Greenhoe

2016 Jul 4

10:12am UTC

page 55

CHAPTER 2. STOCHASTIC PROCESSING ON WEIGHTED GRAPHS (ℝ, |⋅|, ≤)

1 30

1 30

6 1 50

5 4

⚅ ⚄

⚂

𝖷(⋅)

3

1 50

1 30

3 5

2

⚁

1 30

𝖪 3 5

1 20

⚀

1

3

𝖸(⋅)

𝙂 ⚃

6 5

4

2 1

1 20

1 10

1 10

Figure 2.5: weighted die mappings (Example 2.14 page 54) physical distance 𝖽(⚃, ⚂) = 2 on the die from ⚃ to ⚂ is twice as much as it is to any of the other four die faces. If we retain use of the real line but replace the traditional expected value 𝖤(𝖷) with ̈ ̈ the outcome expected value 𝖤(𝖷), a small but significant improvement is made (𝖵𝖺𝗋(𝖷; 𝖤)̈ ≈ 1.143 < ̈ 1.43 = 𝖵𝖺𝗋(𝖷; 𝖤)). Arguably a better choice still is to abandon the real line altogether in favor of ̈ the isomorphic structure 𝖪 and the statistic 𝖤(𝖸) = {4} evaluated on 𝖪, yielding not only an intü ̈ itively better result but also a variance 𝖵𝖺𝗋(𝖸; 𝖤) that is more than 4 times smaller than that of 𝖤(𝖷) ̈ ̈ (𝖵𝖺𝗋(𝖸; 𝖤)̈ ≈ 0.337 < 1.43 = 𝖵𝖺𝗋(𝖷; 𝖤)). ✎PROOF: ̇ ∁(𝙂) = {⚃}


𝖤(𝖷) ≜

𝑥𝖯(𝑥) 𝖽𝑥


𝑥𝖯(𝑥)

by definition of 𝖯

=

∫ ℝ

∑

𝑥∈ℤ

=1×

1 1 1 3 1 1 +2× +3× +4× +5× +6× 10 20 30 5 50 30

1 900 (1 × 30 + 2 × 15 + 3 × 10 + 4 × 180 + 5 × 6 + 6 × 10) = =3 300 300 ̈ 𝖵𝖺𝗋(𝖷; 𝖤) = 𝖵𝖺𝗋(𝖷) by Theorem 2.1 page 47 =

≜ =

∫ ℝ

[𝑥 − 𝖤(𝖷)]2 𝖯(𝑥)


[𝑥 − 𝖤(𝖷)]2 𝖯(𝑥)


∑

𝑥∈ℤ

1 1 1 3 1 1 (1 − 3)2 + (2 − 3)2 + (3 − 3)2 + (4 − 3)2 + (5 − 3)2 + (6 − 3)2 10 20 30 5 50 30 1 429 143 (120 + 15 + 0 + 180 + 24 + 90) = = = = 1.43 300 300 100 ̈ 𝖤(𝖷) ≜ arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) by definition of 𝖤̈ (Definition 2.14 page 46) =

𝑥∈ℝ

𝑦∈ℝ

≜ arg min max |𝑥 − 𝑦|𝖯(𝑦) 𝑥∈ℝ

by definition standard metric on real line (Definition 1.35 page 21)

𝑦∈ℝ

2.4 1.8

= arg min 𝑥∈ℝ

1 |𝑥 − 1| 10 3 { |𝑥 − 4| 5

25 ≈ {3.5714} 7} ̈ ̈ 𝑥) 𝖯(𝑥) 𝖵𝖺𝗋(𝖷; 𝖤)̈ ≜ 𝖽2 𝖤(𝖷), ∑ ( ={

≤𝑥≤ for 25 7 otherwise

23 5

𝖿 (𝑥) ≜ max |𝑥 − 𝑦|𝖯(𝑦) 𝑦∈ℝ

/ /

9 25 9 35

25 7

23 5

𝑥

because 𝖿 (𝑥) is minimized at argument 𝑥 =

25 7


𝑥∈ℝ

=

∑

𝑥∈ℝ 2016 Jul 4

𝖽2 (

25 , 𝑥 𝖯(𝑥) 7 )

10:12am UTC

̈ by 𝖤(𝖷) result A book concerning symbolic sequence processing

Daniel J. Greenhoe

version 0.50

page 56


2 1 2 1 2 1 23 2 1 2 1 25 25 25 25 25 25 − 1) + ( − 2) + ( − 3) + ( − 4) + ( − 5) + ( − 6) 7 10 7 20 7 30 7 5 7 50 7 30 16805 3361 3361 = = = ≈ 1.143 49 ∗ 300 49 ∗ 60 2940 ̇ ̈ 𝖤(𝖸) = 𝖸(∁(𝙂) because 𝙂 and 𝙃 are isomorphic )

=(

= 𝖸({⚃})


= {4}

by definition of 𝖸

101 ̈ ̇ 𝖵𝖺𝗋(𝖸; 𝖤)̈ = 𝖵𝖺𝗋(𝙂) = ≈ 0.337 300


✏ Spinner examples 3

2

4 𝖷(⋅)

𝖯(4)= 1 6

𝖽(2,3)=1

𝖽(5,6)=1

𝙃

6

2 𝖽(6 ,1) =

1

1

𝖸(⋅)

𝖯(2)= 1 6

1 )=

,2 𝖽(1

𝖯(➄)= 1 6

𝖯(➅)= 1 6

,➃ )

=1

𝖽(

➄

➂

𝖯(➂)= 1 6

𝙂

➅

➁ 𝖽(➅

,➀ )

=1

𝖯(1)= 1 6

1 )=

➀

➁ ➀,

5 𝖹(⋅)

𝖯(➁)= 1 6

𝖯(6)= 1 6

6

𝖽(0 ,5) = 3 2

0

3 6)= 2

, 𝖽(0

𝖽(6 ,

𝖽(

4

=1

,5) 𝖽(4

𝖯(5)= 1 6

1)=

1

𝙆 𝖽(3 ,4) =

1

3 3)= 2

, 𝖽(0 0 𝖯(0)= 𝖽(0 /6 ,2) = 3 2

𝖽(0,1)= 3 2

𝖯(6)= 1 6

𝖯(3)= 1 6

3

𝖽(➂

𝖽(0,4)= 3 2

1

5

1 ➃

)=

➄ ➃,

𝖽(5,6)=1

𝖽(3 ,4) =

𝖽(➁,➂)=1

4

=1

𝖯(4)= 1 6

𝖯(➃)= 1 6

𝖽(➄,➅)=1

,5) 𝖽(4

𝖯(5)= 1 6

(ℤ, |⋅|, ≤)

6

5

1

3

2 1 )=

,2 𝖽(1

𝖯(3)= 1 6

𝖽(2,3)=1

1

𝖯(2)= 1 6

𝖯(1)= 1 6

𝖯(➀)= 1 6

𝖶(⋅)

(ℝ, |⋅|, ≤) 1

2

3

4

5

6

Figure 2.6: Six value fair spinner with assorted random variable mappings (Example 2.15 page 56) Example 2.15 (spinner mappings). A six value board game spinner has a cyclic structure as illustrated in Figure 2.6 (page 56). Again, the order and metric geometry of the real line mapped to by the random variable 𝖷 is very dissimilar to that of the outcome subspace that it is supposed to represent. Therefore, statistical inferences based on 𝖷 will likely result in values that are arguably unacceptable. Both random variables 𝖸 and 𝖹 map to structures in which order and metric geometry are preserved. The mappings yield the following statistics: ̇ geometry of 𝙂: ∁(𝙂) = {➀, ➁, ➂, ➃, ➄, ➅} 35 ≈ 2.917 ̈ traditional statistics on real line: 𝖤(𝖶) = 3.5 𝖵𝖺𝗋(𝖶; 𝖤) = 12 ̈ ̈ ≈ 2.917 outcome subspace statistics on real line: 𝖤(𝖶) = {3.5} 𝖵𝖺𝗋(𝖶; 𝖤)̈ = 35 12 ̈ ̈ outcome subspace statistics on integer line: 𝖤(𝖷) = {3, 4} 𝖵𝖺𝗋(𝖷; 𝖤)̈ = 20 ≈ 1.667 12 ̈ ̈ ̈ outcome subspace stats. on isomorphic structure: 𝖤(𝖸) = {1, 2, 3, 4, 5, 6} 𝖵𝖺𝗋(𝖸; 𝖤) = 0 ̈ ̈ outcome subspace stats. on extended structure: 𝖤(𝖹) = {0} 𝖵𝖺𝗋(𝖹; 𝖤)̈ = 49 = 2.25 ✎PROOF: ̇ ∁(𝙂) = {➀, ➁, ➂, ➃, ➄, ➅}


𝖤(𝖶) ≜


∑

𝑥𝖯(𝑥)

𝑥∈ℝ

7 = 3.5 2 ̈ 𝖵𝖺𝗋(𝖶; 𝖤) = 𝖵𝖺𝗋(𝖷) =

version 0.50

by fair die example Example 2.10 (page 49) by Theorem 2.1 page 47 A book concerning symbolic sequence processing

Daniel J. Greenhoe

2016 Jul 4

10:12am UTC

page 57


≜

∑

[𝑥 − 𝖤(𝖷)]2 𝖯(𝑥) 𝖽𝑥


𝑥∈ℝ

35 ≈ 2.917 12 ̈ 𝖤(𝖶) = 𝖤(𝖶) =

by fair die example Example 2.10 (page 49) because on real line, 𝖯 is symmetric, and by Theorem 2.2 page 47

= {3.5} ̈ ̈ 𝖵𝖺𝗋(𝖶; 𝖤)̈ = 𝖵𝖺𝗋(𝖶; 𝖤)

by 𝖤(𝖶) result ̈ because 𝖤(𝖶) = 𝖤(𝖶)

35 ≈ 2.917 12 ̈ 𝖤(𝖷) ≜ arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) =

𝑥∈ℤ

𝑦∈ℤ

≜ arg min max |𝑥 − 𝑦| 𝑥∈ℤ

𝑦∈ℤ

1 6

by fair die example Example 2.10 (page 49) by fair die example Example 2.10 (page 49)

= 𝖸[{➀, ➁, ➂, ➃, ➄, ➅}] = {1, 2, 3, 4, 5, 6} ̈ ̇ 𝖵𝖺𝗋(𝖸; 𝖤)̈ = 𝖵𝖺𝗋(𝙂) =0 𝑦∈𝖧

= arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑦∈𝖧⧵{0}


because 𝙂 and 𝖧 are isomorphic under mapping 𝖸 ̇ by ∁(𝙂) result by definition of 𝖸

̈ 𝖤(𝖹) ≜ arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑥∈𝖧

by definition of 𝖤̈ (Definition 2.14 page 46) by definition of integer line (Definition 1.36 page 21) and 𝙂

= {3, 4} 5 ̈ 𝖵𝖺𝗋(𝖷; 𝖤)̈ = ≈ 1.667 3 ̇ ̈ 𝖤(𝖸) = 𝖸[∁(𝙂) ]

𝑥∈𝖧

̈ by 𝖵𝖺𝗋(𝖶; 𝖤) result

𝑦∈𝖧⧵{0}

1 6

by spinner outcome subspace example (Example 2.4 page 35) by definition of 𝖤̈ (Definition 2.14 page 46) because 𝖯(0) = 0 by definition of 𝙂

1 because 𝖿 (𝑥) = 𝑥 is strictly isotone and by Lemma 1.7 (page 17) 6 𝑥∈𝖧 𝑦∈𝖧⧵{0} 012321 3 𝖽(1, 1) ⋯ 𝖽(1, 6) ⎧ 101232⎫ ⎧ 3 ⎫ ⎧ ⎫ ⎧ 𝖽(2, 1) ⋯ 𝖽(2, 6) ⎫ ⎪ 210123⎪ ⎪ 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 321012⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 𝖽(3, 1) ⋯ 𝖽(3, 6) ⎪ = arg min max ⎨ 𝖽(4, 1) ⋯ 𝖽(4, 6) ⎬ = arg min max ⎨ = arg min ⎨ 3 ⎬ = ⎨ ⎬ ⎬ 𝑦∈𝖧 𝑥∈𝖧 𝑦∈𝖧⧵{0} ⎪ 2 3 2 1 0 1 ⎪ 𝑥∈𝖧 𝑥∈𝖧 ⎪ 3 ⎪ ⎪ ⎪ ⎪ 𝖽(5, 1) ⋯ 𝖽(5, 6) ⎪ 1 2 3 2 1 0 ⎪ 𝖽(6, 1) ⋯ 𝖽(6, 6) ⎪ ⎪ 3 3 3 3 3 3⎪ ⎪ 33 ⎪ ⎪ ⎪ ⎩ 𝖽(0, 1) ⋯ 𝖽(0, 6) ⎭ ⎩ 2 2 2 2 2 2⎭ ⎩ 2 ⎭ ⎩ 0 ⎭ ̇ ̈ ̈ (Definition 2.14 page 46) 𝖵𝖺𝗋(𝖹) ≜ 𝖽2 ∁(𝙂), 𝑥) 𝖯(𝑥) by definition of 𝖵𝖺𝗋 ∑ ( = arg min max 𝖽(𝑥, 𝑦)

𝑥∈𝖧

=

∑

𝖽2 ({0}, 𝑥) 𝖯(𝑥)


𝑥∈𝖧

=

3 21 3 21 9 3 21 = | 𝖧⧵{0} | = 6 (2) 6 (2) 6 = 4 ∑ (2) 6 𝑥∈𝖧⧵{0}

✏ Example 2.16 (weighted spinner mappings). Let 𝙂 be weighted spinner outcome subspace (Example 2.5 page 37) with random variable mappings as illustrated in Figure 2.7 (page 58). This yields the following statistics: ̇ geometry of 𝙂: ∁(𝙂) = {➀, ➅} ̈ traditional statistics on real line (ℝ, |⋅|, ≤): 𝖤(𝖷) = 3.5 𝖵𝖺𝗋(𝖶; 𝖤) = 174 ≈ 4.25 ̈ ̈ outcome subspace statistics on real line (ℝ, |⋅|, ≤): 𝖤(𝖷) = {3.5} 𝖵𝖺𝗋(𝖶; 𝖤)̈ = 174 ≈ 4.25 ̈ ̈ outcome subspace statistics on isomorphic structure 𝙃 : 𝖤(𝖸) = {1, 6} 𝖵𝖺𝗋(𝖸; 𝖤)̈ = 53 ≈ 1.67 37 = 1.85 ̈ ̈ outcome subspace statistics on continuous structure 𝙆 : 𝖤(𝖹) = {0.5} 𝖵𝖺𝗋(𝖹; 𝖤)̈ = 20 ̈ Note that based on the variance values, the statistic 𝖤(𝖹) on the continuous ring 𝙆 is arguably a ̈ much better statistic than 𝖤(𝖷) on the (continuous) real line (ℝ, |⋅|, ≤). 2016 Jul 4

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2.2. RANDOM VARIABLES ON OUTCOME SUBSPACES 1 10

4

1 10

3

5

6

1

➄

2

𝖹(⋅)

𝙂 3 10

1

➅

➁ ➀

3

5

➂ 10

𝖸(⋅)

𝖧

4

➃

𝖪 2

6

1 10

3 10

1

𝖷(⋅) 1

2

4

3

6

5

(ℝ, |⋅|, ≤)

Figure 2.7: weighted spinner mappings (Example 2.16 page 57) ✎PROOF: ̇ ∁(𝙂) = {1, 6} 𝖤(𝖷) ≜

∑

by weighted spinner outcome subspace example (Example 2.5 page 37)

𝑥𝖯(𝑥)

𝑥∈ℤ

3 1 1 1 1 1 +2× +3× +4× +5× +6× 10 10 10 10 10 10 1 35 7 = (1 × 3 + 2 + 3 + 4 + 5 + 6 × 3) = = = 3.5 10 10 2 ̈ 𝖵𝖺𝗋(𝖷; 𝖤) = 𝖵𝖺𝗋(𝖷) by Theorem 2.1 page 47 =1×

≜

∑

[𝑥 − 𝖤(𝖷)]2 𝖯(𝑥)


𝑥∈ℤ

7 2 3 7 2 1 7 2 1 7 2 1 7 2 1 7 2 3 = (1 − ) + (2 − ) + (3 − ) + (4 − ) + (5 − ) + (6 − ) 2 10 2 10 2 10 2 10 2 10 2 10 1 5 2 3 2 1 2 1 2 3 2 5 2 = − × 3 + (− ) + (− ) + ( ) + ( ) + ( ) × 3 ] 10 [( 2 ) 2 2 2 2 2 1 170 17 = [75 + 9 + 1 + 1 + 9 + 75] = = = 4.25 40 40 4 ̈ 𝖤(𝖷) = 𝖤(𝖷) because on real line, 𝖯 is symmetric, and by Theorem 2.2 page 47 = {3.5} ̈ ̈ 𝖵𝖺𝗋(𝖷; 𝖤)̈ = 𝖵𝖺𝗋(𝖷; 𝖤)

by 𝖤(𝖷) result ̈ by 𝖤(𝖷) and 𝖤(𝖷) results

17 = 4.25 4 ̇ ̈ 𝖤(𝖸) = 𝖸[∁(𝙂) ] =

̈ by 𝖵𝖺𝗋(𝖷; 𝖤) result

= 𝖸[{➀, ➅}]

because 𝙂 and 𝖧 are isomorphic under 𝖸 ̇ by ∁(𝙂) result

= {1, 6} ̈ ̇ 𝖵𝖺𝗋(𝖸; 𝖤)̈ = 𝖵𝖺𝗋(𝙂)

by definition of 𝖸 because 𝙂 and 𝖧 are isomorphic under 𝖸

5 ≈ 1.667 3 ̈ 𝖤(𝖹) ≜ arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) =

𝑥∈𝖪

𝑦∈𝖪

by weighted spinner outcome subspace example (Example 2.5 page 37) by definition of 𝖤̈ (Definition 2.14 page 46) 0.9 0.75

𝖿 (𝑥) ≜ max 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑦∈𝖪

0.3 0.25

= 0.5 ̈ 𝖵𝖺𝗋(𝖹; 𝖤)̈ ≜

∑

0 0.5 1

̈ 𝖽2 (𝖤(𝖹), 𝑥) 𝖯(𝑥)

2

3 3.5 4

5

6 0.5

𝑥


𝑥∈𝙆

=

1 𝖽2 ( , 𝑥) 𝖯(𝑥) 2 𝑥∈𝙆 ∑


1 2 3 3 2 1 5 2 1 5 2 1 3 2 1 1 2 3 =( ) +( ) +( ) +( ) +( ) +( ) 2 10 2 10 2 10 2 10 2 10 2 10 version 0.50


Daniel J. Greenhoe

2016 Jul 4

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page 59


=

1 74 37 (3 + 9 + 25 + 25 + 9 + 3) = = = 1.85 40 40 20

✏

Pseudo-random number generator (PRNG) examples

2 9

1 9

0

1 9

1 9

1

1 9

0 8

1 9

7

2

4

(ℤ, |⋅|, ≤)

1 9

1 9

𝗌(𝑥𝑛 )

0

𝙂

2 9

1 2

𝖹(⋅)

3

2 9

1 9

1 9

1 2

5 6

3 9

1 9

1 9

4 𝙂9

1 9

2 9

3

𝖸(⋅)

2

3 1 9

1

1 9

𝖧 2 9

4

2 9

3

3 9

1 9

0 4 3 9

𝖷(⋅) 0

1

2

3

4

2 9

1 9

1 9

2 9

3 9

(ℝ, |⋅|, ≤)

Figure 2.8: LCG mappings to linear (𝖷), non-linear discrete (𝖸) and non-linear continuous (𝖹) ordered metric spaces (Example 2.17 page 59)

Example 2.17 (LCG mappings, standard ordering). The equation 𝑥𝑛+1 = (7𝑥𝑛 + 5) mod 9 with 𝑥0 = 1 is a linear congruential (LCG) pseudo-random number generator (PRNG) that has full period 5 of 9 values. These 9 values can be mapped, using a surjective (Definition 1.14 page 7) function 𝗌 ∈ 𝙂 𝙂9 to the 5 element set {0, 1, 2, 3, 4} to “shape” the distribution from a uniform distribution to non-uniform:6 . 𝑛 0 1 2 3 4 5 6 7 8 9 10 11 ⋯ 𝑥𝑛 1 3 8 7 0 5 4 6 2 1 3 8 ⋯ 𝑦𝑛 ≜ 𝗌(𝑥𝑛 ) 1 3 0 2 4 1 3 4 4 1 3 0 ⋯ Let 𝙂 be the outcome subspace and 𝖷, 𝖸, and 𝖹 be the outcome random variables illustrated in Figure 2.8 (page 59). This yields the following statistics: ̇ geometry of 𝙂9 : ∁(𝙂) = {0, 1, 2, 3, 4, 5, 6, 7, 8} ̇ geometry of 𝙂: ∁(𝙂) = {4} ̈ traditional statistics on real line: 𝖤(𝖷) = 73 ≈ 2.333 𝖵𝖺𝗋(𝖷; 𝖤) = 229 ≈ 2.444 ̈ ̈ = 2.4} 𝖵𝖺𝗋(𝖷; 𝖤)̈ = 551 ≈ 2.449 outcome subspace statistics on real line: 𝖤(𝖷) = { 12 5 225 ̈ ̈ outcome subspace statistics on integer line: 𝖤(𝖸) = {2, 3} 𝖵𝖺𝗋(𝖸; 𝖤)̈ = 169 ≈ 1.778 ̈ ̈ {4} outcome subspace statistics on isomorphic structure: 𝖤(𝖹) = 𝖵𝖺𝗋(𝖹; 𝖤)̈ = 43 ≈ 1.333 ̈ ̈ Note that unlike the statistics 𝖤(𝖷) and 𝖤(𝖷) on the real line, the statistic 𝖤(𝖹) on the isomorphic structure 𝙆 yields the maximally likely result, and a much smaller variance as well. 5

📃 Hull and Dobell (1962), 📘 Jr. and Gentle (1980) page 137 ⟨Theorem 6.1⟩, 📘 Severance (2001) page 86 ⟨HullDobell Theorem⟩ 6 The sequence ⦅1, 3, 0, 2, 4, 1, 3, 0, 2, 4, 1, 3, ⋯⦆ is generated by the equation 𝑦𝑛+1 = (𝑦𝑛 + 2) mod 5 with 𝑦0 = 1 2016 Jul 4

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Daniel J. Greenhoe

version 0.50

page 60


✎PROOF: ̇ ) ≜ arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) ∁(𝙂 9 𝑥∈𝙂9

= arg min max 𝖽(𝑥, 𝑦) 𝑥∈𝙂9


𝑦∈𝙂9

𝑦∈𝙂9

1 9

by definition of 𝙂9 1 because 𝜙(𝑥) = 𝑥 is strictly isotone and by Lemma 1.7 page 17 9


𝑦∈𝙂9

= arg min {4, 4, 4, 4, 4, 4, 4, 4, 4} because the maximum distance in 𝙂9 from any 𝑥 is 4 𝑥∈𝙂9

= {0, 1, 2, ⋯ , 8}

because the distances for values of 𝑥 in 𝙂9 are the same

̇ ∁(𝙂) = {4} 𝖤(𝖷) ≜

∑

by weighted ring outcome subspace example (Example 2.6 page 39) 𝑥𝖯(𝑥)


𝑥∈ℝ

2 1 1 2 3 21 7 +1× +2× +3× +4× = = ≈ 2.333 9 9 9 9 9 9 3 ̈ 𝖵𝖺𝗋(𝖷; 𝖤) = 𝖵𝖺𝗋(𝖷) by Theorem 2.1 page 47 =0×

≜

∑

[𝑥 − 𝖤(𝖷)]2 𝖯(𝑥)


𝑥∈ℝ

7 22 7 21 = (0 − ) + (1 − ) + (2 − 3 9 3 9 ̈ 𝖤(𝖷) ≜ arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑥∈ℝ

𝑦∈ℝ

≜ arg min max |𝑥 − 𝑦|𝖯(𝑦) 𝑥∈ℝ

7 21 7 22 7 2 3 198 22 + 3 − + 4 − = = ≈ 2.457 3) 9 ( 3) 9 ( 3) 9 81 9 by definition of 𝖤̈ (Definition 2.14 page 46) by definition usual metric on real line (Definition 1.35 page 21)

𝑦∈ℝ

𝖿(𝑥) ≜

max

𝑦∈(ℝ,|⋅|,≤)

|𝑥 − 𝑦|𝖯(𝑦)

/

8 15

|𝑥 − 4|𝖯(4) for 𝑥 ≤ 12 5 = arg min 𝑥∈ℝ { |𝑥 − 0|𝖯(0) otherwise 12 = 2.4 5} ̈ ̈ 𝖵𝖺𝗋(𝖷; 𝖤)̈ ≜ 𝖽2 𝖤(𝖷), 𝑥) 𝖯(𝑥) ∑ (

𝑥

/

12 5

because expression is minimized at argument 𝑥 =

={

12 5


𝑥∈ℝ

=

∑

𝑥∈ℝ

𝖽2 (

12 , 𝑥 𝖯(𝑥) 5 )


22 21 21 22 23 12 12 12 12 12 − 0 ) + ( − 1) + ( − 2 ) + ( − 3) + ( − 4) 5 9 5 9 5 9 5 9 5 9 1 551 2 2 2 2 2 = 2(12 − 0) + 1(12 − 5) + 1(12 − 10) + 2(12 − 15) + 3(12 − 20) = ≈ 2.449 25 × 9 225

=(

̈ 𝖤(𝖸) by definition of 𝖤̈ (Definition 2.14 page 46)

≜ arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑥∈𝖧

𝑦∈𝖧

= arg min max |𝑥 − 𝑦|𝖯(𝑦) 𝑥∈𝖧

by definition of integer line (Definition 1.36 page 21)

𝑦∈𝖧

0×2 ⎧ 1⎪ 1 × 2 = arg min max ⎨ 2 × 2 𝑦∈𝖧 9 𝑥∈𝖧 ⎪ 3×2 ⎩ 4×2 2 ̈ ̈ 𝖵𝖺𝗋(𝖸; 𝖤)̈ ≜ 𝖽 𝖤(𝖸), 𝑥) 𝖯(𝑥) ∑ (

1×1 0×1 1×1 2×1 3×1

12 ⎧ ⎫ ⎪ 1⎪ 9 ⎬ = arg min 9 ⎨ 6 𝑥∈𝖧 ⎪ ⎪ 6 ⎭ ⎩ 8 ̈ (Definition 2.14 page 46) by definition of 𝖵𝖺𝗋

2×1 1×1 0×1 1×1 2×1

3×2 2×2 1×2 0×2 1×2

4×3 3×3 2×3 1×3 0×3

⎫ ⎪ ⎬ ⎪ ⎭

⎧ ⎫ 1⎪ ⎪ = arg min ⎨ 2 ⎬ 9⎪ 3 ⎪ 𝑥∈𝖧 ⎩ ⎭

𝑥∈ℤ

=

∑

𝖽2 ({2, 3}, 𝑥) 𝖯(𝑥)


𝑥∈ℤ

version 0.50


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2016 Jul 4

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page 61


2 1 1 2 3 16 + 12 × + 02 × + 0 2 × + 12 × = ≈ 1.778 9 9 9 9 9 9 ̇ ̈ 𝖤(𝖹) = 𝖹[∁(𝙂) because 𝙂 and 𝖧 are isomorphic under 𝖹 ] ̇ = {4} by ∁(𝙂) result = 22 ×

̈ 𝖵𝖺𝗋(𝖹; 𝖤)̈ ≜

∑

̈ 𝖽2 (𝖤(𝖹), 𝑥) 𝖯(𝑥)


𝖽2 ({4}, 𝑥) 𝖯(𝑥)


𝑥∈𝖧

=

∑

𝑥∈𝖧

2 1 1 2 3 12 4 + 22 × + 22 × + 1 2 × + 02 × = = ≈ 1.333 9 9 9 9 9 9 3

= 12 ×

✏ 1 9

1 9

8

7 1 9

3

1 9

2 9

0 𝙂9

1

1 9

4 1 9

6

1 9

3 𝙂

1 9

2 9

0

𝗌(𝑥𝑛 )

5 2

2 9

2 9

1 9

1

2 4

1 9

3 0

𝖹(⋅)

𝖧 1 9

1

1 9

2

3 9

4 3 9

1 9

Figure 2.9: sequentially ordered LCG mappings (Example 2.18 page 61) Example 2.18 (LCG mappings, sequential ordering). In Example 2.17 (page 59), the structures 𝙂9 , 𝙂, and 𝖧 were ordered as a standard ring of integers (0 < 1 < 2 < ⋯ < 7 < 8 < 0 for 𝙂9 ). In this current example, as illustrated in Figure 2.9 (page 61), these structures are ordered as they appear in the sequences generated by 𝑥𝑛+1 = (7𝑥𝑛 + 5) mod 9 and 𝗌 (see Example 2.17 for sequence description). This yields the following statistics: ̇ geometry of 𝙂9 : ∁(𝙂) = {0, 1, 2, 3, 4, 5, 6, 7, 8} ̇ geometry of 𝙂: ∁(𝙂) = {3} ̈ ̈ outcome subspace statistics on isomorphic structure: 𝖤(𝖹) = {3} 𝖵𝖺𝗋(𝖹; 𝖤)̈ = 109 ≈ 1.111 Note that a change in ordering structure (from standard ring ordering to sequential ordering) yields ̈ ̈ a change in statistics (𝖤(𝖹) = {3} as opposed to 𝖤(𝖹) = {4}). Intuitively, the sequential ordering of Example 2.18 should yield a better estimate than that of Example 2.17, because it more closely matches the way the PRNG produces a sequence. This intuition is also supported by the variance ̈ ̈ values (𝖵𝖺𝗋(𝖹) = 12/9 for standard ring ordering, 𝖵𝖺𝗋(𝖹) = 10 for sequential ordering). However, coun9 terintuitively, the sequential ordering no longer yields the maximally likely result of {4}. ✎PROOF: ̇ ) ∁(𝙂 9 = {0, 1, 2, ⋯ , 8} ̇∁(𝙂) = {3} ̇ ̈ 𝖤(𝖹) = 𝖹[∁(𝙂) ] = 𝖹[{3}] = {3} ̈ ̇ 𝖵𝖺𝗋(𝖹; 𝖤)̈ = 𝖵𝖺𝗋(𝙂) 10 = ≈ 1.111 9 2016 Jul 4

10:12am UTC

by LCG mappings standard ordering example (Example 2.17 page 59) by Example 2.7 (page 41) because 𝙂 and 𝖧 are isomorphic under 𝖹 ̇ by ∁(𝙂) result by definition of 𝖹 because 𝙂 and 𝖧 are isomorphic under 𝖹 by Example 2.7 (page 41) A book concerning symbolic sequence processing

Daniel J. Greenhoe

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page 62


✏ 1 9

1 9

8

7 1 9

3

1 9

2 9

0 𝙂9

1

1 9

4 1 9

6

3 𝙂

1 9

1 9

2 9

0

𝗌(𝑥𝑛 )

5 2

2 9

2 9

1 9

1

1 9

3 0

𝖹(⋅)

2

𝖧 1 9

4

1

1 9

2

3 9

4 3 9

1 9

Figure 2.10: LCG mappings to linear (𝖷), non-linear discrete (𝖸) and non-linear continuous (𝖹) ordered metric spaces (Example 2.19 page 62) Example 2.19 (LCG mappings, sequential directed graph). Let 𝙂, 𝖧 and 𝖹 be a illustrated in Figure 2.10 (page 62). In Example 2.18 (page 61), the outcome values were ordered sequentially like a PRNG, but the metrics were commutative, which is unlike a PRNG. In this example, the outcomes are assigned quasi-metrics (Definition D.6 page 155, Remark 1.5 page 20) that are non-commutative. For example in the shaped sequence 𝗌(𝑥𝑛 ) = ⦅… , 3, 4, 4, 1, 3, …⦆, the “distance” from 3 to 4 is 𝖽(3, 4) = 1, but from 4 to 3 is 𝖽(4, 3) = 2. This yields the following statistics: ̇ geometry of 𝙂9 : ∁(𝙂) = {0, 1, 2, 3, 4, 5, 6, 7, 8} ̇ geometry of 𝙂: ∁(𝙂) = {3} ̈ ̈ outcome subspace statistics on isomorphic structure: 𝖤(𝖹) = {3} 𝖵𝖺𝗋(𝖹; 𝖤)̈ = 43 ≈ 1.333 ̈ Note that this technique yields the same estimate 𝖤(𝖹) = {3} as Example 2.18, but with a larger variance. ✎PROOF: ̇ ) ≜ arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) ∁(𝙂 9 𝑥∈𝙂9

𝑦∈𝙂9


𝑦∈𝙂9


𝑦∈𝙂9

1 9

by definition of ∁̇ (Definition 2.3 page 28) by definition of 𝙂9 1 because 𝜙(𝑥) = 𝑥 is strictly isotone and by Lemma 1.7 page 17 9

= arg min {8, 8, 8, 8, 8, 8, 8, 8, 8} because the maximum distance in 𝙂9 from any 𝑥 is 8 𝑥∈𝙂9

= {0, 1, 2, ⋯ , 8} ̇ ∁(𝙂) = {3} ̇ ̈ 𝖤(𝖹) = 𝖹[∁(𝙂) ] = 𝖹[{3}] = {3} ̇ ̈ ̇ ∁(𝙂)) 𝖵𝖺𝗋(𝖹; 𝖤)̈ = 𝖵𝖺𝗋( =

4 ≈ 1.333 3

because the distances for values of 𝑥 in 𝙂9 are the same by Example 2.8 (page 43) because 𝙂 and 𝖧 are isomorphic under 𝖹 ̇ by ∁(𝙂) result by definition of 𝖹 because 𝙂 and 𝖧 are isomorphic under 𝖹 by Example 2.8 (page 43)

✏ Genomic signal processing (GSP) examples Example 2.20 (DNA to linear structures). Genomic Signal Processing (GSP) analyzes biological sequences called genomes. These sequences are constructed over a set of 4 symbols that are comversion 0.50


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2016 Jul 4

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page 63

CHAPTER 2. STOCHASTIC PROCESSING ON WEIGHTED GRAPHS 1 4

(ℝ, |⋅|, ≤)

1

(ℤ, |⋅|, ≤)

𝘛

𝘈

3 2

1 4

3

𝙂 𝖷(⋅)

𝖸(⋅)

0 1 4

𝘊

𝘎

2 1 0

1 4

Figure 2.11: DNA random variable mappings to real line and integer line (Example 2.20 page 62) 𝘛 ◻, 𝘊 and ◻, 𝘎 each of which corresponds to a nucleobase (adenine, thymine, 𝘈 ◻, monly referred to as ◻, cytosine, and guanine, respectively).7 A typical genome sequence contains a large number of sym𝘛 ◻, 𝘊 ◻}, 𝘎 𝘈 ◻, bols (about 3 billion for humans, 29751 for the SARS virus).8 Let 𝙂 ≜ ({◻, 𝖽, ∅, 𝖯) be the outcome subspace (Definition 2.1 page 27) generated by a genome where 𝖽 is the discrete metric (Definition D.8 𝘛 = 𝖯(◻) 𝘊 = 𝘈 = 𝖯(◻) page 157), ≤= ∅ indicates a completely unordered set (Definition 1.20 page 8), and 𝖯(◻) 1 𝘎 = (uniformly distributed). Let 𝖧 ≜ (ℝ, |⋅|, ≤) be the real line (Definition 1.35 page 21). This yields the 𝖯(◻) 4 following statistics: ̇ 𝘛 ◻, 𝘊 ◻} 𝘎 𝘈 ◻, geometry of 𝙂: ∁(𝙂) = {◻, ̈ traditional statistics on real line: 𝖤(𝖷) = 1.5 𝖵𝖺𝗋(𝖷; 𝖤) = 54 = 1.25 ̈ ̈ outcome subspace statistics on real line: 𝖤(𝖷) = 1.5 𝖵𝖺𝗋(𝖷; 𝖤)̈ = 54 = 1.25 ̈ ̈ outcome subspace statistics on integer line: 𝖤(𝖸) = {1, 2} 𝖵𝖺𝗋(𝖸; 𝖤)̈ = 12 = 0.5 𝘛 ◻ 𝘊 and ◻ 𝘎 in general again have an order structure and a metric geometry (Re𝘈 ◻, The symbols ◻, mark 2.2 page 32) that is fundamentally dissimilar from that mapped to by the random variables 𝖷 and 𝖸. Therefore, statistical inferences made using these random variables will likely lead to results that arguably have little relationship with intuition or reality.

✎PROOF: ̇ 𝘛 ◻, 𝘊 ◻} 𝘎 𝘈 ◻, ∁(𝙂) = {◻,

by Example 2.9 page 45

𝖤(𝖷) ≜



𝑥𝖯(𝑥)

by definition of 𝖯 and Proposition 1.2 page 23

=

∫ ℝ

∑

𝑥∈ℤ

=

1 𝑥 ∑ 4 𝑥∈{0,1,2,3}

by definitions of 𝙂, 𝖧 and 𝖷

1 6 3 = (0 + 1 + 2 + 3) = = = 1.5 4 4 2 ̈ 𝖵𝖺𝗋(𝖷; 𝖤) = 𝖵𝖺𝗋(𝖷) by Theorem 2.1 page 47 = =

∫ ℝ

[𝑥 − 𝖤(𝖷)]2 𝖯(𝑥)


[𝑥 − 𝖤(𝖷)]2 𝖯(𝑥)


∑

𝑥∈ℤ

=

1 3 2 𝑥 − ∑[ 4 𝑥∈𝖧 2]


7


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1 3 2 3 2 3 2 3 2 0 − + 1 − + 2 − + 3 − ( ( ( 4 [( 2) 2) 2) 2) ] 1 20 5 2 2 2 2 = [(0 − 3) + (2 − 3) + (4 − 3) + (6 − 3) ] = 16 = 4 = 1.25 2 4⋅2 ̈ 𝖤(𝖷) = 𝖤(𝖷) because on real line, 𝖯 is symmetric, and by Theorem 2.2 page 47 3 = = 1.5 by 𝖤(𝖷) result 2 ̈ 𝖤(𝖷) ≜ arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) by definition of 𝖤̈ (Definition 2.14 page 46). (alternate proof ) =

𝑥∈ℝ

𝑦∈ℝ

= arg min max |𝑥 − 𝑦|𝖯(𝑦) 𝑥∈ℝ




by definition usual metric on real line

𝑦∈ℝ

max

|𝑥 − 𝑦|

max

|𝑥 − 𝑦|

𝑦∈{0,1,2,3}

𝑦∈{0,1,2,3}

1 4

by definition of 𝙂 1 because 𝖿(𝑥) = 𝑥 is strictly isotone and by Lemma 1.7 (page 17) 4

|𝑥 − 3| for 𝑥 ≤ 32 { |𝑥 − 0| otherwise }

1 3 = { } = 1 = 1.5 2 2 ̈ ̈ 𝖵𝖺𝗋(𝖷; 𝖤)̈ = 𝖵𝖺𝗋(𝖷; 𝖤)

𝑥∈ℤ

= arg min 𝑥∈ℤ

by definition of integer line (Definition 1.36 page 21)

𝑦∈ℤ

max

|𝑥 − 𝑦|

max

|𝑥 − 𝑦|

𝑦∈{0,1,2,3}

3 2


𝑦∈ℤ

𝑦∈{0,1,2,3}

|𝑥 − 𝑦|


= arg min max |𝑥 − 𝑦|𝖯(𝑦) = arg min

/

32

max

𝑦∈{0,1,2,3}

𝑥

̈ because 𝖤(𝖷) = 𝖤(𝖷)

5 4 ̈ 𝖤(𝖸) ≜ arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑥∈ℤ

𝖿(𝑥) ≜

/

32

because expression is minimized at argument 𝑥 =

=

𝑥∈ℤ

3

1 4

0 1 2 ⎧ ⎪ 1 0 1 = arg min max ⎨ 2 1 0 𝑥∈{0,1,2,3} 𝑦∈{0,1,2,3} ⎪ ⎩ 3 2 1 ̈ ̈ 𝖵𝖺𝗋(𝖸; 𝖤)̈ ≜ 𝖽2 𝖤(𝖸), 𝑥) 𝖯(𝑥) ∑ (

3 2 1 0

1 because 𝜙(𝑥) = 𝑥 is strictly isotone and by Lemma 1.7 page 17 4 3 ⎧ ⎫ ⎧ ⎪ 2 ⎫ ⎪ ⎪ 1 ⎫ ⎪ ⎪ ⎬ = arg min ⎨ 2 ⎬ = arg min ⎨ 2 ⎬ 𝑥∈{0,1,2,3} ⎪ 𝑥∈{0,1,2,3} ⎪ ⎪ ⎩ ⎪ ⎩ 3 ⎪ ⎭ ⎭ ⎭ ̈ by definition of 𝖵𝖺𝗋 (Definition 2.14 page 46)

𝑥∈ℤ

=

∑

𝖽2 ({1, 2}, 𝑥) 𝖯(𝑥)


𝑥∈ℤ

= |0 − 1|2 ×

1 1 1 1 1 + |1 − 1|2 × + |2 − 2|2 × + |3 − 2|2 × = = 0.5 4 4 4 4 2

✏ Example 2.21 (GSP to complex plane). A possible solution for the GSP problem (Example 2.20 page 62) is to map 𝘛 ◻, 𝘊 ◻} 𝘎 to the complex plane (Example 1.10 page 21) rather than the real line 𝘈 ◻, {◻, (Definition 1.35 page 21) such that (see also illustration to the right) 𝘛 𝘈 𝑎 ≜ 𝖷(◻) = −1 + 𝑖 𝑡 ≜ 𝖷(◻) = 1+𝑖 𝘊 𝘎 𝑐 ≜ 𝖷(◻) = −1 − 𝑖 𝑔 ≜ 𝖷(◻) = 1 − 𝑖. However, this solution also is arguably unsatisfactory for two reasons:

𝘈

𝘛

𝘊

𝘎

𝑎

•

•

•

•

𝑡

𝖷(⋅) 𝑐

𝑔

1. The order structures are dissimilar. Note that 𝘊 and ◻ 𝘈 are incomparable (Definition 1.20 page 8). 𝑐 < 𝑎, but ◻ version 0.50


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2016 Jul 4

10:12am UTC

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2. The metric geometries are dissimilar. Let 𝖽 be the discrete metric and 𝗉 the usual metric in ℂ. Note that 𝘛 = 𝖽(◻, 𝘊 = 𝖽(◻, 𝘎 = 1, but 𝘈 ◻) 𝘈 ◻) 𝘈 ◻) 𝖽(◻, 𝗉 (𝑎, 𝑡) = |𝑎 − 𝑡| = 2 ≠ 2√2 = |𝑎 − 𝑔| = 𝗉 (𝑎, 𝑔).

𝖽(0 ,

𝑡)= 1

/2

/2

)= 1

1

)=

,𝑐 𝖽(0

𝖽(𝘛

𝑐

𝖯(𝘎)= 1 4

𝖯(𝑐)= 1 4

𝖧

/

𝖯(𝘊 )= 1 4

𝘎

1 2

𝖽(𝘊 ,𝘎)=1

𝖯(0)= 0 4

)=

𝘊

0

,𝑔

𝖽(𝘈,𝘊 )=1

𝖷(⋅)

𝖽(0

,𝘊

/

∑

1 2

̈ 𝖵𝖺𝗋(𝖷) ≜

= 𝑎)

= {0}

𝑡

𝑔 𝖯(𝑔)= 1 4

because 𝖯(0) = 0 by definition of 𝙂 1 because 𝜙(𝑥) = 𝑥 is strictly isotone and by Lemma 1.7 page 17 4

𝑥∈𝖧⧵{0} 𝑦∈𝖧⧵{0}

𝖽(1, 1) 𝖽(2, 1) 𝖽(3, 1) 𝖽(4, 1) 𝖽(0, 1)

𝑎 , 𝖽(0

1 4

= arg min max 𝖽(𝑥, 𝑦) ⎧ ⎪ = arg min max ⎨ 𝑦∈𝖧 𝑥∈𝖧 ⎪ ⎩

𝘛

𝙂 𝖽(𝘛 ,𝘎 )=1

𝑥∈𝖧⧵{0} 𝑦∈𝖧⧵{0}

𝖯(𝑡)= 1 4

1

𝑦∈𝖧⧵{0}

= arg min max 𝖽(𝑥, 𝑦)

𝖯(𝑎)= 1 4


𝑦∈𝖧

= arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑥∈𝖧

𝘈

𝖯(𝘛 )= 1 4

)=

𝑥∈𝖧

𝖽(𝘈,𝘛 )=1

𝘎

̈ 𝖤(𝖧) ≜ arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦)

𝖯(𝘈)= 1 4

, 𝖽(𝘈

Example 2.22 (DNA mapping with extended range). Example 2.20 (page 62) presented a mapping from a DNA structure to a linearly ordered lattices, but the order and metric geometry was not preserved. In this example, a different structure is used that does preserve both order and metric geometry (see illustration to the right). This yields the following statistics: ̈ ̈ 𝖤(𝖷) = {0} 𝖵𝖺𝗋(𝖷) = 14 ✎PROOF:

𝖽(1, 2) 𝖽(2, 2) 𝖽(3, 2) 𝖽(4, 2) 𝖽(0, 2)

0 1 1 ⎧ 1 0 1 ⎫ ⎪ ⎪ ⎨ 1 1 0 ⎬ = arg min max 𝑦∈𝖧 𝑥∈𝖧 ⎪ 11 11 11 ⎪ ⎭ ⎩ 2 2 2 because expression is minimized at 𝑥 = {0} 𝖽(1, 3) 𝖽(2, 3) 𝖽(3, 3) 𝖽(4, 3) 𝖽(0, 3)

𝖽(1, 4) 𝖽(2, 4) 𝖽(3, 4) 𝖽(4, 4) 𝖽(0, 4)

̈ 𝖽2 (𝖤(𝖷), 𝑥) 𝖯(𝑥)


𝖽2 ({0}, 𝑥) 𝖯(𝑥)


1 1 1 0 1 2

1 ⎫ ⎧ 1 ⎫ ⎪ ⎪ 1 ⎪ ⎬ = arg min ⎨ ⎬ 𝑥∈𝖧 ⎪ 1 ⎪ ⎪ 1 ⎩ 2 ⎭ ⎭

𝑥∈𝖧

=

∑

𝑥∈𝖧

=

1 21 1 21 1 21 1 = | 𝖧⧵{0} |( ) = 4( ) = ( ) ∑ 2 4 2 4 2 4 4 𝑥∈𝖧⧵{0}

✏ Example 2.23 (GSP with Markov model). Markov probability models have often been used in genomic signal processing (GSP). A change in the statistics in the sequence may in some cases mean a change in function of the genomic sequence (DNA code). Finding such a change in statistics then is very useful in identifying functions of segments of genomic sequences. Let 𝙂 be an outcome subspace (Definition 2.1 page 27) representing a Markov model of depth 2 for a genomic sequence as illustrated in Figure 2.12 (page 66), with joint and conditional probabilities computed over a finite window. Let 𝙃 be an outcome subspace isomorphic to 𝙂, and 𝖷 be a random variable mapping ̈ 𝙂 to 𝖧. A change in the value of the statistic 𝖤(𝖷) over the window then may indicate a change in function within the genomic sequence. 2016 Jul 4

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Daniel J. Greenhoe

version 0.50

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2.3. OPERATIONS ON OUTCOME SUBSPACES

𝘛𝘛

𝘛𝘊

𝘊𝘊

𝘛𝘎

𝘛𝘈

𝘊𝘛

𝘈𝘛

𝘊𝘈

𝘈𝘊

𝘊𝘎

𝘈𝘎

𝘎𝘊

𝘎𝘛

𝘈𝘈

𝘎𝘈

𝘎𝘎

Figure 2.12: DNA with depth-2 Markov modelling (Example 2.23 page 65)

2.3 Operations on outcome subspaces 2.3.1 Summation Example 2.24 (pair of dice outcome subspace). A pair of real dice has a structure as illustrated in Figure 2.13 (page 67). The values represent the standard sum of die faces and thus range from 2 to 12. The table in the figure provides the metric distances between summed values based on the number of edges that must be transversed to move from the first value to the second value. Alternatively, the distance is the number of times the dice must be rotated 90 degrees to move from the first value being face up to the second value being face up. This structure is also illustrated in the undirected graph on the right in Figure 2.13, along with each value's standard probability. ✎PROOF: ̇ ∁(𝙂) ≜ arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑥∈𝙂

𝑦∈𝙂


𝖽(2, 2)𝖯(2) 𝖽(2, 3)𝖯(3) 𝖽(2, 4)𝖯(4) ⎧ ⎪ 𝖽(3, 2)𝖯(2) 𝖽(3, 3)𝖯(3) 𝖽(3, 4)𝖯(4) = arg min max ⎨ ⋮ ⋮ ⋮ 𝑦∈𝙂 𝑥∈𝙂 ⎪ ⎩ 𝖽(12, 2)𝖯(2) 𝖽(12, 3)𝖯(3) 𝖽(12, 4)𝖯(4) 8

⋯ 𝖽(2, 12)𝖯(12) ⋯ 𝖽(3, 12)𝖯(12) ⋱ ⋮ ⋯ 𝖽(12, 12)𝖯(12)

⎫ ⎪ ⎬ ⎪ ⎭

Many many thanks to Katie L. Greenhoe and Jonathan J. Greenhoe for help computing these values.

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2016 Jul 4

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page 67

CHAPTER 2. STOCHASTIC PROCESSING ON WEIGHTED GRAPHS 2 36 1 6 1 6

1 6

1 6

⚅

⚄

⚂

⚃

1 6

+

⚁ 1 6

⚀

1 6

1 6

1 6

3 ⚅

⚄

⚂

⚃

1 6

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

real die (Example 2.2 page 31)

real die (Example 2.2 page 31)

2 0 1 1 1 1 2 2 2 2 3 4

3 1 0 1 1 1 1 2 2 2 2 3

2 1 36

1 6

⚀

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

2 3 4 5 6 7 8 9 10 11 12

4

=

⚁ 1 6

2 36

4 1 1 0 1 1 1 1 2 2 2 2

5 1 1 1 0 1 1 1 1 2 2 2

6 1 1 1 1 0 1 1 1 1 2 2

3 36

11 3 36

6 36

7

4 36

4 36

9

5

8 2 2 1 1 1 1 0 1 1 1 1

12 1 36

8

6 5 36

7 2 1 1 1 1 0 1 1 1 1 2

10

5 36

9 10 11 12 2 2 3 4 2 2 2 3 2 2 2 2 1 2 2 2 1 1 2 2 1 1 1 2 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0

Figure 2.13: metrics based on number of die edges for a pair of real dice (Example 2.24 page 66)

⎧ ⎪ ⎪ ⎪ 1⎪ = arg min max ⎨ 𝑦∈𝙂 36 𝑥∈𝙂 ⎪ ⎪ ⎪ ⎪ ⎩ ⎧ ⎪ ⎪ ⎪ 1⎪ = arg min max ⎨ 𝑦∈𝙂 36 𝑥∈𝙂 ⎪ ⎪ ⎪ ⎪ ⎩ ∁𝖺̇ (𝙂) ≜ arg min 𝑥∈𝙂

∑

0×1 1×1 1×1 1×1 1×1 2×1 2×1 2×1 2×1 3×1 4×1 0 2 1 0 1 2 1 2 1 2 2 2 2 4 2 4 2 4 3 4 4 6

1×2 0×2 1×2 1×2 1×2 1×2 2×2 2×2 2×2 2×2 3×2 3 4 3 4 0 4 3 0 3 4 3 4 3 4 6 4 6 8 6 8 6 8

1×3 1×3 0×3 1×3 1×3 1×3 1×3 2×3 2×3 2×3 2×3 5 12 5 6 5 6 5 6 0 6 5 0 5 6 5 6 5 6 10 6 10 12

1×4 1×4 1×4 0×4 1×4 1×4 1×4 1×4 2×4 2×4 2×4 10 10 5 5 5 5 0 5 5 5 5


1×5 2×6 2×5 2×4 2×3 3×2 4×1 1×5 1×6 2×5 2×4 2×3 2×2 3×1 1×5 1×6 1×5 2×4 2×3 2×2 2×1 1×5 1×6 1×5 1×4 2×3 2×2 2×1 0×5 1×6 1×5 1×4 1×3 2×2 2×1 1×5 0×6 1×5 1×4 1×3 1×2 2×1 1×5 1×6 0×5 1×4 1×3 1×2 1×1 1×5 1×6 1×5 0×4 1×3 1×2 1×1 1×5 1×6 1×5 1×4 0×3 1×2 1×1 2×5 1×6 1×5 1×4 1×3 0×2 1×1 2×5 2×6 1×5 1×4 1×3 1×2 0×1 8 6 6 4 12 ⎧ 10 ⎫ 8 6 4 3 ⎫ ⎪ ⎪ ⎪ 8 6 4 2 ⎪ ⎪ 8 ⎪ 4 6 4 2 ⎪ ⎪ 6 ⎪ 4 3 4 2 ⎪ 1⎪ 6 ⎪ 4 3 2 2 ⎬ = arg min ⎨ 5 ⎬ = {7} 36 ⎪ 6 ⎪ 𝑥∈𝙂 4 3 2 1 ⎪ 0 3 2 1 ⎪ ⎪ 6 ⎪ 4 0 2 1 ⎪ ⎪ 8 ⎪ 4 3 0 1 ⎪ ⎪ 10 ⎪ 4 3 2 0 ⎭ ⎩ 12 ⎭ ̇ by definition of ∁ (Definition 2.4 page 28)

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

𝖺

𝑦∈𝙂

0 + 2 + 3 + 4 + 5 + 12 + 10 + 8 + 6 + 6 + 4 60 ⎧ 1 + 0 + 3 + 4 + 5 + 6 + 10 + 8 + 6 + 4 + 3 ⎫ ⎧ 50 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪1+2+0+4+ 5 + 6 + 5 +8+6+4+2⎪ ⎪ 43 ⎪ ⎪1+2+3+0+ 5 + 6 + 5 +4+6+4+2⎪ ⎪ 38 ⎪ 1 + 2 + 3 + 4 + 0 + 6 + 5 + 4 + 3 + 4 + 2 ⎪ ⎪ 1 ⎪ 34 ⎪ 1 = arg min ⎨ 2 + 2 + 3 + 4 + 5 + 0 + 5 + 4 + 3 + 2 + 2 ⎬ = arg min ⎨ 32 ⎬ = {7} 36 ⎪ 2 + 4 + 3 + 4 + 5 + 6 + 0 + 4 + 3 + 2 + 1 ⎪ 36 ⎪ 34 ⎪ 𝑥∈𝙂 𝑥∈𝙂 2 + 4 + 6 + 4 + 5 + 6 + 5 + 0 + 3 + 2 + 1 ⎪ 38 ⎪ ⎪ ⎪ 2 + 4 + 6 + 8 + 5 + 6 + 5 + 4 + 0 + 2 + 1 ⎪ 43 ⎪ ⎪ ⎪ ⎪ 50 ⎪ ⎪ 3 + 4 + 6 + 8 + 10 + 6 + 5 + 4 + 3 + 0 + 1 ⎪ ⎩ 60 ⎭ ⎩ 4 + 6 + 6 + 8 + 10 + 12 + 5 + 4 + 3 + 2 + 0 ⎭ 2 ̈ ̈ ̈ (Definition 2.14 page 46) 𝖵𝖺𝗋(𝙂) ≜ 𝖽 𝖤(𝙂), 𝑥)] 𝖯(𝑥) by definition of 𝖵𝖺𝗋 ∑[ ( 𝑥∈𝙂

2016 Jul 4

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Daniel J. Greenhoe

version 0.50

page 68


=

∑


[𝖽(7, 𝑥)]2 𝖯(𝑥)

𝑥∈𝙂

1 2 2 × 1 + 1 2 × 2 + 1 2 × 3 + 12 × 4 + 1 2 × 5 + 0 2 × 6 + 12 × 5 + 1 2 × 4 + 1 2 × 3 + 1 2 × 2 + 2 2 × 1 ) 36 ( 1 = (4 + 2 + 3 + 4 + 5 + 0 + 5 + 4 + 3 + 2 + 4) 36 36 = 36 =1

=

✏ The next two examples are examples of sums of outcome subspaces (Definition 2.1 page 27): Example 2.25 68 (sum of dice pair) and Example 2.27 page 71 (sum of spinner pair).

page

12 11 10 9 8 7 6 5 4 3 2

(ℝ, |⋅|, ≤)

𝙃

𝙂 2 36

2 36

3 4 𝖷(⋅)

2 1 36

3 36

3

11 3 36

6 36

10

7

4 36

9 6 5 36

10

4 4 36

5

11

12 1 36

8

𝖸(⋅)

7

2

12 9

5 6

8

5 36

Figure 2.14: pair of dice mappings (Example 2.14 page 54) Example 2.25 (pair of dice and hypothesis testing). Let 𝙂 be the pair of dice outcome subspace 𝙂 (Example 2.24 page 66), 𝖷 ∈ (ℝ, |⋅|, ≤) an outcome random variable mapping from 𝙂 to the real line 𝙂 (Definition 1.35 page 21), and 𝖷 ∈ 𝖧 a mapping to a structure 𝖧 that is isomorphic to 𝙂, as illustrated in Figure 2.14 (page 68). This yields the following statistics: ̇ ̇ geometry of 𝙂: ∁(𝙂) = ∁𝖺̇ (𝙂) = {7} 𝖵𝖺𝗋(𝙂) = 1 ̈ traditional statistics on real line (ℝ, |⋅|, ≤): 𝖤(𝖷) = 7 𝖵𝖺𝗋(𝖶; 𝖤) = 35 ≈ 5.833 6 35 ̈ ̈ ̈ outcome subspace statistics on real line (ℝ, |⋅|, ≤): 𝖤(𝖷) = {7} 𝖵𝖺𝗋(𝖶; 𝖤) = 6 ≈ 5.833 ̈ ̈ outcome subspace statistics on isomorphic structure 𝙃 : 𝖤(𝖸) = {7} 𝖵𝖺𝗋(𝖸; 𝖤)̈ = 1 Although the expected values of both outcome subspaces are the essentially the same (7 and {7}), the isomorphic structure 𝖧 yields a much smaller variance (a much smaller expected error). This is significant in statistical applications such as hypothesis testing. Suppose for example we have two pair of real dice (Example 2.2 page 31), one pair being made of two uniformly distributed die and one pair of weighted die. We want to know which pair is the uniform die. So we roll each pair one time. Suppose the outcome of the first pair is 11 and the the outcome of the second pair is 6. Which pair is more likely to be the uniform pair? Using traditional statistical analysis, the answer is the second pair, because it is closer to the expected value (0.414 standard deviations as opposed to 1.656 standard deviations). However, this result is deceptive, because as can be seen in Figure 2.13 (page 67), the distance from the expected value to the values 11 and 6 are the same (𝖽(7, 11) = 𝖽(7, 6) = 1 = 1 standard deviation). So arguably the outcome of the single roll test would contribute nothing to a good decision algorithm. ✎PROOF: ̇ ∁(𝙂) = {7}


̈ 𝖵𝖺𝗋(𝙂) =1 version 0.50

by Example 2.24 (page 66) A book concerning symbolic sequence processing

Daniel J. Greenhoe

2016 Jul 4

10:12am UTC

page 69


𝖤(𝖷) ≜

𝑥𝖯(𝑥) 𝖽𝑥 ∫ 𝑥∈ℝ 1 𝑥36𝖯(𝑥) ≜ ∑ 36 𝑥∈ℤ

by definition of 𝖤 (Definition 1.40 page 22) by definition of 𝖯

1 (2 × 1 + 3 × 2 + 4 × 3 + 5 × 4 + 6 × 5 + 7 × 6 + 8 × 5 + 9 × 4 + 10 × 3 + 11 × 2 + 12 × 1) 36 252 = =7 36 ̈ 𝖵𝖺𝗋(𝖷; 𝖤) = 𝖵𝖺𝗋(𝖷) by Theorem 2.1 page 47 =

= =

∫ 𝑥∈ℝ ∑

[𝑥 − 𝖤(𝖷)]2 𝖯(𝑥) 𝖽𝑥


[𝑥 − 𝖤(𝖷)]2 𝖯(𝑥)


𝑥∈ℤ

=

∑

(𝑥 − 7)2

𝑥∈ℤ

1 36


25 × 1 + 16 × 2 + 9 × 3 + 4 × 4 + 1 × 5 35 = ≈ 5.833 36 6 ̈ 𝖤(𝖷) = 𝖤(𝖷) by Theorem 2.2 (page 47) =2

= {7} ̈ ̈ 𝖵𝖺𝗋(𝖷; 𝖤)̈ = 𝖵𝖺𝗋(𝖷; 𝖤)

by 𝖤(𝖷) result ̈ because 𝖤(𝖷) ≡ 𝖤(𝖷)

35 ≈ 5.833 6 ̇ ̈ 𝖤(𝖸) = 𝖸[∁(𝙂) ] =


= 𝖸[{7}]

because 𝙂 and 𝖧 are isometric under 𝖸 ̇ by ∁(𝙂) result

= {7} ̈ ̈ ̇ 𝖵𝖺𝗋(𝖸; 𝖤) = 𝖵𝖺𝗋(𝙂)

by definition of 𝖸 because 𝙂 and 𝖧 are isometric under 𝖸 ̇ by 𝖵𝖺𝗋(𝙂) result

=1

✏ 3 36

4 36 1 6

1 6

➃

1 6

➄

➂

➅

➁ ➀

1 6

1 6

+ 1 6

1 6

1 6

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ spinner

➄

➂

➅

➁ ➀

=

spinner

2 0 1 2 3 2 1 2 3 4 3 2

3 1 0 1 2 3 2 1 2 3 4 3

4 2 1 0 1 2 3 2 1 2 3 4

5 36

6

2 6 36

12

1 6

6 2 3 2 1 0 1 2 3 2 1 2

7 1 2 3 2 1 0 1 2 3 2 1

8 2 1 2 3 2 1 0 1 2 3 2

1 36

8

5 36

9

11 2 36

(Example 2.4 page 35)

5 3 2 1 0 1 2 3 2 1 2 3

3

7 1 36

1 6

2 36

4 5

1 6

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(Example 2.4 page 35)

2 3 4 5 6 7 8 9 10 11 12

➃

1 6

10 3 36

4 36

9 10 11 12 3 4 3 2 2 3 4 3 1 2 3 4 2 1 2 3 3 2 1 2 2 3 2 1 1 2 3 2 0 1 2 3 1 0 1 2 2 1 0 1 3 2 1 0

Figure 2.15: metrics based on a pair of spinners (Example 2.26 page 70) 2016 Jul 4

10:12am UTC


Daniel J. Greenhoe

version 0.50

page 70


Example 2.26 (pair of spinners). A pair of spinners (Example 2.4 page 35) has a structure as illustrated in Figure 2.15 (page 69). The values represent the standard sum of spinner positions (1, 2, … , 6) and thus range from 2 to 12. The table in the figure provides the metric distances between summed values based on how many positions one must traverse to get from one value to the next (in which ever direction is shortest). This structure is also illustrated in the undirected graph in the upper right of Figure 2.15, along with each value's standard probability. ✎PROOF: ̇ ∁(𝙂) ≜ arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑥∈𝙂

𝑦∈𝙂


𝖽(2, 2)𝖯(2) 𝖽(2, 3)𝖯(3) 𝖽(2, 4)𝖯(4) ⋯ ⎧ ⎪ 𝖽(3, 2)𝖯(2) 𝖽(3, 3)𝖯(3) 𝖽(3, 4)𝖯(4) ⋯ = arg min max ⎨ ⋮ ⋮ ⋮ ⋱ 𝑦∈𝙂 𝑥∈𝙂 ⎪ ⎩ 𝖽(12, 2)𝖯(2) 𝖽(12, 3)𝖯(3) 𝖽(12, 4)𝖯(4) ⋯ 0×1 1×2 2×3 3×4 2×5 1×6 ⎧ 1×1 0×2 1×3 2×4 3×5 2×6 ⎪ ⎪ 2×1 1×2 0×3 1×4 2×5 3×6 ⎪ 3×1 2×2 1×3 0×4 1×5 2×6 1 ⎪ 2×1 3×2 2×3 1×4 0×5 1×6 = arg min max ⎨ 1 × 1 2 × 2 3 × 3 2 × 4 1 × 5 0 × 6 𝑦∈𝙂 36 𝑥∈𝙂 ⎪ 2×1 1×2 2×3 3×4 2×5 1×6 ⎪ 3×1 2×2 1×3 2×4 3×5 2×6 ⎪ 4×1 3×2 2×3 1×4 2×5 3×6 ⎪ 3×1 4×2 3×3 2×4 1×5 2×6 ⎩ 2×1 3×2 4×3 3×4 2×5 1×6 0 2 6 12 10 6 10 12 12 6 ⎧ 1 0 3 8 15 12 5 8 9 8 ⎪ ⎪ 2 2 0 4 10 18 10 4 6 6 ⎪ 3 4 3 0 5 12 15 8 3 4 1 ⎪ 2 6 6 4 0 6 10 12 6 2 = arg min max ⎨ 1 4 9 8 5 0 5 8 9 4 𝑦∈𝙂 36 𝑥∈𝙂 ⎪ 2 2 6 12 10 6 0 4 6 6 ⎪ 3 4 3 8 15 12 5 0 3 4 ⎪ 4 6 6 4 10 18 10 4 0 2 ⎪ 3 8 9 8 5 12 15 8 3 0 ⎩ 2 6 12 12 10 6 10 12 6 2 ∁𝖺̇ (𝙂) ≜ arg min 𝑥∈𝙂

∑


𝖽(2, 12)𝖯(12) ⎪ 𝖽(3, 12)𝖯(12) ⎫ ⎬ ⋮ 𝖽(12, 12)𝖯(12) ⎪ ⎭ 2×5 3×4 4×3 3×2 2×1 1×5 2×4 3×3 4×2 3×1 2×5 1×4 2×3 3×2 4×1 3×5 2×4 1×3 2×2 3×1 2×5 3×4 2×3 1×2 2×1 1×5 2×4 3×3 2×2 1×1 0×5 1×4 2×3 3×2 2×1 1×5 0×4 1×3 2×2 3×1 2×5 1×4 0×3 1×2 2×1 3×5 2×4 1×3 0×2 1×1 2×5 3×4 2×3 1×2 0×1 2 12 ⎧ 15 ⎫ 3 ⎫ ⎪ ⎪ ⎪ 4 ⎪ ⎪ 18 ⎪ 3 ⎪ ⎪ 15 ⎪ 2 ⎪ 1 ⎪ 12 ⎪ 1 ⎬ = arg min ⎨ 9 ⎬ = {7} 36 ⎪ 12 ⎪ 𝑥∈𝙂 2 ⎪ 3 ⎪ ⎪ 15 ⎪ 2 ⎪ ⎪ 18 ⎪ 1 ⎪ ⎪ 15 ⎪ 0 ⎭ ⎩ 12 ⎭ by definition of ∁̇ (Definition 2.4 page 28)

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

𝖺

𝑦∈𝙂

0 + 2 + 6 + 12 + 10 + 6 + 10 + 12 + 12 + 6 + 2 78 ⎧ 1 + 0 + 3 + 8 + 15 + 12 + 5 + 8 + 9 + 8 + 3 ⎫ ⎧ 72 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ 2 + 2 + 0 + 4 + 10 + 18 + 10 + 4 + 6 + 6 + 4 ⎪ ⎪ 66 ⎪ ⎪ 3 + 4 + 3 + 0 + 5 + 12 + 15 + 8 + 3 + 4 + 3 ⎪ ⎪ 60 ⎪ 2 + 6 + 6 + 4 + 0 + 6 + 10 + 12 + 6 + 2 + 2 ⎪ 1 ⎪ 56 ⎪ 1⎪ = arg min ⎨ 1 + 4 + 9 + 8 + 5 + 0 + 5 + 8 + 9 + 4 + 1 ⎬ = arg min ⎨ 54 ⎬ = {7} 36 ⎪ 2 + 2 + 6 + 12 + 10 + 6 + 0 + 4 + 6 + 6 + 2 ⎪ 36 ⎪ 56 ⎪ 𝑥∈𝙂 𝑥∈𝙂 ⎪ 60 ⎪ ⎪ 3 + 4 + 3 + 8 + 15 + 12 + 5 + 0 + 3 + 4 + 3 ⎪ ⎪ 66 ⎪ ⎪ 4 + 6 + 6 + 4 + 10 + 18 + 10 + 4 + 0 + 2 + 2 ⎪ ⎪ 72 ⎪ ⎪ 3 + 8 + 9 + 8 + 5 + 12 + 15 + 8 + 3 + 0 + 1 ⎪ ⎩ 78 ⎭ ⎩ 2 + 6 + 12 + 12 + 10 + 6 + 10 + 12 + 6 + 2 + 0 ⎭ ̈ ̈ ̈ (Definition 2.14 page 46) 𝖵𝖺𝗋(𝙂) ≜ 𝖽 𝖤(𝙂), 𝑥)]2 𝖯(𝑥) by definition of 𝖵𝖺𝗋 ∑[ ( 𝑥∈𝙂

=

∑


[𝖽(7, 𝑥)]2 𝖯(𝑥)

𝑥∈𝙂

1 2 1 × 1 + 22 × 2 + 3 2 × 3 + 2 2 × 4 + 12 × 5 + 0 2 × 6 + 1 2 × 5 + 2 2 × 4 + 2 2 × 3 + 2 2 × 2 + 1 2 × 1 ) 36 ( 1 = (2 + 8 + 27 + 16 + 5 + 0 + 5 + 16 + 12 + 8 + 1) 36 7 100 25 = = 2 ≈ 2.778 = 36 9 9 =

✏ version 0.50


Daniel J. Greenhoe

2016 Jul 4

10:12am UTC

page 71

CHAPTER 2. STOCHASTIC PROCESSING ON WEIGHTED GRAPHS (ℝ, |⋅|, ≤)

12 11 10 9 8 7 6 5 4 3 2

3 36

4 36

2 36

4

4

3

5 5 36

𝖧

𝙂

6

2

𝖷(⋅)

1 36

6 36

12

10 3 36

2

8

7 8

12

5 36

9

11 2 36

6 𝖸(⋅)

7 1 36

3

5

9

11 10

4 36

Figure 2.16: pair of spinner mappings (Example 2.27 page 71) Example 2.27 (pair of spinner and hypothesis testing). Let 𝙂 be a pair of spinners (Example 2.26 page 70), 𝖷 a random variable mapping to the real line (Definition 1.35 page 21), and 𝖸 a random variable mapping to an ordered metric space (Definition 1.34 page 20) that is isomorphic to 𝙂 under 𝖸, as illustrated in Figure 2.16 (page 71). This yields the following statistics: ̇ ̇ geometry of 𝙂: ∁(𝙂) = ∁𝖺̇ (𝙂) = {7} 𝖵𝖺𝗋(𝙂) = 259 ≈ 2.778 ̈ traditional statistics on real line (ℝ, |⋅|, ≤): 𝖤(𝖷) = 7 𝖵𝖺𝗋(𝖶; 𝖤) = 356 ≈ 5.833 ̈ ̈ outcome subspace statistics on real line (ℝ, |⋅|, ≤): 𝖤(𝖷) = {7} 𝖵𝖺𝗋(𝖶; 𝖤)̈ = 356 ≈ 5.833 ̈ ̈ outcome subspace statistics on isomorphic structure 𝙃 : 𝖤(𝖸) = {7} 𝖵𝖺𝗋(𝖸; 𝖤)̈ = 1 ̈ Although the expected value of both outcome subspaces are the same (𝖤(𝖷) = 𝖤(𝖸) = 7), the isomorphic outcome subspace 𝙃 yields a much smaller variance (a much smaller expected error). This is significant in statistical applications such as hypothesis testing. Suppose for example we have two pair of spinners (Example 2.4 page 35), one pair being made of two uniformly distributed spinners, and one pair of weighted spinners. We want to estimate which is which. So we spin each pair one time. Suppose the outcome of the first pair is 12 and the the outcome of the second pair is 10. Which pair is more likely to be the uniform pair? Using traditional statistical analysis, the answer is the second pair, because it is closer to the expected value (𝖽(7, 10) = |7 − 10| = 3 = 1.8 standard deviations as opposed to 𝖽(7, 12) = |7 − 12| = 5 = 3 standard deviations). However, this result is deceptive, because as can be seen in the table in Figure 2.15 (page 69), 12 is acually closer to the expected value in 𝙂 than is 10 (𝖽(7, 12) = 1 < 3 = 𝖽(7, 10)). So arguably the better choice, based on this one trial, is the first pair. ✎PROOF: ̇ ∁(𝙂) = {7} ∁̇ (𝙂) = {7} 𝖺

25 ̇ 𝖵𝖺𝗋(𝙂) = ≈ 2.778 9 252 =7 𝖤(𝖷) = 36 35 𝖵𝖺𝗋(𝖷) = ≈ 5.833 6 ̈ 𝖤(𝖷) = 𝖤(𝖷) = {7} ̈ ̈ ̈ 𝖵𝖺𝗋(𝖷; 𝖤) = 𝖵𝖺𝗋(𝖷; 𝖤) = 𝖵𝖺𝗋(𝖷) 35 = ≈ 5.833 6 ̇ ̈ 𝖤(𝖸) = 𝖸[∁(𝙂) ] 2016 Jul 4

10:12am UTC

by Example 2.26 (page 70) by Example 2.26 (page 70) by Example 2.26 (page 70) by Example 2.25 (page 68) by Example 2.25 (page 68) because on real line, 𝖯 is symmetric, and by Theorem 2.2 page 47 by 𝖤(𝖷) result ̈ because 𝖤(𝖷) = 𝖤(𝖷) by Theorem 2.1 page 47

because 𝙂 and 𝖧 are isomorphic under 𝖸 A book concerning symbolic sequence processing

Daniel J. Greenhoe

version 0.50

page 72


= 𝖸[{7}]


= {7} ̈ ̈ ̈ 𝖵𝖺𝗋(𝖸; 𝖤) = 𝖵𝖺𝗋(𝙂)

by definition of 𝖷 because 𝙂 and 𝖧 are isomorphic under 𝖸

=1


✏ 2 1.5 1 1 3

−3

−2

−1

0

1

2

8 3

𝑥 3

4

5

6

7

8

9

Figure 2.17: real line addition arg min𝑥 max𝑦 calculation graph (Example 2.28 page 72) Example 2.28 (linear addition). Let 𝖷 be a random variable (Definition 2.13 page 46) mapping to a real line ordered metric space (Definition 1.35 page 21) resulting in probability values of 𝖯(0) = 𝖯(2) = 21 , and 𝖯(𝑥) = 0 otherwise. Let 𝖸 be a random variable mapping to a real line ordered metric space resulting in probability values of 𝖯(0) = 𝖯(1) = 14 , 𝖯(2) = 12 , and 𝖯(𝑥) = 0 otherwise. Let 𝖹 ≜ 𝖷 + 𝖸 be the random variable mapping to the outcome subspace (Definition 2.1 page 27) induced by adding 𝖷 and 𝖸 resulting in probabilities 𝑧 0 1 2 3 4 , and 𝖯(𝑧) = 0 otherwise. 𝖯(𝑧) 18 18 38 18 28 Note that although the traditional expectation 𝖤 (Definition 1.40 page 22) distributes over addition such that 𝖤(𝖷 + 𝖸) = 18 = 1 + 54 = 𝖤(𝖷) + 𝖤(𝖸), 8 the alternative expecation 𝖤̈ (Definition 2.14 page 46) does not: ̈ + 𝖸) = 8 ≠ 7 = 1 + 4 = 𝖤(𝖷) ̈ ̈ 𝖤(𝖷 + 𝖤(𝖸). 3 3 3 ✎PROOF: 𝖤(𝖷) = =

∫ 𝑥∈ℝ ∑





1 1 +2× 2 2


𝑥∈ℤ

=0× =1 𝖤(𝖸) = =

∫ 𝑦∈ℝ ∑

𝑦𝖯(𝑦) 𝖽𝑦


𝑦𝖯(𝑦) 𝖽𝑦


1 1 1 +1× +2× 4 4 2


𝑦∈ℤ

=0× = 𝖤(𝖹) = =

5 4 ∫ 𝑧∈ℝ ∑

𝑧𝖯(𝑧)

𝑧𝖯(𝑧)

by definition of 𝖤 (Definition 1.40 page 22) by definition of 𝖯 and Proposition 1.2 page 23

𝑧∈ℤ

version 0.50


Daniel J. Greenhoe

2016 Jul 4

10:12am UTC

page 73


=0×

1 1 3 1 2 +1× +2× +3× +4× 8 8 8 8 8


9 4 ̈ 𝖤(𝖷) = arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) =

𝑥∈ℝ


𝑦∈ℝ

= arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑥∈ℤ


by definition of 𝖯(𝑥)

𝑦∈ℤ

|2 − 𝑥| 12 { |𝑥| 12

for 𝑥 ≤ 1 otherwise }

= {1}

because max(𝑥) is minimized at 𝑥 = 1 by definition of 𝖤̈ (Definition 2.14 page 46)

̈ 𝖤(𝖸) = arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑥∈ℝ

𝑦∈ℝ




𝑦∈ℤ

|2 − 𝑥| 12 { |𝑥| 14

for 43 ≥ 𝑥 ≥ 4 } otherwise

4 ={ } 3 ̈ 𝖤(𝖹) = arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑥∈ℝ

because max(𝑦) is minimized at 𝑦 = by definition of 𝖤̈ (Definition 2.14 page 46)

𝑦∈ℝ



𝑦∈ℤ

|𝑥 − 2| 83 ⎧ ⎪ = arg min ⎨ |𝑥 − 4| 28 𝑥∈ℤ ⎪ 1 ⎩ |𝑥| 8 8 ={ } 3

4 3

for −2 ≥ 𝑥 ≥ 3 for −2 ≤ 𝑥 ≤ 38 for 38 ≤ 𝑥 ≤ 3

⎫ ⎪ ⎬ ⎪ ⎭ because max(𝑧) is minimized at 𝑧 =

8 (see Figure 2.17 page 72) 3

✏

2.3.2 Multiplication 1 10

4

1 10

3

5

6

1

➄

2

3 10

1

𝖹(⋅)

𝙂 ➅

➁ ➀

3

5

➂ 10

𝖸(⋅)

𝖧

4

➃

𝖪 2

6

1 10

3 10

1

𝖷(⋅) 1

2

3

4

5

6

(ℝ, |⋅|, ≤)

Figure 2.18: pair of spinner mappings (Example 2.29 page 73) Example 2.29 (ring multiplication). Let 𝖷 ∈ 𝖧𝙂 be a random variable where 𝙂 is the weighted spinners illustrated in Figure 2.18 (page 73). Note that, in agreement with Corollary 2.1 (page 48), ̈ ̈ 𝖤(2𝖷) = {4} = {2 × 5 mod 6} = 2{5} mod 6 = 2𝖤(𝖷) mod 6 . ✎PROOF: ̇ ∁(𝙂) ≜ arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑥∈𝙃 2016 Jul 4

10:12am UTC

𝑦∈𝙃


by definition of ∁̇ (Definition 2.3 page 28) Daniel J. Greenhoe

version 0.50

page 74


⎧ ⎪ 1⎪ = arg min max ⎨ 𝑦∈𝙃 10 𝑥∈𝙃 ⎪ ⎪ ⎩ = {5} ∁𝖺̇ (𝙂) ≜ arg min 𝑥∈𝙃

∑

0×1 1×1 2×1 3×1 2×1 1×1

1×1 0×1 1×1 2×1 3×1 2×1

2×1 1×1 0×1 1×1 2×1 3×1

3×1 2×1 1×1 0×1 1×1 2×1

2×1 3×1 2×1 1×1 0×1 1×1

1×5 2×5 3×5 2×5 1×5 0×5

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

⎧ ⎪ 1⎪ = arg min ⎨ 10 ⎪ 𝑥∈𝙃 ⎪ ⎩

5 10 15 10 5 3

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

by definition of ∁𝖺̇ (Definition 2.4 page 28)


𝑦∈𝙃

0 × 1+1 × 1+2 × 1+3 × 1+2 × 1+1 × 5 ⎧1 × 1+0 × 1+1 × 1+2 × 1+3 × 1+2 × 5 ⎪ 1 ⎪2 × 1+1 × 1+0 × 1+1 × 1+2 × 1+3 × 5 = arg min ⎨ 3 × 1 + 2 × 1 + 1 × 1 + 0 × 1 + 1 × 1 + 2 × 5 10 𝑥∈𝙃 ⎪2 × 1+3 × 1+2 × 1+1 × 1+0 × 1+1 × 5 ⎪ ⎩1 × 1+2 × 1+3 × 1+2 × 1+1 × 1+0 × 5

⎫ ⎧ ⎪ ⎪ ⎪ 1⎪ ⎬ = arg min 10 ⎨ 𝑥∈𝙃 ⎪ ⎪ ⎪ ⎪ ⎩ ⎭

13 17 21 17 13 9

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

= {5} ̇ ̈ 𝖤(𝖷) = 𝖷[∁(𝙂) ]

because 𝙂 and 𝙃1 are isomorphic ̇ by ∁(𝙂) result

= 𝖷[{5}] = {5}

by definition of 𝖷 by definition of 𝖤̈ (Definition 2.14 page 46)

̈ 𝖤(2𝖷) ≜ arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑥∈𝙃2

𝑦∈𝙃2

= arg min max 𝑥∈𝙃2

𝑦∈𝙃2

0×2 1×2 1×6 1 1×2 0×2 1×6 10 { 1 × 2 1 × 2 0 × 6 }

= arg min 𝑥∈𝙃2

6 1 6 10 { 2 }

= {4}

✏

2.3.3 Metric transformation It is possible to use a metric transform (Definition D.11 page 158) to transform the structure of an outcome subspace (Definition 2.1 page 27) into a completely different outcome subspace. This is demonstrated in Example 2.30 (page 75)–Example 2.32 (page 76). Naturally, by doing so one can sometimes even change the geometric centers (Definition 2.3 page 28, Definition 2.4 page 28) of the outcome subspaces, and hence also the statistics of random variables that map to/from them. This is demonstrated in Example 2.32 (page 76)–Example 2.33 (page 76). Theorem 2.4. Let 𝜙 be a METRIC PRESERVING FUNCTION (Definition D.11 page 158). Let 𝙂 ≜ (𝛺, ≤,̇ 𝖽,̇ 𝖯̇ ) and 𝙃 be OUTCOME SUBSPACEs (Definition 2.1 page 27). (1). 𝜙(𝙃 ) = 𝙂 and ̇ ) = ∁(𝙂) ̇ (2). 𝜙 is STRICTLY ISOTONE and ⟹ ∁(𝙃 { (3). 𝖯 is uniform } ✎PROOF: ̇ ) = 𝜙 ∁(𝙃 ̇ ) ∁(𝙃 [ ] ≜ arg min max 𝜙[𝖽(𝑥, 𝑦)]𝖯(𝑦) 𝑥∈𝙂

𝑦∈𝙂

= arg min max 𝜙[𝖽(𝑥, 𝑦)] 𝑥∈𝙂

𝑦∈𝙂


̇ = ∁(𝙂)

𝑦∈𝙂

by hypothesis (1) by definition of ∁̇ (Definition 2.3 page 28) by hypothesis (3) and Lemma 1.7 page 17 by hypothesis (2) and Lemma 1.7 page 17 by definition of ∁̇ (Definition 2.3 page 28)

✏ version 0.50


Daniel J. Greenhoe

2016 Jul 4

10:12am UTC

page 75


⎞ ⎛ ⎟ ⎜ 1 6 ⚅ ⎟ ⎜ 1 1 ⎟ ⎜ 6⚄ 6 ⚂ ⎟ ⎜ 𝜙⎜ ⎟ ⚁ ⎜ 1⚃ ⎟ 1 6 ⎜ 6 ⎟ ⚀ 1 6 ⎜ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⎟ ⎜ ⎟ ⎝ real die (Example 2.2 page 31) ⎠

=

𝜙∘

⎛ ⎞ ⎜ ⎟ 1 ⎜ ⎟ 6 ➃ 1 ⎜ 16 ⎟ 6 ➄ ➂ ⎜ ⎟ ⎜ ⎟ 𝗀⎜ ⎟ ➅ ➁ 1 ⎜ 16 ⎟ 6 ➀ 1 ⎜ ⎟ 6 ⎜⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⎟ ⎜ ⎟ ⎝ spinner (Example 2.4 page 35) ⎠

=

⎛ ⎞ ⎜ ⎟ 1 6 ⚅ ⎜ 1 ⎟ 1 ⎜ 6⚄ ⎟ 6 ⚂ ⎜ ⎟ ⎜ ⎟ ⚁ ⎜ 1⚃ ⎟ 1 6 ⎜ 6 ⎟ ⚀ 1 6 ⎜ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⎟ ⎜ ⎟ ⎝ fair die (Example 2.1 page 29) ⎠

Figure 2.19: discrete metric preserving function 𝜙 on outcome subspaces (Example 2.30 page 75) Example 2.30 (discrete metric transform on outcome subspaces). Let 𝗀 be a function (a pullback function Theorem 1.7 page 19) such that 𝗀(➀) = ⚀, 𝗀(➁) = ⚁, 𝗀(➂) = ⚂, 𝗀(➃) = ⚅, 𝗀(➄) = ⚄, and 𝗀(➅) = ⚃. Then under the dicrete metric preserving function 𝜙 (Example D.7 page 160) the real die outcome subspace (Example 2.2 page 31) becomes the fair die outcome subspace (Example 2.1 page 29), and under 𝜙 ∘ 𝗀 the spinner outcome subspace (Example 2.4 page 35) also becomes the fair die outcome subspace, as illustrated in Figure 2.19 (page 75). This yields the following geometric statistics: ̇ ̇ ) = {⚀, ⚁, ⚂, ⚃, ⚄, ⚅} . ∁(𝙂) = ∁(𝙃 ⎞ ⎛ ⎞ ⎛ ⎟ ⎜ ⎜ ⎟ 1 1 ⎟ ⎜ ⎜ ⎟ 10 10 ➃ ➃ 1 1 ⎜ 104 ⎜ 104 10 ⎟ 10 ⎟ ➄ ➂ ➄ ➂ ⎟ ⎜ ⎟ ⎜ ⎟. ⎜ ⎜ ⎟ 𝜙1 ⎜ =⎜ ⎟ ⎟ ➅ ➁ 2 ➅ ➁ 2 ⎜ 101 ⎜ 101 10 ⎟ 10 ⎟ ➀ 1 ➀ 1 ⎜ ⎟ ⎜ ⎟ 10 10 ⎜⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⎟ ⎜⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⎟ ⎜ ⎟ ⎜ ⎟ spinner wagon wheel ⎝ ⎠ ⎝ ⎠

𝜙2 ∘

⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ 1 1 ⎜ ⎟ ⎜ 10 10 ⚅ ⎟ ➃ 1 4 1 ⎜ 104 10 ⎟ 10 ⎟ ⎜ 10 ⚂ ➄ ➂ ⎜ ⎟ ⎜ ⚄ ⎟ ⎟=⎜ 𝗀⎜⎜ ⎟. ⎟ ⎜ ⚃ ⚁ ➅ ➁ ⎟ 2 1 2 ⎜ 101 10 ⎟ 10 10 ⎟ ⎜ ⚀ ➀ 1 1 ⎜ ⎟ 10 10 ⎜⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⎟ ⎜ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⎟ ⎟ ⎜ ⎟ ⎜ weighted die ⎠ spinner ⎝ ⎠ ⎝

Figure 2.20: Example D.11 (page 160) metric preserving function 𝜙1 and Example D.8 (page 160) metric preserving function 𝜙2 on spinner outcome subspace (Example 2.31 page 75, Example 2.32 page 76) Example 2.31. Let 𝜙1 be the metric preserving function defined in Example D.11 (page 160). Then under 𝜙1 , the spinner outcome subspace (Example 2.4 page 35) becomes what is here called the wagon wheel output subspace, as illustrated on the left in Figure 2.20 (page 75). Let 𝙂 be the spinner outcome subspace and 𝙃 the wagon wheel outcome subspace. This yields the following geometric statistics: ̇ ̇ ) = {➄} . ∁(𝙂) = {➃, ➅} ∁(𝙃 Note that the metric transform 𝜙1 also moves the outcome center from one that is not maximally likely, to one that is. ✎PROOF: ̇ ∁(𝙂) ≜ arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑥∈𝙂

𝑦∈𝙂

⎧ ⎪ 1⎪ = arg min max ⎨ 𝑦∈𝙂 10 𝑥∈𝙂 ⎪ ⎪ ⎩

0×1 1×1 2×1 3×1 2×1 1×1

̇ ) ≜ arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) ∁(𝙃 𝑥∈𝙂 2016 Jul 4

𝑦∈𝙂

10:12am UTC

by definition of ∁̇ (Definition 2.3 page 28) 1×2 0×2 1×2 2×2 3×2 2×2

2×1 1×1 0×1 1×1 2×1 3×1

3×1 2×1 1×1 0×1 1×1 2×1

2×4 3×4 2×4 1×4 0×4 1×4

1×1 2×1 3×1 2×1 1×1 0×1

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

⎧ ⎪ 1⎪ = arg min ⎨ 10 ⎪ 𝑥∈𝙂 ⎪ ⎩

8 12 8 4 6 4

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

⎫ ⎧ ⎪ ⎪ ⎪ ⎪ =⎨ ➃ ⎬ ⎪ ⎪ ⎪ ⎩ ➅ ⎪ ⎭



Daniel J. Greenhoe

version 0.50

page 76


⎧ ⎪ 1⎪ = arg min max ⎨ 𝑦∈𝙃 10 𝑥∈𝙃 ⎪ ⎪ ⎩

0×1 1×1 2×1 1×1 2×1 1×1

1×2 0×2 1×2 2×2 1×2 2×2

2×1 1×1 0×1 1×1 2×1 1×1

1×1 2×1 1×1 0×1 1×1 2×1

2×4 1×4 2×4 1×4 0×4 1×4

1×1 2×1 1×1 2×1 1×1 0×1

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

⎧ ⎪ 1⎪ = arg min ⎨ 10 ⎪ 𝑥∈𝙃 ⎪ ⎩

8 4 8 4 2 4

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

⎧ ⎫ ⎪ ⎪ ⎪ ⎪ =⎨ ⎬ ⎪ ➄ ⎪ ⎪ ⎪ ⎩ ⎭

✏ Example 2.32. Let 𝜙2 be the metric preserving function defined in Example D.8 (page 160). Let 𝗀 be the function defined in Example 2.30 (page 75). Then under under 𝜙2 ∘ 𝗀, the spinner outcome subspace (Example 2.4 page 35) becomes the weighted die outcome subspace (Example 2.3 page 33), as illustrated on the right in Figure 2.20 (page 75). Let 𝙂 be the spinner outcome subspace and 𝙃 the weighted die outcome subspace. These structures have the following geometric statistics: ̇ ̇ ) = {➀, ➂, ➃, ➄, ➅} . ∁(𝙂) = {➃, ➅} ∁(𝙃 Note that in Example 2.31 (page 75), the metric transform 𝜙1 results in a smaller (smaller cardinal̇ ̇ ity Definition 1.13 page 7) center (| ∁(𝙂) | = 2 > 1 = | ∁(𝙃 ) |). But here, the metric transform 𝜙2 ∘ 𝗀 results ̇ ̇ in a larger center. (| ∁(𝙂) | = 2 < 5 = | ∁(𝙃 ) |). ✎PROOF: ̇ ∁(𝙂) = {➃, ➅} ̇∁(𝙃 ) ≜ arg min max 𝖽(𝑥, 𝑦) 𝖯(𝑦) 𝑥∈𝙂

by Example 2.31 (page 75) by definition of ∁̇ (Definition 2.3 page 28)

𝑦∈𝙂

⎧ ⎪ 1⎪ = arg min max ⎨ 𝑦∈𝙃 10 𝑥∈𝙃 ⎪ ⎪ ⎩

0×1 1×1 1×1 1×1 1×1 2×1

1×2 0×2 1×2 1×2 2×2 1×2

1×1 1×1 0×1 2×1 1×1 1×1

1×1 1×1 2×1 0×1 1×1 1×1

1×4 2×4 1×4 1×4 0×4 1×4

2×1 1×1 1×1 1×1 1×1 0×1

⎧ ⎫ ⎪ ⎪ 1⎪ ⎪ ⎬ = arg min 10 ⎨ 𝑥∈𝙃 ⎪ ⎪ ⎪ ⎪ ⎩ ⎭

4 8 4 4 4 4

⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎬=⎨ ⎪ ⎪ ⎪ ⎭ ⎪ ⎩

➀ ➂ ➃ ➄ ➅

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

✏ 2 1.5 1 ≈ 0.716 1 3

−3

−2

−1

0

1

2

7 3

8 3

𝑥 3

4

5

6

7

8

9

Figure 2.21: real line addition arg min𝑥 max𝑦 calculation graph (Example 2.33 page 76) ̈ Example 2.33 (linear addition with metric transform). Example 2.28 (page 72) gave the result 𝖤(𝖷+ 7 4 8 ̈ ̈ + 𝖤(𝖸)). Again, 𝖸) = 3 rather than the perhaps more desirable result of 3 (which equals 1 + 3 = 𝖤(𝖷) we adjust the geometric statistics of an outcome subspace by use of a metric preserving function (Definition D.11 page 158). In particular, we use the power transform/snowflake transform (Example D.4 page 159) ln 2 ̈ + 𝖸) = 7 , as illustrated in Figure 2.21 (page 76). 𝖿(𝑥) = 𝑥𝑎 . If we let 𝑎 = ln 7−ln ≈ 2.0600427, then 𝖤(𝖷 3 5

version 0.50


Daniel J. Greenhoe

2016 Jul 4

10:12am UTC

CHAPTER 3 SYMBOLIC SEQUENCE PROCESSING ON RN

❝…those

who assert that the mathematical sciences say nothing of the beautiful or the good are in error. For these sciences say and prove a great deal about them; if they do not expressly mention them, but prove attributes which are their results or definitions, it is not true that they tell us nothing about them. The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree.❞ Aristotle 384 BC – 322 BC, Greek philosopher

1

3.1 Outcome subspace sequences 3.1.1 Definitions Definition 3.1. Let 𝔻1 and 𝔻2 be CONVEX SUBSETs (Definition 1.25 page 9) of ℤ. Let 𝔻 ≜ (⋀ 𝔻1 − ⋁ 𝔻2 − 1 ∶ ⋁ 𝔻1 − ⋀ 𝔻2 + 1). Let ⦅𝑥𝑛 ⦆𝔻1 and ⦅𝑦𝑛 ⦆𝔻2 be SEQUENCEs over an OUTCOME SUBSPACE (𝛺, ≤,̇ 𝖽,̇ 𝖯̇ ). The outcome subspace sequence metric 𝗉 (⦅𝑥𝑛 ⦆ , ⦅𝑦𝑛 ⦆) is defined as ̇ ⎧ 𝖽(𝑥𝑛 , 𝑦𝑛 ) if 𝑛 ∈ 𝔻1 and 𝑛 ∈ 𝔻2 ⎫ ⎪ 1 if 𝑛 ∈ 𝔻1 but 𝑛 ∉ 𝔻2 ⎪ 𝗉 (⦅𝑥𝑛 ⦆ , ⦅𝑦𝑛 ⦆) ≜ 𝖿(𝑛) where 𝖿(𝑛) ≜ ⎨ ∀𝑛∈𝔻 ∑ 1 if 𝑛 ∉ 𝔻1 but 𝑛 ∈ 𝔻2 ⎬ 𝑛∈𝔻 ⎪ ⎪ otherwise ⎩ 0 ⎭ Proposition 3.1. Let ⦅𝑥𝑛 ⦆𝔻1 and ⦅𝑦𝑛 ⦆𝔻2 be SEQUENCEs over an OUTCOME SUBSPACE (𝛺, ≤,̇ 𝖽,̇ 𝖯̇ ). Let 𝗉 be the OUTCOME SUBSPACE SEQUENCE METRIC. 𝔻1 ∩ 𝔻2 ≠ ∅ ⟹ 𝗉 is a METRIC ✎PROOF:

This follows from the Fréchet product metric (Proposition D.9 page 160). In particular, 𝗉 is a sum of ̇ 𝑛 , 𝑦𝑛 ) and the discrete metric (Definition D.8 page 157). ✏ metrics that include the metrics 𝖽(𝑥

In standard signal processing, the autocorrelation of a sequence ⦅𝑥𝑛 ⦆𝑛∈𝔻 is another sequence ⦅𝑦𝑛 ⦆𝑛∈𝔻 defined as 𝑦𝑛 ≜ ∑𝑚∈ℤ 𝑥𝑚 𝑥𝑚−𝑛 . However, this definition requires that the sequence ⦅𝑥𝑛 ⦆𝑛∈𝔻 be con1

quote: 📘 Aristotle (330BC?) ⟨Book XIII Part 3⟩ image: http://upload.wikimedia.org/wikipedia/commons/9/98/Sanzio_01_Plato_Aristotle.jpg

77

page 78

3.1. OUTCOME SUBSPACE SEQUENCES

structed over a field. In an outcome subspace sequence, we in general do not have a field; for example, in a die outcome subspace, the expressions ⚀ + ⚁ and ⚀ × ⚁ are undefined. This paper offers an alternative definition (next) for autocorrelation that uses the distance 𝖽̇ and that does not require a field. Definition 3.2. Let ⦅𝑥𝑛 ⦆𝑛∈𝔻 and ⦅𝑦𝑛 ⦆𝑛∈𝔻 be SEQUENCEs over the OUTCOME SUBSPACE (𝛺, ≤,̇ 𝖽,̇ 𝖯̇ ). Let 𝗉 (⦅𝑥𝑛 ⦆ , ⦅𝑦𝑛 ⦆) be the outcome subspace sequence metric (Definition 3.1 page 77). The cross-correlation 𝑅𝗑𝗒 (𝑛) of ⦅𝑥𝑛 ⦆ and ⦅𝑦𝑛 ⦆ and the autocorrelation 𝑅𝗑𝗑 (𝑛) of ⦅𝑥𝑛 ⦆ are defined as 𝑅𝗑𝗒 (𝑛) ≜ − 𝗉 ⦅𝑥 ⦆ , ⦅𝑦𝑚 ⦆) (cross-correlation 𝑅𝗑𝗒 (𝑛) of ⦅𝑥𝑛 ⦆ and ⦅𝑦𝑛 ⦆) ∑ ( 𝑚−𝑛 𝑚∈ℤ

𝑅𝗑𝗑 (𝑛) ≜ −

𝗉 ⦅𝑥 ⦆ , ⦅𝑥𝑚 ⦆) ∑ ( 𝑚−𝑛

(autocorrelation 𝑅𝗑𝗑 (𝑛) of ⦅𝑥𝑛 ⦆)

𝑚∈ℤ

Moreover, the 𝘔 -offset autocorrelation of ⦅𝑥𝑛 ⦆ and ⦅𝑦𝑛 ⦆ is here defined as 𝑅𝗑𝗑 (𝑛) + 𝘔 page 25).

(Definition 1.44

3.1.2 Examples of symbolic sequence statistics Example 3.1 (fair die sequence). Consider the pseudo-uniformly distributed fair die 2 page 28) sequence generated by the C code where 'A' represents ⚀ , 1 # include < s t d l i b . h> 2 ... 'B' represents ⚁ , 3 srand ( 0 x5EED ) ; ⋮ 4 f o r ( n= 0 ; n ... srand ( 0 x5EED ) ; f o r ( n= 0 ; n 0} 2 √𝑎𝑥2 + 𝑏𝑥 + 𝑐 𝖽𝑥 = 2𝑎𝑥 + 𝑏 √𝑎𝑥2 + 𝑏𝑥 + 𝑐 + 4𝑎𝑐 − 𝑏 ln 2𝑎𝑥 + 𝑏 + 2√𝑎(𝑎𝑥2 + 𝑏𝑥 + 𝑐) + 𝜀 ) ( } {∫ 4𝑎 8𝑎3/2

✎PROOF: 1. lemma: (first equality by the product rule, and the second equality by the chain rule) d 2𝑎𝑥 + 𝑏 √ 2 2𝑎𝑥 + 𝑏 d√ 2 d 2𝑎𝑥 + 𝑏 √ 2 𝑎𝑥 + 𝑏𝑥 + 𝑐 ] = ( 𝑎𝑥 + 𝑏𝑥 + 𝑐 ) + ( 𝑎𝑥 + 𝑏𝑥 + 𝑐 ) [( ) )( d𝑥 4𝑎 4𝑎 d𝑥 d𝑥 4𝑎 )( =( =

2𝑎𝑥 + 𝑏 2𝑎𝑥 + 𝑏 2𝑎 + ( )(√𝑎𝑥2 + 𝑏𝑥 + 𝑐 ) ) 4𝑎 4𝑎 ( 2√𝑎𝑥2 + 𝑏𝑥 + 𝑐 )

(2𝑎𝑥 + 𝑏)2 + 4𝑎(𝑎𝑥2 + 𝑏𝑥 + 𝑐) 8𝑎√𝑎𝑥2 + 𝑏𝑥 + 𝑐 4𝑎 𝑥 + 4𝑎𝑥𝑏 + 𝑏2 + 4𝑎2 𝑥2 + 4𝑎𝑏𝑥 + 4𝑎𝑐 2 2

= =

8𝑎√𝑎𝑥2 + 𝑏𝑥 + 𝑐 8𝑎2 𝑥2 + 8𝑎𝑏𝑥 + 𝑏2 + 4𝑎𝑐 8𝑎√𝑎𝑥2 + 𝑏𝑥 + 𝑐

2. lemma: If (2𝑎𝑥 + 𝑏 + 2√𝑎(𝑎𝑥2 + 𝑏𝑥 + 𝑐)) > 0 then d 4𝑎𝑐 − 𝑏2 ln (2𝑎𝑥 + 𝑏 + 2√𝑎(𝑎𝑥2 + 𝑏𝑥 + 𝑐)) ] d𝑥 [ 8𝑎3/2 =

d 4𝑎𝑐 − 𝑏2 ln 2𝑎𝑥 + 𝑏 + 2√𝑎(𝑎𝑥2 + 𝑏𝑥 + 𝑐))] [ 3/2 ) d𝑥 ( ( 8𝑎

by linearity of

=

2√𝑎(2𝑎𝑥 + 𝑏) 4𝑎𝑐 − 𝑏2 1 2𝑎 + ( 8𝑎3/2 )( 2𝑎𝑥 + 𝑏 + 2√𝑎(𝑎𝑥2 + 𝑏𝑥 + 𝑐) )( 2√𝑎𝑥2 + 𝑏𝑥 + 𝑐 )

by chain rule

=

d d𝑥

(4𝑎𝑐 − 𝑏2 )[2𝑎√𝑎𝑥2 + 𝑏𝑥 + 𝑐 + √𝑎(2𝑎𝑥 + 𝑏)] 8𝑎3/2 (2𝑎𝑥 + 𝑏 + 2√𝑎(𝑎𝑥2 + 𝑏𝑥 + 𝑐))√𝑎𝑥2 + 𝑏𝑥 + 𝑐

=

(4𝑎𝑐 − 𝑏2 )[√𝑎(2𝑎𝑥 + 𝑏) + 2𝑎√𝑎𝑥2 + 𝑏𝑥 + 𝑐 ] 2 2 (8𝑎√𝑎𝑥 + 𝑏𝑥 + 𝑐 )[√𝑎(2𝑎𝑥 + 𝑏) + 2𝑎√𝑎𝑥 + 𝑏𝑥 + 𝑐 ]

=

4𝑎𝑐 − 𝑏2 8𝑎√𝑎𝑥2 + 𝑏𝑥 + 𝑐

6

📘 Gradshteyn and Ryzhik (2007) page 94 ⟨2.25 Forms containing √𝑎 + 𝑏𝑥 + 𝑐𝑥2 , 2.26 Forms containing 1 √𝑎 + 𝑏𝑥 + 𝑐𝑥2 and integral powers of 𝑥⟩ 📘 Jeffrey (1995) page 160 ⟨4.3.4 Integrands containing (𝑎+𝑏𝑥+𝑐𝑥2 ) /2 ⟩, 📘 Jeffrey 1 and Dai (2008) pages 172–173 ⟨4.3.4 Integrands containing (𝑎 + 𝑏𝑥 + 𝑐𝑥2 ) /2 ⟩, 2016 Jul 4

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Daniel J. Greenhoe

version 0.50

page 112

A.3. CALCULATION

3. lemma: If (2𝑎𝑥 + 𝑏 + 2√𝑎(𝑎𝑥2 + 𝑏𝑥 + 𝑐)) < 0 then d 4𝑎𝑐 − 𝑏2 ln |2𝑎𝑥 + 𝑏 + 2√𝑎(𝑎𝑥2 + 𝑏𝑥 + 𝑐)| ] d𝑥 [ 8𝑎3/2 =

d 4𝑎𝑐 − 𝑏2 ln −2𝑎𝑥 − 𝑏 − 2√𝑎(𝑎𝑥2 + 𝑏𝑥 + 𝑐))] ( 8𝑎3/2 )[ d𝑥 (

by linearity of

=

2√𝑎(2𝑎𝑥 + 𝑏) 4𝑎𝑐 − 𝑏2 1 2𝑎 + ( 8𝑎3/2 )( −2𝑎𝑥 − 𝑏 − 2√𝑎(𝑎𝑥2 + 𝑏𝑥 + 𝑐) )( 2√𝑎𝑥2 + 𝑏𝑥 + 𝑐 )

by chain rule

=

d d𝑥

(4𝑎𝑐 − 𝑏2 )[2𝑎√𝑎𝑥2 + 𝑏𝑥 + 𝑐 + √𝑎(2𝑎𝑥 + 𝑏)] 8𝑎3/2 (−2𝑎𝑥 − 𝑏 − 2√𝑎(𝑎𝑥2 + 𝑏𝑥 + 𝑐))√𝑎𝑥2 + 𝑏𝑥 + 𝑐

=

−(4𝑎𝑐 − 𝑏2 )[√𝑎(2𝑎𝑥 + 𝑏) + 2𝑎√𝑎𝑥2 + 𝑏𝑥 + 𝑐 ] 2 2 (8𝑎√𝑎𝑥 + 𝑏𝑥 + 𝑐 )[√𝑎(2𝑎𝑥 + 𝑏) + 2𝑎√𝑎𝑥 + 𝑏𝑥 + 𝑐 ]

=

−(4𝑎𝑐 − 𝑏2 ) 8𝑎√𝑎𝑥2 + 𝑏𝑥 + 𝑐

4. Complete the proof: If (2𝑎𝑥 + 𝑏 + 2√𝑎(𝑎𝑥2 + 𝑏𝑥 + 𝑐)) > 0 then d 2𝑎𝑥 + 𝑏 √ 2 4𝑎𝑐 − 𝑏2 𝑎𝑥 + 𝑏𝑥 + 𝑐 + ln (2𝑎𝑥 + 𝑏 + 2√𝑎(𝑎𝑥2 + 𝑏𝑥 + 𝑐)) ] d𝑥 [ 4𝑎 8𝑎3/2 d d 2𝑎𝑥 + 𝑏 √ 2 4𝑎𝑐 − 𝑏2 𝑎𝑥 + 𝑏𝑥 + 𝑐 ] + ln 2𝑎𝑥 + 𝑏 + 2√𝑎(𝑎𝑥2 + 𝑏𝑥 + 𝑐)) = [( ) ] d𝑥 4𝑎 d𝑥 [( 8𝑎3/2 ) ( = =

8𝑎2 𝑥2 + 8𝑎𝑏𝑥 + 𝑏2 + 4𝑎𝑐 8𝑎√𝑎𝑥2 + 𝑏𝑥 + 𝑐 8𝑎2 𝑥2 + 8𝑎𝑏𝑥 + 𝑏2 + 4𝑎𝑐

+ +

4𝑎𝑐 − 𝑏2 d ln 2𝑎𝑥 + 𝑏 + 2√𝑎(𝑎𝑥2 + 𝑏𝑥 + 𝑐)) ] d𝑥 [( 8𝑎3/2 ) (

by item (1)

4𝑎𝑐 − 𝑏2

by item (2)

8𝑎√𝑎𝑥2 + 𝑏𝑥 + 𝑐 8𝑎√𝑎𝑥2 + 𝑏𝑥 + 𝑐 2 8𝑎(𝑎𝑥 + 𝑏𝑥 + 𝑐) = = √𝑎𝑥2 + 𝑏𝑥 + 𝑐 2 8𝑎√𝑎𝑥 + 𝑏𝑥 + 𝑐 5. Note that simply forcing the agruement of ln to be positive as in7 2𝑎𝑥+𝑏 √𝑎𝑥2 + 𝑏𝑥 + 𝑐 + 4𝑎𝑐−𝑏2 ln 2𝑎𝑥 + 𝑏 + 2√𝑎(𝑎𝑥2 + 𝑏𝑥 + 𝑐) + 𝜀 3/2 4𝑎 | | 8𝑎

is not a solution to

∫ √𝑎𝑥2

+ 𝑏𝑥 + 𝑐 𝖽𝑥 when (2𝑎𝑥 + 𝑏 + 2√𝑎(𝑎𝑥2 + 𝑏𝑥 + 𝑐)) < 0:

4𝑎𝑐 − 𝑏2 d 2𝑎𝑥 + 𝑏 √ 2 𝑎𝑥 + 𝑏𝑥 + 𝑐 + ln |2𝑎𝑥 + 𝑏 + 2√𝑎(𝑎𝑥2 + 𝑏𝑥 + 𝑐)| ] d𝑥 [ 4𝑎 8𝑎3/2 d 2𝑎𝑥 + 𝑏 √ 2 d 4𝑎𝑐 − 𝑏2 2 = 𝑎𝑥 + 𝑏𝑥 + 𝑐 + ] d𝑥 [( 8𝑎3/2 ) ln |2𝑎𝑥 + 𝑏 + 2√𝑎(𝑎𝑥 + 𝑏𝑥 + 𝑐)|] d𝑥 [( 4𝑎 ) = =

8𝑎2 𝑥2 + 8𝑎𝑏𝑥 + 𝑏2 + 4𝑎𝑐 8𝑎√𝑎𝑥2 + 𝑏𝑥 + 𝑐 8𝑎2 𝑥2 + 8𝑎𝑏𝑥 + 𝑏2 + 4𝑎𝑐

+ −

d 4𝑎𝑐 − 𝑏2 ln 2𝑎𝑥 + 𝑏 + 2√𝑎(𝑎𝑥2 + 𝑏𝑥 + 𝑐)| ] d𝑥 [( 8𝑎3/2 ) | 4𝑎𝑐 − 𝑏2

by item (1) by item (3)

8𝑎√𝑎𝑥2 + 𝑏𝑥 + 𝑐 8𝑎√𝑎𝑥2 + 𝑏𝑥 + 𝑐 2 2 8𝑎(𝑎𝑥 + 𝑏𝑥) + 2𝑏 8𝑎(𝑎𝑥2 + 𝑏𝑥 + 𝑐) + 2𝑏2 − 8𝑎𝑐 = = 8𝑎√𝑎𝑥2 + 𝑏𝑥 + 𝑐 8𝑎√𝑎𝑥2 + 𝑏𝑥 + 𝑐 7

The solution ln |⋯| is used in 📘 Jeffrey (1995) page 160 and 📘 Jeffrey and Dai (2008) pages 172–173.

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2016 Jul 4

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APPENDIX A. LAGRANGE ARC DISTANCE

2𝑏2 − 8𝑎𝑐

= √𝑎𝑥2 + 𝑏𝑥 + 𝑐 +

8𝑎√𝑎𝑥2

≠ √𝑎𝑥2 + 𝑏𝑥 + 𝑐

for 𝑏2 ≠ 4𝑎𝑐

+ 𝑏𝑥 + 𝑐

6. Note further that constraining 𝑎 > 0 is also not a solution8 because it does not guarantee that the argument 𝑢 of ln(𝑢) will be positive. Take for example 𝑎 = 1, 𝑏 = −3, 𝑐 = 3 and 𝑥 = 1. Then 2𝑎𝑥 + 𝑏 + 2√𝑎(𝑎𝑥2 + 𝑏𝑥 + 𝑐) = 2 ⋅ 1 ⋅ 1 − 3 + 2√1(1 ⋅ 1 − 3 ⋅ 1 + 3) = −1 < 0 .

✏ Theorem A.1. Let 𝖱(𝑝, 𝑞), 𝑟𝑝 , 𝑟𝑞 , and 𝜙 be as defined in Definition A.1 (page 110). Let 𝜌 ≜ 𝑟𝑞 − 𝑟𝑝 . If 𝑟𝑝 ≠ 0, 𝑟𝑞 ≠ 0 and 𝜙 ≠ 0 then 𝖱(𝑝, 𝑞) = 𝑟𝑞 √(𝑟𝑞 𝜙)2 + 𝜌2 − 𝑟𝑝 √(𝑟𝑝 𝜙)2 + 𝜌2 2𝜌

+

|𝜌| ln 𝑟 𝜌𝜙 + |𝜌|√(𝑟𝑞 𝜙)2 + 𝜌2 − ln 𝑟𝑝 𝜌𝜙 + |𝜌|√(𝑟𝑝 𝜙)2 + 𝜌2 ) ( )] 2𝜙 [ ( 𝑞

✎PROOF: 1. Let 𝛾 ≜ 𝑟𝑝 𝜙. 2. lemmas: 𝜌𝜙 + 𝛾 = (𝑟𝑞 − 𝑟𝑝 )𝜙 + 𝑟𝑝 𝜙 = 𝑟𝑞 𝜙 𝜌2 𝜙2 + 2𝜌𝛾𝜙 + (𝛾 2 + 𝜌2 ) = (𝑟𝑞 − 𝑟𝑝 )2 𝜙2 + 2(𝑟𝑞 − 𝑟𝑝 )(𝑟𝑝 𝜙)𝜙 + (𝑟𝑝 𝜙)2 + 𝜌2 = (𝑟2𝑝 + 𝑟2𝑞 − 2𝑟𝑝 𝑟𝑞 )𝜙2 + 2(𝑟𝑞 − 𝑟𝑝 )(𝑟𝑝 𝜙)𝜙 + (𝑟𝑝 𝜙)2 + 𝜌2 = (𝑟2𝑝 + 𝑟2𝑞 )𝜙2 − 2(𝑟𝑝 𝜙)2 + (𝑟𝑝 𝜙)2 + 𝜌2 = (𝑟𝑞 𝜙)2 + 𝜌2 3. lemma: 2𝜌2 𝜃 + 2𝜌𝛾 + 2√𝜌2 (𝜌2 𝜃 2 + 2𝜌𝛾𝜃 + (𝛾 2 + 𝜌2 )) > 0. Proof: 2𝜌2 𝜃 + 2𝜌𝛾 + 2√𝜌2 (𝜌2 𝜃 2 + 2𝜌𝛾𝜃 + (𝛾 2 + 𝜌2 )) > 2𝜌2 𝜃 + 2𝜌𝛾 + 2√𝜌2 𝛾 2

because √𝑥 is strictly monotonically increasing

= 2𝜌2 𝜃 + 2𝜌𝛾 + 2|𝜌|𝛾

because 𝛾 > 0

2

= 2𝜌 𝜃 + 2𝛾(𝜌 + |𝜌|) ≥0 4. Completing the proof… 𝖱(𝑝, 𝑞) 𝜃=𝜙

≜

∫ 𝜃=0

𝖽𝑟 2 2 (𝜃) + 𝑟 √ ( 𝖽𝜃 ) 𝖽𝜃 𝑟𝑞 − 𝑟𝑝

𝜙

=

∫ 0 √[( 𝜙

= = 8

𝜙

2

𝜃 + 𝑟𝑝 + [ ) ] 2

(𝑟𝑞 − 𝑟𝑝 )𝜃 + 𝑟𝑝 𝜙

√[ ∫ 0

𝜙

by def. of 𝖱(𝑝, 𝑞)

]

+

𝜙

𝑟𝑞 − 𝑟𝑝 [

2

𝑟𝑞 − 𝑟𝑝

𝜙

]

𝖽𝜃

by def. of 𝗋(𝜃)

2

]

𝖽𝜃

1 𝜙 2 2 2 2 √(𝑟𝑞 − 𝑟𝑝 ) 𝜃 + 2(𝑟𝑞 − 𝑟𝑝 )(𝑟𝑝 𝜙)𝜃 + (𝑟𝑝 𝜙) + (𝑟𝑞 − 𝑟𝑝 ) 𝖽𝜃 𝜙∫ 0

The 𝑎 > 0 constraint is used in 📘 Gradshteyn and Ryzhik (2007) page 94

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Daniel J. Greenhoe

version 0.50

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A.4. PROPERTIES

=

1 𝜙 𝜙∫ 0

=

1 2𝜌2 𝜃 + 2𝜌𝛾 √𝜌2 𝜃 2 + 2𝜌𝛾𝜃 + (𝛾 2 + 𝜌2 ) 𝜙[ 4𝜌2 +

2 + 𝜌2 ) 𝖽𝜃 𝜌2 𝜃 2 + 2𝜌𝛾 ⏟ ⏟ 𝜃 + (𝛾 ⏟⏟⏟ 𝑏 √𝑎 𝑐

[by item (3) and

𝜃=𝜙 4𝜌2 (𝛾 2 + 𝜌2 ) − (2𝜌𝛾)2 2 2 (𝜌2 𝜃 2 + 2𝜌𝛾𝜃 + (𝛾 2 + 𝜌2 )) √ ln 2𝜌 𝜃 + 2𝜌𝛾 + 2 𝜌 ( )] 8|𝜌|3 𝜃=0

by Lemma A.1]

=

𝜃=𝜙 |𝜌| 1 𝜌𝜃 + 𝛾 √𝜌2 𝜃 2 + 2𝜌𝛾𝜃 + (𝛾 2 + 𝜌2 ) + ln (2𝜌2 𝜃 + 2𝜌𝛾 + 2|𝜌|√𝜌2 𝜃 2 + 2𝜌𝛾𝜃 + (𝛾 2 + 𝜌2 )) ]𝜃=0 𝜙 [ 2𝜌 2

=

(𝜌𝜙 + 𝛾)√𝜌2 𝜙2 + 2𝜌𝛾𝜙 + (𝛾 2 + 𝜌2 ) |𝜌| + ln 2𝜌2 𝜙 + 2𝜌𝛾 + 2|𝜌|√𝜌2 𝜙2 + 2𝜌𝛾𝜙 + (𝛾 2 + 𝜌2 )) 2𝜌𝜙 2𝜙 ( [ ] −

𝛾√𝛾 2 + 𝜌2 |𝜌| + ln 2𝜌𝛾 + 2|𝜌|√𝛾 2 + 𝜌2 ) 2𝜙 ( [ 2𝜌𝜙 ]

𝑟𝑞 𝜙√(𝑟𝑞 𝜙)2 + 𝜌2 − 𝑟𝑝 𝜙√(𝑟𝑝 𝜙)2 + 𝜌2

=

2𝜌𝜙 +

=

|𝜌| ln(2) + ln 𝜌2 𝜙 + 𝜌𝛾 + |𝜌|√(𝑟𝑞 𝜙)2 + 𝜌2 − ln(2) − ln (𝜌𝛾 + |𝜌|√𝛾 2 + 𝜌2 ) ( ) ] 2𝜙 [

𝑟𝑞 √(𝑟𝑞 𝜙)2 + 𝜌2 − 𝑟𝑝 √(𝑟𝑝 𝜙)2 + 𝜌2 2𝜌

+

by item (2)


✏

A.4 Properties A.4.1 Arc function R(p,q) properties If we really want the Langrange arc distance 𝖽(𝑝, 𝑞) to be an extension of the spherical metric, then 𝖱(𝑝, 𝑞) must equal 𝑟𝑝 𝜙 (the arc length between 𝑝 and 𝑞 on a circle centered at the origin) when 𝑟𝑝 = 𝑟𝑞 . This is in fact the case, as demonstrated next. Proposition A.1 (𝖱(𝑝, 𝑞) on spherical surface). Let 𝖱(𝑝, 𝑞), 𝑟𝑝 , 𝑟𝑞 , and 𝜙 be defined as in Definition A.1. 𝑟𝑝 = 𝑟𝑞 ≠ 0 { 𝜙 ≠ 0

and

}

⟹

{𝖱(𝑝, 𝑞) = 𝑟𝑝 𝜙}

✎PROOF: 1. lemma:

lim

𝜌→0

version 0.50

𝑟𝑞 √(𝑟𝑞 𝜙)2 + 𝜌2 − 𝑟𝑝 √(𝑟𝑝 𝜙)2 + 𝜌2 2𝜌

= lim

𝑟𝑞 √(𝑟𝑞 𝜙)2 + 0 − 𝑟𝑝 √(𝑟𝑝 𝜙)2 + 0

2𝜌 𝜙 (𝑟𝑞 − 𝑟𝑝 )(𝑟𝑞 + 𝑟𝑝 ) = lim 𝜌→0 ( 2 ) 𝑟𝑞 − 𝑟𝑝 𝜌→0

= 𝑟𝑝 𝜙 A book concerning symbolic sequence processing

Daniel J. Greenhoe

2 2 𝜙 𝑟𝑞 − 𝑟𝑝 𝜌→0 ( 2 ) 𝑟𝑞 − 𝑟𝑝

= lim = lim

𝜙

(𝑟𝑞 + 𝑟𝑝 )

𝜌→0 ( 2 )

2016 Jul 4

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page 115


2. lemma: 𝜌 ln 𝑟 𝜌𝜙 + |𝜌|√(𝑟𝑞 𝜙)2 + 𝜌2 − ln 𝑟𝑝 𝜌𝜙 + |𝜌|√(𝑟𝑝 𝜙)2 + 𝜌2 ) ( )] 2𝜙 [ ( 𝑞 𝜌 = lim+ ln 𝑟𝑞 𝜌𝜙 + 𝜌√(𝑟𝑞 𝜙)2 − ln 𝑟𝑝 𝜌𝜙 + 𝜌√(𝑟𝑝 𝜙)2 ) ( )] 𝜌→0 2𝜙 [ ( lim

𝜌→0+

𝜌 ln(𝜌) + ln 𝑟𝑞 𝜙 + √(𝑟𝑞 𝜙)2 − ln(𝜌) − ln 𝑟𝑝 𝜙 + √(𝑟𝑝 𝜙)2 ( ) ( )] 𝜌→0 2𝜙 [ 𝜌 = lim+ + ln 𝑟𝑞 𝜙 + √(𝑟𝑞 𝜙)2 − ln 𝑟𝑝 𝜙 + √(𝑟𝑝 𝜙)2 ( ) ( )] 𝜌→0 2𝜙 [ = lim+

=0 3. lemma: 𝜌 ln 𝑟𝑞 𝜌𝜙 + |𝜌|√(𝑟𝑞 𝜙)2 + 𝜌2 − ln 𝑟𝑝 𝜌𝜙 + |𝜌|√(𝑟𝑝 𝜙)2 + 𝜌2 ) ( )] 𝜌→0 2𝜙 [ ( 𝜌 = lim− ln −𝑟𝑞 |𝜌|𝜙 + |𝜌|√(𝑟𝑞 𝜙)2 − ln −𝑟𝑝 |𝜌|𝜙 + |𝜌|√(𝑟𝑝 𝜙)2 ) ( )] 𝜌→0 2𝜙 [ ( lim−

𝜌 ln (𝜌) + ln −𝑟𝑞 𝜙 + √(𝑟𝑞 𝜙)2 − ln (𝜌) − ln −𝑟𝑝 𝜙 + √(𝑟𝑝 𝜙)2 ( ) ( )] 𝜌→0 2𝜙 [ 𝜌 = lim− ln −𝑟𝑞 𝜙 + √(𝑟𝑞 𝜙)2 − ln −𝑟𝑝 𝜙 + √(𝑟𝑝 𝜙)2 ) ( )] 𝜌→0 2𝜙 [ ( = lim−

=0 4. lemma: By item (2), item (3), and by continuity … 𝜌 lim ln 𝑟𝑞 𝜌𝜙 + |𝜌|√(𝑟𝑞 𝜙)2 + 𝜌2 − ln 𝑟𝑝 𝜌𝜙 + |𝜌|√(𝑟𝑝 𝜙)2 + 𝜌2 =0 ) ( )] 𝜌→0 2𝜙 [ ( 5. Completing the proof … 𝑟𝑞 √(𝑟𝑞 𝜙)2 + 𝜌2 − 𝑟𝑝 √(𝑟𝑝 𝜙)2 + 𝜌2

lim 𝖱(𝑝, 𝑞) = lim

𝜌→0

𝜌→0 [

+

2𝜌

𝜌 ln 𝑟 𝜌𝜙 + |𝜌|√(𝑟𝑞 𝜙)2 + 𝜌2 − ln 𝑟𝑝 𝜌𝜙 + |𝜌|√(𝑟𝑝 𝜙)2 + 𝜌2 ) ( )] 2𝜙 ( 𝑞

by Theorem A.1

𝜌 ln 𝑟𝑞 𝜌𝜙 + |𝜌|√(𝑟𝑞 𝜙)2 + 𝜌2 − ln 𝑟𝑝 𝜌𝜙 + |𝜌|√(𝑟𝑝 𝜙)2 + 𝜌2 by item (1) ) ( )] 𝜌→0 2𝜙 [ (

= 0 + lim =0+0

by item (4)

=0

✏ Later in Theorem A.3 (page 119), we want to prove that the Langrange arc distance (Definition A.1 page 110) 𝖽(𝑝, 𝑞) is indeed, as its name suggests, a distance function (Definition B.1 page 127). Proposition A.2 (symmetry) and Proposition A.4 (positivity) will help. Meanwhile, Proposition A.4 will itself receive help from Proposition A.3 (monotonicity). Proposition A.2 (symmetry of 𝖱). Let 𝖱(𝑝, 𝑞) be defined as in Definition A.1 (page 110). 𝖱(𝑝, 𝑞) = 𝖱(𝑞, 𝑝) ∀𝑝,𝑞∈ℝ𝘕 ( SYMMETRIC) ✎PROOF: 1. dummy variable: Let 𝜇 ≜ 𝜙 − 𝜃 which implies 𝜃 = 𝜙 − 𝜇 and 𝖽𝜃 = − 𝖽𝜇. 2016 Jul 4

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A.4. PROPERTIES

2. lemma: 𝗋(𝜇; 𝑞, 𝑝) ≜ 𝗋(𝜙 − 𝜃; 𝑞, 𝑝) 𝑟𝑝 − 𝑟𝑞 𝑟𝑞 − 𝑟 𝑝 𝑟𝑞 − 𝑟𝑝 = (𝜙 − 𝜃) + 𝑟𝑞 = 𝜃 + (𝑟𝑝 − 𝑟𝑞 ) + 𝑟𝑞 = 𝜃 + 𝑟𝑝 ( 𝜙 ) ( 𝜙 ) ( 𝜙 )

by item (1)

= 𝗋(𝜃; 𝑝, 𝑞) 3. Completing the proof… 𝜃=𝜙

𝖱(𝑝, 𝑞) =

∫ 𝜃=0 √

𝗋2 (𝜃; 𝑝, 𝑞) +

𝖽𝑟(𝜃; 𝑝, 𝑞) 2 𝖽𝜃 [ ] 𝖽𝜃

𝜙−𝜇=𝜙

=

∫ 𝜙−𝜇=0 √

𝗋2 (𝜇; 𝑞, 𝑝) +

𝖽𝑟(𝜇; 𝑞, 𝑝) 2 (− 𝖽𝜇) by item (1) and item (2) [ − 𝖽𝜇 ]

𝜇=0

=−

∫ 𝜇=𝜙

𝗋2 (𝜇; 𝑞, 𝑝) + [ √

𝜇=𝜙

=

∫ 𝜇=0 √

𝗋2 (𝜇; 𝑞, 𝑝) +

𝖽𝑟(𝜇; 𝑞, 𝑝) 2 𝖽𝜇 ] 𝖽𝜇

𝖽𝑟(𝜇; 𝑞, 𝑝) 2 𝖽𝜇 ] [ 𝖽𝜇

by the Second Fundamental Theorem of Calculus 9

= 𝖱(𝑞, 𝑝)

✏ Proposition A.3 (monotonicity of 𝖱). Let 𝖱(𝑝, 𝑞) and 𝜙 be defined as in Definition A.1 (page 110). Let 𝜙1 be the polar angle between the point pair (𝑝1 , 𝑞1 ) in ℝ𝘕 and Let 𝜙2 the polar angle between the point pair (𝑝2 , 𝑞2 ) in ℝ𝘕 . {𝜙1 < 𝜙2 } ⟹ {𝖱(𝑝1 , 𝑞1 ) < 𝖱(𝑝2 , 𝑞2 )} ∀𝜙1 ,𝜙2 ∈(0∶𝜋] (strictly monotonically increasing IN 𝜙) ✎PROOF: 2

𝜙 d𝗋(𝜃) d d 𝖱(𝑝, 𝑞) ≜ 𝗋2 (𝜃) + 𝖽𝜃 [ √ d𝜙 d𝜙 ∫ d𝜃 ] 0

by Definition A.1 (page 110) 2

2

=

=

=

𝜙 𝑟𝑞 − 𝑟𝑝 𝑟𝑞 − 𝑟𝑝 d 𝜃 + 𝑟𝑝 + 𝖽𝜃 ] [ 𝜙 ] d𝜙 ∫ 𝜙 ) 0 √[(

𝑟𝑞 − 𝑟𝑝 √[(

√

𝑟2𝑞 +

𝜙

2

𝜙 + 𝑟𝑝 + ) ] [

𝑟𝑞 − 𝑟𝑝 [

𝜙

2

𝑟𝑞 − 𝑟𝑝 𝜙

]

by the First Fundamental Theorem of Calculus 10

2

]

>0 ⟹ 𝖱(𝑝, 𝑞) is strictly monotonically increasing in 𝜙

✏ Proposition A.4 (positivity of 𝖱). Let 𝖱(𝑝, 𝑞) and 𝜙 be defined as in Definition A.1 (page 110). 𝜙 ∈ (0 ∶ 𝜋] ⟹ { 𝖱(𝑝, 𝑞) > 0 ∀𝑝,𝑞∈ℝ𝘕 ( POSITIVE) } } { (𝜙 ≠ 0) 9 10

📘 Hijab (2016) page 170 ⟨Theorem 4.4.3 Second Fundamental Theorem of Calculus⟩, 📘 Amann and Escher (2008) page 31 ⟨Theorem 4.13 The second fundamental theorem of calculus⟩ 📘 Schechter (1996) page 674 ⟨25.15⟩, 📘 Haaser and Sullivan (1991) page 218

version 0.50


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2016 Jul 4

10:12am UTC

page 117


✎PROOF: 2

𝜙 d𝗋(𝜃) d 𝖱(𝑝, 𝑞) ≜ 𝗋2 (𝜃) + 𝖽𝜃 [ √ ∫ d𝜙 d𝜃 ] 0

=

∫√

𝗋2 (𝜃) +

by Definition A.1 (page 110)

| | d𝗋(𝜃) 2 | d𝗋(𝜃) 2 | 𝖽𝜃 − 𝗋2 (𝜃) + 𝖽𝜃 [ d𝜃 ] | [ d𝜃 ] | ∫√ |𝜙 |0

>0

by the Second Fundamental Theorem of Calculus by Proposition A.3 (page 116)

✏ 𝘕

For the sake of continuity at the origin of ℝ , one might hope that it doesn't matter which “direction” the points 𝑝 or 𝑞 approach the origin when computing the limit of 𝖱(𝑝, 𝑞). This however is not the case, as demonstrated next and illustrated to the right and in Example A.1 (page 118). In fact, the limits very much depend on 𝜙…resulting in a discontinuity at the origin, as demonstrated in Theorem A.2 (page 119).

≈ 1.9𝑟𝑞 ≈ 1.3𝑟𝑞 𝑟𝑞

lim 𝖱(𝑝, 𝑞)

𝑟𝑝 →0

(0,0)

/

𝜋2

𝜋

𝜙

Proposition A.5 (limit cases of 𝖱). Let 𝖱(𝑝, 𝑞), 𝑟𝑝 , 𝑟𝑞 , and 𝜙 be defined as in Definition A.1 (page 110). ln (𝜙 + √𝜙2 + 1) ⎤ 𝑟𝑞 ⎡ ⎢√𝜙2 + 1 + ⎥ lim 𝖱(𝑝, 𝑞) = 𝑟𝑝 →0 2⎢ 𝜙 ⎥ ⎣ ⎦ 2 ln (𝜙 + √𝜙 + 1) ⎤ 𝑟𝑝 ⎡ ⎢√𝜙2 + 1 + ⎥ lim 𝖱(𝑝, 𝑞) = 𝑟𝑞 →0 2⎢ 𝜙 ⎥ ⎣ ⎦

∀𝑝,𝑞∈ℝ𝘕 ⧵(0,0,⋯,0), 𝜙≠0

(𝑝 approaching origin)

∀𝑝,𝑞∈ℝ𝘕 ⧵(0,0,⋯,0), 𝜙≠0

(𝑞 approaching origin)

✎PROOF: lim 𝖱(𝑝, 𝑞)

𝑟𝑝 →0

= lim

𝑟𝑞 √(𝑟𝑞 𝜙)2 + 𝜌2 − 𝑟𝑝 √(𝑟𝑝 𝜙)2 + 𝜌2 2𝜌

𝑟𝑝 →0

+


by Theorem A.1 (page 113) = =

𝑟𝑞 √(𝑟𝑞 𝜙)2 + 𝑟2𝑞 − 0 2𝑟𝑞 𝑟𝑞 √𝜙2 + 1 2

+

+

|𝑟𝑞 | ln 𝑟2 𝜙 + |𝑟𝑞 |√(𝑟𝑞 𝜙)2 + 𝑟2𝑞 − ln 0 + |𝑟𝑞 |√0 + 𝑟2𝑞 ) ( )] 2𝜙 [ ( 𝑞

by lim operation 𝑟𝑝 →0

𝑟𝑞

ln 𝑟2 𝜙 + 𝑟2𝑞 √𝜙2 + 1) − ln (𝑟2𝑞 )] 2𝜙 [ ( 𝑞

ln(𝑟2𝑞 ) + ln (𝜙 + √𝜙2 + 1) − ln (𝑟2𝑞 ) ⎤ 𝑟 ⎡ ln (𝜙 + √𝜙2 + 1) ⎤ 𝑟𝑞 ⎡ 𝑞 2 2 ⎢ ⎥ ⎢ ⎥ √𝜙 + 1 + √𝜙 + 1 + = = 2⎢ 𝜙 2⎢ 𝜙 ⎥ ⎥ ⎣ ⎦ ⎣ ⎦ lim 𝖱(𝑝, 𝑞) = lim 𝖱(𝑞, 𝑝)

𝑟𝑞 →0

𝑟𝑞 →0

ln (𝜙 + √𝜙2 + 1) ⎤ 𝑟𝑝 ⎡ 2 ⎢ ⎥ √𝜙 + 1 + = 2⎢ 𝜙 ⎥ ⎣ ⎦

by Proposition A.2

by previous result

✏ 2016 Jul 4

10:12am UTC


Daniel J. Greenhoe

version 0.50

page 118

A.4. PROPERTIES

Example A.1. Let 𝖱(𝑝, 𝑞), 𝜙, and 𝑟𝑞 be defined as in Definition A.1. If 𝜙 = 0

𝑦

then lim 𝖱[𝑝, (𝑟𝑞 , 0)] = 𝑟𝑞 𝑝→0

If 𝜙 = 𝜋/2 then lim 𝖱[𝑝, (𝑟𝑞 , 0)] = 𝑟𝑞 × (1.323652 ⋯) ≈ 1.3𝑟𝑞 𝑝→0 If 𝜙 = 𝜋

𝑟𝑞

then lim 𝖱[𝑝, (𝑟𝑞 , 0)] = 𝑟𝑞 × (1.944847 ⋯) ≈ 1.9𝑟𝑞

𝑝

b

𝑥 𝑞

𝑝→0

✎PROOF: 2 ⎡ ln 𝜋2 + √( 𝜋2 ) + 1 ⎤ 𝑟𝑞 ⎢ ( )⎥ 𝜋 2 𝖱(𝑝, 𝑞)|𝜙=𝜋/2 = ⎢√( ) + 1 + 𝜋 ⎥ 2 2 2 ⎢ ⎥ ⎣ ⎦ 2 ln (𝜋 + √𝜋 + 1) ⎤ 𝑟𝑞 ⎡ ⎥ 𝖱(𝑝, 𝑞)|𝜙=𝜋 = ⎢√𝜋 2 + 1 + 2⎢ 𝜋 ⎥ ⎣ ⎦

= 𝑟𝑞 × (1.323652 ⋯)

= 𝑟𝑞 × (1.944847 ⋯)

✏

A.4.2 Distance function d(p,q) properties The Langrange arc distance 𝖽(𝑝, 𝑞) is defined in two parts: one part being the Euclidean distance 1/𝜋 𝑟 − 𝑟 and the second part the length of the arc 1/𝜋 𝖱(𝑝, 𝑞). There is risk in creating a multipart def| 𝑞 𝑝| inition…with the possible consequences being discontinuity at the boundary of the parts. Proposition A.6 (next) demonstrates that when 𝑟𝑝 ≠ 0 and 𝑟𝑞 ≠ 0, there is continuity as 𝜙 → 0. However, Theorem A.3 (page 119) demonstrates that in general for values of 𝜙 > 0, 𝖽(𝑝, 𝑞) is discontinuous at the origin. Proposition A.6. Let 𝖱(𝑝, 𝑞), 𝑟𝑝 , 𝑟𝑞 , 𝜙, and (0, 0, ⋯ , 0) be defined as in Definition A.1 (page 110). (A). lim 𝖱(𝑝, 𝑞) = |𝑟𝑞 − 𝑟𝑝 | = 𝜋𝖽(𝑝, 𝑞) when 𝜙 = 0 𝜙→0

(B). (C).

lim lim 𝖱(𝑝, 𝑞) = lim lim 𝖱(𝑝, 𝑞) = 𝑟𝑞 = 𝜋𝖽((0, 0, ⋯ , 0) , 𝑞)

𝜙→0 𝑟𝑝 →0

𝑟𝑝 →0 𝜙→0

lim lim 𝖱(𝑝, 𝑞) = lim lim 𝖱(𝑝, 𝑞) = 𝑟𝑝 = 𝜋𝖽(𝑝, (0, 0, ⋯ , 0))

𝜙→0 𝑟𝑞 →0

𝑟𝑞 →0 𝜙→0

✎PROOF: 𝜙

𝖽𝑟 2 𝖽𝜃 𝑟2 (𝜃) + ( √ 𝜙→0 ∫ 𝖽𝜃 ) 0

lim 𝖱(𝑝, 𝑞) ≜ lim

𝜙→0

𝑟𝑞 − 𝑟𝑝

𝜙

= lim

𝜙→0 ∫ 0

√[(

𝜙

by definition of 𝖱(𝑝, 𝑞) (Definition A.1 page 110) 2

𝜃 + 𝑟𝑝 + [ ) ]

2

𝑟𝑞 − 𝑟𝑝 𝜙

]

𝖽𝜃

by definition of 𝗋(𝜃) (Definition A.1 page 110)

1 𝜙 2 2 √[(𝑟𝑞 − 𝑟𝑝 )𝜃 + 𝑟𝑝 𝜙] + [𝑟𝑞 − 𝑟𝑝 ] 𝖽𝜃 ] 𝜙→0 [ 𝜙 ∫ 0

= lim

=

=

lim𝜙→0

d d𝜙

𝜙

∫0 √[(𝑟𝑞 − 𝑟𝑝 )𝜃 + 𝑟𝑝 𝜙]2 + [𝑟𝑞 − 𝑟𝑝 ]2 𝖽𝜃 lim𝜙→0

d 𝜙 d𝜙

lim𝜙→0 √[(𝑟𝑞 − 𝑟𝑝 )𝜙 + 𝑟𝑝 𝜙]2 + [𝑟𝑞 − 𝑟𝑝 ]2 1

by L'Hôpital's rule

by First Fundamental Theorem of Calculus

= √[𝑟𝑞 − 𝑟𝑝 ]2 version 0.50


Daniel J. Greenhoe

2016 Jul 4

10:12am UTC

page 119


= |𝑟𝑞 − 𝑟𝑝 | = 𝜋 𝖽(𝑝, 𝑞)|𝜙=0

by Definition A.1 (page 110)

ln (𝜙 + √𝜙2 + 1) ⎤ 𝑟𝑞 ⎡ 2 ⎢ ⎥ √𝜙 + 1 + lim lim 𝖱(𝑝, 𝑞) = lim by Proposition A.5 (page 117) 𝜙→0 𝑟𝑝 →0 𝜙→0 2 ⎢ 𝜙 ⎥ ⎣ ⎦ d ⎡ lim ln (𝜙 + √𝜙2 + 1) ⎤ 𝑟𝑞 ⎢ 𝜙→0 d𝜙 ⎥ = ⎢√02 + 1 + by L'Hôpital's rule ⎥ d 2⎢ ⎥ lim 𝜙 𝜙→0 d𝜙 ⎣ ⎦ 2𝜙 ⎡ ⎤ 1+ 𝑟𝑞 ⎢ 2√𝜙2 + 1 ⎥ 𝑟𝑞 ⎥ = [1 + 1/1] = 𝑟𝑞 = 𝜋𝖽((0, 0, ⋯ , 0) , 𝑞) = ⎢1 + lim 𝜙→0 2⎢ 2 √𝜙2 + 1 ⎥ 𝜙 + ⎢ ⎥ ⎣ ⎦ by Proposition A.2 (page 115)

lim lim 𝖱(𝑝, 𝑞) = lim lim 𝖱(𝑞, 𝑝)

𝜙→0 𝑟𝑞 →0

𝜙→0 𝑟𝑞 →0

= 𝑟𝑝 = 𝜋𝖽(𝑝, (0, 0, ⋯ , 0)) lim lim 𝖱(𝑝, 𝑞) = lim |𝑟𝑞 − 𝑟𝑝 | 𝑟 →0

𝑟𝑝 →0 𝜙→0

by previous result by (A)

𝑝

= 𝑟𝑞 = 𝜋𝖽((0, 0, ⋯ , 0) , 𝑞) lim lim 𝖱(𝑝, 𝑞) = lim |𝑟𝑞 − 𝑟𝑝 | 𝑟 →0

𝑟𝑞 →0 𝜙→0

by (A)

𝑞

= 𝑟𝑝 = 𝜋𝖽(𝑝, (0, 0, ⋯ , 0))

✏ Theorem A.2. Let the LAGRANGE ARC DISTANCE 𝖽(𝑝, 𝑞) and ORIGIN be defined as in Definition A.1. The function 𝖽(𝑝, 𝑞) is DISCONTINUOUS at the ORIGIN of ℝ𝘕 , but is CONTINUOUS everywhere else in ℝ𝘕 . ✎PROOF: 1. Proof for when 𝑝 and 𝑞 are not at the origin and 𝜙 ≠ 0: (a) In this case, 𝖽(𝑝, 𝑞) = 1/𝜋𝖱(𝑝, 𝑞). (b) 𝖱(𝑝, 𝑞) is continuous everywhere in its domain because its solution, as given by Theorem A.1 (page 113), consists entirely of continuous functions such as ln(𝑥), |𝑥|, etc. (c) Therefore, in this case, 𝖽(𝑝, 𝑞) is also continuous. 2. Proof for when 𝑝 and 𝑞 are not at the origin and 𝜙 = 0: This follows from (A) of Proposition A.6 (page 118). 3. Proof for discontinuity at origin: This follows from Proposition A.5 (page 117), where it is demonstrated that the limit of 𝖱(𝑝, 𝑞) is very much dependent on the “direction” from which 𝑝 or 𝑞 approaches the origin. For an illustration of this concept, see Example A.1 (page 118).

✏ Theorem A.3. Let 𝖽(𝑝, 𝑞) be LAGRANGE ARC DISTANCE (Definition A.1 page 110). The function 𝖽(𝑝, 𝑞) is a DISTANCE FUNCTION (Definition B.1 page 127). In particular, (1). 𝖽(𝑝, 𝑞) ≥ 0 ∀𝑝,𝑞∈ℝ𝘕 ( NON-NEGATIVE) and 𝘕 (2). 𝖽(𝑝, 𝑞) = 0 ⟺ 𝑝 = 𝑞 ∀𝑝,𝑞∈ℝ ( NONDEGENERATE) and (3). 𝖽(𝑝, 𝑞) = 𝖽(𝑞, 𝑝) ∀𝑝,𝑞∈ℝ𝘕 ( SYMMETRIC) 2016 Jul 4

10:12am UTC


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version 0.50

page 120

A.4. PROPERTIES

✎PROOF:

The Langrange arc distance (Definition A.1 page 110) is simply the Euclidean metric (Definition D.10 page 157) if 𝑝 or 𝑞 is at the origin, or if 𝜙 = 0. In this case, (1)–(3) are satisfied automatically because all metrics have these properties (Definition D.7 page 155). What is left to prove is that 𝖱(𝑝, 𝑞) has these properties when 𝑝 and 𝑞 are not at the origin and 𝜙 ≠ 0. 1. Proof that 𝖱(𝑝, 𝑞) ≥ 0: If 𝜙 = 0, then the Euclidean metric is used. For any 𝜙 > 0, 𝖱(𝑝, 𝑞) > 0, as demonstrated by Proposition A.4 (page 116). 2. Proof that 𝑝 = 𝑞 ⟹ 𝖱(𝑝, 𝑞) = 0: If 𝑝 = 𝑞, then 𝜙 = 0, and the Euclidean metric is used, not 𝖱(𝑝, 𝑞). 3. Proof that 𝖱(𝑝, 𝑞) = 0 ⟹ 𝑝 = 𝑞: If 𝖽(𝑝, 𝑞) = 0 and 𝜙 = 0, then the Euclidean metric is used. If 𝖽(𝑝, 𝑞) = 0 and 𝜙 > 0, then 𝖱(𝑝, 𝑞) never equals 0 anyways, as demonstrated by Proposition A.4 (page 116). 4. Proof that 𝖱(𝑝, 𝑞) = 𝖱(𝑞, 𝑝): This is demonstrated by Proposition A.2 (page 115).

✏ The Lagrange arc distance is not a metric because in general the triangle inequality property does not hold (next theorem). Furthermore, the Lagrange arc distance does not induce a norm because it is not translation invariant (the translation invariant property is a necessary condition for a metric to induce a norm, Theorem D.5 page 154), and balls in a Lagrange arc distance space are in general not convex (balls are always convex in a normed linear space, Theorem D.4 page 152). For more details about distance spaces, see APPENDIX B (page 125). Theorem A.4. In the LAGRANGE ARC DISTANCE SPACE (𝑋, 𝖽) over a field 𝔽 (1). 𝖽(𝑝, 𝑟) ≰ 𝖽(𝑝, 𝑞) + 𝖽(𝑞, 𝑟) ∀𝑝,𝑞,𝑟∈𝑋 ( TRIANGLE INEQUALITY FAILS) (2). 𝖽(𝑝 + 𝑟, 𝑞 + 𝑟) ≠ 𝖽(𝑝, 𝑞) ∀𝑝,𝑞,𝑟∈𝑋 ( NOT TRANSLATION INVARIANT) (3). 𝖽(𝛼𝑝, 𝛼𝑞) = |𝛼| 𝖽(𝑝, 𝑞) ∀𝑝,𝑞,𝑟∈𝑋, 𝛼∈ℝ ( HOMOGENEOUS) (4). 𝖽 does not induce a norm (5). balls in (𝑋, 𝖽) are in general NOT CONVEX ✎PROOF: 1. Proof that the triangle inequality property fails to hold in (𝑋, 𝖽): Consider the following case11 … 𝖽(𝑝, 𝑟) ≜ 𝖽((1, 0) , (−0.5, 0)) = 0.767324 ⋯ 1 ≰ 0.756406 ⋯ = 0.692330 ⋯ + 0.064076 ⋯ 𝑞 = 𝖽((1, 0) , (−0.5, 0.2)) + 𝖽((−0.5, 0.2) , (−0.5, 0)) 𝑝 ≜ 𝖽(𝑝, 𝑞) + 𝖽(𝑞, 𝑟) 𝑟 −1 1 ⟹ triangle inequality fails in (𝑋, 𝖽) b b

b

2. Proof that (𝑋, 𝖽) is not translation invariant:12 Let 𝑟 ≜ (1/2, 1/2). Then… 1 𝑞

𝑞+𝑟 b

b

b

𝑝+𝑟 b

𝑝

−1

1

𝖽(𝑝 + 𝑟, 𝑞 + 𝑟) ≜ 𝖽((1, 12 ) , ( 21 , 1)) = 0.229009 ⋯ ≠ = 𝖽(( 12 , 0) , (0, 12 )) ≜ 𝖽(𝑝, 𝑞) ⟹ (𝑋, 𝖽) is not translation invariant

1 2

3. Proof that (𝑋, 𝖽) is homogeneous: 𝛼𝑞 1 𝑞

b

𝛼𝑝

b

b

−1

b

𝑝 1

Let 𝑟𝛼𝑝 be the magnitude of 𝛼𝑝 ≜ ⦅𝛼𝑥1 , 𝛼𝑥2 , ⋯ , 𝛼𝑥𝘕 ⦆. Let 𝑟𝛼𝑞 be the magnitude of 𝛼𝑞 ≜ ⦅𝛼𝑦1 , 𝛼𝑦2 , ⋯ , 𝛼𝑦𝘕 ⦆. Let 𝜙𝛼 be the polar angle between 𝛼𝑝 and 𝛼𝑞.

11

💾 See experiment log file “lab_larc_distances_R2.xlg” generated by the program “ssp.exe”.

12


version 0.50


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2016 Jul 4

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page 121


(a) If 𝑟𝑝 = 0 or 𝑟𝑞 = 0 or 𝜙 = 0 then 𝖽(𝑝, 𝑞) is the Euclidean metric, which is homogeneous. 1 2

𝘕

(b) lemmas:

𝑟𝛼𝑝 ≜

2 [𝛼𝑥𝑛 ] ∑ ( 1 ) 𝘕

𝑟𝛼𝑞 ≜

𝘕

= |𝛼|

∑

𝑥2𝑛

≜ |𝛼|𝑟𝑝

𝑦2𝑛

≜ |𝛼|𝑟𝑞

1

1 2

𝘕

2 [𝛼𝑦𝑛 ] ) (∑ 1

= |𝛼|

∑ 1

𝘕

𝘕

1 1 [𝛼𝑥𝑛 ][𝛼𝑦𝑛 ] = arccos 𝑥𝑛 𝑦𝑛 ≜ 𝜙 ∑ ( 𝑟𝛼𝑝 𝑟𝛼𝑞 𝑛=1 ) ( 𝑟𝑝 𝑟𝑞 ∑ ) 𝑛=1 𝑟𝛼𝑞 − 𝑟𝛼𝑝 𝛼𝑟𝑞 − 𝛼𝑟𝑝 𝗋(𝜃; 𝛼𝑝, 𝛼𝑞) ≜ 𝜃 + 𝑟𝛼𝑝 = 𝜃 + 𝛼𝑟𝑝 = 𝛼𝗋(𝜃; 𝑝, 𝑞) ( 𝜙𝛼 ) ( ) 𝜙 𝜙𝛼 ≜ arccos

(c) If 𝖽(𝑝, 𝑞) is not the Euclidean metric then … 𝜋𝖽(𝛼𝑝, 𝛼𝑞) ≜ 𝖱(𝛼𝑝, 𝛼𝑞) ≜

𝜙𝛼

√

∫ 0

[𝗋(𝜃; 𝛼𝑝, 𝛼𝑞)]2 +

by definition of 𝖽 (Definition A.1 page 110) d𝗋(𝜃; 𝛼𝑝, 𝛼𝑞) 2 𝖽𝜃 [ ] d𝜃

by definition of 𝖱 (Definition A.1 page 110)

𝜙

=

2 2 + d 𝛼𝗋(𝜃; 𝑝, 𝑞) 𝑝, 𝑞)] [𝛼𝗋(𝜃; √ [ d𝜃 ] 𝖽𝜃 ∫ 0

by item (3b)

𝜙

= |𝛼|

𝜙

2 2 + d 𝗋(𝜃; 𝑝, 𝑞) 𝑝, 𝑞)] [𝗋(𝜃; √ [ d𝜃 ] 𝖽𝜃 ∫ 0

by linearity of

≜ |𝛼|𝖱(𝑝, 𝑞)

∫ 0

𝖽𝜃 operator

by definition of 𝖱 (Definition A.1 page 110)

4. Proof that 𝖽 does not induce a norm on 𝑋: This follows directly from item (2) and Theorem D.5 (page 154). 5. Proof that balls (Definition B.4 page 128) in 𝖽 are in general not convex (Definition 1.25 page 9): This is demonstrated graphically in Figure A.2 (page 122) and Figure A.3 (page 124). For an algebraic demonstration, consider the following:13 (a) Let 𝖡((0, 1) , 1) be the unit ball in (ℝ2 , 𝖽) centered at (0, 1). (b) Let 𝑝 ≜ (−0.70, −1.12), 𝑞 ≜ (0.70, −1.12), 𝑟 ≜ (0, −1.12), and 𝜆 = 1/2. (c) Then 𝖽((0, 1) , 𝑝) = 0.959536 ⋯ < 1 𝖽((0, 1) , 𝑞) = 0.959536 ⋯ < 1 𝖽((0, 1) , 𝑟) = 1.060688 ⋯ > 1

⟹ 𝑝 ∈ 𝖡((0, 1) , 1) ⟹ 𝑞 ∈ 𝖡((0, 1) , 1) ⟹ 𝑟 ∉ 𝖡((0, 1) , 1)

and BUT

(d) This implies that the set 𝖡((0, 1) , 1) is not convex because 1 1 (−0.70, −1.12) + (1 − ) (0.70, −1.12) 2 2 = (0, −1.12)

𝜆𝑝 + (1 − 𝜆)𝑞 ≜

by item (5b)

≜𝑟

by item (5b)

∉ 𝖡((0, 1) , 1)

by item (5c)

⟹ the set 𝖡((0, 1) , 1) is not convex

by Definition 1.25 page 9

✏ Remark A.1 (Lagrange arc distance versus Euclidean metric). As is implied by the metric balls illustrated in Figure A.2 (page 122) and Figure A.3 (page 124), the Lagrange arc distance 𝖽 and Euclidean metric 𝗉 are similar in the sense that they often lead to the same results14 in determining which of the two points 𝑞1 or 𝑞2 is “closer” to a point 𝑝. But in some cases the two metrics lead to two different 13


14

For empirical evidence of this, see 📃 Greenhoe (2016c).

2016 Jul 4

10:12am UTC


Daniel J. Greenhoe

version 0.50

page 122

A.5. EXAMPLES

𝖽(( 0, 1), 𝖽(( 0, 1), 𝖽(( 0, 1), 𝖽(( 1, 0), 𝖽(( 1, 0), 𝖽(( −1, 0), 𝖽(( 0, 1), 𝖽(( 0, 1), 𝖽(( 0, 1),

2 1 b

b

b

b

1

2

b

−2

−1 −1 b

−2 b

( 1, 0)) ( −1, 0)) ( 0, −1)) ( 0, −1)) ( −1, 0)) ( 0, −1)) ( 2, 0)) ( 0, −2)) ( −2, 1))

= = = = = = = = =

/ / 1 1/2 1 1/2 0.8167968 ⋯ 1.5346486 ⋯ 0.6966032 ⋯ 12 12

Figure A.1: Lagrange arc distance examples in ℝ2

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Figure A.2: Lagrange arc distance unit balls, and dashed 𝜋1 -scaled Euclidean metric unit balls results. One such case is illustrated as follows:15 𝖽(𝑝, 𝑞1 ) ≜ 𝖽((1, 0) , (−0.5, 0)) = 0.767324 ⋯ 𝑞2 1 𝖽(𝑝, 𝑞2 ) ≜ 𝖽((1, 0) , (−0.5, 0.75)) = 0.654039 ⋯ 𝑝 𝗉 (𝑝, 𝑞1 ) ≜ 𝗉 ((1, 0) , (−0.5, 0)) = 1.5 𝑞1 −1 1 𝗉 (𝑝, 𝑞2 ) ≜ 𝗉 ((1, 0) , (−0.5, 0.75)) = √(1.5)2 + (0.75)2 = 1.677050 ⋯ That is, 𝑞2 is closer than 𝑞1 to 𝑝 with respect to the Lagrange arc distance, but 𝑞1 is closer than 𝑞2 to 𝑝 with respect to the Euclidean metric. b

b

b

A.5 Examples Example A.2 (Lagrange arc distance in ℝ2 ). Figure A.1 (page 122) illustrates the Lagrange arc distance on some pairs of points in ℝ2 .16 Example A.3 (Lagrange arc distance in ℝ3 ). Some examples of Lagrange arc distances in ℝ3 are given in Table A.1 (page 123).17 15


16


17


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𝖽(( 0, 1, 0), 𝖽(( 0, 1, 0), 𝖽(( 0, 1, 0), 𝖽(( 0, 1, 0), 𝖽(( 0, 1, 0), 𝖽(( 1, 0, 0), 𝖽(( 1, 0, 0), 𝖽(( 1, 0, 0), 𝖽(( 1, 0, 0), 𝖽(( 0, 0, 1), 𝖽(( 0, 0, 1), 𝖽(( 0, 0, 1), 𝖽(( 0, 0, −1), 𝖽(( 0, 0, −1), 𝖽(( −1, 0, 0), 𝖽(( 0, 1, 0), 𝖽(( 0, 1, 0), 𝖽(( 0, 1, 0), 𝖽(( 0, 1, 0), 𝖽(( 1, 1, 1),

( 1, 0, 0)) = 1/2 ( 0, 0, 1)) = 1/2 ( 0, 0, −1)) = 1/2 ( −1, 0, 0)) = 1/2 ( 0, −1, 0)) = 1 ( 0, 0, 1)) = 1/2 ( 0, 0, −1)) = 1/2 ( −1, 0, 0)) = 1 ( 0, −1, 0)) = 1/2 ( 0, 0, −1)) = 1 ( −1, 0, 0)) = 1/2 ( 0, −1, 0)) = 1/2 ( −1, 0, 0)) = 1/2 ( 0, −1, 0)) = 1/2 ( 0, −1, 0)) = 1/2 ( 2, 0, 0)) = 0.816796 ⋯ ( 0, −2, 0)) = 1.534648 ⋯ ( −2, 1, 0)) = 0.696603 ⋯ ( −1, 0, −1)) = 0.617920 ⋯ ( −1/2, 1/4, −2)) = 1.366268 ⋯

𝜙 𝜙 𝜙 𝜙 𝜙 𝜙 𝜙 𝜙 𝜙 𝜙 𝜙 𝜙 𝜙 𝜙 𝜙 𝜙 𝜙 𝜙 𝜙 𝜙

= = = = = = = = = = = = = = = = = ≈ = ≈

/ 𝜋/2 𝜋/2 𝜋/2 𝜋 𝜋/2 𝜋/2 𝜋 𝜋/2 𝜋 𝜋/2 𝜋/2 𝜋/2 𝜋/2 𝜋/2 𝜋/2 𝜋 1.107 𝜋 2.2466 𝜋2

90∘ 90∘ 90∘ 90∘ 180∘ 90∘ 90∘ 180∘ 90∘ 180∘ 90∘ 90∘ 90∘ 90∘ 90∘ 90∘ 180∘ 63∘ 90∘ 128.72∘

Table A.1: Some examples of Lagrange arc distances in ℝ3 (see Example A.3 page 122) Example A.4 (Lagrange arc distance balls in ℝ2 ). Some unit balls in ℝ2 in with respect to the Lagrange arc distance are illustrated in Figure A.2 (page 122). Example A.5 (Lagrange arc distance balls in ℝ3 ). Some unit balls in ℝ3 with respect to the Lagrange arc distance are illustrated in Figure A.3 (page 124).

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A.5. EXAMPLES

2

0

𝑧

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0

−2 −2 2

−2 0 𝑥

2

−2

0 2

0 𝑦

−2

𝑥

centered at (0, 0, 0)

0 2

𝑦

−2

centered at (0, 0, −1)

0

−2

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𝑧

−2

−4 −4 2

−2 0 𝑥

2

−2

0 2

0 𝑦

−2

𝑥


0 2

𝑦

−2


−8

𝑧

𝑧

−4 −10

−6 −12 2

−2 0 𝑥

2

−2

0 2

−2

0 𝑦


𝑥

0 2

−2

𝑦


Figure A.3: unit Lagrange arc distance balls in ℝ3

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❝ The Epicureans are wont to ridicule this theorem, saying it is evident even to an ass and

needs no proof; it is as much the mark of an ignorant man, they say, to require persuasion of evident truths as to believe what is obscure without question.…That the present theorem is known to an ass they make out from the observation that, if straw is placed at one extremity of the sides, an ass in quest of provender will make his way along the one side and not by way of the two others. ❞ Proclus Lycaeus (412 – 485 AD), Greek philosopher, commenting on the Epicureans view of the triangle inequality property. 1

B.1 Introduction and summary Metric spaces provide a framework for analysis and have several very useful properties. Many of these properties follow in part from the triangle inequality. However, there are several applications2 in which the triangle inequality does not hold but in which we would still like to perform analysis. So the questions that natually follow are: Q1. What happens if we remove the triangle inequality all together? Q2. What happens if we replace the triangle inequality with a generalized relation? A distance space is a metric space without the triangle inequality constraint. Section B.2 introduces distance spaces and demonstrates that some properties commonly associated with metric spaces also hold in any distance space: 1

📘 Lycaeus (circa 450), page 251 references for applications in which the triangle inequality may not hold: 📃 Maligranda and Orlicz (1987) page 54 ⟨“pseudonorm”⟩, 📃 Lin (1998) ⟨“similarity measures”, Table 6⟩, 📃 Veltkamp and Hagedoorn (2000) ⟨“shape similarity measures”⟩, 📃 Veltkamp (2001) ⟨“shape matching”⟩, 📃 Costa et al. (2004) ⟨“network distance estimation”⟩, 📃 Burstein et al. (2005) page 287 ⟨distance matrices for “genome phylogenies”⟩, 📘 Jiménez and Yukich (2006) page 224 ⟨“statistical distances”⟩, 📃 Szirmai (2007) page 388 ⟨“geodesic ball”⟩, 📃 Crammer et al. (2007) page 326 ⟨“decisiontheoretic learning”⟩, 📃 Crammer et al. (2008) page 1758 ⟨“approximate triangle inequality”⟩, 📃 Vitányi (2011) page 2455 ⟨“information distance”⟩ 2

125

page 126 D1. D2. D3. D4. D5. D6.

B.1. INTRODUCTION AND SUMMARY

∅ and 𝑋 are open the intersection of a finite number of open sets is open the union of an arbitrary number of open sets is open every Cauchy sequence is bounded any subsequence of a Cauchy sequence is also Cauchy the Cantor Intersection Theorem holds

(Theorem B.2 page 128) (Theorem B.2 page 128) (Theorem B.2 page 128) (Proposition B.1 page 131) (Proposition B.2 page 131) (Theorem B.5 page 132)

The following five properties (M1–M5) do hold in any metric space. However, the examples from Section B.2 listed below demonstrate that the five properties do not hold in all distance spaces: M1. the metric function is continuous fails to hold in Example B.1–Example B.3 M2. open balls are open fails to hold in Example B.1 and Example B.2 M3. the open balls form a base for a topology fails to hold in Example B.1 and Example B.2 M4. the limits of convergent sequences are unique fails to hold in Example B.1 M5. convergent sequences are Cauchy fails to hold in Example B.2 Hence, Section B.2 answers question Q1. APPENDIX C begins to answer question Q2 by first introducing a new function, called the power 1 triangle function in a distance space (𝑋, 𝖽), as 𝜏(𝑝, 𝜎; 𝑥, 𝑦, 𝑧; 𝖽) ≜ 2𝜎 [ 21 𝖽𝑝 (𝑥, 𝑧) + 12 𝖽𝑝 (𝑧, 𝑦)] 𝑝 for some (𝑝, 𝜎) ∈ ℝ∗ × ℝ. APPENDIX C then goes on to use this function to define a new relation, called the power triangle inequality in (𝑋, 𝖽), and defined as ▵ (𝑝, 𝜎; 𝖽) ≜ {(𝑥, 𝑦, 𝑧) ∈ 𝑋 3 |𝖽(𝑥, 𝑦) ≤ 𝜏(𝑝, 𝜎; 𝑥, 𝑦, 𝑧; 𝖽)}. ○ The power triangle inequality is a generalized form of the triangle inequality in the sense that the two inequalities coincide at (𝑝, 𝜎) = (1, 1). Other special values include (1, 𝜎) yielding the relaxed triangle inequality (and its associated near metric space) and (∞, 𝜎) yielding the σ-inframetric inequality (and its associated σ-inframetric space). Collectively, a distance space with a power triangle inequality is herein called a power distance space and denoted (𝑋, 𝖽, 𝑝, 𝜎).3 The power triangle function, at 𝜎 = 12 , is a special case of the power mean with 𝘕 = 2 and 𝜆1 = 𝜆2 = 12 . Power means have the elegant properties of being continuous and monontone with respect to a free parameter 𝑝. From this it is easy to show that the power triangle function is also continuous and monontone with respect to both 𝑝 and 𝜎. Special values of 𝑝 yield operators coinciding with maximum, minimum, mean square, arithmetic mean, geometric mean, and harmonic mean. Power means are briefly described in APPENDIX D.3.3.4 Section C.2 investigates the properties of power distance spaces. In particular, it shows for what values of (𝑝, 𝜎) the properties M1–M5 hold. Here is a summary of the results in a power distance space (𝑋, 𝖽, 𝑝, 𝜎), for all 𝑥, 𝑦, 𝑧 ∈ 𝑋: 1 ∗ + (M1) holds for any (𝑝, 𝜎) ∈ (ℝ ⧵{0}) × ℝ such that 2𝜎 = 2 𝑝 (Theorem C.5 page 144) 1 ∗ + (M2) holds for any (𝑝, 𝜎) ∈ (ℝ ⧵{0}) × ℝ such that 2𝜎 ≤ 2 𝑝 (Corollary C.6 page 142) 1 ∗ + (M3) holds for any (𝑝, 𝜎) ∈ (ℝ ⧵{0}) × ℝ such that 2𝜎 ≤ 2 𝑝 (Corollary C.5 page 141) ∗ + (M4) holds for any (𝑝, 𝜎) ∈ ℝ × ℝ (Theorem C.6 page 144) ∗ + (M5) holds for any (𝑝, 𝜎) ∈ ℝ × ℝ (Theorem C.3 page 143) APPENDIX D.4 briefly introduces topological spaces. The open balls of any metric space form a base for a topology. This is largely due to the fact that in a metric space, open balls are open. Because of this, in metric spaces it is convenient to use topological structure to define and exploit analytic concepts such as continuity, convergence, closed sets, closure, interior, and accumulation point. For example, in a metric space, the traditional definition of defining continuity using open balls and 3

power triangle inequality: Definition C.3 page 138; power distance space: Definition C.2 page 138; examples of power distance space: Definition C.4 page 138; 4 power triangle function: Definition C.1 (page 137); power mean: Definition D.15 (page 166); power mean is continuous and monontone: Theorem D.14 (page 166); power triangle function is continuous and monontone: Corollary C.1 (page 138); Special values of 𝑝: Corollary C.2 (page 139), Corollary D.2 (page 169)

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APPENDIX B. DISTANCE SPACES

the topological definition using open sets, coincide with each other. Again, this is largely because the open balls of a metric space are open.5 However, this is not the case for all distance spaces. In general, the open balls of a distance space are not open, and they are not a base for a topology. In fact, the open balls of a distance space are a base for a topology if and only if the open balls are open. While the open sets in a distance space do induce a topology, it's open balls may not.6 A distance space (Definition B.1 page 127) can be defined as a metric space (Definition D.7 page 155) without the triangle inequality constraint. Much of the material in this section about distance spaces is standard in metric spaces. However, this paper works through this material again to demonstrate “how far we can go”, and can't go, without the triangle inequality.

B.2 Fundamental structure of distance spaces B.2.1 Definitions Definition B.1. 7 A function 𝖽 in the set ℝ𝑋×𝑋 (Definition 1.6 page 6) is a distance if 1. 𝖽(𝑥, 𝑦) ≥ 0 ∀𝑥,𝑦∈𝑋 ( NON-NEGATIVE) and 2. 𝖽(𝑥, 𝑦) = 0 ⟺ 𝑥 = 𝑦 ∀𝑥,𝑦∈𝑋 ( NONDEGENERATE) and 3. 𝖽(𝑥, 𝑦) = 𝖽(𝑦, 𝑥) ∀𝑥,𝑦∈𝑋 ( SYMMETRIC) The pair (𝑋, 𝖽) is a distance space if 𝖽 is a DISTANCE on a set 𝑋. Definition B.2. 8 Let (𝑋, 𝖽) be a DISTANCE SPACE and 𝟚𝑋 be the POWER SET of 𝑋 (Definition 1.8 page 6). The 0 for 𝐴 = ∅ diameter in (𝑋, 𝖽) of a set 𝐴 ∈ 𝟚𝑋 is diam 𝐴 ≜ { sup {𝖽(𝑥, 𝑦) |𝑥, 𝑦 ∈ 𝐴} otherwise Definition B.3. 9 Let (𝑋, 𝖽) be a DISTANCE SPACE. Let 𝟚𝑋 be the POWER SET (Definition 1.8 page 6) of 𝑋. A set 𝐴 is bounded in (𝑋, 𝖽) if 𝐴 ∈ 𝟚𝑋 and diam 𝐴 < ∞.

B.2.2 Properties Theorem B.1. 10 Let ⦅𝑥𝑛 ⦆𝑛∈ℤ be a SEQUENCE in a DISTANCE SPACE (𝑋, 𝖽). The DISTANCE SPACE (𝑋, 𝖽) does not necessarily have all the nice properties that a METRIC SPACE (Definition D.7 page 155) has. In particular, note the following: 5

open ball: Definition B.4 page 128; metric space: Definition D.7 page 155; base: Definition D.17 page 170; topology: Definition D.16 page 170; open: Definition B.5 page 128; continuity in topological space: Definition D.19 page 171; convergence in distance space: Definition B.7 page 130; convergence in topological space: Definition D.20 page 172; closed set: Definition D.16 page 170; closure, interior, accumulation point: Definition D.18 page 171; coincidence in all metric spaces and some power distance spaces: Theorem C.2 page 142; 6 if and only if statement: Theorem B.3 page 130; open sets of a distance space induce a topology: Corollary B.1 page 129; 7 📃 Menger (1928) page 76 ⟨“Abstand 𝑎 𝑏 definiert ist…” (distance from 𝑎 to 𝑏 is defined as…”)⟩, 📃 Wilson (1931b) page 361 ⟨§1., “distance”, “semi-metric space”⟩, 📃 Blumenthal (1938) page 38, 📘 Blumenthal (1953) page 7 ⟨“DEFINITION 5.1. A distance space is called semimetric provided…”⟩, 📃 Galvin and Shore (1984) page 67 ⟨“distance function”⟩, 📘 Laos (1998) page 118 ⟨“distance space”⟩, 📘 Khamsi and Kirk (2001) page 13 ⟨“semimetric space”⟩, 📃 Bessenyei and Pales (2014) page 2 ⟨“semimetric space”⟩, 📘 Deza and Deza (2014) page 3 ⟨“distance (or dissimilarity)”⟩ 8 in metric space: 📘 Hausdorff (1937), page 166, 📘 Copson (1968), page 23, 📘 Michel and Herget (1993), page 267, 📘 Molchanov (2005) page 389 9 in metric space: 📘 Thron (1966), page 154 ⟨definition 19.5⟩, 📘 Bruckner et al. (1997) page 356 10 📃 Greenhoe (2016b) 2016 Jul 4

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B.3. OPEN SETS IN DISTANCE SPACES

𝖽 is a DISTANCE in (𝑋, 𝖽) 𝖡 is an OPEN BALL in (𝑋, 𝖽) 𝑩 is the set of all OPEN BALLs in (𝑋, 𝖽) ⦅𝑥𝑛 ⦆ is CONVERGENT in (𝑋, 𝖽) ⦅𝑥𝑛 ⦆ is CONVERGENT in (𝑋, 𝖽)

1. 2. 3. 4. 5.

⟹ / ⟹ / ⟹ / ⟹ / ⟹ /

𝖽 is CONTINUOUS in (𝑋, 𝖽) 𝖡 is OPEN in (𝑋, 𝖽) 𝑩 is a BASE for a topology on 𝑋 limit is UNIQUE ⦅𝑥𝑛 ⦆ is CAUCHY in (𝑋, 𝖽)

(Example B.3 page 135). (Example B.2 page 134). (Example B.2 page 134). (Example B.1 page 133). (Example B.2 page 134).

B.3 Open sets in distance spaces B.3.1 Definitions Definition B.4. 12 Let (𝑋, 𝖽) be a DISTANCE SPACE (Definition B.1 page 127). An open ball centered at 𝑥 with radius 𝑟 is the set 𝖡(𝑥, 𝑟) ≜ {𝑦 ∈ 𝑋|𝖽(𝑥, 𝑦) < 𝑟} . A closed ball centered at 𝑥 with radius 𝑟 is the set 𝖡 (𝑥, 𝑟) ≜ {𝑦 ∈ 𝑋|𝖽(𝑥, 𝑦) ≤ 𝑟} . Definition B.5. Let (𝑋, 𝖽) be a DISTANCE SPACE. Let 𝑋⧵𝐴 be the SET DIFFERENCE of 𝑋 and a set 𝐴. A set 𝑈 is open in (𝑋, 𝖽) if 𝑈 ∈ 𝟚𝑋 and for every 𝑥 in 𝑈 there exists 𝑟 ∈ ℝ+ such that 𝖡(𝑥, 𝑟) ⊆ 𝑈 . A set 𝑈 is an open set in (𝑋, 𝖽) if 𝑈 is OPEN in (𝑋, 𝖽). A set 𝐷 is closed in (𝑋, 𝖽) if (𝑋 ⧵𝐷) is OPEN. A set 𝐷 is a closed set in (𝑋, 𝖽) if 𝐷 is CLOSED in (𝑋, 𝖽).

B.3.2 Properties Theorem B.2. 13 Let (𝑋, 𝖽) be a DISTANCE SPACE. Let 𝘕 be any (finite) positive integer. Let 𝛤 be a SET possibly with an uncountable number of elements. 1. 𝑋 is OPEN. 2. ∅ is OPEN. 𝘕

each element in {𝑈𝑛 |𝑛=1,2,…,𝘕 }

3.

is OPEN

⋂

𝑈𝑛 is OPEN.

⋃

𝑈𝛾 is OPEN.

𝑛=1

each element in {𝑈𝛾 ∈ 𝟚𝑋 |𝛾 ∈ 𝛤 } is OPEN

4.

⟹ ⟹

𝛾∈𝛤

✎PROOF: 1. Proof that 𝑋 is open in (𝑋, 𝖽): (a) By definition of open set 𝖡(𝑥, 𝑟) ⊆ 𝑋.

(Definition B.5 page 128),

𝑋 is open ⟺ ∀𝑥 ∈ 𝑋

∃𝑟

such that

(b) By definition of open ball (Definition B.4 page 128), it is always true that 𝖡(𝑥, 𝑟) ⊆ 𝑋 in (𝑋, 𝖽). (c) Therefore, 𝑋 is open in (𝑋, 𝖽). 2. Proof that ∅ is open in (𝑋, 𝖽): (a) By definition of open set


∅ is open ⟺ ∀𝑥 ∈ 𝑋

∃𝑟

such that

𝖡(𝑥, 𝑟) ⊆ ∅.

(b) By definition of empty set ∅ (Definition 1.1 page 5), this is always true because no 𝑥 is in ∅. (c) Therefore, ∅ is open in (𝑋, 𝖽). 11

Heath (1961) page 810 ⟨THEOREM⟩, 📃 Galvin and Shore (1984) page 71 ⟨2.3 LEMMA⟩ in metric space: 📘 Aliprantis and Burkinshaw (1998), page 35 13 in metric space: 📘 Dieudonné (1969), pages 33–34, 📘 Rosenlicht (1968) page 39 📃

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3. Proof that ⋃ 𝑈𝛾 is open in (𝑋, 𝖽): (a) By definition of open set (Definition B.5 page 128), ⋃ 𝑈𝛾 is open ⟺ ∀𝑥 ∈ ⋃ 𝑈𝛾 such that 𝖡(𝑥, 𝑟) ⊆ ⋃ 𝑈𝛾 .

∃𝑟

(b) If 𝑥 ∈ ⋃ 𝑈𝛾 , then there is at least one 𝑈 ∈ ⋃ 𝑈𝛾 that contains 𝑥. (c) By the left hypothesis in (4), that set 𝑈 is open and so for that 𝑥, ∃𝑟

such that

𝖡(𝑥, 𝑟) ⊆ 𝑈 ⊆ ⋃ 𝑈𝛾 .

(d) Therefore, ⋃ 𝑈𝛾 is open in (𝑋, 𝖽). 4. Proof that 𝑈1 and 𝑈2 are open ⟹ 𝑈1 ∩ 𝑈2 is open: (a) By definition of open set (Definition B.5 page 128), 𝑈1 ∩ 𝑈2 is open ⟺ ∀𝑥 ∈ 𝑈1 ∩ 𝑈2 such that 𝖡(𝑥, 𝑟) ⊆ 𝑈1 ∩ 𝑈2 .

∃𝑟

(b) By the left hypothesis above, 𝑈1 and 𝑈2 are open; and by the definition of open sets page 128), there exists 𝑟1 and 𝑟2 such that 𝖡(𝑥, 𝑟1 ) ⊆ 𝑈1 and 𝖡(𝑥, 𝑟2 ) ⊆ 𝑈2 .

(Definition B.5

(c) Let 𝑟 ≜ min {𝑟1 , 𝑟2 }. Then 𝖡(𝑥, 𝑟) ⊆ 𝑈1 and 𝖡(𝑥, 𝑟) ⊆ 𝑈2 . (d) By definition of set intersection ∩ then, 𝖡(𝑥, 𝑟) ⊆ 𝑈1 ∩ 𝑈2 . (e) By definition of open set


𝑈1 ∩ 𝑈2 is open.

5. Proof that ⋂𝘕𝑛=1 𝑈𝑛 is open (by induction): (a) Proof for 𝘕 = 1 case: ⋂𝘕𝑛=1 𝑈𝑛 = ⋂1𝑛=1 𝑈𝑛 = 𝑈1 is open by hypothesis. (b) Proof that 𝘕 case ⟹ 𝘕 + 1 case: 𝘕 +1

⎛𝘕 ⎞ 𝑈𝑛 = ⎜ 𝑈𝑛 ⎟ ∩ 𝑈𝘕 +1 ⋂ ⎜⋂ ⎟ 𝑛=1 ⎝𝑛=1 ⎠ ⟹ open

by property of

⋂

by “𝘕 case” hypothesis and (4) lemma page 129

✏ Corollary B.1. Let (𝑋, 𝖽) be a DISTANCE SPACE. The set TOPOLOGY on 𝑋, and (𝑋, 𝑻 ) is a TOPOLOGOGICAL SPACE.

𝑻 ≜ {𝑈 ∈ 𝟚𝑋 |𝑈 is OPEN in (𝑋, 𝖽)}

is a

✎PROOF:

This follows directly from the definition of an open set (Definition B.5 page 128), Theorem B.2 (page 128), and the definition of topology (Definition D.16 page 170). ✏

Of course it is possible to define a very large number of topologies even on a finite set with just a handful of elements;14 and it is possible to define an infinite number of topologies even on a linearly ordered infinite set like the real line ( ℝ, ≤).15 Be that as it may, Definition B.6 (next definition) defines a single but convenient topological space in terms of a distance space. Note that every metric space conveniently and naturally induces a topological space because the open balls of the metric space form a base for the topology. This is not the case for all distance spaces. But if the open balls of a distance space are all open, then those open balls induce a topology (next theorem).16 14

For a finite set 𝑋 with 𝑛 elements, there are 29 topologies on 𝑋 if 𝑛 = 3, 6942 topologies on 𝑋 if 𝑛 = 5, and and 8,977,053,873,043 (almost 9 trillion) topologies on 𝑋 if 𝑛 = 10. References: 💻 Sloane (2014) ⟨http://oeis.org/ A000798⟩, 📘 Brown and Watson (1996), page 31, 📘 Comtet (1974) page 229, 📘 Comtet (1966), 📘 Chatterji (1967), page 7, 📘 Evans et al. (1967), 📘 Krishnamurthy (1966), page 157 15 For examples of topologies on the real line, see the following: 📘 Adams and Franzosa (2008) page 31 ⟨”six topologies on the real line”⟩, 📘 Salzmann et al. (2007) pages 64–70 ⟨Weird topologies on the real line⟩, 📘 Murdeshwar (1990) page 53 ⟨“often used topologies on the real line”⟩, 📘 Joshi (1983) pages 85–91 ⟨§4.2 Examples of Topological Spaces⟩ 16 metric space: Definition D.7 page 155; open ball: Definition B.4 page 128; base: Definition D.17 page 170; topology: Definition D.16 page 170; not all open balls are open in a distance space: Example B.1 (page 133) and Example B.2 (page 134); 2016 Jul 4

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Definition B.6. Let (𝑋, 𝖽) be a DISTANCE SPACE. The set 𝑻 ≜ {𝑈 ∈ 𝟚𝑋 |𝑈 is OPEN in (𝑋, 𝖽)} is the topology induced by (𝑋, 𝖽) on 𝑋. The pair (𝑋, 𝑻 ) is called the topological space induced by (𝑋, 𝖽). For any distance space (𝑋, 𝖽), no matter how strange, there is guaranteed to be at least one topological space induced by (𝑋, 𝖽)—and that is the indiscrete topological space (Example D.12 page 170) because for any distance space (𝑋, 𝖽), ∅ and 𝑋 are open sets in (𝑋, 𝖽) (Theorem B.2 page 128). Theorem B.3. Let 𝑩 be the set of all OPEN BALLs in a DISTANCE SPACE (𝑋, 𝖽). ⟺ {every OPEN BALL in 𝑩 is OPEN} {𝑩 is a BASE for a TOPOLOGY} ✎PROOF: every open ball in 𝑩 is open ⟹ for every 𝑥 in 𝐵𝑦 ∈ 𝑩 there exists 𝑟 ∈ ℝ+ such that 𝖡(𝑥, 𝑟) ⊆ 𝐵𝑦 by definition of open (Definition B.5 page 128) ⟹

for every 𝑥 ∈ 𝑋 and for every 𝐵𝑦 ∈ 𝑩 containing 𝑥, { there exists 𝐵𝑥 ∈ 𝑩 such that } 𝑥 ∈ 𝐵 𝑥 ⊆ 𝐵𝑦 .

because ∀ (𝑥, 𝑟) ∈ 𝑋 × ℝ+ , 𝖡(𝑥, 𝑟) ⊆ 𝑋

⟹ 𝑩 is a base for 𝑻

by Theorem D.16 page 170

⟹

for every 𝑥 ∈ 𝑋 and for every 𝑈 ⊆ 𝑻 containing 𝑥, { there exists 𝐵𝑥 ∈ 𝑩 such that } 𝑥 ∈ 𝐵𝑥 ⊆ 𝑈 .

by Theorem D.16 page 170

⟹

for every 𝑥 ∈ 𝑋 and for every 𝐵𝑦 ∈ 𝑩 ⊆ 𝑻 containing 𝑥, { there exists 𝐵𝑥 ∈ 𝑩 such that } 𝑥 ∈ 𝐵 𝑥 ⊆ 𝐵𝑦 .

by definition of base (Definition D.17 page 170)

⟹

for every 𝑥 ∈ 𝐵𝑦 ∈ 𝑩 ⊆ 𝑻 , { there exists 𝐵𝑥 ∈ 𝑩 such that

𝑥 ∈ 𝐵 𝑥 ⊆ 𝐵𝑦 . }

⟹ every open ball in 𝑩 is open

by definition of open (Definition B.5 page 128)

✏

B.4 Sequences in distance spaces B.4.1 Definitions Definition B.7. 17 Let ⦅𝑥𝑛 ∈ 𝑋⦆𝑛∈ℤ be a SEQUENCE in a DISTANCE SPACE (𝑋, 𝖽). The sequence ⦅𝑥𝑛 ⦆ converges to a limit 𝑥 if for any 𝜀 ∈ ℝ+ , there exists 𝘕 ∈ ℤ such that for all 𝑛 > 𝘕 , 𝖽(𝑥𝑛 , 𝑥) < 𝜀. This condition can be expressed in any of the following forms: 1. The limit of the sequence ⦅𝑥𝑛 ⦆ is 𝑥. 3. lim ⦅𝑥𝑛 ⦆ = 𝑥. 𝑛→∞ 2. The sequence ⦅𝑥𝑛 ⦆ is convergent with limit 𝑥. 4. ⦅𝑥𝑛 ⦆ → 𝑥. A SEQUENCE that converges is convergent. Definition B.8. 18 Let ⦅𝑥𝑛 ∈ 𝑋⦆𝑛∈ℤ be a SEQUENCE in a DISTANCE SPACE (𝑋, 𝖽). The sequence ⦅𝑥𝑛 ⦆ is a Cauchy sequence in (𝑋, 𝖽) if for every 𝜀 ∈ ℝ+ , there exists 𝘕 ∈ ℤ such that ∀𝑛, 𝑚 > 𝘕 , 𝖽(𝑥𝑛 , 𝑥𝑚 ) < 𝜀 ( CAUCHY CONDITION). Definition B.9. 19 Let ⦅𝑥𝑛 ∈ 𝑋⦆𝑛∈ℤ be a SEQUENCE in a DISTANCE SPACE (𝑋, 𝖽). The sequence ⦅𝑥𝑛 ∈ 𝑋⦆𝑛∈ℤ is complete in (𝑋, 𝖽) if ⟹ ⦅𝑥𝑛 ⦆ is CAUCHY in (𝑋, 𝖽) ⦅𝑥𝑛 ⦆ is CONVERGENT in (𝑋, 𝖽). 17

in metric space: 📘 Rosenlicht (1968) page 45, 📘 Giles (1987) page 37 ⟨3.2 Definition⟩, 📘 Khamsi and Kirk (2001) page 13 ⟨Definition 2.1⟩ “→” symbol: 📘 Leathem (1905) page 13 ⟨section III.11⟩ 18 in metric space: 📘 Apostol (1975) page 73 ⟨4.7⟩, 📘 Rosenlicht (1968) page 51 19 in metric space: 📘 Rosenlicht (1968) page 52

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B.4.2 Properties Proposition B.1. 20 Let ⦅𝑥𝑛 ∈ 𝑋⦆𝑛∈ℤ be a SEQUENCE in a DISTANCE SPACE (𝑋, 𝖽). ⟹ { ⦅𝑥𝑛 ⦆ is CAUCHY in (𝑋, 𝖽) } { ⦅𝑥𝑛 ⦆ is BOUNDED in (𝑋, 𝖽) } ✎PROOF: ⦅𝑥𝑛 ⦆ is Cauchy ⟹ for every 𝜀 ∈ ℝ+ , ∃𝘕 ∈ ℤ ⟹ ∃𝘕 ∈ ℤ

such that

⟹ ∃𝘕 ∈ ℤ

such that

∀𝑛, 𝑚 > 𝘕 , 𝖽(𝑥𝑛 , 𝑥𝑚 ) < 𝜀 (Definition B.8 page 130) ∀𝑛, 𝑚 > 𝘕 , 𝖽(𝑥𝑛 , 𝑥𝑚 ) < 1 (arbitrarily choose 𝜀 ≜ 1) such that

∀𝑛, 𝑚 ∈ ℤ, 𝖽(𝑥𝑛 , 𝑥𝑚+1 ) < max {{1} ∪ {𝖽(𝑥𝑝 , 𝑥𝑞 ) |𝑝, 𝑞 ≯ 𝑁 }} (by Definition B.3 page 127)

⟹ ⦅𝑥𝑛 ⦆ is bounded

✏ Proposition B.2. 21 Let ⦅𝑥𝑛 ∈ 𝑋⦆𝑛∈ℤ be a SEQUENCE in a DISTANCE SPACE (𝑋, 𝖽). Let 𝖿 ∈ ℤℤ (Definition 1.6 page 6) be a STRICTLY MONOTONE function such that 𝖿(𝑛) < 𝖿(𝑛 + 1). ⟹ 𝑥𝖿(𝑛) ⦆𝑛∈ℤ is CAUCHY ⦅𝑥 ⦅ 𝑛 ⦆𝑛∈ℤ is CAUCHY ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ sequence is CAUCHY

subsequence is also CAUCHY

✎PROOF: ⦅𝑥𝑛 ⦆𝑛∈ℤ is Cauchy ⟹ for any given 𝜀 > 0, ∃𝘕 ⟹ for any given 𝜀 > 0, ∃𝘕

such that ′

such that

∀𝑛, 𝑚 > 𝘕 , 𝖽(𝑥𝑛 , 𝑥𝑚 ) < 𝜀

by Definition B.8 page 130

′

∀𝖿(𝑛), 𝖿 (𝑚) > 𝘕 , 𝖽 (𝑥𝖿(𝑛) , 𝑥𝖿(𝑚) ) < 𝜀 by Definition B.8 page 130

⟹ ⦅𝑥𝖿(𝑛) ⦆𝑛∈ℤ is Cauchy

✏ Theorem B.4. 22 Let (𝑋, 𝖽) be a DISTANCE SPACE. Let 𝐴− be the CLOSURE (Definition D.18 page 171) of a 𝐴 in a TOPOLOGICAL SPACE INDUCED BY (𝑋, 𝖽). 1. LIMITs are UNIQUE in (𝑋, 𝖽) (Definition B.7 page 130) and ⟹ 𝐴 is CLOSED in (𝑋, 𝖽) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ { 2. (𝐴, 𝖽) is COMPLETE in (𝑋, 𝖽) (Definition B.9 page 130) } 𝐴 = 𝐴−

✎PROOF: 1. Proof that 𝐴 ⊆ 𝐴− : by Lemma D.5 page 171 2. Proof that 𝐴− ⊆ 𝐴 (proof that 𝑥 ∈ 𝐴− ⟹ 𝑥 ∈ 𝐴): (a) Let 𝑥 be a point in 𝐴− (𝑥 ∈ 𝐴− ). (b) Define a sequence of open balls ⦅𝖡(𝑥, 11 ) , 𝖡 (𝑥, 12 ) , 𝖡 (𝑥, 13 ) , …⦆. (c) Define a sequence of points ⦅𝑥1 , 𝑥2 , 𝑥3 , …⦆ such that 𝑥𝑛 ∈ 𝖡 (𝑥𝑛 , 1𝑛 ) ∩ 𝐴. (d) Then ⦅𝑥𝑛 ⦆ is convergent in 𝑋 with limit 𝑥 by Definition B.7 page 130 (e) and ⦅𝑥𝑛 ⦆ is Cauchy in 𝐴 by Definition B.8 page 130. (f) By the hypothesis 2, ⦅𝑥𝑛 ⦆ is therefore also convergent in 𝐴. Let this limit be 𝑦. Note that 𝑦 ∈ 𝐴. 20

in metric space: 📘 Giles (1987) page 49 ⟨Theorem 3.30⟩ in metric space: 📘 Rosenlicht (1968) page 52 22 in metric space: 📘 Kubrusly (2001) page 128 ⟨Theorem 3.40⟩, 📘 Haaser and Sullivan (1991) page 75 ⟨6⋅10, 6⋅11 Propositions⟩, 📘 Bryant (1985) page 40 ⟨Theorem 3.6, 3.7⟩, 📘 Sutherland (1975) pages 123–124 21

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(g) By hypothesis 1, limits are unique, so 𝑦 = 𝑥. (h) Because 𝑦 ∈ 𝐴 (item (2f)) and 𝑦 = 𝑥 (item (2g)), so 𝑥 ∈ 𝐴. (i) Therefore, 𝑥 ∈ 𝐴− ⟹ 𝑥 ∈ 𝐴 and 𝐴− ⊆ 𝐴.

✏ Proposition B.3. 23 Let ⦅𝑥𝑛 ⦆𝑛∈ℤ be a sequence in a DISTANCE SPACE (𝑋, 𝖽). Let 𝖿 ∶ ℤ → ℤ be a strictly increasing function such that 𝖿 (𝑛) < 𝖿 (𝑛 + 1). ⟹ 𝑥𝖿(𝑛) ⦆𝑛∈ℤ → 𝑥 ⦅𝑥 ⦅ 𝑛 ⦆𝑛∈ℤ → 𝑥 ⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ sequence converges to limit 𝑥

subsequence converges to the same limit 𝑥

✎PROOF: ⦅𝑥𝑛 ⦆𝑛∈ℤ → 𝑥 ⟹ ∀𝜀 > 0, ∃𝘕

∀𝑛 > 𝘕 , 𝖽(𝑥𝑛 , 𝑥) < 𝜀 such that ∀𝖿(𝑛) > 𝖿 (𝑁), 𝖽(𝑥𝖿 (𝑛) , 𝑥) < 𝜀

such that

⟹ ∀𝜀 > 0, ∃𝖿 (𝑁) ⟹ ⦅𝑥𝖿(𝑛) ⦆𝑛∈ℤ → 𝑥

by Theorem C.2 page 142 by Theorem C.2 page 142

✏ Theorem B.5 (Cantor intersection theorem). 24 Let (𝑋, 𝖽) be a DISTANCE SPACE (Definition B.1 page 127), ⦅𝐴𝑛 ⦆𝑛∈ℤ a SEQUENCE with each 𝐴𝑛 ∈ 𝟚𝑋 , and | 𝐴 | the number of elements in 𝐴. and (Definition B.9 page 130) ⎫ ⎧ 1. (𝑋, 𝖽) is COMPLETE ⎪ 2. 𝐴𝑛 is CLOSED ∀𝑛∈ℕ (Definition D.16 page 170) and ⎪ 𝐴𝑛 = 1 ⟹ ⎬ ⎨ 3. diam 𝐴 ≥ diam 𝐴 and | } {| ⋂ 𝑛 𝑛+1 ∀𝑛∈ℕ (Definition B.2 page 127) 𝑛∈ℕ ⎪ ⎪ 4. diam B.7 page 130) → 0 (Definition ⦅𝐴 ⦆ ⎩ ⎭ 𝑛 𝑛∈ℤ ✎PROOF: 1. Proof that | ⋂𝑛∈ℤ 𝐴𝑛 | < 2: (a) Let 𝐴 ≜ ∩𝐴𝑛 . (b) 𝑥 ≠ 𝑦 and {𝑥, 𝑦} ∈ 𝐴 ⟹ 𝖽(𝑥, 𝑦) > 0 and {𝑥, 𝑦} ⊆ 𝐴𝑛 ∀𝑛 (c) ∃𝑛

diam 𝐴𝑛 < 𝖽(𝑥, 𝑦) by left hypothesis 4

such that

(d) ⟹ ∃𝑛

sup {𝖽(𝑥, 𝑦) |𝑥, 𝑦 ∈ 𝐴𝑛 } < 𝖽(𝑥, 𝑦)

such that

(e) This is a contradiction, so {𝑥, 𝑦} ∉ 𝐴 and | ⋂ 𝐴𝑛 | < 2. 2. Proof that | ∩𝐴𝑛 | ≥ 1: (a) Let 𝑥𝑛 ∈ 𝐴𝑛 and 𝑥𝑚 ∈ 𝐴𝑚 (b) ∀𝜀, ∃𝘕 ∈ ℕ

such that

𝐴𝑁 < 𝜀

(c) ∀𝑚, 𝑛 > 𝘕 , 𝑥𝑛 ∈ 𝐴𝑛 ⊆ 𝐴𝑁 and 𝑥𝑚 ∈ 𝐴𝑚 ⊆ 𝐴𝑁 (d) 𝖽(𝑥𝑛 , 𝑥𝑚 ) ≤ diam 𝐴𝑁 < 𝜀 ⟹ {𝑥𝑛 } is a Cauchy sequence (e) Because {𝑥𝑛 } is complete, 𝑥𝑛 → 𝑥. (f) ⟹ 𝑥 ∈ (𝐴𝑛 )− = 𝐴𝑛 (g) ⟹ | 𝐴𝑛 | ≥ 1

✏ 23 24

in metric space: 📘 Rosenlicht (1968) page 46 in metric space: 📘 Davis (2005), page 28, 📘 Hausdorff (1937), page 150

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Definition B.10. 25 Let (𝑋, 𝖽) be a DISTANCE SPACE. Let 𝐶 be the set of all CONVERGENT sequences in (𝑋, 𝖽). The DISTANCE FUNCTION 𝖽 is continuous in (𝑋, 𝖽) if ⦅𝑥𝑛 ⦆ , ⦅𝑦𝑛 ⦆ ∈ 𝐶

⟹

lim ⦅𝖽(𝑥𝑛 , 𝑦𝑛 )⦆ = 𝖽 lim ⦅𝑥𝑛 ⦆ , lim ⦅𝑦𝑛 ⦆ . (𝑛→∞ ) 𝑛→∞ A DISTANCE FUNCTION is discontinuous if it is not CONTINUOUS. 𝑛→∞

Remark B.1. Rather than defining continuity of a distance function in terms of the sequential characterization of continuity as in Definition B.10 (previous), we could define continuity using an inverse image characterization of continuity” (Definition B.6 page 130). Assuming an equivalent topological space is used for both characterizations, the two characterizations are equivalent (Theorem D.20 page 173). In fact, one could construct an equivalence such as the following: ⟹ ⦅𝑥𝑛 ⦆ , ⦅𝑦𝑛 ⦆ ∈ 𝐶 ⎧ ⎫ ⎪ ⎪ ⎫ ⎧ 2 𝑋 ⎪ ⎪ 𝖽 is continuous in ℝ ⎪ lim ⦅𝖽(𝑥𝑛 , 𝑦𝑛 )⦆ = 𝖽 lim ⦅𝑥𝑛 ⦆ , lim ⦅𝑦𝑛 ⦆ ⎪ ⟺ (𝑛→∞ ) ⎬ 𝑛→∞ ⎬ ⎨ (Definition D.19 page 171) ⎨ 𝑛→∞ ⎪ ⎪ ⎪ ⎪ (Definition D.20 page 172) ⎩ (inverse image characterization of continuity) ⎭ ⎪ ⎪ ⎩ (sequential characterization of continuity) ⎭ 2 Note that just as ⦅𝑥𝑛 ⦆ is a sequence in 𝑋, so the ordered pair (⦅𝑥𝑛 ⦆ , ⦅𝑦𝑛 ⦆) is a sequence in 𝑋 . The remainder follows from Theorem D.20 (page 173). However, use of the inverse image characterization is somewhat troublesome because we would need a topology on 𝑋 2 , and we don't immediately have one defined and ready to use. In fact, we don't even immediately have a distance space on 𝑋 2 defined or even open balls in such a distance space. The result is, for the scope of this paper, it is arguably not worthwhile constructing the extra structure, but rather instead this paper uses the sequential characterization as a definition (as in Definition B.10).

B.5 Examples Similar distance functions and several of the observations for the examples in this section can be found in 📘 Blumenthal (1953) pages 8–13. In a metric space, all open balls are open, the open balls form a base for a topology, the limits of convergent sequences are unique, and the metric function is continuous. In the distance space of the next example, none of these properties hold. Example B.1. 26 Let (𝑥, 𝑦) be an ordered pair in ℝ2 . Let (𝑎 ∶ 𝑏) be an open interval and (𝑎 ∶ 𝑏] a halfopen interval in ℝ. Let |𝑥| be the absolute value of 𝑥 ∈ ℝ. The function 𝖽(𝑥, 𝑦) ∈ ℝℝ×ℝ such that 𝑦 ∀ (𝑥, 𝑦) ∈ {4} × (0 ∶ 2] (vertical half-open interval) ⎫ ⎧ ⎪ ⎪ ∀ (𝑥, 𝑦) ∈ (0 ∶ 2] × {4} (horizontal half-open interval) ⎬ is a distance on ℝ. 𝖽(𝑥, 𝑦) ≜ ⎨ 𝑥 ⎪ ⎪ (Euclidean) ⎩ |𝑥 − 𝑦| otherwise ⎭ Note some characteristics of the distance space (ℝ, 𝖽): 1. (ℝ, 𝖽) is not a metric space because 𝖽 does not satisfy the triangle inequality: 𝖽(0, 4) ≜ |0 − 4| = 4 ≰ 2 = |0 − 1| + 1 ≜ 𝖽(0, 1) + 𝖽(1, 4) 2. Not every open ball in (ℝ, 𝖽) is open. For example, the open ball 𝖡(3, 2) is not open because 4 ∈ 𝖡(3, 2) but for all 0 < 𝜀 < 1 𝖡(4, 𝜀) = (4 − 𝜀 ∶ 4 + 𝜀) ∪ (0 ∶ 𝜀) ⊈ (1 ∶ 5) = 𝖡(3, 2) 25

📘 Blumenthal (1953) page 9 ⟨DEFINITION 6.3⟩ A similar distance function 𝖽 and item (4) page 134 can in essence be found in 📘 Blumenthal (1953) page 8. Definitions for Example B.1: (𝑥, 𝑦): Definition 1.3 (page 5); (𝑎 ∶ 𝑏) and (𝑎 ∶ 𝑏]: Definition 1.23 (page 9); |𝑥|: Definition 1.24 (page 9); ℝℝ×ℝ : Definition 1.6 (page 6); distance: Definition B.1 (page 127); open ball: Definition B.4 (page 128); open: Definition B.5 (page 128); base: Definition D.17 (page 170); topology: Definition D.16 (page 170); open set: Definition B.5 (page 128); topological space induced by (ℝ, 𝖽): Definition B.6 (page 130); discontinuous: Definition B.10 (page 133); 26

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3. The open balls of (ℝ, 𝖽) do not form a base for a topology on ℝ. This follows directly from item (2) and Theorem B.3 (page 130). 4. In the distance space (ℝ, 𝖽), limits are not unique; For example, the sequence ⦅1/𝑛⦆∞ 1 converges both to the limit 0 and the limit 4 in (ℝ, 𝖽): lim 𝖽 (𝑥𝑛 , 0) ≜ lim 𝖽(1/𝑛, 0) ≜ lim |1/𝑛 − 0| = 0 ⟹ ⦅1/𝑛⦆ → 0 𝑛→∞ 𝑛→∞ 𝑛→∞ lim 𝖽 (𝑥𝑛 , 4) ≜ lim 𝖽(1/𝑛, 4) ≜ lim ⦅1/𝑛⦆ =0 ⟹ ⦅1/𝑛⦆ → 4 𝑛→∞

𝑛→∞

𝑛→∞

5. The topological space (𝑋, 𝑻 ) induced by (ℝ, 𝖽) also yields limits of 0 and 4 for the sequence ⦅1/𝑛⦆∞ 1 , just as it does in item (4). This is largely due to the fact that, for small 𝜀, the open balls 𝖡(0, 𝜀) and 𝖡(4, 𝜀) are open. 𝖡(0, 𝜀) is open ⟹ ⟺ 𝖡(4, 𝜀) is open ⟹ ⟺

for each 𝑈 ∈ 𝑻 that contains 0, ∃𝘕 ∈ ℕ such that 1/𝑛 ∈ 𝑈 ∀𝑛 > 𝘕 by definition of convergence (Definition D.20 page 172) ⦅1/𝑛⦆ → 0 for each 𝑈 ∈ 𝑻 that contains 4, ∃𝘕 ∈ ℕ such that 1/𝑛 ∈ 𝑈 ∀𝑛 > 𝘕 by definition of convergence (Definition D.20 page 172) ⦅1/𝑛⦆ → 4

6. The distance function 𝖽 is discontinuous (Definition B.10 page 133): lim ⦅𝖽(1 − 1/𝑛, 4 − 1/𝑛)⦆ = lim ⦅|(1 − 1/𝑛) − (4 − 1/𝑛)|⦆ = |1 − 4| = 3 ≠ 4 = 𝖽(0, 4)

𝑛→∞

𝑛→∞

= 𝖽 lim ⦅1 − 1/𝑛⦆ , lim ⦅4 − 1/𝑛⦆ (𝑛→∞ ) 𝑛→∞ In a metric space, all convergent sequences are also Cauchy. However, this is not the case for all distance spaces, as demonstrated next: 27

The function 𝖽(𝑥, 𝑦) ∈ ℝℝ×ℝ such that |𝑥 − 𝑦| for 𝑥 = 0 or 𝑦 = 0 or 𝑥 = 𝑦 (Euclidean) 𝖽(𝑥, 𝑦) ≜ { 1 } otherwise (discrete) Note some characteristics of the distance space (ℝ, 𝖽):

Example B.2.

is a distance on ℝ.

1. (𝑋, 𝖽) is not a metric space because the triangle inequality does not hold: 𝖽( 14 , 12 ) = 1 ≰ 34 = | 14 − 0| + |0 − 12 | = 𝖽( 14 , 0) + 𝖽(0, 21 ) 2. The open ball 𝖡( 14 , 12 ) is not open because for any 𝜀 ∈ ℝ+ , no matter how small, 𝖡(0, 𝜀) = (−𝜀 ∶ +𝜀) ⊈ {0, 14 } = {𝑥 ∈ 𝑋|𝖽( 14 , 𝑥) < 12 } ≜ 𝖡( 41 , 12 ) 3. Even though not all the open balls are open, it is still possible to have an open set in (𝑋, 𝖽). For example, the set 𝑈 ≜ {1, 2} is open: 𝖡(1, 1) ≜ {𝑥 ∈ 𝑋|𝖽(1, 𝑥) < 1} = {1} ⊆ {1, 2} ≜ 𝑈 𝖡(2, 1) ≜ {𝑥 ∈ 𝑋|𝖽(2, 𝑥) < 1} = {2} ⊆ {1, 2} ≜ 𝑈 4. By item (2) and Theorem B.3 (page 130), the open balls of (ℝ, 𝖽) do not form a base for a topology on ℝ. 5. Even though the open balls in (ℝ, 𝖽) do not induce a topology on 𝑋, it is still possible to find a set of open sets in (𝑋, 𝖽) that is a topology. For example, the set {∅, {1, 2}, ℝ} is a topology on ℝ. 27

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6. In (ℝ, 𝖽), limits of convergent sequences are unique: lim |𝑥𝑛 − 0| = 0 for 𝑥 = 0 OR ⎫ ⎧ ⎪ ⎪ ⟹ lim 𝖽 (𝑥𝑛 , 𝑥) = ⎨ ⦅𝑥𝑛 ⦆ → 𝑥 |𝑥 − 𝑥| = 0 for constant ⦅𝑥𝑛 ⦆ for 𝑛 > 𝘕 OR ⎬ 𝑛→∞ ⎪ ⎪ 1 ≠ 0 otherwise ⎩ ⎭ which says that there are only two ways for a sequence to converge: either 𝑥 = 0 or the sequence eventually becomes constant (or both). Any other sequence will diverge. Therefore we can say the following: (a) If 𝑥 = 0 and the sequence is not constant, then the limit is unique and 0. (b) If 𝑥 = 0 and the sequence is constant, then the limit is unique and 0. (c) If 𝑥 ≠ 0 and the sequence is constant, then the limit is unique and 𝑥. (d) If 𝑥 ≠ 0 and the sequence is not constant, then the sequence diverges and there is no limit. 7. In (ℝ, 𝖽), a convergent sequence is not necessarily Cauchy. For example, (a) the sequence ⦅1/𝑛⦆𝑛∈ℕ is convergent with limit 0: lim 𝖽(1/𝑛, 0) = lim 1/𝑛 = 0 𝑛→∞

𝑛→∞

(b) However, even though ⦅ / ⦆ is convergent, it is not Cauchy: lim 𝖽(1/𝑛, 1/𝑚) = 1 ≠ 0 1𝑛

𝑛,𝑚→∞

8. The distance function 𝖽 is discontinuous in (𝑋, 𝖽): lim ⦅𝖽(1/𝑛, 2 − 1/𝑛)⦆ = 1 ≠ 2 = 𝖽(0, 2) = 𝖽 lim ⦅1/𝑛⦆ , lim ⦅2 − 1/𝑛⦆ . ) (𝑛→∞ 𝑛→∞

𝑛→∞

28

The function 𝖽(𝑥, 𝑦) ∈ ℝℝ×ℝ such that 2|𝑥 − 𝑦| ∀ (𝑥, 𝑦) ∈ {(0, 1) , (1, 0)} (dilated Euclidean) 𝖽(𝑥, 𝑦) ≜ { |𝑥 − 𝑦| otherwise } (Euclidean) Note some characteristics of the distance space (ℝ, 𝖽): Example B.3.

is a distance on ℝ.

1. (ℝ, 𝖽) is not a metric space because 𝖽 does not satisfy the triangle inequality: 𝖽(0, 1) ≜ 2|0 − 1| = 2 ≰ 1 = |0 − 1/2| + |1/2 − 1| ≜ 𝖽(0, 1/2) + 𝖽(1/2, 1) 2. The function 𝖽 is discontinuous: lim ⦅𝖽(1 − 1/𝑛, 1/𝑛)⦆ ≜ lim ⦅|1 − 1/𝑛 − 1/𝑛|⦆ = 1 ≠ 2 = 2|0 − 1| ≜ 𝖽(0, 1) = 𝖽 lim ⦅1 − 1/𝑛⦆ , lim ⦅1/𝑛⦆ . (𝑛→∞ ) 𝑛→∞ 𝑛→∞

𝑛→∞

3. In (𝑋, 𝖽), open balls are open: (a) 𝗉 (𝑥, 𝑦) ≜ |𝑥 − 𝑦| is a metric and thus all open balls in that do not contain both 0 and 1 are open. (b) By Example D.3 (page 159), 𝗊 (𝑥, 𝑦) ≜ 2|𝑥 − 𝑦| is also a metric and thus all open balls containing 0 and 1 only are open. (c) The only question remaining is with regards to open balls that contain 0, 1 and some other element(s) in ℝ. But even in this case, open balls are still open. For example: 𝖡(−1, 2) = (−1 ∶ 2) = (−1 ∶ 1) ∪ (1 ∶ 2) Note that both (−1 ∶ 1) and (1 ∶ 2) are open, and thus by Theorem B.2 (page 128), 𝖡(−1, 2) is open as well. 4. By item (3) and Theorem B.3 (page 130), the open balls of (ℝ, 𝖽) do form a base for a topology on ℝ. 28

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5. In (𝑋, 𝖽), the limits of convergent sequences are unique. This is demonstrated in Example C.3 (page 146) using additional structure developed in APPENDIX C. 6. In (𝑋, 𝖽), convergent sequences are Cauchy. This is also demonstrated in Example C.3 (page 146). The distance functions in Example B.1 (page 133)–Example B.3 (page 135) were all discontinuous. In the absence of the triangle inequality and in light of these examples, one might try replacing the triangle inequality with the weaker requirement of continuity. However, as demonstrated by the next example, this also leads to an arguably disastrous result. Example B.4. 29 The function 𝖽 ∈ ℝℝ×ℝ such that 𝖽(𝑥, 𝑦) ≜ (𝑥 − 𝑦)2 is a distance on ℝ. Note some characteristics of the distance space (ℝ, 𝖽): 1. (ℝ, 𝖽) is not a metric space because the triangle inequality does not hold: 𝖽(0, 2) ≜ (0 − 2)2 = 4 ≰ 2 = (0 − 1)2 + (1 − 2)2 ≜ 𝖽(0, 1) + 𝖽(1, 2) 2. The distance function 𝖽 is continuous in (𝑋, 𝖽). This is demonstrated in the more general setting of APPENDIX C in Example C.4 (page 147). 3. Calculating the length of curves in (𝑋, 𝖽) leads to a paradox:30 (a) Partition [0 ∶ 1] into 2𝘕 consecutive line segments connected at the points 1 2 3 2𝘕 −1 1 ⦅0, 2𝘕 , 2𝘕 , 2𝘕 , … , 2𝘕 , 1⦆ (b) Then the distance, as measured by 𝖽, between any two consecutive points is 2 1 𝖽(𝑝𝑛 , 𝑝𝑛+1 ) ≜ (𝑝𝑛 − 𝑝𝑛+1 )2 = ( 21𝘕 ) = 22𝘕 (c) But this leads to the paradox that the total length of [0 ∶ 1] is 0: 2𝘕 −1

lim

𝘕 →∞

2𝘕 1 1 = lim = lim =0 ∑ 22𝘕 𝘕 →∞ 22𝘕 𝘕 →∞ 2𝘕 𝑛=0

29

📘 Blumenthal (1953) pages 12–13, 📘 Laos (1998) pages 118–119 This is the method of “inscribed polygons” for calculating the length of a curve and goes back to Archimedes: 📘 Brunschwig et al. (2003) page 26, 📘 Walmsley (1920), page 200 ⟨§158⟩, 30

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APPENDIX C POWER DISTANCE SPACES

❝Man muss immer generalisieren.❞ ❝One should always generalize.❞ attributed to Karl Gustav Jakob Jacobi (1804–1851), mathematician

1

C.1 Definitions This paper introduces a new relation called the power triangle inequality (Definition C.2 page 138). It is a generalization of other common relations, including the triangle inequality (Definition C.3 page 138). The power triangle inequality is defined in terms of a function herein called the power triangle function (next definition). This function is a special case of the power mean with 𝘕 = 2 and 𝜆1 = 𝜆2 = 21 (Definition D.15 page 166). Power means have the attractive properties of being continuous and strictly monontone with respect to a free parameter 𝑝 ∈ ℝ∗ (Theorem D.14 page 166). This fact is inherited and exploited by the power triangle inequality (Corollary C.1 page 138). Definition C.1. Let (𝑋, 𝖽) be a DISTANCE SPACE (Definition B.1 page 127). Let ℝ+ be the set of all POSITIVE REAL NUMBERS and ℝ∗ be the set of EXTENDED REAL NUMBERS (Definition 1.2 page 5). The power triangle function 𝜏 on (𝑋, 𝖽) is defined as 1 𝜏(𝑝, 𝜎; 𝑥, 𝑦, 𝑧; 𝖽) ≜ 2𝜎 [ 21 𝖽𝑝 (𝑥, 𝑧) + 12 𝖽𝑝 (𝑧, 𝑦)] 𝑝 ∀(𝑝,𝜎)∈ℝ∗ ×ℝ⊢ , 𝑥,𝑦,𝑧∈𝑋 Remark C.1. 2 In the field of probabilistic metric spaces, a function called he triangle function was introduced by Sherstnev in 1962. However, the power triangle function as defined in this present paper is not a special case of (is not compatible with) the triangle function of Sherstnev. Another definition of triangle function has been offered by Bessenyei in 2014 with special cases of 𝛷(𝑢, 𝑣) ≜ 1 𝑐(𝑢 + 𝑣) and 𝛷(𝑢, 𝑣) ≜ (𝑢𝑝 + 𝑣𝑝 ) 𝑝 , which are similar to the definition of power triangle function offered in this present paper. 1

quote: 📘 Davis and Hersh (1999) page 134 image: http://en.wikipedia.org/wiki/Carl_Gustav_Jacobi 2 📘 Sherstnev (1962), page 4, 📘 Schweizer and Sklar (1983) page 9 ⟨(1.6.1)–(1.6.4)⟩, 📃 Bessenyei and Pales (2014) page 2

137

page 138

C.2. PROPERTIES

Definition C.2. Let (𝑋, 𝖽) be a DISTANCE SPACE. Let 𝟚𝑋𝑋𝑋 be the set of all trinomial RELATIONs (Defini▵ (𝑝, 𝜎; 𝖽) in 𝟚𝑋𝑋𝑋 is a power triangle inequality on (𝑋, 𝖽) if tion 1.5 page 6) on 𝑋. A relation ○ ▵ (𝑝, 𝜎; 𝖽) ≜ {(𝑥, 𝑦, 𝑧) ∈ 𝑋 3 |𝖽(𝑥, 𝑦) ≤ 𝜏(𝑝, 𝜎; 𝑥, 𝑦, 𝑧; 𝖽)} ○ for some (𝑝, 𝜎) ∈ ℝ∗ × ℝ+ . The tupple (𝑋, 𝖽, 𝑝, 𝜎) is a power distance space and 𝖽 a power distance or power distance function ▵ (𝑝, 𝜎; 𝖽) holds. if (𝑋, 𝖽) is a DISTANCE SPACE in which the TRIANGLE RELATION ○ The power triangle function can be used to define some standard inequalities (next definition). See Corollary C.2 (page 139) for some justification of the definitions. Definition C.3. ▵ ( ∞, 𝜎/2; 𝖽 1. ○ ▵ ( ∞, 12 ; 𝖽 2. ○ ▵ ( 2, √2/2; 𝖽 3. ○ ▵ ( 1, 𝜎; 𝖽 4. ○ ▵ ( 1, 1; 𝖽 5. ○

3

▵ (𝑝, 𝜎; 𝖽) be a POWER TRIANGLE INEQUALITY on a DISTANCE SPACE (𝑋, 𝖽). Let ○ ▵ ( 1/2, 2; 𝖽 ) is the square mean root inequality ) is the σ-inframetric inequality 6. ○ ▵ ( 0, 12 ; 𝖽 ) is the geometric inequality ) is the inframetric inequality 7. ○ ▵ ( −1, 14 ; 𝖽 ) is the harmonic inequality ) is the quadratic inequality 8. ○ ▵ ( −∞, 12 ; 𝖽 ) is the minimal inequality ) is the relaxed triangle inequality 9. ○ ) is the triangle inequality

Definition C.4. 4 Let (𝑋, 𝖽) be a DISTANCE SPACE (Definition B.1 page 127). 1. (𝑋, 𝖽) is a metric space if the TRIANGLE INEQUALITY 2. (𝑋, 𝖽) is a near metric space if the RELAXED TRIANGLE INEQUALITY 3. (𝑋, 𝖽) is an inframetric space if the INFRAMETRIC INEQUALITY 4. (𝑋, 𝖽) is a σ-inframetric space if the σ-INFRAMETRIC INEQUALITY

holds in 𝑋. holds in 𝑋. holds in 𝑋. holds in 𝑋.

C.2 Properties C.2.1 Relationships of the power triangle function Corollary C.1. Let 𝜏(𝑝, 𝜎; 𝑥, 𝑦, 𝑧; 𝖽) be the POWER TRIANGLE FUNCTION (Definition C.1 page 137) in the DISTANCE SPACE (Definition B.1 page 127) (𝑋, 𝖽). Let (ℝ, |⋅|, ≤) be the ORDERED METRIC SPACE with the usual ordering relation ≤ and usual metric |⋅| on ℝ. The function 𝜏(𝑝, 𝜎; 𝑥, 𝑦, 𝑧; 𝖽) is CONTINUOUS and STRICTLY MONOTONE in (ℝ, |⋅|, ≤) with respect to both the variables 𝑝 and 𝜎. ✎PROOF: 1. Proof that 𝜏(𝑝, 𝜎; 𝑥, 𝑦, 𝑧; 𝖽) is continuous and strictly monotone with respect to 𝑝: This follows directly from Theorem D.14 (page 166). 3

📃 Bessenyei and Pales (2014) page 2, 📃 Czerwik (1993) page 5 ⟨b-metric; (1),(2),(5)⟩, 📃 Fagin et al. (2003b), 📃 Fagin et al. (2003a) ⟨Definition 4.2 (Relaxed metrics)⟩, 📃 Xia (2009) page 453 ⟨Definition 2.1⟩, 📘 Heinonen (2001) page 109 ⟨14.1 Quasimetric spaces.⟩, 📘 Kirk and Shahzad (2014) page 113 ⟨Definition 12.1⟩, 📘 Deza and Deza (2014) page 7, 📃 Hoehn and Niven (1985) page 151, 📘 Gibbons et al. (1977) page 51 ⟨square-mean-root (SMR) (2.4.1)⟩, 📘 Euclid (circa 300BC) ⟨triangle inequality—Book I Proposition 20⟩ 4 metric space: 📘 Dieudonné (1969), page 28, 📘 Copson (1968), page 21, 📘 Hausdorff (1937) page 109, 📘 Fréchet (1928), 📃 Fréchet (1906) page 30 near metric space: 📃 Czerwik (1993) page 5 ⟨b-metric; (1),(2),(5)⟩, 📃 Fagin et al. (2003b), 📃 Fagin et al. (2003a) ⟨Definition 4.2 (Relaxed metrics)⟩, 📃 Xia (2009) page 453 ⟨Definition 2.1⟩, 📘 Heinonen (2001) page 109 ⟨14.1 Quasimetric spaces.⟩, 📘 Kirk and Shahzad (2014) page 113 ⟨Definition 12.1⟩, 📘 Deza and Deza (2014) page 7

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APPENDIX C. POWER DISTANCE SPACES

2. Proof that 𝜏(𝑝, 𝜎; 𝑥, 𝑦, 𝑧; 𝖽) is continuous and strictly monotone with respect to 𝜎: 1

𝑝 1 1 𝜏(𝑝, 𝜎; 𝑥, 𝑦, 𝑧; 𝖽) ≜ 2𝜎 [ 𝖽𝑝 (𝑥, 𝑧) + 𝖽𝑝 (𝑧, 𝑦)] ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 2 2

by definition of 𝜏 (Definition C.1 page 137)

𝖿(𝑝, 𝑥, 𝑦, 𝑧)

= 2𝜎𝖿 (𝑝, 𝑥, 𝑦, 𝑧)

where 𝖿 is defined as above

⟹ 𝜏 is affine with respect to 𝜎 ⟹ 𝜏 is continuous and strictly monotone with respect to 𝜎:

✏ Corollary C.2. Let 𝜏(𝑝, 𝜎; 𝑥, 𝑦, 𝑧; 𝖽) be the POWER TRIANGLE FUNCTION in the DISTANCE SPACE (Defini(𝑋, 𝖽). ( MAXIMUM, corresponds to INFRAMETRIC SPACE) ⎧ 2𝜎 max {𝖽(𝑥, 𝑧), 𝖽(𝑧, 𝑦)} 1 for 𝑝 = ∞, ⎪ 2𝜎 1/2𝖽2 (𝑥, 𝑧) + 1/2𝖽2 (𝑧, 𝑦) 2 for 𝑝 = 2, ( QUADRATIC MEAN) ] [ ⎪ for 𝑝 = 1, ( ARITHMETIC MEAN, corresponds to NEAR METRIC SPACE) ⎪ 𝜎[𝖽(𝑥, 𝑧) + 𝖽(𝑧, 𝑦)] 𝜏(𝑝, 𝜎; 𝑥, 𝑦, 𝑧; 𝖽) = ⎨ 𝑦) for 𝑝 = 0 ( GEOMETRIC MEAN) ⎪ 2𝜎√𝖽(𝑥, 𝑧) √𝖽(𝑧, −1 1 1 ⎪ 4𝜎 [ 𝖽(𝑥,𝑧) + 𝖽(𝑧,𝑦) ] for 𝑝 = −1 ( HARMONIC MEAN) ⎪ 2𝜎 min {𝖽(𝑥, 𝑧), 𝖽(𝑧, 𝑦)} for 𝑝 = −∞, ( MINIMUM) ⎩

tion B.1 page 127)

✎PROOF:

✏

These follow directly from Theorem D.14 (page 166).

Corollary C.3. Let (𝑋, 𝖽) be a DISTANCE SPACE. −1 1 + 1 2𝜎 min {𝖽(𝑥, 𝑧), 𝖽(𝑧, 𝑦)} ≤ 4𝜎 [ 𝖽(𝑥,𝑧) ≤ 2𝜎√𝖽(𝑥, 𝑧) √𝖽(𝑧, 𝑦) 𝖽(𝑧,𝑦) ] ≤ 𝜎[𝖽(𝑥, 𝑧) + 𝖽(𝑧, 𝑦)] ≤ 2𝜎 max {𝖽(𝑥, 𝑧), 𝖽(𝑧, 𝑦)} ✎PROOF:

✏

These follow directly from Corollary D.2 (page 169).

C.2.2 Properties of power distance spaces The power triangle inequality property of a power distance space axiomatically endows a metric with an upper bound. Lemma C.1 (next) demonstrates that there is a complementary lower bound somewhat similar in form to the power triangle inequality upper bound. In the special case where 1 2𝜎 = 2 𝑝 , the lower bound helps provide a simple proof of the continuity of a large class of power 1 distance functions (Theorem C.5 page 144). The inequality 2𝜎 ≤ 2 𝑝 is a special relation in this text and appears repeatedly in this appendix; it appears as an inequality in Lemma C.2 (page 141), Corollary C.5 (page 141) and Corollary C.6 (page 142), and as an equality in Lemma C.1 (next) and Theorem C.5 (page 144). It is plotted in Figure C.1 (page 140). Lemma C.1. 5 Let (𝑋, 𝖽, 𝑝, 𝜎) be a POWER TRIANGLE TRIANGLE SPACE (Definition C.2 page 138). Let |⋅| be the ABSOLUTE VALUE function (Definition 1.24 page 9). Let max {𝑥, 𝑦} be the maximum and min {𝑥, 𝑦} the minimum of any 𝑥, 𝑦 ∈ ℝ∗ . Then, for all (𝑝, 𝜎) ∈ ℝ∗ × ℝ+ , 𝑝 𝑝 𝑝 𝑝 𝑝 1. 𝖽 (𝑥, 𝑦) ≥ max 0, 2 𝑝 𝖽 (𝑥, 𝑧) − 𝖽 (𝑧, 𝑦) , 2 𝑝 𝖽 (𝑦, 𝑧) − 𝖽 (𝑧, 𝑥) (2𝜎) { (2𝜎) } ∀𝑥,𝑦,𝑧∈𝑋 and 2.

𝖽(𝑥, 𝑦) ≥ |𝖽(𝑥, 𝑧) − 𝖽(𝑧, 𝑦)|

1

if 𝑝 ≠ 0 and 2𝜎 = 2 𝑝

∀𝑥,𝑦,𝑧∈𝑋.

✎PROOF: 5

in metric space ((𝑝, 𝜎) = (1, 1)): 📘 Dieudonné (1969) page 28, 📘 Michel and Herget (1993) page 266, 📘 Berberian (1961) page 37 ⟨Theorem II.4.1⟩ 2016 Jul 4

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C.2. PROPERTIES 𝜎 geometric inequality at (𝑝, 𝜎) = (0, 12 )

4

2nd order inflection point at (− 12 ln 2, 12 𝑒−2 )

3

triangle inequality at (𝑝, 𝜎) = (1, 1)

2

quadratic mean inequality at 2, √2 ( 2 )

harmonic inequality at (𝑝, 𝜎) = (−1, 14 )

square mean root (SMR) inequality at ( 12 , 2)

1

1

to inframetric inequality at (𝑝, 𝜎) = (∞, 12 ) 𝑝

1 2

to minimal inequality at (𝑝, 𝜎) = (∞, 12 )

−3

1

Figure C.1: 𝜎 = 12 (2 𝑝 ) = 2 𝑝 −1 or 𝑝 =

−2

−1

0

1

2

3

ln 2 (see Lemma C.1 page 139, Lemma C.2 page 141, Corollary C.6 page 142, Corollary C.5 ln(2𝜎)

page 141, and Theorem C.5 page 144).

1. lemma:

2 𝖽𝑝 (𝑥, 𝑧) (2𝜎)𝑝

− 𝖽𝑝 (𝑧, 𝑦) ≤ 𝖽𝑝 (𝑥, 𝑦)

∀ (𝑝, 𝜎) ∈ ℝ∗ × ℝ+ : Proof:

1 𝑝 2 𝑝 2 2𝜎 [1/2𝖽𝑝 (𝑥, 𝑦) + 1/2𝖽𝑝 (𝑦, 𝑧)] 𝑝 ] − 𝖽𝑝 (𝑧, 𝑦) 𝖽 (𝑥, 𝑧) − 𝖽𝑝 (𝑧, 𝑦) ≤ [ 𝑝 𝑝 (2𝜎) (2𝜎) 2(2𝜎)𝑝 1 𝑝 = /2𝖽 (𝑥, 𝑦) + 1/2𝖽𝑝 (𝑦, 𝑧)] − 𝖽𝑝 (𝑧, 𝑦) (2𝜎)𝑝 [ = [𝖽𝑝 (𝑥, 𝑦) + 𝖽𝑝 (𝑦, 𝑧)] − 𝖽𝑝 (𝑦, 𝑧)

by power triangle inequality

by symmetric property of 𝖽

𝑝

= 𝖽 (𝑥, 𝑦) 2. Proof for (𝑝, 𝜎) ∈ ℝ∗ × ℝ+ case: 2 𝖽𝑝 (𝑥, 𝑧) − 𝖽𝑝 (𝑧, 𝑦) by (1) lemma 𝖽𝑝 (𝑥, 𝑦) ≥ (2𝜎) 𝑝 2 𝖽𝑝 (𝑦, 𝑧) − 𝖽𝑝 (𝑧, 𝑥) by commutative property of 𝖽 and (1) lemma 𝖽𝑝 (𝑥, 𝑦) = 𝖽𝑝 (𝑦, 𝑥) ≥ (2𝜎) 𝑝 𝖽𝑝 (𝑥, 𝑦) ≥ 0 by non-negative property of 𝖽 (Definition B.1 page 127) 1

The rest follows because 𝗀(𝑥) ≜ 𝑥 𝑝 is strictly monotone in ℝℝ . 1

3. Proof for 2𝜎 = 2 𝑝 case: 1

𝖽(𝑥, 𝑦) ≥ max

𝑝 2 𝑝 2 𝑝 0, 𝖽 (𝑥, 𝑧) − 𝖽𝑝 (𝑧, 𝑦) , 𝖽 (𝑦, 𝑧) − 𝖽𝑝 (𝑧, 𝑥) 𝑝 𝑝 { (2𝜎) } (2𝜎)

by item (2) (page 140) 1

= max {0, 𝖽(𝑥, 𝑧) − 𝖽(𝑧, 𝑦) , 𝖽(𝑦, 𝑧) − 𝖽(𝑧, 𝑥)}

by 2𝜎 = 2 𝑝 hypothesis ⟺

= max {0, (𝖽(𝑥, 𝑧) − 𝖽(𝑧, 𝑦)), −(𝖽(𝑥, 𝑧) − 𝖽(𝑧, 𝑦))}


2 =1 (2𝜎)𝑝

= |(𝖽(𝑥, 𝑧) − 𝖽(𝑧, 𝑦))|

✏ Theorem C.1. Let (𝑋, 𝖽, 𝑝, 𝜎) be a POWER DISTANCE SPACE (Definition C.2 page 138). Let 𝖡 be an OPEN BALL ∗ + (Definition B.4 page 128) on (𝑋, 𝖽). Then for all (𝑝, 𝜎) ∈ (ℝ ⧵{0}) × ℝ , 1 + 1. ∃𝑟𝑞 ∈ ℝ such that A. 2𝜎 ≤ 2 𝑝 and ⟹ ⟹ { B. 𝑞 ∈ 𝖡(𝜃, 𝑟) } { } { B. 𝑞 ∈ 𝖡(𝜃, 𝑟) } 𝖡(𝑞, 𝑟𝑞 ) ⊆ 𝖡(𝜃, 𝑟) ✎PROOF: 1. lemma: 𝑞 ∈ 𝖡(𝜃, 𝑟) ⟺ 𝖽(𝜃, 𝑞) < 𝑟

by definition of open ball (Definition B.4 page 128)

⟺ 0 < 𝑟 − 𝖽(𝜃, 𝑞) ⟺ ∃𝑟𝑞 ∈ ℝ+ version 0.50

such that

by field property of real numbers 0 < 𝑟𝑞 < 𝑟 − 𝖽(𝜃, 𝑞) by The Archimedean Property 6


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2. Proof that (𝐴), (𝐵) ⟹ (1): 𝖡(𝑞, 𝑟𝑞 ) ≜ {𝑥 ∈ 𝑋|𝖽(𝑞, 𝑥) < 𝑟𝑞 } 𝑝

𝑝 𝑟𝑞 𝑝

𝑝

𝑝

= {𝑥 ∈ 𝑋|𝖽 (𝑞, 𝑥)
0

≜ 𝖡(𝑞, 𝑟𝑞 )

by definition of open ball (Definition B.4 page 128)

⊆ 𝖡(𝜃, 𝑟)

by hypothesis 2

✏ Corollary C.4. Let (𝑋, 𝖽, 𝑝, 𝜎) be a POWER DISTANCE SPACE. Then for all (𝑝, 𝜎) ∈ (ℝ∗ ⧵{0}) × ℝ+ , 1 𝑝 ⟹ {every OPEN BALL in (𝑋, 𝖽) is OPEN} {2𝜎 ≤ 2 } ✎PROOF:

This follows from Theorem C.1 (page 140) and Theorem B.3 (page 130).

✏

Corollary C.5. Let (𝑋, 𝖽, 𝑝, 𝜎) be a POWER DISTANCE SPACE. Let 𝑩 be the set of all OPEN BALLs in (𝑋, 𝖽). Then for all (𝑝, 𝜎) ∈ (ℝ∗ ⧵{0}) × ℝ+ , 1 𝑝 ⟹ { 𝑩 is a BASE for (𝑋, 𝑻 ) } {2𝜎 ≤ 2 } ✎PROOF: 1. The set of all open balls in (𝑋, 𝖽) is a base for (𝑋, 𝑻 ) by Corollary C.4 (page 141) and Theorem D.16 (page 170). 2. 𝑻 is a topology on 𝑋 by Definition D.17 (page 170).

✏ Lemma C.2 (next) demonstrates that every point in an open set is contained in an open ball that is contained in the original open set (see also Figure C.2 page 142). Lemma C.2. Let (𝑋, 𝖽, 𝑝, 𝜎) be a POWER DISTANCE SPACE. Let 𝖡 be an OPEN BALL on (𝑋, 𝖽). Then for all (𝑝, 𝜎) ∈ (ℝ∗ ⧵{0}) × ℝ+ , 1 + 1. ∀𝑥 ∈ 𝑈 , ∃𝑟 ∈ ℝ such that B. 𝑈 is A. 2𝜎 ≤ 2 𝑝 and ⟹ ⟹ { B. 𝑈 is OPEN in (𝑋, 𝖽) } { } { 𝖡(𝑥, 𝑟) ⊆ 𝑈 OPEN in (𝑋, 𝖽) } 6

📘 Aliprantis and Burkinshaw (1998) page 17 ⟨Theorem 3.3 (“The Archimedean Property”) and Theorem 3.4⟩, 📘 Zorich (2004) page 53 ⟨6∘ (“The principle of Archimedes”) and 7∘ ⟩ 2016 Jul 4

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𝑝2 b

𝑝6 b

b

b

b

𝑝1 b

𝑝7

𝑝5 b

𝑝8

𝑝3

𝑝4

b

Figure C.2: open set (see Lemma C.2 page 141) ✎PROOF: 1. Proof that for ((𝐴), (𝐵) ⟹ (1)): 𝑈=

𝖡 𝑥 , 𝑟 |𝖡 𝑥 , 𝑟 ⊆ 𝑈 } ⋃ { ( 𝛾 𝛾) ( 𝛾 𝛾) ⊇ 𝖡(𝑥, 𝑟)

by left hypothesis and Corollary C.5 page 141 because 𝑥 must be in one of those balls in 𝑈

2. Proof that ((𝐵) ⟸ (1)) case: 𝑈=

⋃

{𝑥 ∈ 𝑋|𝑥 ∈ 𝑈 }

by definition of union operation

=

𝖡(𝑥, 𝑟) |𝑥 ∈ 𝑈 and 𝖡(𝑥, 𝑟) ⊆ 𝑈 } ⋃{ ⟹ 𝑈 is open

⋃

by hypothesis (1) by Corollary C.5 page 141 and Corollary B.1 page 129

✏ Corollary C.6. 7 Let (𝑋, 𝖽, 𝑝, 𝜎) be a POWER DISTANCE SPACE. Let 𝖡 be an OPEN BALL on (𝑋, 𝖽). Then for all (𝑝, 𝜎) ∈ (ℝ∗ ⧵{0}) × ℝ+ , 1 𝑝 2𝜎 ≤ 2 ⟹ { every OPEN BALL 𝖡(𝑥, 𝑟) in (𝑋, 𝖽) is OPEN } { } ✎PROOF: The union of any set of open balls is open

by Corollary C.5 page 141

⟹ the union of a set of just one open ball is open ⟹ every open ball is open.

✏ Theorem C.2. 8 Let (𝑋, 𝖽, 𝑝, 𝜎) be a POWER DISTANCE SPACE. Let (𝑋, 𝑻 ) be a TOPOLOGICAL SPACE INDUCED BY (𝑋, 𝖽). Let ⦅𝑥𝑛 ∈ 𝑋⦆𝑛∈ℤ be a sequence in (𝑋, 𝖽). for any 𝜀 ∈ ℝ+ , there exists 𝘕 ∈ ℤ converges to a limit 𝑥 ⟺ ⦅𝑥 ⦆ 𝑛 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ { such that for all 𝑛 > 𝘕 , 𝖽(𝑥𝑛 , 𝑥) < 𝜀 } (Definition D.20 page 172)

✎PROOF: ⦅𝑥𝑛 ⦆ → 𝑥 ⟺ 𝑥𝑛 ∈ 𝑈 ⟺ ∃𝖡(𝑥, 𝜀)

∀𝑈 ∈ 𝑁𝑥 , 𝑛 > 𝘕 such that

𝑥𝑛 ∈ 𝖡(𝑥, 𝜀) ∀𝑛 > 𝘕

⟺ 𝖽 ( 𝑥𝑛 , 𝑥 ) < 𝜀 7 8

by Definition D.20 page 172 by Lemma C.2 page 141 by Definition B.4 page 128

in metric space ((𝑝, 𝜎) = (1, 1)): 📘 Rosenlicht (1968) pages 40–41, 📘 Aliprantis and Burkinshaw (1998) page 35 in metric space: 📘 Rosenlicht (1968) page 45, 📘 Giles (1987) page 37 ⟨3.2 Definition⟩

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✏ In distance spaces (Definition B.1 page 127), not all convergent sequences are Cauchy (Example B.2 page 134). However in a distance space with any power triangle inequality (Definition C.2 page 138), all convergent sequences are Cauchy (next theorem). Theorem C.3. 9 Let (𝑋, 𝖽, 𝑝, 𝜎) be a POWER DISTANCE SPACE. Let 𝖡 be an OPEN BALL on (𝑋, 𝖽). For any (𝑝, 𝜎) ∈ ℝ∗ × ℝ+ , ⦅𝑥𝑛 ⦆ is BOUNDED ⦅𝑥𝑛 ⦆ is CAUCHY ⦅𝑥𝑛 ⦆ is CONVERGENT ⟹ ⟹ } } { in (𝑋, 𝖽) } { in (𝑋, 𝖽) { in (𝑋, 𝖽) ✎PROOF: 1. Proof that convergent ⟹ Cauchy: 𝖽(𝑥𝑛 , 𝑥𝑚 ) ≤ 𝜏(𝑝, 𝜎; 𝑥𝑛 , 𝑥𝑚 , 𝑥) 1 1 ≜ 2𝜎 [ 𝖽𝑝 (𝑥𝑛 , 𝑥) + 𝖽𝑝 (𝑥𝑚 , 𝑥)] 2 2 1 𝑝 1 1 < 2𝜎 [ 𝜀𝑝 + 𝜀𝑝 ] 2 2 = 2𝜎𝜀 ⟹ Cauchy

by definition of power triangle inequality (Definition C.2 page 138) 1 𝑝

by definition of power triangle function (Definition C.1 page 137) by convergence hypothesis (Definition D.20 page 172) by definition of convergence (Definition D.20 page 172) by definition of Cauchy (Definition B.8 page 130)

𝖽(𝑥𝑛 , 𝑥𝑚 ) ≤ 𝜏(∞, 𝜎; 𝑥𝑛 , 𝑥𝑚 , 𝑥)

by definition of power triangle inequality at 𝑝 = ∞

= 2𝜎 max {𝖽(𝑥𝑛 , 𝑥), 𝖽(𝑥𝑚 , 𝑥)}

by Corollary C.2 (page 139)

= 2𝜎 max {𝜀, 𝜀}

by convergent hypothesis (Definition D.20 page 172)

= 2𝜎𝜀

by definition of max

𝖽(𝑥𝑛 , 𝑥𝑚 ) ≤ 𝜏(−∞, 𝜎; 𝑥𝑛 , 𝑥𝑚 , 𝑥)

by definition of power triangle inequality at 𝑝 = −∞

= 2𝜎 min {𝖽(𝑥𝑛 , 𝑥), 𝖽(𝑥𝑚 , 𝑥)}


= 2𝜎 min {𝜀, 𝜀}


= 2𝜎𝜀

by definition of min

𝖽(𝑥𝑛 , 𝑥𝑚 ) ≤ 𝜏(0, 𝜎; 𝑥𝑛 , 𝑥𝑚 , 𝑥)

by definition of power triangle inequality at 𝑝 = 0

= 2𝜎√𝖽(𝑥𝑛 , 𝑥) √𝖽(𝑥𝑚 , 𝑥)


= 2𝜎√𝜀 √𝜀


= 2𝜎𝜀

by property of ℝ

2. Proof that Cauchy ⟹ bounded: by Proposition B.1 (page 131).

✏ Theorem C.4. 10 Let (𝑋, 𝖽, 𝑝, 𝜎) be a POWER DISTANCE SPACE. Let 𝖿 ∈ ℤℤ be a STRICTLY MONOTONE function such that 𝖿(𝑛) < 𝖿(𝑛 + 1). For any (𝑝, 𝜎) ∈ ℝ∗ × ℝ+ 1. ⦅𝑥𝑛 ⦆𝑛∈ℤ is CAUCHY and ⟹ {⦅𝑥𝑛 ⦆𝑛∈ℤ is CONVERGENT.} { 2. ⦅𝑥𝖿(𝑛) ⦆𝑛∈ℤ is CONVERGENT } 9

in metric space: 📘 Giles (1987) page 49 ⟨Theorem 3.30⟩, 📘 Rosenlicht (1968) page 51, 📘 Apostol (1975) pages 72– 73 ⟨Theorem 4.6⟩ 10 in metric space: 📘 Rosenlicht (1968) page 52 2016 Jul 4

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✎PROOF: 𝖽(𝑥𝑛 , 𝑥) = 𝖽(𝑥, 𝑥𝑛 )


≤ 𝜏(𝑝, 𝜎; 𝑥, 𝑥𝑛 , 𝑥𝖿(𝑛) )

by definition of power triangle inequality (Definition C.2 page 138)

1 1 ≜ 2𝜎 [ 𝖽𝑝 (𝑥, 𝑥𝖿(𝑛) ) + 𝖽𝑝 (𝑥𝖿(𝑛) , 𝑥𝑛 )] 2 2 1 𝑝 1 1 = 2𝜎 [ 𝜀 + 𝖽𝑝 (𝑥𝖿(𝑛) , 𝑥𝑛 )] 2 2 1 𝑝 1 1 = 2𝜎 [ 𝜀𝑝 + 𝜀𝑝 ] 2 2 = 2𝜎𝜀

1 𝑝

by definition of power triangle function (Definition C.1 page 137) by left hypothesis 2 by left hypothesis 1

⟹ convergent

by definition of convergent

(Definition D.20 page 172)

✏ Theorem C.5. 11 Let (𝑋, 𝖽, 𝑝, 𝜎) be a POWER DISTANCE SPACE. Let (ℝ, 𝗊) be a metric space of real numbers with the usual metric 𝗊(𝑥, 𝑦) ≜ |𝑥 − 𝑦|. Then 1 𝑝 2𝜎 = 2 ⟹ { 𝖽 is CONTINUOUS in (ℝ, 𝗊) } { } ✎PROOF: |𝖽(𝑥, 𝑦) − 𝖽(𝑥𝑛 , 𝑦𝑛 )| ≤ |𝖽(𝑥, 𝑦) − 𝖽(𝑥𝑛 , 𝑦)| + |𝖽(𝑥𝑛 , 𝑦) − 𝖽(𝑥𝑛 , 𝑦𝑛 )| by triangle inequality of (ℝ, |𝑥 − 𝑦|) = |𝖽(𝑥, 𝑦) − 𝖽(𝑦, 𝑥𝑛 )| + |𝖽(𝑦, 𝑥𝑛 ) − 𝖽(𝑥𝑛 , 𝑦𝑛 )| by commutative property of 𝖽 (Definition B.1 page 127) 1

≤ 𝖽(𝑥, 𝑥𝑛 ) + 𝖽(𝑦, 𝑦𝑛 ) =0

by 2𝜎 = 2 𝑝 and Lemma C.1 (page 139)

as 𝑛 → ∞

✏ In distance spaces and topological spaces, limits of convergent sequences are in general not unique (Example B.1 page 133, Example D.16 page 172). However Theorem C.6 (next) demonstrates that, in a power distance space, limits are unique. Theorem C.6 (Uniqueness of limit). 12 Let (𝑋, 𝖽, 𝑝, 𝜎) be a POWER DISTANCE SPACE. Let 𝑥, 𝑦, ∈ 𝑋 and let ⦅𝑥𝑛 ∈ 𝑋⦆ be an 𝑋-valued sequence. 1. {(⦅𝑥𝑛 ⦆ , ⦅𝑦𝑛 ⦆) → (𝑥, 𝑦)} and ⟹ {𝑥 = 𝑦} { 2. (𝑝, 𝜎) ∈ ℝ∗ × ℝ+ } ✎PROOF: 1. lemma: Proof that for all (𝑝, 𝜎) ∈ ℝ∗ × ℝ+ and for any 𝜀 ∈ ℝ+ , there exists 𝘕 such that 𝖽(𝑥, 𝑦) < 2𝜎𝜀: 𝖽(𝑥, 𝑦) ≤ 𝜏(𝑝, 𝜎; 𝑥, 𝑦, 𝑥𝑛 )

by definition of power triangle inequality (Definition C.2 page 138)

1 1 ≜ 2𝜎 [ 𝖽𝑝 (𝑥, 𝑥𝑛 ) + 𝖽𝑝 (𝑥𝑛 , 𝑦)] 2 2 1 𝑝 1 1 < 2𝜎 [ 𝜀𝑝 + 𝜀𝑝 ] 2 2 = 2𝜎𝜀 𝖽(𝑥, 𝑦) ≤ 𝜏(∞, 𝜎; 𝑥, 𝑦, 𝑥𝑛 )

12

by definition of power triangle function (Definition C.1 page 137) by left hypothesis and for 𝑝 ∈ ℝ∗ ⧵{−∞, 0, ∞} by definition of power triangle inequality at 𝑝 = ∞

= 2𝜎 max {𝖽(𝑥, 𝑥𝑛 ), 𝖽(𝑥𝑛 , 𝑦)} 11

1 𝑝


in metric space ((𝑝, 𝜎) = (1, 1) case): 📘 Berberian (1961) page 37 ⟨Theorem II.4.1⟩ in metric space: 📘 Rosenlicht (1968) page 46, 📘 Thomson et al. (2008) page 32 ⟨Theorem 2.8⟩

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< 2𝜎𝜀

by left hypothesis

𝖽(𝑥, 𝑦) ≤ 𝜏(−∞, 𝜎; 𝑥, 𝑦, 𝑥𝑛 )

by definition of power triangle inequality at 𝑝 = −∞

= 2𝜎 min {𝖽(𝑥, 𝑥𝑛 ), 𝖽(𝑥𝑛 , 𝑦)}


< 2𝜎𝜀

by left hypothesis

𝖽(𝑥, 𝑦) ≤ 𝜏(0, 𝜎; 𝑥, 𝑦, 𝑥𝑛 )

by definition of power triangle inequality at 𝑝 = 0

= 2𝜎√𝖽(𝑥, 𝑥𝑛 ) √𝖽(𝑥𝑛 , 𝑦)


= 2𝜎√𝜀 √𝜀

by left hypothesis

< 2𝜎𝜀

by property of real numbers

2. Proof that 𝑥 = 𝑦 (proof by contradiction): 𝑥 ≠ 𝑦 ⟹ 𝖽(𝑥, 𝑦) ≠ 0

by the nondegenerate property of 𝖽 (Definition B.1 page 127)

⟹ 𝖽(𝑥, 𝑦) > 0 ⟹ ∃𝜀

such that

by non-negative property of 𝖽 (Definition B.1 page 127) 𝖽(𝑥, 𝑦) > 2𝜎𝜀

⟹ contradiction to (1) lemma page 144 ⟹ 𝖽(𝑥, 𝑦) = 0 ⟹ 𝑥=𝑦

✏

C.3 Examples ▵ (𝑝, 𝜎; 𝖽) that holds in every It is not always possible to find a triangle relation (Definition C.2 page 138) ○ distance space (Definition B.1 page 127), as demonstrated by Example C.1 and Example C.2 (next two examples). Example C.1. Let 𝖽(𝑥, 𝑦) ∈ ℝℝ×ℝ be defined such that 𝑦 ∀ (𝑥, 𝑦) ∈ {4} × (0 ∶ 2] (vertical half-open interval) ⎧ ⎪ ∀ (𝑥, 𝑦) ∈ (0 ∶ 2] × {4} (horizontal half-open interval) 𝖽(𝑥, 𝑦) ≜ ⎨ 𝑥 ⎪ (Euclidean) ⎩ |𝑥 − 𝑦| otherwise Note the following about the pair (ℝ, 𝖽):

⎫ ⎪ ⎬. ⎪ ⎭

1. By Example B.1 (page 133), (ℝ, 𝖽) is a distance space, but not a metric space—that is, the tri▵ (1, 1; 𝖽) does not hold in (ℝ, 𝖽). angle relation ○ 2. Observe further that (ℝ, 𝖽) is not a power distance space. In particular, the triangle relation ▵ (𝑝, 𝜎; 𝖽) does not hold in (ℝ, 𝖽) for any finite value of 𝜎 (does not hold for any 𝜎 ∈ ℝ+ ): ○ 1

𝖽(0, 4) = 4 ≰ 0 = lim 2𝜎𝜀 = lim 2𝜎[1/2|0 − 𝜀|𝑝 + 1/2𝜀𝑝 ] 𝑝 𝜀→0

𝜀→0

1

▵ (𝑝, 𝜎; 0, 4, 𝜀; 𝖽) ≜ lim 2𝜎[1/2𝖽𝑝 (0, 𝜀) + 1/2𝖽𝑝 (𝜀, 4)] 𝑝 ≜ lim ○ 𝜀→0

𝜀→0

Example C.2. Let 𝖽(𝑥, 𝑦) ∈ ℝℝ×ℝ be defined such that |𝑥 − 𝑦| for 𝑥 = 0 or 𝑦 = 0 or 𝑥 = 𝑦 (Euclidean) 𝖽(𝑥, 𝑦) ≜ . { 1 } otherwise (discrete) Note the following about the pair (ℝ, 𝖽): 1. By Example B.2 (page 134), (ℝ, 𝖽) is a distance space, but not a metric space—that is, the tri▵ (1, 1; 𝖽) does not hold in (ℝ, 𝖽). angle relation ○ 2016 Jul 4

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▵ (𝑝, 𝜎; 𝖽) 2. Observe further that (ℝ, 𝖽) is not a power distance space—that is, the triangle relation ○ does not hold in (ℝ, 𝖽) for any value of (𝑝, 𝜎) ∈ ℝ∗ × ℝ+ . ▵ (𝑝, 𝜎; 𝖽) does not hold for any (𝑝, 𝜎) ∈ {∞} × ℝ+ : (a) Proof that ○ lim 𝖽(1/𝑛, 1/𝑚) ≜ 1 ≰ 0 = 2𝜎 max {0, 0}

by definition of 𝖽

𝑛,𝑚→∞

= 2𝜎 lim max {𝖽(1/𝑛, 0) , 𝖽(0, 1/𝑚)}


𝑛,𝑚→∞

1

≥ lim 2𝜎[1/2𝖽𝑝 (1/𝑛, 0) + 1/2𝖽𝑝 (0, 1/𝑚)] 𝑝


≜ lim 𝜏(𝑝, 𝜎, 1/𝑛, 1/𝑚, 0)

by definition of 𝜏 (Definition C.1 page 137)

𝑛,𝑚→∞ 𝑛,𝑚→∞

▵ (𝑝, 𝜎; 𝖽) does not hold for any (𝑝, 𝜎) ∈ ℝ∗ ×ℝ+ : By Corollary C.1 (page 138), the (b) Proof that ○ triangle function (Definition C.1 page 137) 𝜏(𝑝, 𝜎; 𝑥, 𝑦, 𝑧; 𝖽) is continuous and strictly monotone in ▵ (𝑝, 𝜎; 𝖽) fails to hold (ℝ, |⋅|, ≤) with respect to the variable 𝑝. Item 2a demonstrates that ○ at the best case of 𝑝 = ∞, and so by Corollary C.1, it doesn't hold for any other value of 𝑝 ∈ ℝ∗ either. Example C.3. Let 𝖽 be a function in ℝℝ×ℝ such that 2|𝑥 − 𝑦| ∀ (𝑥, 𝑦) ∈ {(0, 1) , (1, 0)} (dilated Euclidean) 𝖽(𝑥, 𝑦) ≜ . { |𝑥 − 𝑦| otherwise } (Euclidean) Note the following about the pair (ℝ, 𝖽): 1. By Example B.3 (page 135), (ℝ, 𝖽) is a distance space, but not a metric space—that is, the tri▵ (1, 1; 𝖽) does not hold in (ℝ, 𝖽). angle relation ○ 2. But observe further that (ℝ, 𝖽, 1, 2) is a power distance space: ▵ (1, 2; 𝖽) (Definition C.2 page 138) holds for all (𝑥, 𝑦) ∈ {(0, 1) , (1, 0)}: (a) Proof that ○ 𝖽(1, 0) = 𝖽(0, 1) ≜ 2|0 − 1| = 2 ≤ 2 ≤ 2(|0 − 𝑧| + |𝑧 − 1|)

by definition of 𝖽 by definition of |⋅| (Definition 1.24 page 9)

∀𝑧∈ℝ 1

= 2𝜎(1/2|0 − 𝑧|𝑝 + 1/2|𝑧 − 1|𝑝 ) 𝑝 ≜ 2𝜎(1/2𝖽𝑝 (0, 𝑧) + 𝖽𝑝 (𝑧, 1))

1 𝑝

∀𝑧∈ℝ ∀𝑧∈ℝ

≜ 𝜏(1, 2; 0, 1, 𝑧)

for (𝑝, 𝜎) = (1, 2) for (𝑝, 𝜎) = (1, 2) and by definition of 𝖽 by definition of 𝜏 (Definition C.1 page 137)

▵ (1, 2; 𝖽) holds for all other (𝑥, 𝑦) ∈ ℝ∗ × ℝ+ : (b) Proof that ○ 𝖽(𝑥, 𝑦) ≜ 2|𝑥 − 𝑦| ≤ (|𝑥 − 𝑧| + |𝑧 − 𝑦|)

by definition of 𝖽 by property of Euclidean metric spaces 1

= 2𝜎(1/2|0 − 𝑧|𝑝 + 1/2|𝑧 − 1|𝑝 ) 𝑝

for (𝑝, 𝜎) = (1, 1)

≜ 𝜏(1, 1; 𝑥, 𝑦, 𝑧) ≤ 𝜏(1, 2; 𝑥, 𝑦, 𝑧)

by definition of 𝜏 (Definition C.1 page 137) by Corollary C.1 (page 138)

3. In (𝑋, 𝖽), the limits of convergent sequences are unique. This follows directly from the fact that (ℝ, 𝖽, 1, 2) is a power distance space (item (2) page 146) and by Theorem C.6 page 144. 4. In (𝑋, 𝖽), convergent sequences are Cauchy. This follows directly from the fact that (ℝ, 𝖽, 1, 2) is a power distance space (item (2) page 146) and by Theorem C.3 page 143. version 0.50


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Example C.4. Let 𝖽 be a function in ℝℝ×ℝ such that 𝖽(𝑥, 𝑦) ≜ (𝑥 − 𝑦)2 . Note the following about the pair (ℝ, 𝖽): 1. It was demonstrated in Example B.4 (page 136) that (ℝ, 𝖽) is a distance space, but that it is not a metric space because the triangle inequality does not hold. 2. However, the tuple (ℝ, 𝖽, 𝑝, 𝜎) is a power distance space (Definition C.2 page 138) for any (𝑝, 𝜎) ∈ ℝ∗ × [2 ∶ ∞): In particular, for all 𝑥, 𝑦, 𝑧 ∈ ℝ, the power triangle inequality (Definition C.2 page 138) must hold. The “worst case” for this is when a third point 𝑧 is exactly “halfway between” 𝑥 and 𝑦 in 𝖽(𝑥, 𝑦); that is, when 𝑧 = 𝑥+𝑦 : 2 (𝑥 − 𝑦)2 ≜ 𝖽(𝑥, 𝑦) ≤ 𝜏(𝑝, 𝜎; 𝑥, 𝑦, 𝑧; 𝖽)

by definition of 𝖽 by definition power triangle inequality 1

≜ 2𝜎[1/2𝖽𝑝 (𝑥, 𝑧) + 1/2𝖽𝑝 (𝑧, 𝑦)] 𝑝

by definition 𝜏 (Definition C.1 page 137)

≜ 2𝜎 [1/2(𝑥 − 𝑧)2𝑝 + 1/2(𝑧 − 𝑦)2𝑝 ]

1 𝑝

= 2𝜎 [1/2|𝑥 − 𝑧|2𝑝 + 1/2|𝑧 − 𝑦|2𝑝 ]

1 𝑝

by definition of 𝖽 because (𝑥)2 = |𝑥|2 for all 𝑥 ∈ ℝ 1

2𝑝 𝑝 𝑥 + 𝑦 2𝑝 1 𝑥 + 𝑦 𝑥+𝑦 = 2𝜎 / |𝑥 − + / 2 − 𝑦 because 𝑧 = is the “worst case” scenario | | | [ ] 2 2 2 𝑦 − 𝑥 2𝑝 1 𝑥 − 𝑦 2𝑝 1𝑝 = 2𝜎 [1/2| + /2| 2 | 2 | ] 1 𝑥 − 𝑦 2𝑝 𝑝 2𝜎 = 2𝜎 [| = |𝑥 − 𝑦|2 2 | ] 4 ⟹ (𝑝, 𝜎) ∈ ℝ∗ × [2 ∶ ∞) 12

3. The power distance function 𝖽 is continuous in (ℝ, 𝖽, 𝑝, 𝜎) for any (𝑝, 𝜎) such that 𝜎 ≥ 2 and 1 2𝜎 = 𝑝 𝑝 . This follows directly from Theorem C.5 (page 144).

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❝Dirichlet

alone, not I, nor Cauchy, nor Gauss knows what a completely rigorous proof is. Rather we learn it first from him. When Gauss says he has proved something it is clear; when Cauchy says it, one can wager as much pro as con; when Dirichlet says it, it is certain.❞ Carl Gustav Jacob Jacobi (1804–1851), Jewish-German mathematician

1

D.1 Linear spaces D.1.1 Structure 2

Definition D.1. 2 Let (𝔽 , +, ⋅, 0, 1) be a FIELD. Let 𝑋 be a set, let + be an OPERATOR in 𝑋 𝑋 , and let ⊗ be an operator in 𝑋 𝔽 ×𝑋 . The structure 𝞨 ≜ (𝑋, +, ⋅, (𝔽 , ∔, ⨰)) is a linear space over (𝔽 , +, ⋅, 0, 1) if 1. ∃𝟘 ∈ 𝑋 such that 𝒙 + 𝟘 = 𝒙 ∀𝒙∈𝑋 (+ IDENTITY) ∗ 2. ∃𝒚 ∈ 𝑋 such that 𝒙 + 𝒚 = 𝟘 ∀𝒙∈𝑋 (+ INVERSE) 3. (𝒙 + 𝒚) + 𝒛 = 𝒙 + (𝒚 + 𝒛) ∀𝒙,𝒚,𝒛∈𝑋 (+ is ASSOCIATIVE) 4. 𝒙+𝒚 = 𝒚+𝒙 ∀𝒙,𝒚∈𝑋 (+ is COMMUTATIVE) 5. 1⋅𝒙 = 𝒙 ∀𝒙∈𝑋 (⋅ IDENTITY) 6. 𝛼 ⋅ (𝛽 ⋅ 𝒙) = (𝛼 ⋅ 𝛽) ⋅ 𝒙 ∀𝛼,𝛽∈𝑆 and 𝒙∈𝑋 (⋅ ASSOCIATES with ⋅) 7. 𝛼 ⋅ (𝒙 + 𝒚) = (𝛼 ⋅ 𝒙) + (𝛼 ⋅ 𝒚) ∀𝛼∈𝑆 and 𝒙,𝒚∈𝑋 (⋅ DISTRIBUTES over +) 8. (𝛼 + 𝛽) ⋅ 𝒙 = (𝛼 ⋅ 𝒙) + (𝛽 ⋅ 𝒙) ∀𝛼,𝛽∈𝑆 and 𝒙∈𝑋 (⋅ PSEUDO-DISTRIBUTES over +) Definition D.2. 3 Let 𝞨 ≜ (𝑋, +, ⋅, (𝔽 , ∔, ⨰)) be a LINEAR SPACE (Definition D.1 page 149). A set 𝐷 ⊆ 𝑋 is convex in 𝞨 if 𝜆𝒙 + (1 − 𝜆)𝒚 ∈ 𝐷 ∀𝒙,𝒚∈𝐷 and ∀𝜆∈(0,1) A set is concave in 𝞨 if it is NOT CONVEX in 𝞨. 1

quote:

http://lagrange.math.trinity.edu/aholder/misc/quotes.shtml 📘 Schubring (2005), page 558 image: http://en.wikipedia.org/wiki/Carl_Gustav_Jakob_Jacobi 2 📘 Kubrusly (2001) pages 40–41 ⟨Definition 2.1 and following remarks⟩, 📘 Haaser and Sullivan (1991), page 41, 📘 Halmos (1948), pages 1–2, 📘 Peano (1888a) ⟨Chapter IX⟩, 📘 Peano (1888b), pages 119–120, 📃 Banach (1922) pages 134–135 3 📘 Mitrinović et al. (2010) page 1, 📘 van de Vel (1993) pages 5–6, 📘 Bollobás (1999), page 2

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D.1.2 Metric Linear Spaces Metric space stucture can be added to a linear space resulting in a metric linear space (next definition). One key difference between metric linear spaces and normed linear spaces is that the balls in a normed linear space (Definition D.4 page 151) are always convex (Definition D.2 page 149); this is not true for all metric linear spaces (Theorem D.4 page 152).4 Definition D.3. 5 Let 𝞨 ≜ (𝑋, +, ⋅, (𝔽 , ∔, ⨰), 𝖽). The tuple 𝞨 is a metric linear space if 1. if (𝑋, +, ⋅, (𝔽 , ∔, ⨰)) is a LINEAR SPACE and 𝙓 2. 𝖽 is a METRIC in ℝ and 6 3. 𝖽(𝒙 + 𝒛, 𝒚 + 𝒛) = 𝖽(𝒙, 𝒚) ∀𝒙,𝒚,𝒛∈𝑋 ( TRANSLATION INVARIANT) and 4. 𝛼𝑛 → 𝛼 and 𝒙𝑛 → 𝒙 ⟹ 𝛼𝑛 𝒙𝑛 → 𝛼𝒙 Theorem D.1.

7

Let 𝞨 ≜ (𝑋, +, ⋅, (𝔽 , ∔, ⨰), 𝖽) be a metric linear space. 𝖡(𝜃, 𝑟) ∈ 𝞨 is CONVEX 𝖽(𝜽, 𝜆𝒙 + (1 − 𝜆)𝒚) ≤ 𝜆𝖽(𝜽, 𝒙) + (1 − 𝜆)𝖽(𝜽, 𝒚) ⟹ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ { ∀𝜽∈𝑋, 𝑟∈ℝ+ } 𝖽 is a CONVEX function

✎PROOF: 𝖽(𝜽, 𝜆𝒙 + (1 − 𝜆)𝒚) ≤ 𝜆𝖽(𝜽, 𝒙) + (1 − 𝜆)𝖽(𝜽, 𝒚) ≤ 𝜆𝑟 + (1 − 𝜆)𝑟

by convexity hypothesis ∀𝒙, 𝒚 ∈ 𝖡(𝜃, 𝑟)

=𝑟 ⟹ 𝜆𝒙 + (1 − 𝜆)𝒚 ∈ 𝖡(𝜃, 𝑟)

∀𝒙, 𝒚 ∈ 𝖡(𝜃, 𝑟)

⟹ 𝖡(𝜃, 𝑟) ∈ (𝙓 , 𝖽) is convex

∀𝜃 ∈ 𝙓

✏ Theorem D.2. 8 Let (𝑋, +, ⋅, (ℝ, ∔, ⨰), 𝖽) be a real metric linear space. 1. 𝖽(𝒙 + 𝒛, 𝒚 + 𝒛) = 𝖽(𝒙, 𝒚) ∀𝒙,𝒚,𝒛∈𝙓 ( TRANSLATION INVARIANT) { 2. 𝖽(𝜆𝒙, 𝜆𝒚) = 𝜆 𝖽(𝒙, 𝒚) ∀𝒙,𝒚∈𝙓 , 𝜆∈[0,1] ( HOMOGENEOUS) ⟹ {𝖡(𝜃, 𝑟) ∈ (𝙓 , 𝖽) is CONVEX ∀𝜽∈𝙓 , 𝑟∈ℝ+ }

and

}

✎PROOF: 𝖽(𝜽, 𝜆𝒙 + (1 − 𝜆)𝒚) = 𝖽(𝟘, 𝜆𝒙 + (1 − 𝜆)𝒚 − 𝜽)

by translation invariance hypothesis

= 𝖽(𝟘, 𝜆(𝒙 − 𝜽) + (1 − 𝜆)(𝒚 − 𝜽)) ≤ 𝖽(𝟘, 𝜆(𝒙 − 𝜽)) + 𝖽(𝜆(𝒙 − 𝜽), 𝜆(𝒙 − 𝜽) + (1 − 𝜆)(𝒚 − 𝜽))

by subadditive property

= 𝖽(𝟘, 𝜆(𝒙 − 𝜽)) + 𝖽(𝟘, 𝟘 + (1 − 𝜆)(𝒚 − 𝜽))


= 𝜆𝖽(𝟘, 𝒙 − 𝜽) + (1 − 𝜆)𝖽(𝟘, 𝒚 − 𝜽)

by homogeneous hypothesis

= 𝜆𝖽(𝜽, 𝒙) + (1 − 𝜆)𝖽(𝜽, 𝒚)


4

📘 Bruckner et al. (1997) page 478 📘 Maddox (1989) page 90, 📘 Bruckner et al. (1997) page 477 ⟨Definition 12.3⟩, 📘 Rolewicz (1985) page 1, 📘 Loève (1977) page 79 6 Some authors do not require the translation invariant property for the definition of the metric linear space, as indicated by the following references: 📘 Maddox (1989) page 90 ⟨“Some authors…do not include translation invariance in the def5

inition of metric linear space, since they use a theorem of Kakutani to show that a non-translation invariant metric may be replaced by a translation invariant metric which yields the same topolgy.”⟩, 📘 Friedman (1970) page 125 ⟨Definition 4.1.4⟩, 📃 Dobrowolski and Mogilski (1995) page 86 7 📘 Norfolk (1991), page 5 8 📃 Norfolk (1991) pages 5–6, http://groups.google.com/group/sci.math/msg/a6f0a7924027957d

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≤ 𝜆𝑟 + (1 − 𝜆)𝑟

∀𝒙, 𝒚 ∈ 𝖡(𝜃, 𝑟)

=𝑟 ⟹ 𝜆𝒙 + (1 − 𝜆)𝒚 ∈ 𝖡(𝜃, 𝑟)

∀𝒙, 𝒚 ∈ 𝖡(𝜃, 𝑟)

⟹ 𝖡(𝜃, 𝑟) ∈ (𝙓 , 𝖽) is convex

∀𝜃 ∈ 𝙓

✏

D.1.3 Normed Linear Spaces Definition D.4. 9 Let (𝑋, +, ⋅, (𝔽 , ∔, ⨰)) be a LINEAR SPACE (Definition D.1 page 149) and |⋅| ∈ ℝ𝔽 the ABSOLUTE VALUE function. A functional ‖⋅‖ in ℝ𝙓 is a norm if ‖𝒙‖ ≥ 0 ∀𝒙∈𝑋 ( STRICTLY POSITIVE) 2. ∀𝒙∈𝑋 ( NONDEGENERATE) ‖𝒙‖ = 0 ⟺ 𝒙 = 𝟘 3. ∀𝒙∈𝑋, 𝛼∈ℂ ( HOMOGENEOUS) ‖𝛼𝒙‖ = |𝛼| ‖𝒙‖ 4. ‖𝒙 + 𝒚‖ ≤ ‖𝒙‖ + ‖𝒚‖ ∀𝒙,𝒚∈𝑋 ( SUBADDITIVE/ TRIANGLE INQUALITY). A normed linear space is the tuple (𝑋, +, ⋅, (𝔽 , ∔, ⨰), ‖⋅‖). 1.

and and and

Example D.1 (The usual norm). 10 Let ℝℝ be the set of all functions with domain and range the set of real numbers ℝ. The absolute value |⋅| ∈ ℝℝ is a norm. Example D.2 (𝑙𝑝 norms). Let ⦅𝑥𝑛 ⦆𝑛∈ℤ be a sequence of real numbers. 1 𝑝

‖⦅𝑥𝑛 ⦆‖𝑝 ≜

𝑝 |𝑥𝑛 | ∑ ( )

is a norm for 𝑝 ∈ [1 ∶ ∞]

𝑛∈ℤ

D.1.4 Relationship between metrics and norms Metrics generated by norms Theorem D.3. 11 Let 𝖽 ∈ ℝ𝙓 ×𝙓 be a function on a REAL normed linear space (𝑋, +, ⋅, (ℝ, ∔, ⨰), ‖⋅‖). Let 𝖡(𝒙, 𝑟) ≜ {𝒚 ∈ 𝙓 | ‖𝒚 − 𝒙‖ < 𝑟} be the OPEN BALL (Definition B.4 page 128) of radius 𝑟 centered at a point 𝒙. 𝖽(𝒙, 𝒚) ≜ ‖𝒙 − 𝒚‖ is a metric on 𝙓 The next definition defines this metric formally. Definition D.5. 12 Let (𝑋, +, ⋅, (𝔽 , ∔, ⨰), ‖⋅‖) be a NORMED LINEAR SPACE (Definition D.4 page 151). The metric induced by the norm ‖⋅‖ is the function 𝖽 ∈ ℝ𝙓 such that 𝖽(𝒙, 𝒚) ≜ ‖𝒙 − 𝒚‖ ∀𝒙,𝒚∈𝑋 . Corollary D.1. 13 Let 𝞨 ≜ (𝑋, +, ⋅, (𝔽 , ∔, ⨰), ‖⋅‖) be a NORMED LINEAR SPACE (Definition D.4 page 151). The norm ‖⋅‖ is CONTINUOUS in 𝞨. 9

📃

📘 Aliprantis and Burkinshaw (1998), pages 217–218, 📘 Banach (1932a), page 53, 📘 Banach (1932b), page 33, Banach (1922) page 135 10 📘 Giles (1987) page 3 11 📘 Michel and Herget (1993), page 344, 📘 Banach (1932a) page 53 12 📘 Giles (2000) page 1 ⟨1.1 Definition⟩ 13 📘 Giles (2000) page 2

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Theorem D.4 (next) demonstrates that all open or closed balls in any normed linear space are convex. However, the converse is not true—that is, a metric not generated by a norm may still produce a ball that is convex. Theorem D.4. 14 Let (𝑋, +, ⋅, (𝔽 , ∔, ⨰), 𝖽) be a METRIC LINEAR SPACE (Definition D.3 page 150). Let 𝖡 be an OPEN BALL (Definition B.4 page 128). and ∃ ‖⋅‖ ∈ ℝ𝙓 such that ⎫ ⎧ 1. 𝖡(𝒙, 𝑟) = 𝒙 + 𝐵(0, 𝑟) ⎪ 2. 𝖡(𝟘, 𝑟) = 𝑟 𝖡(𝟘, 1) and ⎪ 𝒚) = ‖𝒚 − 𝒙‖ 𝖽(𝒙, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⎬ ⟹ ⎨ 3. 𝖡(𝒙, 𝑟) is CONVEX and ⎪ ⎪ 𝖽 is generated by a norm ⎭ 4. 𝒙 ∈ 𝖡(𝟘, 𝑟) ⟺ −𝒙 ∈ 𝖡(𝟘, 𝑟) ( SYMMETRIC) ⎩ ✎PROOF: 1. Proof that 𝖽(𝒙 + 𝒛, 𝒚 + 𝑣𝑧) = 𝖽(𝒙, 𝒚) (invariant): 𝖽(𝒙 + 𝒛, 𝒚 + 𝑣𝑧) = ‖(𝒚 + 𝑣𝑧) − (𝒙 + 𝒛)‖

by left hypothesis

= ‖𝒚 − 𝒙‖ = 𝖽(𝒙, 𝒚)

by left hypothesis

2. Proof that 𝖡(𝒙, 𝑟) = 𝒙 + 𝐵(0, 𝑟): 𝖡(𝒙, 𝑟) = {𝒚 ∈ 𝙓 |𝖽(𝒙, 𝒚) < 𝑟}

by definition of open ball 𝐵

= {𝒚 ∈ 𝙓 |𝖽(𝒚 − 𝒚, 𝒚 − 𝒙) < 𝑟}

by right result 1.

= {𝒚 ∈ 𝙓 |𝖽(𝟘, 𝒚 − 𝒙) < 𝑟} = {𝒖 + 𝒙 ∈ 𝙓 |𝖽(𝟘, 𝒖) < 𝑟}

let 𝒖 ≜ 𝒚 − 𝒙

= 𝒙 + {𝒖 ∈ 𝙓 |𝖽(𝟘, 𝒖) < 𝑟} = 𝒙 + 𝐵(0, 𝑟)


3. Proof that 𝖡(𝟘, 𝑟) = 𝑟 𝖡(𝟘, 1): 𝖡(𝟘, 𝑟) = {𝒚 ∈ 𝙓 |𝖽(𝟘, 𝒚) < 𝑟} 1 = {𝒚 ∈ 𝙓 | 𝖽(𝟘, 𝒚) < 1} 𝑟 1 = {𝒚 ∈ 𝙓 | ‖𝒚 − 𝟘‖ < 1} 𝑟 1 1 = {𝒚 ∈ 𝙓 | ‖ 𝒚 − 𝟘‖ < 1} 𝑟 𝑟 1 1 = {𝒚 ∈ 𝙓 |𝖽( 𝟘, 𝒚) < 1} 𝑟 𝑟 = {𝑟𝒖 ∈ 𝙓 |𝖽(𝟘, 𝒖) < 1}


by left hypothesis by homogeneous property of ‖⋅‖ page 151 by left hypothesis 1 let 𝒖 ≜ 𝒚 𝑟

= 𝑟 {𝒖 ∈ 𝙓 |𝖽(𝟘, 𝒖) < 1} = 𝑟 𝐵(𝟘, 1)


4. Proof that 𝖡(𝒑, 𝑟) is convex: We must prove that for any pair of points 𝒙 and 𝒚 in the open ball 𝖡(𝒑, 𝑟), any point 𝜆𝒙 + (1 − 𝜆)𝒚 is also in the open ball. That is, the distance from any point 𝜆𝒙 + (1 − 𝜆)𝒚 to the ball's center 𝒑 must be less 14

📘 Giles (2000) page 2 ⟨1.2 Remarks⟩, 📘 Giles (1987) pages 22–26 ⟨2.4 Theorem, 2.11 Theorem⟩

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than 𝑟. 𝖽(𝒑, 𝜆𝒙 + (1 − 𝜆)𝒚) = ‖𝒑 − 𝜆𝒙 − (1 − 𝜆)𝒚‖ ‖ ‖ ‖ ‖ = ‖⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝜆𝒑 + (1 − 𝜆)𝒑 − 𝜆𝒙 − (1 − 𝜆)𝒚‖ ‖ ‖ ‖ ‖ 𝒑 = ‖𝜆𝒑 − 𝜆𝒙 + (1 − 𝜆)𝒑 − (1 − 𝜆)𝒚‖

by left hypothesis

≤ ‖𝜆𝒑 − 𝜆𝒙‖ + ‖(1 − 𝜆)𝒑 − (1 − 𝜆)𝒚‖

by subadditivity property of ‖⋅‖ page 151

= |𝜆| ‖𝒑 − 𝒙‖ + |1 − 𝜆| ‖𝒑 − 𝒚‖

by homogeneous property of ‖⋅‖ page 151

= 𝜆 ‖𝒑 − 𝒙‖ + (1 − 𝜆) ‖𝒑 − 𝒚‖

because 0 ≤ 𝜆 ≤ 1

≤ 𝜆𝑟 + (1 − 𝜆)𝑟

because 𝒙, 𝒚 are in the ball 𝖡(𝒑, 𝑟)

=𝑟 5. Proof that 𝒙 ∈ 𝖡(𝟘, 𝑟) ⟺ −𝒙 ∈ 𝖡(𝟘, 𝑟) (symmetric): 𝒙 ∈ 𝖡(𝟘, 𝑟) ⟺ 𝒙 ∈ {𝒚 ∈ 𝙓 |𝖽(𝟘, 𝒚) < 𝑟} ⟺ 𝒙 ∈ {𝒚 ∈ 𝙓 | ‖𝒚 − 𝟘‖ < 𝑟}

by definition of open ball 𝐵 by left hypothesis

⟺ 𝒙 ∈ {𝒚 ∈ 𝙓 | ‖𝒚‖ < 𝑟} ⟺ 𝒙 ∈ {𝒚 ∈ 𝙓 | ‖(−1)(−𝒚)‖ < 𝑟} ⟺ 𝒙 ∈ {𝒚 ∈ 𝙓 ||−1| ‖−𝒚‖ < 𝑟}

by homogeneous property of ‖⋅‖ page 151

⟺ 𝒙 ∈ {𝒚 ∈ 𝙓 | ‖−𝒚 − 𝟘‖ < 𝑟} ⟺ 𝒙 ∈ {𝒚 ∈ 𝙓 |𝖽(𝟘, −𝒚) < 𝑟}

by left hypothesis

⟺ 𝒙 ∈ {−𝒖 ∈ 𝙓 |𝖽(𝟘, 𝒖) < 𝑟}

let 𝒖 ≜ −𝒚

⟺ 𝒙 ∈ (− {𝒖 ∈ 𝙓 |𝖽(𝟘, 𝒖) < 𝑟}) ⟺ 𝒙 ∈ (−𝖡(𝟘, 𝑟)) ⟺ −𝒙 ∈ 𝖡(𝟘, 𝑟)

✏ Theorem D.4 (page 152) demonstrates that if a metric 𝖽 in a metric space (𝑋, +, ⋅, (𝔽 , ∔, ⨰), 𝖽) is generated by a norm, then the ball 𝖡(𝑥, 𝑟) in that metric linear space is convex. However, the converse is not true. That is, it is possible for the balls in a metric space (𝙔 , 𝗉) to be convex, but yet the metric 𝗉 not be generated by a norm. Norms generated by metrics Every normed linear space is also a metric linear space (Theorem D.3 page 151). However, the converse is not true—not every metric linear space is a normed linear space. A characterization of metric linear spaces that are normed linear spaces is provided by Theorem D.5 (page 154). Lemma D.1.

15

Let (𝑋, +, ⋅, (𝔽 , ∔, ⨰), 𝖽) be a METRIC LINEAR SPACE. Let ‖𝒙‖ ≜ 𝖽(𝒙, 𝟘) ∀𝒙 ∈ 𝑋. 1. ∀𝒙∈𝑋 and ‖𝒙‖ = ‖−𝒙‖ ⎧ ⎪ 2. = 0 ⟺ 𝒙 = 0 ∀𝒙∈𝑋 and ‖𝒙‖ 𝖽(𝒙 + 𝒛, 𝒚 + 𝒛) = 𝖽(𝒙, 𝒚) ∀𝒙,𝒚,𝒛∈𝑋 ⟹ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⎨ ⎪ ∀𝒙,𝒚∈𝑋 ⎩ 3. ‖𝒙 + 𝒚‖ ≤ ‖𝒙‖ + ‖𝒚‖ TRANSLATION INVARIANT

✎PROOF: 15

📃

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1. Proof that ‖𝒙‖ = ‖−𝒙‖: ‖𝒙‖ = 𝖽(𝒙, 𝟘)

by definition of ‖⋅‖

= 𝖽(𝒙 − 𝒙, 𝟘 − 𝒙)


= 𝖽(𝟘, −𝒙) = ‖−𝒙‖


2a. Proof that ‖𝒙‖ = 0 ⟹ 𝒙 = 0: 0 = ‖𝒙‖

by left hypothesis

= 𝖽(𝒙, 𝟘)


= 𝖽(𝒙, 𝟘)


⟹ 𝒙=𝟘

by property of metrics

2b. Proof that ‖𝒙‖ = 0 ⟸ 𝒙 = 0: ‖𝒙‖ = 𝖽(𝒙, 𝟘)


= 𝖽(𝟘, 𝟘)

by right hypothesis

=0


3. Proof that ‖𝒙 + 𝒚‖ ≤ ‖𝒙‖ + ‖𝒚‖: ‖𝒙 + 𝒚‖ = 𝖽(𝒙 + 𝒚, 𝟘) = 𝖽(𝒙 + 𝒚 − 𝒚, 𝟘 − 𝒚)

by definition of ‖⋅‖ by translation invariance hypothesis

= 𝖽(𝒙, −𝒚) ≤ 𝖽(𝒙, 𝟘) + 𝖽(𝟘, 𝒚)


= 𝖽(𝒙, 𝟘) + 𝖽(𝒚, 𝟘)


= ‖𝒙‖ + ‖𝒚‖


✏ Theorem D.5. 16 Let (𝑋, +, ⋅, (𝔽 , ∔, ⨰)) be a LINEAR SPACE. Let 𝖽(𝒙, 𝒚) ≜ ‖𝒙 − 𝒚‖ ∀𝒙, 𝒚 ∈ 𝑋. 1. 𝖽(𝒙 + 𝒛, 𝒚 + 𝒛) = 𝖽(𝒙, 𝒚) ∀𝒙,𝒚,𝒛∈𝑋 ( TRANSLATION INVARIANT) and ⟺ ‖⋅‖ is a NORM } 2. 𝖽(𝛼𝒙, 𝛼𝒚) = |𝛼|𝖽(𝒙, 𝒚) ∀𝒙,𝒚∈𝑋, 𝛼∈𝔽 ( HOMOGENEOUS) ✎PROOF: 1. Proof of ⟹ assertion: (a) Proof that ‖⋅‖ is strictly positive: This follows directly from the definition of 𝖽. (b) Proof that ‖⋅‖ is nondegenerate: This follows directly from Lemma D.1 (page 153). (c) Proof that ‖⋅‖ is homogeneous: This follows from the second left hypothesis. (d) Proof that ‖⋅‖ satisfies the triangle-inequality: This follows directly from Lemma D.1 (page 153). 2. Proof of ⟸ assertion: 𝖽(𝒙 + 𝒛, 𝒚 + 𝒛) = ‖(𝒙 + 𝒛) − (𝒚 + 𝒛)‖


= ‖𝒙 − 𝒚‖ = 𝖽(𝒙, 𝒚) 𝖽(𝛼𝒙, 𝛼𝒚) = ‖(𝛼𝒙) − (𝛼𝒚)‖

by definition of 𝖽 by definition of 𝖽

= ‖𝛼(𝒙 − 𝒚)‖ = |𝛼| ‖𝒙 − 𝒚‖

by definition of ‖⋅‖ page 151

= |𝛼|𝖽(𝒙, 𝒚)


✏ 16

📘 Bollobás (1999), page 21

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D.2 Metric spaces D.2.1 Algebraic structure Definition D.6. 17 A function 𝖽 ∈ ℝ⊢ 𝑋×𝑋 (Definition 1.6 page 6) is a quasi-metric on a SET 𝑋 if 1. 𝖽(𝑥, 𝑦) ≥ 0 ∀𝑥,𝑦∈𝑋 ( NON-NEGATIVE) and 2. 𝖽(𝑥, 𝑦) = 0 ⟺ 𝑥 = 𝑦 ∀𝑥,𝑦∈𝑋 ( NONDEGENERATE) and 3. 𝖽(𝑥, 𝑦) ≤ 𝖽(𝑥, 𝑧) + 𝖽(𝑧, 𝑦) ∀𝑥,𝑦,𝑧∈𝑋 ( SUBADDITIVE/ ORIENTED TRIANGLE INEQUALITY). The pair (𝑋, 𝖽) is a quasi-metric space if 𝖽 is a QUASI-METRIC on 𝑋. A QUASI-METRIC is also called an asymmetric metric and a directed metric. Definition D.7. 18 Let 𝑋 be a set and ℝ⊢ the set of non-negative real numbers. A function 𝖽 ∈ ℝ⊢ 𝑋×𝑋 is a metric on 𝑋 if 1. 𝖽(𝑥, 𝑦) ≥ 0 ∀𝑥,𝑦∈𝑋 ( NON-NEGATIVE) 2. 𝖽(𝑥, 𝑦) = 0 ⟺ 𝑥 = 𝑦 ∀𝑥,𝑦∈𝑋 ( NONDEGENERATE) 3. 𝖽(𝑥, 𝑦) = 𝖽(𝑦, 𝑥) ∀𝑥,𝑦∈𝑋 ( SYMMETRIC) 4. 𝖽(𝑥, 𝑦) ≤ 𝖽(𝑥, 𝑧) + 𝖽(𝑧, 𝑦) ∀𝑥,𝑦,𝑧∈𝑋 ( SUBADDITIVE/ TRIANGLE INEQUALITY).19 A metric space is the pair (𝑋, 𝖽).

and and and

Actually, it is possible to significantly simplify the definition of a metric to an equivalent statement requiring only half as many conditions. These equivalent conditions (a “characterization”) are stated in Theorem D.6 (next). Theorem D.6 (metric characterization). 𝖽(𝑥, 𝑦) is a metric

⟺

1.

{

2.

20

Let 𝖽 be a function in (ℝ⊢ )𝑋×𝑋 . 𝖽(𝑥, 𝑦) = 0 ⟺ 𝑥 = 𝑦 ∀𝑥,𝑦∈𝑋 𝖽(𝑥, 𝑦) ≤ 𝖽(𝑧, 𝑥) + 𝖽(𝑧, 𝑦) ∀𝑥,𝑦,𝑧∈𝑋

and

✎PROOF:

1. Proof that [𝖽(𝑥, 𝑦) is a metric] ⟹ [(1) and (2)]:

1a. Proof that 𝖽(𝑥, 𝑦) = 0 ⟺ 𝑥 = 𝑦: by left hypothesis 2 (𝖽(𝑥, 𝑦) is nondegenerate)

1b. Proof that 𝖽(𝑥, 𝑦) ≤ 𝖽(𝑧, 𝑥) + 𝖽(𝑧, 𝑦): 𝖽(𝑥, 𝑦) ≤ 𝖽(𝑥, 𝑧) + 𝖽(𝑧, 𝑦) = 𝖽(𝑧, 𝑥) + 𝖽(𝑧, 𝑦)

by right hypothesis 4 (triangle inequality) by right hypothesis 3 (commutative)

2. Proof that [𝖽(𝑥, 𝑦) is a metric] ⟸ [(1) and (2)]: 17

📘 Deza and Deza (2014) pages 6–7, 📘 Deza and Deza (2006) page 4, 📃 Wilson (1931a) page 675 ⟨§1.⟩, 📃 Ribeiro (1943), 📃 Kelly (1963) page 71 ⟨Introduction⟩, 📃 Patty (1967), 📃 Stoltenberg (1969) page 65, 📘 Grabiec et al. (2006) page 3 ⟨Introduction⟩, 📘 Euclid (circa 300BC) ⟨triangle inequality—Book I Proposition 20⟩ 18 📘 Dieudonné (1969), page 28, 📘 Copson (1968), page 21, 📘 Hausdorff (1937) page 109, 📘 Fréchet (1928), 📘 Fréchet (1906) page 30 19 📘 Euclid (circa 300BC) ⟨Book I Proposition 20⟩ 20 📘 Busemann (1955) page 3, 📘 Michel and Herget (1993), page 264, 📘 Giles (1987), page 18 2016 Jul 4

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2a. Proof that 𝖽(𝑥, 𝑦) ≥ 0: 1 ⋅0 2 1 = 𝖽(𝑦, 𝑦) 2 1 = 𝖽(𝑦, 𝑧)| 2 𝑧=𝑦 1 ≤ [𝖽(𝑥, 𝑦) + 𝖽(𝑥, 𝑧)]𝑧=𝑦 2 1 = [𝖽(𝑥, 𝑦) + 𝖽(𝑥, 𝑦)] 2 = 𝖽(𝑥, 𝑦)

0=

by right hypothesis 1


2b. Proof that 𝖽(𝑥, 𝑦) = 0 ⟺ 𝑥 = 𝑦: by right hypothesis 1 2c. Proof that 𝖽(𝑥, 𝑦) = 𝖽(𝑦, 𝑥): 𝖽(𝑥, 𝑦)|𝑧=𝑦 ≤ [𝖽(𝑧, 𝑥) + 𝖽(𝑧, 𝑦)]𝑧=𝑦 :0 = 𝖽(𝑦, 𝑥) + 𝖽(𝑦, 𝑦)


= 𝖽(𝑦, 𝑥)


𝖽(𝑦, 𝑥)|𝑧=𝑥 ≤ [𝖽(𝑧, 𝑦) + 𝖽(𝑧, 𝑥)]𝑧=𝑥 :0 = 𝖽(𝑥, 𝑦) + 𝖽(𝑥, 𝑥)


= 𝖽(𝑥, 𝑦)


2d. Proof that 𝖽(𝑥, 𝑦) ≤ 𝖽(𝑥, 𝑧) + 𝖽(𝑧, 𝑦): 𝖽(𝑥, 𝑦) ≤ 𝖽(𝑧, 𝑥) + 𝖽(𝑧, 𝑦) = 𝖽(𝑥, 𝑧) + 𝖽(𝑧, 𝑦)

by right hypothesis 2 by result 2c

✏ The triangle inequality property stated in the definition of metrics (Definition D.7 page 155) axiomatically endows a metric with an upper bound. Lemma D.2 (next) demonstrates that there is a complementary lower bound similar in form to the triangle-inequality upper bound. Lemma D.2. 21 Let (𝑋, 𝖽) be a METRIC SPACE (Definition D.7 page 155). Let |⋅| be the ABSOLUTE VALUE function (Definition 1.24 page 9). 1. |𝖽(𝑥, 𝑝) − 𝖽(𝑝, 𝑦)| ≤ 𝖽(𝑥, 𝑦) ∀𝑥,𝑦,𝑝∈𝑋 2. 𝖽(𝑥, 𝑝) − 𝖽(𝑝, 𝑦) ≤ 𝖽(𝑥, 𝑦) ∀𝑥,𝑦,𝑝∈𝑋 ✎PROOF: 1. Proof that |𝖽(𝑥, 𝑝) − 𝖽(𝑝, 𝑦)| ≤ 𝖽(𝑥, 𝑦): |𝖽(𝑥, 𝑝) − 𝖽(𝑝, 𝑦)| ≤ |𝖽(𝑥, 𝑦) + 𝖽(𝑦, 𝑝) − 𝖽(𝑝, 𝑦)| by subadditive property (Definition D.7 page 155) = |𝖽(𝑥, 𝑦) + 𝖽(𝑝, 𝑦) − 𝖽(𝑝, 𝑦)| by symmetry property of metrics (Definition D.7 page 155) = |𝖽(𝑥, 𝑦) + 0| = 𝖽(𝑥, 𝑦)

by non-negative property of metrics (Definition D.7 page 155)

2. Proof that 𝖽(𝑥, 𝑝) ≥ 𝖽(𝑝, 𝑦) ⟹ 𝖽(𝑥, 𝑝) − 𝖽(𝑝, 𝑦) ≤ 𝖽(𝑥, 𝑦): 𝖽(𝑥, 𝑝) − 𝖽(𝑝, 𝑦) = |𝖽(𝑥, 𝑝) − 𝖽(𝑝, 𝑦)| ≤ 𝖽(𝑥, 𝑦) 21

by left hypothesis and definition of |⋅| by item (1)

📘 Dieudonné (1969), page 28, 📘 Michel and Herget (1993), page 266

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3. Proof that 𝖽(𝑥, 𝑝) ≤ 𝖽(𝑝, 𝑦) ⟹ 𝖽(𝑥, 𝑝) − 𝖽(𝑝, 𝑦) ≤ 𝖽(𝑥, 𝑦): |𝖽(𝑥, 𝑝) − 𝖽(𝑝, 𝑦)| ≤ 0

by left hypothesis

≤ 𝖽(𝑥, 𝑦)

by non-negative property of metrics (Definition D.7 page 155)

✏ The triangle inequality property stated in the definition of metrics tended from two to any finite number of metrics (next).

(Definition D.7 page 155)

can be ex-

Proposition D.1. 22 Let (𝑋, 𝖽) be a METRIC SPACE (Definition D.7 page 155) and ⦇𝑥𝑛 ∈ 𝑋⦈𝘕1 an 𝘕 -TUPLE (Definition 1.11 page 6) on 𝑋. 𝘕 −1

𝖽(𝑥1 , 𝑥𝘕 ) ≤

𝖽 𝑥 ,𝑥 ∑ ( 𝑛 𝑛+1 )

∀𝘕 ∈ ℕ⧵1

𝑛=1

✎PROOF:

Proof by induction:

Proof that the {𝘕 = 2 case} is true: 2−1

𝖽(𝑥1 , 𝑥2 ) ≤

𝖽 𝑥 ,𝑥 ∑ ( 𝑛 𝑛+1 ) 𝑛=1

Proof for that the {𝘕 case} ⟹ {𝘕 + 1 case}: 𝖽(𝑥1 , 𝑥𝘕 +1 ) ≤ 𝖽(𝑥1 , 𝑥𝘕 ) + 𝖽(𝑥𝘕 , 𝑥𝘕 +1 )

by subadditive property (Definition D.7 page 155)

⎛𝘕 −1 ⎞ ≤⎜ 𝖽 (𝑥𝘕 , 𝑥𝘕 +1 )⎟ + 𝖽(𝑥𝘕 , 𝑥𝘕 +1 ) ⎜∑ ⎟ ⎝ 𝑛=1 ⎠

by {𝘕 case} hypothesis

𝘕

=

𝖽 𝑥 ,𝑥 ∑ ( 𝑛 𝑛+1 ) 𝑛=1

✏ 23

Let 𝑋 be a set and 𝖽 ∈ ℝ𝑋×𝑋 . The function 𝖽 is the discrete metric on ℝ𝑋×𝑋 if 0 if 𝑥 = 𝑦 𝖽(𝑥, 𝑦) ≜ ∀𝑥,𝑦∈𝑋 { 1 otherwise }

Definition D.8.

Definition D.9 (usual metric). The function 𝖽(𝑥, 𝑦) ≜ |𝑥 − 𝑦|

24

Let |⋅| ∈ ℝ⊢ 𝑅 be an ABSOLUTE VALUE function on a RING 𝑅. is a METRIC on ℝ, called the usual metric. 𝘕

Definition D.10. Let 𝑋 be a set and 𝖽 ∈ ℝℝ . The function 𝖽 is the Euclidean metric on ℝ𝘕 if √𝘕 √ 𝖽((𝑥1 , 𝑥2 , ⋯ , 𝑥𝘕 ) , (𝑦1 , 𝑦2 , ⋯ , 𝑦𝘕 )) ≜ √ (𝑥𝑛 − 𝑦𝑛 )2 ∀(𝑥1 ,𝑥2 ,⋯,𝑥𝘕 ), (𝑦1 ,𝑦2 ,⋯,𝑦𝘕 )∈ℝ𝘕 ∑ ⎷ 𝑛=1 22

📘 Dieudonné (1969), page 28 📘 Rosenlicht (1968) page 37 23 📘 Busemann (1955) page 4 ⟨COMMENTS ON THE AXIOMS⟩, 📘 Giles (1987), page 13, 📘 Copson (1968), page 24, 📘 Khamsi and Kirk (2001) page 19 ⟨Example 2.1⟩ 24 📘 Davis (2005) page 16 2016 Jul 4

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D.2. METRIC SPACES

D.2.2 Metric preserving functions ⊢

Definition D.11. 25 Let 𝕄 be the set of all METRIC SPACEs (Definition D.7 page 155) on a set 𝑋. 𝜙 ∈ ℝ⊢ ℝ is a metric preserving function if 𝖽(𝑥, 𝑦) ≜ 𝜙 ∘ 𝗉 (𝑥, 𝑦) is a METRIC on 𝑋 for all (𝑋, 𝗉) ∈ 𝕄 Theorem D.7 (necessary conditions). 𝜙 is a ⎧ ⎪ ⎨ METRIC PRESERVING FUNCTION ⎪ ⎩ (Definition D.11 page 158)

26

Let 𝓡𝜙 be the RANGE of a function 𝜙. −1 1. 𝜙 (0) = {0} ⎫ ⎧ ⎪ ⎪ ⊢ ⟹ ⎬ ⎨ 2. 𝓡𝜙 ⊆ ℝ ⎪ ⎪ ⎩ 3. 𝜙(𝑥 + 𝑦) ≤ 𝜙(𝑥) + 𝜙(𝑦) ⎭

and and (𝜙 is SUBADDITIVE)

⎫ ⎪ ⎬ ⎪ ⎭

✎PROOF: 1. Proof that 𝜙 is a metric preserving function ⟹ 𝜙−1 (0) = {0}: (a) Suppose that the statement is not true and 𝜙−1 (0) = {0, 𝑎}. (b) Then 𝜙(𝑎) = 0 and for some 𝑥, 𝑦 such that 𝑥 ≠ 𝑦 and 𝖽(𝑥, 𝑦) = 𝑎 we have 𝜙 ∘ 𝖽(𝑥, 𝑦) = 𝜙(𝑎) =0 ⟹ 𝜙 ∘ 𝖽 is not a metric ⟹ 𝜙 is not a metric preserving function (c) But this contradicts the original hypothesis, and so it must be that 𝜙−1 (0) = {0}. 2. Proof that 𝓡𝜙 ⊆ ℝ⊢ : 𝓡𝜙 ∘ 𝖽 ⊆ 𝓡𝖽 ⊆ ℝ⊢ 3. Proof that 𝜙 is a metric preserving function ⟹ 𝜙 is subadditive: (a) For 𝜙 to be a metric preserving function, by definition it must work with all metric spaces. (b) So to develop necessary conditions, we can pick any metric space we want (because it is necessary that 𝜙 preserves it as a metric space). (c) For this proof we choose the metric space (ℝ, 𝖽) where 𝖽(𝑥, 𝑦) ≜ |𝑥 − 𝑦| for all 𝑥, 𝑦 ∈ ℝ⊢ : 𝜙(𝑥) + 𝜙(𝑦) = 𝜙(|(𝑥 + 𝑦) − 𝑥|) + 𝜙(|𝑥 − 0|)

by definition of |⋅|

= (𝜙 ∘ 𝖽)(𝑥 + 𝑦, 𝑥) + (𝜙 ∘ 𝖽)(𝑥, 0)


≥ (𝜙 ∘ 𝖽)(𝑥 + 𝑦, 0)

by left hypothesis and Definition D.7 page 155

= 𝜙(|(𝑥 + 𝑦) − 0|)


= 𝜙(𝑥 + 𝑦)

because 𝑥, 𝑦 ∈ ℝ⊢

✏ Theorem D.8 (next theorem) presents some sufficient conditions for a function to be metric preserving. Theorem D.8 (sufficient conditions). 27 Let 𝜙 be a function in ℝℝ . 1. 𝑥 ≥ 𝑦 ⟹ 𝜙(𝑥) ≥ 𝜙(𝑦) ∀𝑥,𝑦∈ℝ⊢ ( ISOTONE) and ⎫ 𝜙 is a METRIC ⎧ ⎧ ⎪ ⎪ ⎪ 2. 𝜙(0) = 0 and ⟹ ⎨ ⎬ ⎨ PRESERVING FUNCTION ⊢ ⎪ ⎪ ⎪ ⎩ 3. 𝜙(𝑥 + 𝑦) ≤ 𝜙(𝑥) + 𝜙(𝑦) ∀𝑥,𝑦∈ℝ ( SUBADDITIVE) ⎩ (Definition D.11 page 158). ⎭

⎫ ⎪ ⎬ ⎪ ⎭

25

📘 Vallin (1999), page 849 ⟨Definition 1.1⟩, 📘 Corazza (1999), page 309, 📘 Deza and Deza (2009) page 80 📘 Corazza (1999), page 310 ⟨Proposition 2.1⟩, 📘 Deza and Deza (2009) page 80 27 📘 Corazza (1999) ⟨Proposition 2.3⟩, 📘 Deza and Deza (2009) page 80, 📘 Kelley (1955) page 131 ⟨Problem C⟩ 26

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✎PROOF: 1. Proof that 𝜙 ∘ 𝖽(𝑥, 𝑦) = 0 ⟹ 𝑥 = 𝑦: 𝜙 ∘ 𝖽(𝑥, 𝑦) = 0 ⟹ 𝖽(𝑥, 𝑦) = 0

by 𝜙 hypothesis 2

⟹ 𝑥=𝑦

by nondegenerate property page 155

2. Proof that 𝜙 ∘ 𝖽(𝑥, 𝑦) = 0 ⟸ 𝑥 = 𝑦: 𝜙 ∘ 𝖽(𝑥, 𝑦) = 𝜙 ∘ 𝖽(𝑥, 𝑥)

by 𝑥 = 𝑦 hypothesis

= 𝜙(0)

by nondegenerate property page 155

=0


3. Proof that 𝜙 ∘ 𝖽(𝑥, 𝑦) ≤ 𝜙 ∘ 𝖽(𝑧, 𝑥) + 𝜙 ∘ 𝖽(𝑧, 𝑦): 𝜙 ∘ 𝖽(𝑥, 𝑦) ≤ 𝜙(𝖽(𝑥, 𝑧) + 𝖽(𝑧, 𝑦))

by 𝜙 hypothesis 1 and triangle inequality page 155

≤ 𝜙(𝖽(𝑧, 𝑥) + 𝖽(𝑧, 𝑦))

by symmetric property of 𝖽 page 155

≤ 𝜙 ∘ 𝖽(𝑧, 𝑥) + 𝜙 ∘ 𝖽(𝑧, 𝑦)


✏ 2

2 𝛼 = 12

1

𝛼 = 12

1 𝑥

0

2

0 1 2 3 4 (A) 𝛼-scaled/dilated

𝑥 𝑥 0 0 1 2 3 4 0 1 2 3 4 (B) power transform/snowflake (C) 𝛼-truncated/radar screen 0

(Example D.3 page 159)



2

2

2

1

1

1

𝑥

0 0

𝛼=1

1

0

1 2 3 4 (D) bounded

0


𝑥 b

1 2 3 (E) discrete

0

4

0


1

2 (F)

3

4


Figure D.1: metric preserving functions Example D.3 (𝛼-scaled metric/dilated metric). 28 Let (𝑋, 𝖽) be a metric space (Definition D.7 page 155). 𝜙(𝑥) ≜ 𝛼𝑥, 𝛼 ∈ ℝ+ is a metric preserving function (Figure D.1 page 159 (A)) ✎PROOF:

The proofs for Example D.3–Example D.8 (page 160) follow from Theorem D.8 (page 158).

✏

Example D.4 (power transform metric/snowflake transform metric). 29 Let (𝑋, 𝖽) be a metric space 𝛼 (Definition D.7 page 155). 𝜙(𝑥) ≜ 𝑥 , 𝛼 ∈ (0 ∶ 1], is a metric preserving function (see Figure D.1 page 159 (B)) Example D.5 (𝛼-truncated metric/radar screen metric). 30 Let (𝑋, 𝖽) be a metric space (Definition D.7 page 155). 𝜙(𝑥) ≜ min {𝛼, 𝑥}, 𝛼 ∈ ℝ+ is a metric preserving function (see Figure D.1 page 159 (C)). ✎PROOF:

✏

by Theorem D.8 (page 158).

28

📘 Deza and Deza (2006) page 44 📘 Deza and Deza (2009) page 81, 📘 Deza and Deza (2006) page 45 30 📘 Giles (1987), page 33, 📘 Deza and Deza (2006) pages 242–243 29

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D.2. METRIC SPACES

Example D.6 (bounded metric). 31 Let (𝑋, 𝖽) be a metric space (Definition D.7 page 155). 𝑥 𝜙(𝑥) ≜ is a metric preserving function (see Figure D.1 page 159 (D)). 1+𝑥 ✎PROOF:

✏

by Theorem D.8 (page 158).

Example D.7 (discrete metric preserving function). 32 Let 𝜙 be a function in ℝℝ . 0 for 𝑥 ≤ 0 𝜙(𝑥) ≜ is a metric preserving function (see Figure D.1 page 159 (E)). { 1 otherwise } ✎PROOF:

✏

by Theorem D.8 page 158.

Example D.8. Let 𝜙 be a function in ℝℝ . 𝑥 for 0 ≤ 𝑥 < 1, 1 for 1 ≤ 𝑥 ≤ 2, 𝜙(𝑥) ≜ } { 𝑥 − 1 for 2 < 𝑥 < 3, 2 for 𝑥 ≥ 3 ✎PROOF:

is a metric preserving function (see Figure D.1 page 159 (F)). ✏

by Theorem D.8 page 158.

2 1 0

2 1 0 bc

b

0 1 2 3 4 5 6 7 (A) (Example D.9 page 160)

2 1 0

0 1 2 3 4 5 6 7 (B) (Example D.10 page 160)

0 1 2 3 4 5 6 7 (C) (Example D.11 page 160)

Figure D.2: non-monotone metric preserving functions Example D.9.

33

Let 𝜙 be a function in ℝℝ . 0 for 𝑥 = 0 𝜙(𝑥) ≜ 1 { 1 + 𝑥+1 for 𝑥 > 0 }

is a metric preserving function (see Figure D.2 page 160 (A)).

Example D.10. 34 Let 𝜙 be a function in ℝℝ . 𝑥 for 𝑥 ≤ 2 𝜙(𝑥) ≜ is a metric preserving function (see Figure D.2 page 160 (B)). 1 { 1 + 𝑥−1 for 𝑥 > 2 } Example D.11. Let 𝜙 be a function in ℝℝ . 𝑥 for 0 ≤ 𝑥 ≤ 2 ⎫ ⎧ ⎪ ⎪ 𝜙(𝑥) ≜ ⎨ −𝑥 + 4 for 2 < 𝑥 < 3 ⎬ is a metric preserving function (see Figure D.2 page 160 (C)). ⎪ ⎪ for 𝑥 ≥ 3 ⎩ 1 ⎭

D.2.3 Product metrics Theorem D.9 (Fréchet product metric). 35 Let 𝑋 be a set. 𝘕 ⎧ ⎧ 1. ⦇𝗉𝑛 ⦈ are METRICS on 𝑋 and ⎫ ⎪ 𝖽(𝑥, 𝑦) = ⎪ 2. 𝛼 ≥ 0 ∀𝑛 = 1, 2, … , 𝘕 and ⎪ 𝛼 𝗉 (𝑥, 𝑦) 𝑛 ∑ 𝑛 𝑛 ⟹ ⎨ ⎬ ⎨ 𝑛=1 ⎪ 3. max {𝛼𝑛 |𝑛=1,2,…,𝘕 } > 0 ⎪ ⎪ is a METRIC on 𝑋 ⎩ ⎭ ⎩

⎫ ⎪ ⎬ ⎪ ⎭

31

📘 Vallin (1999), page 849, 📘 Aliprantis and Burkinshaw (1998) page 39 📘 Corazza (1999), page 311 33 📘 Greenhoe (2015), pages 10–11 ⟨Theorem 4.16⟩ 34 📘 Corazza (1999), page 309, 📘 Doboš (1998), page 25 ⟨Example 1⟩, 📘 Jůza (1956) 35 📘 Deza and Deza (2006) page 47, 📘 Deza and Deza (2009) page 84, 📘 Steen and Seebach (1978) pages 64–65 ⟨Example 37.7⟩, 📘 Isham (1999) page 10 32

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✎PROOF:

1. Proof that 𝑥 = 𝑦 ⟹ 𝖽(𝑥, 𝑦) = 0: 𝘕

𝖽(𝑥, 𝑦) =

∑

𝛼𝑛 𝗉𝑛 (𝑥, 𝑦)


𝛼𝑛 𝗉𝑛 (𝑥, 𝑥)

by left hypothesis

0

by nondegenerate property of metrics (Definition D.7 page 155)

𝑛=1 𝘕

=

∑ 𝑛=1 𝘕

=

∑ 𝑛=1

=0 2. Proof that 𝑥 = 𝑦 ⟸ 𝖽(𝑥, 𝑦) = 0: 0 = 𝖽(𝑥, 𝑦)

by right hypothesis

𝘕

=

∑



𝑛=1

⟹ 𝗉𝑛 (𝑥, 𝑦) = 0 ⟹𝑥=𝑦

∀𝑥, 𝑦 ∈ 𝙓

∀𝑥, 𝑦 ∈ 𝙓

by metric properties page 155 by non-degenerate property of metrics page 155

3. Proof that 𝖽(𝑥, 𝑦) ≤ 𝖽(𝑧, 𝑥) + 𝖽(𝑧, 𝑦): 𝘕

𝖽(𝑥, 𝑦) =

∑



𝛼𝑛 [𝗉𝑛 (𝑥, 𝑧) + 𝗉𝑛 (𝑧, 𝑦) ]

by subadditive property (Definition D.7 page 155)

𝛼𝑛 [𝗉𝑛 (𝑧, 𝑥) + 𝗉𝑛 (𝑧, 𝑦) ]

by symmetry property (Definition D.7 page 155)

𝑛=1 𝘕

≤

∑ 𝑛=1 𝘕

=

∑ 𝑛=1 𝘕

=

∑

𝘕

𝛼𝑛 𝗉𝑛 (𝑧, 𝑥) +

𝑛=1

= 𝖽(𝑧, 𝑥) + 𝖽(𝑧, 𝑦)

∑

𝛼𝑛 𝗉𝑛 (𝑧, 𝑦)

𝑛=1


✏ Theorem D.10 (Power mean metrics). Let 𝑋 be a set. Let ⦇𝑥𝑛 ∈ 𝑋⦈𝘕1 and ⦇𝑦𝑛 ∈ 𝑋⦈𝘕1 be 𝘕 -tuples on 1 𝑟 𝘕 ⎫ ⎧ ⎧ 1. 𝗉 is a METRIC on 𝑋 and ⎫ 𝑟 ⎪ ⎪ 𝖽(⦇𝑥𝑛 ⦈ , ⦇𝑦𝑛 ⦈) ≜ ⎪ ⎪ 𝘕 , 𝑟 ∈ ∶ ∞] 𝜆 𝗉 (𝑥 , 𝑦 ) [1 𝑛 𝑛 𝑋. ⎨ ⟹ ⎨ ∑ 𝑛 ⎬ ⎬ ( ) 𝑛=1 ⎪ 2. ∑ 𝜆𝑛 = 1 ⎪ ⎪ ⎪ ⎩ 𝑛=1 ⎭ ⎩ is a METRIC on 𝑋. ⎭ Moreover, if 𝑟 = ∞, then 𝖽(⦇𝑥𝑛 ⦈ , ⦇𝑦𝑛 ⦈) = max 𝗉 (𝑥𝑛 , 𝑦𝑛 ). 𝑛=1,…,𝘕 ✎PROOF: 2016 Jul 4

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version 0.50

page 162

D.2. METRIC SPACES

1. Proof that ⦇𝑥𝑛 ⦈ = ⦇𝑦𝑛 ⦈ ⟹ 𝖽(⦇𝑥𝑛 ⦈ , ⦇𝑦𝑛 ⦈) = 0 for 𝑟 ∈ [1 ∶ ∞): 1 𝑟

𝘕

𝖽(⦇𝑥𝑛 ⦈ , ⦇𝑦𝑛 ⦈) ≜

(∑ 𝑛=1

𝜆𝑛 𝗉𝑟 (𝑥𝑛 , 𝑦𝑛 ) ) 1 𝑟

𝘕

=

𝜆𝑛 𝗉𝑟 (𝑥𝑛 , 𝑥𝑛 ) ) 𝑛=1

by ⦇𝑥𝑛 ⦈ = ⦇𝑦𝑛 ⦈ hypothesis

(∑

1 𝑟

𝘕

=


0 (∑ 𝑛=1 )

because 𝗉 is nondegenerate

=0 2. Proof that ⦇𝑥𝑛 ⦈ = ⦇𝑦𝑛 ⦈ ⟸ 𝖽(⦇𝑥𝑛 ⦈ , ⦇𝑦𝑛 ⦈) = 0 for 𝑟 ∈ [1 ∶ ∞): 0 = 𝖽(⦇𝑥𝑛 ⦈ , ⦇𝑦𝑛 ⦈)

by 𝖽(⦇𝑥𝑛 ⦈ , ⦇𝑦𝑛 ⦈) = 0 hypothesis 1 𝑟

𝘕

≜

(∑ 𝑛=1

𝜆𝑛 𝗉𝑟 (𝑥𝑛 , 𝑦𝑛 ) )


1

⟹ (𝗉 (𝑥𝑛 , 𝑦𝑛 )) 𝑟 = 0 for 𝑛 = 1, 2, … , 𝘕 ⟹ ⦇𝑥𝑛 ⦈ = ⦇𝑦𝑛 ⦈

because 𝗉 is non-negative because 𝗉 is nondegenerate

3. Proof that 𝖽 satisfies the triangle inequality property for 𝑟 = 1: 1 𝑟

𝘕

𝖽(⦇𝑥𝑛 ⦈ , ⦇𝑦𝑛 ⦈) ≜

(∑ 𝑛=1

𝑟


𝜆𝑛 𝗉 (𝑥𝑛 , 𝑦𝑛 ) )

𝘕

=

∑

𝜆𝑛 𝗉 (𝑥𝑛 , 𝑦𝑛 )

by 𝑟 = 1 hypothesis

𝜆𝑛 [𝗉 (𝑧𝑛 , 𝑥𝑛 ) + 𝗉 (𝑧𝑛 , 𝑦𝑛 )]

by triangle inequality

𝑛=1 𝘕

≤

∑ 𝑛=1 𝘕

=

𝘕

∑

𝜆𝑛 𝗉 (𝑧𝑛 , 𝑥𝑛 ) +

𝑛=1

∑ 1 𝑟

𝘕

=

𝜆𝑛 𝗉 (𝑧𝑛 , 𝑦𝑛 )

𝑛=1

(∑ 𝑛=1

𝜆𝑛 𝗉𝑟 (𝑧𝑛 , 𝑥𝑛 )

)

1 𝑟

𝘕

+

(∑ 𝑛=1

𝜆𝑛 𝗉𝑟 (𝑧𝑛 , 𝑦𝑛 ) )

≜ 𝖽(⦇𝑧𝑛 ⦈ , ⦇𝑥𝑛 ⦈) + 𝖽(⦇𝑧𝑛 ⦈ , ⦇𝑦𝑛 ⦈)

by 𝑟 = 1 hypothesis by definition of 𝖽

4. Proof that 𝖽 satisfies the triangle inequality property for 𝑟 ∈ (1 ∶ ∞): 𝖽(⦇𝑥𝑛 ⦈ , ⦇𝑦𝑛 ⦈) 1 𝑟

𝘕

≜

(∑ 𝑛=1

𝜆𝑛 𝗉𝑟 (𝑥𝑛 , 𝑦𝑛 )


) 1 𝑟

𝘕

≤

(∑ 𝑛=1 𝘕

= version 0.50

𝜆𝑛 [𝗉(𝑧𝑛 , 𝑥𝑛 ) + 𝗉(𝑧𝑛 , 𝑦𝑛 )]𝑟 1 𝑟

1 𝑟

by subaddtive property (Definition D.7 page 155)

) 𝑟

1 𝑟

𝜆 𝗉(𝑧𝑛 , 𝑥𝑛 ) + 𝜆𝑛 𝗉(𝑧𝑛 , 𝑦𝑛 ) ]) [ 𝑛 (∑ 𝑛=1 A book concerning symbolic sequence processing

by subaddtive property (Definition D.7 page 155) Daniel J. Greenhoe

2016 Jul 4

10:12am UTC

page 163

APPENDIX D. SOME MATHEMATICAL TOOLS 𝘕

≤

𝑟

1 𝑟

𝘕

𝑟

1 𝑟

1 𝑟

𝜆 𝗉(𝑧𝑛 , 𝑥𝑛 ) + 𝜆 𝗉(𝑧𝑛 , 𝑦𝑛 ) [ 𝑛 ]) [ 𝑛 ]) (∑ (∑ 𝑛=1 𝑛=1 1 𝑟

𝘕

≤

1 𝑟

(∑ 𝑛=1

𝘕

by Minkowski's inequality

1 𝑟

𝜆𝑛 𝗉𝑟 (𝑧𝑛 , 𝑥𝑛 ) + 𝜆𝑛 𝗉𝑟 (𝑧𝑛 , 𝑦𝑛 ) ∑ ) ( 𝑛=1 )

≜ 𝖽(⦇𝑧𝑛 ⦈ , ⦇𝑥𝑛 ⦈) + 𝖽(⦇𝑧𝑛 ⦈ , ⦇𝑦𝑛 ⦈)


5. Proof for the 𝑟 = ∞ case: (a) Proof that 𝖽(⦇𝑥𝑛 ⦈ , ⦇𝑦𝑛 ⦈) = max ⦇𝑥𝑛 ⦈: by Theorem D.14 page 166 (b) Proof that ⦇𝑥𝑛 ⦈ = ⦇𝑦𝑛 ⦈ ⟹ 𝖽(⦇𝑥𝑛 ⦈ , ⦇𝑦𝑛 ⦈) = 0: 𝖽(⦇𝑥𝑛 ⦈ , ⦇𝑦𝑛 ⦈) ≜ max {𝗉 (𝑥𝑛 , 𝑦𝑛 ) |𝑛 = 1, 2, … , 𝘕 }


= max {𝗉 (𝑥𝑛 , 𝑥𝑛 ) |𝑛 = 1, 2, … , 𝘕 }

by ⦇𝑥𝑛 ⦈ = ⦇𝑦𝑛 ⦈ hypothesis

=0


(c) Proof that ⦇𝑥𝑛 ⦈ = ⦇𝑦𝑛 ⦈ ⟸ 𝖽(⦇𝑥𝑛 ⦈ , ⦇𝑦𝑛 ⦈) = 0: 0 = 𝖽(⦇𝑥𝑛 ⦈ , ⦇𝑦𝑛 ⦈)

by 𝖽(⦇𝑥𝑛 ⦈ , ⦇𝑦𝑛 ⦈) = 0 hypothesis

≜ max {𝗉 (𝑥𝑛 , 𝑦𝑛 ) |𝑛 = 1, 2, … , 𝘕 }


⟹ 𝗉 (𝑥𝑛 , 𝑦𝑛 ) = 0 for 𝑛 = 1, 2, … , 𝘕 ⟹ ⦇𝑥𝑛 ⦈ = ⦇𝑦𝑛 ⦈


(d) Proof that 𝖽 satisfies the triangle inequality property: 𝖽(⦇𝑥𝑛 ⦈ , ⦇𝑦𝑛 ⦈) ≜ max {𝗉 (𝑥𝑛 , 𝑦𝑛 ) |𝑛 = 1, 2, … , 𝘕 }


≤ max {𝗉 (𝑥𝑛 , 𝑧𝑛 ) + 𝗉 (𝑧𝑛 , 𝑦𝑛 ) |𝑛 = 1, 2, … , 𝘕 }

by subadditive property

≤ max {𝗉 (𝑥𝑛 , 𝑧𝑛 ) |𝑛 = 1, 2, … , 𝘕 } + max {𝗉 (𝑧𝑛 , 𝑦𝑛 ) |𝑛 = 1, 2, … , 𝘕 } by non-negative property = max {𝗉 (𝑧𝑛 , 𝑥𝑛 ) |𝑛 = 1, 2, … , 𝘕 } + max {𝗉 (𝑧𝑛 , 𝑦𝑛 ) |𝑛 = 1, 2, … , 𝘕 } by symmetry property ≜ 𝖽(⦇𝑧𝑛 ⦈ , ⦇𝑥𝑛 ⦈) + 𝖽(⦇𝑧𝑛 ⦈ , ⦇𝑦𝑛 ⦈)


6. And so by Theorem D.6 (page 155), 𝖽 is a metric for 𝑟 ∈ [1 ∶ ∞].

✏

D.3 Sums ❝I think that it was Harald Bohr who remarked to me that “all analysts spend half

their time hunting through the literature for inequalities which they want to use and cannot prove.” ❞ G.H. Hardy (1877–1947) in his “Presidential Address” to the London Mathematical Society on November 8, 1928, about a remark that he thought to be from Harald Bohr (1887–1951), Danish mathematician and pictured to the left. 36

36

quote: image:

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D.3. SUMS

D.3.1 Summation Definition D.12. 37 Let + be an addition operator on a tuple ⦇𝑥𝑛 ⦈𝘕𝑚 . The summation of ⦇𝑥𝑛 ⦈ from index 𝑚 to index 𝑁 with respect to + is for 𝘕 < 𝑚 𝘕 ⎧ 0 ⎪ 𝘕 −1 𝑥 ≜ ∑ 𝑛 ⎨ for 𝘕 ≥ 𝑚 𝑛=𝑚 ⎪ (∑ 𝑥𝑛 ) + 𝑥𝘕 ⎩ 𝑛=𝑚 Theorem D.11 (Generalized associative property). + is ASSOCIATIVE 𝘓

38

Let + be an addition operator on a tuple ⦇𝑥𝑛 ⦈𝘕𝑚 .

⟹ 𝘓

𝘕

𝘔

𝘕

𝘔

𝑥 for 𝑚 < 𝘓 < 𝘔 ≤ 𝘕 𝑥 + 𝑥 + 𝑥 = 𝑥 + 𝑥 + ∑ 𝑛 ∑ 𝑛 ) (∑ 𝑛 ∑ 𝑛 ) ∑ 𝑛 (∑ 𝑛 𝑛=𝑚 𝑛=𝑚 𝑛=𝘔 +1 𝑛=𝘓+1 𝑛=𝘔 +1 𝑛=𝘓+1 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝘕

∑

is ASSOCIATIVE

𝑛=𝑚

✎PROOF: 𝘕

1. Proof for 𝘕 < 𝑚 case:

∑

𝑥𝑛 = 0.

𝑛=𝑚

⎛𝑚−1 ⎞ 𝑥 ⎟ + 𝑥 𝑚 = 0 + 𝑥𝑚 = 𝑥 𝑚 . 𝑥𝑛 = ⎜ ∑ 𝑛⎟ ∑ ⎜𝑛=𝑚 𝑛=𝑚 ⎝ ⎠ 𝑚

2. Proof for 𝑁 = 𝑚 case:

𝑚+1

3. Proof for 𝑁 = 𝑚 + 1 case:

∑

𝑚

𝑥𝑛 =

𝑛=𝑚

∑ (𝑛=𝑚

𝑥𝑛

)

+ 𝑥𝑚+1 = 𝑥𝑚 + 𝑥𝑚+1

4. Proof for 𝑁 = 𝑚 + 2 case: ⎛𝑚+1 ⎞ 𝑥 =⎜ 𝑥 ⎟ + 𝑥𝑚+2 ∑ 𝑛 ⎜∑ 𝑛⎟ 𝑛=𝑚 ⎝𝑛=𝑚 ⎠ = (𝑥𝑚 + 𝑥𝑚+1 ) + 𝑥𝑚+2 = 𝑥𝑚 + (𝑥𝑚+1 + 𝑥𝑚+2 )

𝑚+2

by Definition D.12 page 164 by item (3) by left hypothesis

5. Proof that 𝑁 case ⟹ 𝑁 + 1 case: 𝘕 +1

∑

𝘕

𝑥𝑛 =

𝑛=𝑚

𝑥 + 𝑥𝘕 +1 ∑ 𝑛) (𝑛=𝑚 ⏟⏟⏟⏟⏟

by Definition D.12 page 164

associative 𝘓

=

𝘔

∑ (𝑛=𝑚

𝑥𝑛 +

∑ (𝑛=𝘓+1

𝘓

= 37 38

∑ ((𝑛=𝑚

𝘕

𝑥𝑛 +

∑

𝘔

𝑥𝑛 +

∑

𝑛=𝘓+1

𝑥𝑛

𝑛=𝘔 +1

))

+ 𝑥𝘕 +1

𝘕

𝑥𝑛

)

+

∑

𝑛=𝘔 +1

𝑥𝑛

)

+ 𝑥𝘕 +1

📘 Berberian (1961) page 8 ⟨Definition I.3.1⟩, 📃 Fourier (1820) page 280 ⟨“∑” notation⟩ 📘 Berberian (1961) pages 9–10 ⟨Theorem I.3.1⟩

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APPENDIX D. SOME MATHEMATICAL TOOLS 𝘓

=

𝘔

∑ (𝑛=𝑚

𝑥𝑛 +

𝑛=𝘓+1

𝘓

=

∑

𝘕

𝑥𝑛

𝘔

∑ (𝑛=𝑚

𝑥𝑛 +

∑

𝑛=𝘓+1

𝑥𝑛

)

+

∑ (𝑛=𝘔 +1

𝑥𝑛 + 𝑥𝘕 +1

)

⎛ 𝘕 +1 ⎞ +⎜ 𝑥𝑛 ⎟ ∑ ⎟ ) ⎜𝑛=𝘔 +1 ⎝ ⎠

✏

D.3.2 Convexity Definition D.13. 39 A function 𝖿 ∈ ℝℝ is convex if 𝖿 (𝜆𝑥 + [1 − 𝜆]𝑦) ≤ 𝜆𝖿(𝑥) + (1 − 𝜆) 𝖿 (𝑦) ∀𝑥, 𝑦 ∈ ℝ and ∀𝜆 ∈ (0 ∶ 1) A function 𝗀 ∈ ℝℝ is strictly convex if 𝗀(𝜆𝑥 + [1 − 𝜆]𝑦) = 𝜆𝗀(𝑥) + (1 − 𝜆) 𝗀(𝑦) ∀𝑥, 𝑦 ∈ 𝐷, 𝑥 ≠ 𝑦, and ∀𝜆 ∈ (0 ∶ 1) ℝ A function 𝖿 ∈ ℝ is concave if −𝖿 is CONVEX. A function 𝖿 ∈ ℝℝ is affine if 𝖿 is CONVEX and CONCAVE. Theorem D.12 (Jensen's Inequality). 40 Let 𝖿 ∈ ℝℝ be a function. 𝘕 𝘕 ⎧ 1. 𝖿 is CONVEX and ⎫ ⎪ ⎪ 𝘕 ⟹ 𝖿 𝜆 𝑥 ≤ 𝜆 𝖿 𝑥 ⎬ ⎨ 2. ∑ 𝑛 𝑛) ∑ 𝑛 ( 𝑛) 𝜆 = 1 { ( 𝑛 𝑛=1 𝑛=1 ⎪ ⎪ ∑ ⎩ 𝑛=1 ⎭

∀𝑥𝑛 ∈ 𝐷, 𝘕 ∈ ℕ

}

D.3.3 Power means Definition D.14. 41 The ⦇𝜆𝑛 ⦈𝘕1 weighted 𝜙-mean of a tuple ⦇𝑥𝑛 ⦈𝘕1 is defined as 𝘕

𝖬𝜙 (⦇𝑥𝑛 ⦈) ≜ 𝜙

−1

(∑ 𝑛=1

𝜆𝑛 𝜙(𝑥𝑛 )

) ⊢

where 𝜙 is a CONTINUOUS and STRICTLY MONOTONIC function in ℝℝ 𝘕

and

⦇𝜆𝑛 ⦈𝘕𝑛=1

is a sequence of weights for which

∑

𝜆𝑛 = 1.

𝑛=1

Lemma D.3. 42 Let 𝖬𝜙 (⦇𝑥𝑛 ⦈) be the ⦇𝜆𝑛 ⦈𝘕1 weighted 𝜙-mean and 𝖬𝜓 (⦇𝑥𝑛 ⦈) the ⦇𝜆𝑛 ⦈𝘕1 weighted 𝜓mean of a tuple ⦇𝑥𝑛 ⦈𝘕1 . 𝜙𝜓 −1 is CONVEX and 𝜙 is INCREASING ⟹ 𝖬𝜙 (⦇𝑥𝑛 ⦈) ≥ 𝖬𝜓 (⦇𝑥𝑛 ⦈) 𝜙𝜓 −1 is CONVEX and 𝜙 is DECREASING ⟹ 𝖬𝜙 (⦇𝑥𝑛 ⦈) ≤ 𝖬𝜓 (⦇𝑥𝑛 ⦈) 𝜙𝜓 −1 is CONCAVE and 𝜙 is INCREASING ⟹ 𝖬𝜙 (⦇𝑥𝑛 ⦈) ≤ 𝖬𝜓 (⦇𝑥𝑛 ⦈) 𝜙𝜓 −1 is CONCAVE and 𝜙 is DECREASING ⟹ 𝖬𝜙 (⦇𝑥𝑛 ⦈) ≥ 𝖬𝜓 (⦇𝑥𝑛 ⦈) One of the most well known inequalities in mathematics is Minkowski's Inequality. In 1946, H.P. Mulholland submitted a result that generalizes Minkowski's Inequality to an equal weighted ɸ39

📘 Simon (2011) page 2, 📘 Barvinok (2002) page 2, 📘 Bollobás (1999), page 3, 📘 Jensen (1906), page 176 📘 Mitrinović et al. (2010) page 6, 📘 Bollobás (1999) page 3, 📃 Jensen (1906) pages 179–180 41 📘 Bollobás (1999) page 5 42 📘 Pečarić et al. (1992) page 107, 📘 Bollobás (1999) page 5, 📘 Hardy et al. (1952) page 75 40

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D.3. SUMS

mean.43 And Milovanović and Milovanovć (1979) generalized this even further to a weighted ɸmean (next). Theorem D.13. 44 Let 𝜙 be a function in ℝℝ . 1. 𝜙 is CONVEX and 2. 𝜙 is STRICTLY MONOTONE { 3. 𝜙(0) = 0 and 4. log ∘𝜙 ∘ exp is CONVEX 𝘕

⟹

and

𝘕

} 𝘕

𝜙−1 𝜆𝑛 𝜙(𝑥𝑛 + 𝑦𝑛 ) ≤ 𝜙−1 𝜆𝑛 𝜙(𝑥𝑛 ) + 𝜙−1 𝜆𝑛 𝜙(𝑦𝑛 ) ∑ ∑ { ( 𝑛=1 ) ( 𝑛=1 ) (∑ )} 𝑛=1

Definition D.15. 45 Let 𝖬𝜙(𝑥;𝑝) (⦇𝑥𝑛 ⦈) be the ⦇𝜆𝑛 ⦈𝘕1 weighted 𝜙-mean of a NON-NEGATIVE tuple ⦇𝑥𝑛 ⦈𝘕1 . A mean 𝖬𝜙(𝑥;𝑝) (⦇𝑥𝑛 ⦈) is a power mean with parameter 𝑝 if 𝜙(𝑥) ≜ 𝑥𝑝 . That is, 1 𝑝

𝘕

𝖬𝜙(𝑥;𝑝) (⦇𝑥𝑛 ⦈) =

(∑

𝜆𝑛 (𝑥𝑛 )𝑝

𝑛=1

)

Theorem D.14. 46 Let 𝖬𝜙(𝑥;𝑝) (⦇𝑥𝑛 ⦈) be the POWER MEAN with parameter 𝑝 of an 𝘕 -tuple ⦇𝑥𝑛 ⦈𝘕1 in which the elements are NOT all equal. 1 𝑝

𝘕

𝖬𝜙(𝑥;𝑝) (⦇𝑥𝑛 ⦈) ≜

(∑

𝜆𝑛 (𝑥𝑛 )𝑝

is CONTINUOUS and STRICTLY MONOTONE in ℝ∗ .

) max ⦇𝑥𝑛 ⦈ for 𝑝 = +∞

𝑛=1

⎧ 𝑛=1,2,…,𝘕 ⎪ 𝘕 ⎪ 𝜆 𝖬𝜙(𝑥;𝑝) (⦇𝑥𝑛 ⦈) = ⎨ 𝑥𝑛 𝑛 for 𝑝 = 0 ∏ ⎪ 𝑛=1 ⎪ min ⦇𝑥𝑛 ⦈ for 𝑝 = −∞ ⎩ 𝑛=1,2,…,𝘕 ✎PROOF: 1. Proof that 𝑀𝜙(𝑥;𝑝) is strictly monotone in 𝑝: (a) Let 𝑝 and 𝑠 be such that −∞ < 𝑝 < 𝑠 < ∞. 𝑝

𝑠 (b) Let 𝜙𝑝 ≜ 𝑥𝑝 and 𝜙𝑠 ≜ 𝑥𝑠 . Then 𝜙𝑝 𝜙−1 𝑠 =𝑥 .

(c) The composite function 𝜙𝑝 𝜙−1 𝑠 is convex or concave depending on the values of 𝑝 and 𝑠: 𝑝 < 0 (𝜙𝑝 decreasing) 𝑝 > 0 (𝜙𝑝 increasing) 𝑠0

convex convex

(not possible) concave

(d) Therefore by Lemma D.3 (page 165), −∞ < 𝑝 < 𝑠 < ∞ ⟹ 𝖬𝜙(𝑥;𝑝) (⦇𝑥𝑛 ⦈) < 𝖬𝜙(𝑥;𝑠) (⦇𝑥𝑛 ⦈). 2. Proof that 𝖬𝜙(𝑥;𝑝) is continuous in 𝑝 for 𝑝 ∈ ℝ⧵0: The sum of continuous functions is continuous. For the cases of 𝑝 ∈ {−∞, 0, ∞}, see the items that follow. 43

📃 Minkowski (1910) page 115, 📃 Mulholland (1950), 📘 Hardy et al. (1952) ⟨Theorem 24⟩, 📃 Tolsted (1964) page 7, Maligranda (1995) page 258, 📘 Carothers (2000), page 44, 📘 Bullen (2003) page 179 44 📃 Milovanović and Milovanović (1979), 📘 Bullen (2003) page 306 ⟨Theorem 9⟩ 45 📘 Bullen (2003) page 175, 📘 Bollobás (1999) page 6 46 📘 Bullen (2003) pages 175–177 ⟨see also page 203⟩, 📘 Bollobás (1999) pages 6–8, 📃 Bullen (1990) page 250, 📃 Besso (1879), 📃 Bienaymé (1840) page 68, 📃 Brenner (1985) page 160 📃

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APPENDIX D. SOME MATHEMATICAL TOOLS −1

3. Lemma: 𝖬𝜙(𝑥;−𝑝) (⦇𝑥𝑛 ⦈) = {𝖬𝜙(𝑥;𝑝) (⦇𝑥−1 𝑛 ⦈)} . Proof: 1

−1 {𝖬𝜙(𝑥;𝑝) (⦇𝑥𝑛 ⦈)}

−1

−1

𝑝 𝘕 ⎧ ⎫ ⎪ ⎪ −1 𝑝 =⎨ 𝜆𝑛 (𝑥𝑛 ) ⎬ ∑ ( 𝑛=1 ) ⎪ ⎪ ⎩ ⎭

by definition of 𝖬𝜙

1 −𝑝

𝘕 −𝑝

=

𝜆 𝑛 ( 𝑥𝑛 )

(∑ 𝑛=1

)

= 𝖬𝜙(𝑥;−𝑝) (⦇𝑥𝑛 ⦈)


4. Proof that lim 𝖬𝜙 (⦇𝑥𝑛 ⦈) = max ⦇𝑥𝑛 ⦈: 𝑝→∞

𝑛∈ℤ

(a) Let 𝑥𝑚 ≜ max ⦇𝑥𝑛 ⦈ 𝑛∈ℤ

(b) Note that lim 𝖬𝜙 ≤ max ⦇𝑥𝑛 ⦈ because 𝑝→∞

𝑛∈ℤ

1 𝑝

𝘕 𝑝

lim 𝖬𝜙 (⦇𝑥𝑛 ⦈) = lim 𝜆 𝑥𝑛 𝑝→∞ (∑ 𝑛 ) 𝑛=1


𝑝→∞

1 𝑝

𝘕

≤

𝑝 lim 𝜆 𝑥𝑚 𝑝→∞ (∑ 𝑛 ) 𝑛=1

by definition of 𝑥𝑚 in item (4a) and because 𝜙(𝑥) ≜ 𝑥𝑝 and 𝜙−1 are both increasing or both decreasing

1

⎛ ⎞𝑝 𝘕 ⎜ ⎟ 𝑝 = lim ⎜𝑥𝑚 𝜆𝑛 ⎟ ∑ ⎟ 𝑝→∞ ⎜ ⏟⎟ ⎜ 𝑛=1 ⎝ 1 ⎠

because 𝑥𝑚 is a constant

1

𝑝

= lim (𝑥𝑚 ⋅ 1) 𝑝 𝑝→∞ = 𝑥𝑚 = max ⦇𝑥𝑛 ⦈

by definition of 𝑥𝑚 in item (4a)

𝑛∈ℤ

(c) But also note that lim 𝖬𝜙 ≥ max ⦇𝑥𝑛 ⦈ because 𝑝→∞

𝑛∈ℤ

1 𝑝

𝘕 𝑝 𝜆 𝑥𝑛 𝑝→∞ (∑ 𝑛 ) 𝑛=1

lim 𝖬𝜙 (⦇𝑥𝑛 ⦈) = lim

𝑝→∞

𝑝

1

≥ lim (𝑤𝑚 𝑥𝑚 ) 𝑝 𝑝→∞ 1 𝑝

by definition of 𝖬𝜙 by definition of 𝑥𝑚 in item (4a) and because 𝜙(𝑥) ≜ 𝑥𝑝 and 𝜙−1 are both increasing or both decreasing

𝑝 𝑝

= lim 𝑤𝑚 𝑥𝑚 𝑝→∞

= 𝑥𝑚 = max ⦇𝑥𝑛 ⦈ 𝑛∈ℤ

by definition of 𝑥𝑚 in item (4a)

(d) Combining items (b) and (c) we have lim 𝖬𝜙 = max ⦇𝑥𝑛 ⦈. 2016 Jul 4

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𝑛∈ℤ

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D.3. SUMS

5. Proof that lim 𝖬𝜙 (⦇𝑥𝑛 ⦈) = min ⦇𝑥𝑛 ⦈: 𝑝→−∞

𝑛∈ℤ

lim 𝖬𝜙(𝑥;𝑝) (⦇𝑥𝑛 ⦈) = lim 𝖬𝜙(𝑥;−𝑝) (⦇𝑥𝑛 ⦈)

by change of variable 𝑝

𝑝→∞

𝑝→−∞

= lim {𝖬𝜙(𝑥;𝑝) (⦇𝑥−1 𝑛 ⦈)} 𝑝→∞ = lim

𝑝→∞

=

−1

by Lemma in item (3) page 167

1 𝖬𝜙(𝑥;𝑝) (⦇𝑥−1 𝑛 ⦈) lim𝑝→∞ 1

by property of lim 47

lim𝑝→∞ 𝖬𝜙(𝑥;𝑝) (⦇𝑥−1 𝑛 ⦈) 1 = max ⦇𝑥−1 𝑛 ⦈

by item (4)

𝑛∈ℤ

1

=

−1

min ⦇𝑥 ⦈ ( 𝑛∈ℤ 𝑛 ) = min ⦇𝑥𝑛 ⦈ 𝑛∈ℤ

𝘕

6. Proof that lim 𝖬𝜙 (⦇𝑥𝑛 ⦈) = 𝑝→0

∏

𝜆

𝑥𝑛 𝑛 :

𝑛=1

lim 𝖬𝜙 (⦇𝑥𝑛 ⦈) = lim exp {ln {𝖬𝜙 (⦇𝑥𝑛 ⦈)}}

𝑝→0

𝑝→0

1

𝑝 𝘕 ⎫ ⎧ ⎪ ⎪⎫ ⎪ ⎧ ⎪ 𝑝 = lim exp ⎨ln ⎨ 𝜆𝑛 (𝑥𝑛 ) ⎬⎬ ∑ 𝑝→0 ( ) ⎪ ⎪ ⎩ ⎪ ⎩ 𝑛=1 ⎭ ⎭⎪


𝘕

⎧ 𝜕 ⎫ 𝑝 ⎪ ln ∑ 𝜆𝑛 (𝑥𝑛 ) ⎪ )⎪ ⎪ 𝜕𝑝 ( 𝑛=1 = exp ⎨ ⎬ 𝜕 ⎪ ⎪ 𝑝 𝜕𝑝 ⎪ ⎪ ⎩ ⎭𝑝=0

by l'Hôpital's rule48

𝘕

𝘕

⎧ ⎫ 𝜕 𝑝 ⎪ ∑ 𝜆𝑛 exp (ln (𝑥𝑛 )) ⎪ ⎪ 𝑛=1 𝜕𝑝 ⎪ = exp ⎨ ⎬ 𝘕 ⎪ ⎪ 𝜆 ⎪ ⎪ ∑ 𝑛 ⎩ ⎭𝑝=0 𝑛=1

⎧ 𝜕 𝑝 ⎫ ⎪ ∑ 𝜆𝑛 (𝑥𝑛 ) ⎪ ⎪ ⎪ 𝑛=1 𝜕𝑝 = exp ⎨ ⎬ 𝘕 ⎪ ⎪ 𝑝 ⎪ ∑ 𝜆𝑛 (𝑥𝑛 ) ⎪ ⎩ 𝑛=1 ⎭𝑝=0 𝘕

⎧ ⎫ 𝜕 ⎪ ∑ 𝜆𝑛 exp (𝑟 ln (𝑥𝑛 )) ⎪ ⎪ 𝑛=1 𝜕𝑝 ⎪ = exp ⎨ ⎬ 1 ⎪ ⎪ ⎪ ⎪ ⎩ ⎭𝑝=0

𝘕

= exp

𝘕

= exp

𝜆𝑛 exp {𝑝 ln 𝑥𝑛 } ln (𝑥𝑛 ) {∑ } 𝑛=1 𝘕

= 47

𝜆 exp ln (𝑥𝑛 𝑛 ) ∑ { 𝑛=1 }

{∑ 𝑛=1

𝜆𝑛

𝜕 exp (𝑝 ln (𝑥𝑛 )) 𝜕𝑝 }

𝑝=0

𝘕

= exp 𝑝=0

{∑ 𝑛=1

𝜆𝑛 ln (𝑥𝑛 ) }

𝘕 𝘕 ⎧ ⎫ 𝜆𝑛 ⎪ 𝜆 ⎪ = exp ⎨ln 𝑥 = 𝑥𝑛 ∏ 𝑛⎬ ∏ 𝑛 ⎪ ⎪ ⎩ 𝑛=1 ⎭ 𝑛=1

📘 Rudin (1976) page 85 ⟨4.4 Theorem⟩

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✏ 𝘕

Corollary D.2.

Let ⦇𝑥𝑛 ⦈𝘕1 be a tuple. Let ⦇𝜆𝑛 ⦈𝘕1 be a tuple of weighting values such that

49

∑

𝜆𝑛 = 1.

𝑛=1

−1

𝘕

min ⦇𝑥𝑛 ⦈ ≤

𝘕

1 𝜆𝑛 ≤ ∑ ( 𝑛=1 𝑥𝑛 ) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

𝘕 𝜆

𝑥𝑛 ≤ 𝜆𝑥 ∏ 𝑛 ∑ 𝑛 𝑛 𝑛=1 𝑛=1 ⏟⏟⏟ ⏟ geometric mean

harmonic mean

≤ max ⦇𝑥𝑛 ⦈

arithmetic mean

✎PROOF: 1. These five means are all special cases of the power mean 𝖬𝜙(𝑥∶𝑝) (Definition D.15 page 166): 𝑝 = ∞: max ⦇𝑥𝑛 ⦈ 𝑝 = 1: arithmetic mean 𝑝 = 0: geometric mean 𝑝 = −1: harmonic mean 𝑝 = −∞: min ⦇𝑥𝑛 ⦈ 2. The inequalities follow directly from Theorem D.14 (page 166). 3. Generalized AM-GM inequality: If one is only concerned with the arithmetic mean and geometric mean, their relationship can be established directly using Jensen's Inequality: 𝘕

∑

𝘕

𝜆𝑛 𝑥𝑛 = 𝑏log𝑏 (∑𝑛=1 𝜆𝑛 𝑥𝑛 )

𝑛=1

𝘕

≥ 𝑏(∑𝑛=1 𝜆𝑛 log𝑏 𝑥𝑛 ) 𝘕

=

by Jensen's Inequality (Theorem D.12 page 165) 𝘕

∏

𝑏(𝜆𝑛 log𝑏 𝑥𝑛 ) =

𝑛=1

∏

𝘕

𝑏(log𝑏 𝑥𝑛 )𝜆𝑛 =

𝑛=1

∏

𝜆

𝑥𝑛 𝑛

𝑛=1

✏

D.3.4 Inequalities Lemma D.4 (Young's Inequality). 50 𝑥𝑝 𝑦𝑞 𝑥𝑦 < + with 1𝑝 + 1𝑞 = 1 ∀1 < 𝑝 < ∞, 𝑥, 𝑦 ≥ 0, but 𝑦 ≠ 𝑥𝑝−1 𝑝 𝑞 𝑥𝑝 𝑦𝑞 + with 1𝑝 + 1𝑞 = 1 ∀1 < 𝑝 < ∞, 𝑥, 𝑦 ≥ 0, and 𝑦 = 𝑥𝑝−1 𝑥𝑦 = 𝑝 𝑞 Theorem D.15 (Minkowski's Inequality for sequences). 𝘕 -tuples. 1 𝑝

𝘕

𝘕

1 𝑝

𝘕

51

Let ⦇𝑥𝑛 ∈ ℂ⦈𝘕1 and ⦇𝑦𝑛 ∈ ℂ⦈𝘕1 be complex

1 𝑝

𝑝 𝑝 𝑝 ≤ + |𝑥𝑛 | |𝑦𝑛 | |𝑥𝑛 + 𝑦𝑛 | ∑ ∑ ∑ ( 𝑛=1 ) ( 𝑛=1 ) ( 𝑛=1 )

∀1 < 𝑝 < ∞

48

📘 Rudin (1976) page 109 ⟨5.13 Theorem⟩ 📘 Bullen (2003) page 71, 📘 Bollobás (1999) page 5, 📘 Cauchy (1821) pages 457–459 ⟨Note II, theorem 17⟩, 📃 Jensen (1906) page 183, 📃 Hoehn and Niven (1985) page 151 50 📃 Young (1912) page 226, 📘 Hardy et al. (1952) ⟨Theorem 24⟩, 📃 Tolsted (1964) page 5, 📃 Maligranda (1995) page 257, 📘 Carothers (2000), page 43 51 📘 Minkowski (1910), page 115, 📘 Hardy et al. (1952) ⟨Theorem 24⟩, 📃 Maligranda (1995) page 258, 📘 Tolsted (1964), page 7, 📘 Carothers (2000), page 44, 📘 Bullen (2003) page 179 49

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D.4. TOPOLOGICAL SPACES

D.4 Topological Spaces ❝Nevertheless I should not pass over in silence the fact that today the feeling among

mathematicians is beginning to spread that the fertility of these abstracting methods is approaching exhaustion. The case is this: that all these nice general concepts do not fall into our laps by themselves. But definite concrete problems were first conquered in their undivided complexity, singlehanded by brute force, so to speak. Only afterwards the axiomaticians came along and stated: Instead of breaking the door with all your might and bruising your hands, you should have constructed such and such a key of skill, and by it you would have been able to open the door quite smoothly. But they can construct the key only because they are able, after the breaking in was successful, to study the lock from within and without. Before you can generalize, formalize, and axiomatize, there must be a mathematical substance.❞ Hermann Weyl (1885–1955); mathematician, theoretical physicist, and philosopher

52

Definition D.16. 53 Let 𝛤 be a set with an arbitrary (possibly uncountable) number of elements. Let 𝟚𝑋 be the POWER SET of a set 𝑋 (Definition 1.8 page 6). A family of sets 𝑻 ⊆ 𝟚𝑋 is a topology on 𝑋 if 1. ∅ ∈ 𝑻 and 2. 𝑋 ∈ 𝑻 and 3. 𝑈 , 𝑉 ∈ 𝑻 ⟹ 𝑈 ∩ 𝑉 ∈ 𝑻 and 𝑈 ∈𝑻 . 4. {𝑈𝛾 |𝛾 ∈ 𝛤 } ⊆ 𝑻 ⟹ ⋃ 𝛾 𝛾∈𝛤

The ordered pair (𝑋, 𝑻 ) is a TOPOLOGICAL SPACE if 𝑻 is a TOPOLOGY on 𝑋. A set 𝑈 is open in (𝑋, 𝑻 ) if 𝑈 is any element of 𝑻 . A set 𝐷 is closed in (𝑋, 𝑻 ) if 𝐷𝖼 is OPEN in (𝑋, 𝑻 ). Just as the power set 𝟚𝑋 and the set {∅, 𝑋} are algebras of sets on a set 𝑋, so also are these sets topologies on 𝑋 (next example): Example D.12. 54 Let 𝒯 (𝑋) be the set of topologies on a set 𝑋 and 𝟚𝑋 the power set (Definition 1.8 page 6) on 𝑋. {∅, 𝑋} is a topology in 𝒯 (𝑋) (indiscrete topology or trivial topology) 𝟚𝑋 is a topology in 𝒯 (𝑋) (discrete topology) Definition D.17. 1. 𝑩 ⊆ 𝑻 2. ∀𝑈 ∈ 𝑻 ,

55

Let (𝑋, 𝑻 ) be a TOPOLOGICAL SPACE. A set 𝑩 ⊆ 𝟚𝑋 is a base for 𝑻 if and

∃{𝐵𝛾 ∈ 𝑩 } such that 𝑈 =

⋃

𝐵𝛾

𝛾

Let (𝑋, 𝑻 ) be a TOPOLOGICAL SPACE. Let 𝑩 be a subset of 𝟚𝑋 such that 𝑩 ⊆ 𝟚𝑋 . For every 𝑥 ∈ 𝑋 and for every OPEN SET 𝑈 containing 𝑥, ⟺ {𝑩 is a BASE for 𝑻 } { there exists 𝐵𝑥 ∈ 𝑩 such that } 𝑥 ∈ 𝐵𝑥 ⊆ 𝑈 .

Theorem D.16.

56

Let (𝑋, 𝑻 ) be a TOPOLOGICAL SPACE (Definition D.16 page 170) and 𝑩 ⊆ 𝟚𝑋 . 1. 𝑥 ∈ 𝑋 ⟹ ∃𝐵𝑥 ∈ 𝑩 such that 𝑥 ∈ 𝐵𝑥 𝑩 is a base for (𝑋, 𝑻 ) ⟺ { 2. 𝐵1 , 𝐵2 ∈ 𝑩 ⟹ 𝐵1 ∩ 𝐵2 ∈ 𝑩

Theorem D.17.

57

and

52

quote: 📃 Weyl (1935) page 14 ⟨H. Weyl, quoting himself from “a conference on topology and abstract algebra as two ways of mathematical understanding, in 1931”⟩. image: https://en.wikipedia.org/wiki/File:Hermann_Weyl_ ETH-Bib_Portr_00890.jpg: “This work is free and may be used by anyone for any purpose.” 53 📘 Munkres (2000) page 76, 📘 Riesz (1909), 📘 Hausdorff (1914), 📃 Tietze (1923), 📘 Hausdorff (1937) page 258 54 📘 Munkres (2000), page 77, 📘 Kubrusly (2011) page 107 ⟨Example 3.J⟩, 📘 Steen and Seebach (1978) pages 42–43 ⟨II.4⟩, 📘 DiBenedetto (2002) page 18 55 📘 Joshi (1983) page 92 ⟨(3.1) Definition⟩, 📘 Davis (2005) page 46 ⟨Definition 4.15⟩ 56 📘 Joshi (1983) pages 92–93 ⟨(3.2) Proposition⟩, 📘 Davis (2005) page 46 57 📘 Bollobás (1999) page 19

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Example D.13. 58 Let (𝑋, 𝖽) be a metric space. The set 𝑩 ≜ {𝖡(𝑥, 𝑟) |𝑥 ∈ 𝑋, 𝑟 ∈ ℕ} (the set of all open balls in (𝑋, 𝖽)) is a base for a topology on (𝑋, 𝖽). Example D.14 (the standard topology on the real line). 59 The set 𝑩 ≜ {(𝑎 ∶ 𝑏)|𝑎, 𝑏 ∈ ℝ, 𝑎 < 𝑏} is a base for the metric space (ℝ, |𝑏 − 𝑎|) (the usual metric space on ℝ). Definition D.18. 60 Let (𝑋, 𝑻 ) be a TOPOLOGICAL SPACE (Definition D.16 page 170). Let 𝟚𝑋 be the POWER SET of 𝑋. The set 𝐴− is the closure of 𝐴 ∈ 𝟚𝑋 if 𝐴− ≜ 𝐷 ∈ 𝟚𝑋 |𝐴 ⊆ 𝐷 and 𝐷 is CLOSED} . ⋂{ 𝑋 The set 𝐴∘ is the interior of 𝐴 ∈ 𝟚𝑋 if 𝐴∘ ≜ {𝑈 ∈ 𝟚 |𝑈 ⊆ 𝐴 and 𝑈 is OPEN} . ⋃ A point 𝑥 is a closure point of 𝐴 if 𝑥 ∈ 𝐴− . A point 𝑥 is an interior point of 𝐴 if 𝑥 ∈ 𝐴∘ . A point 𝑥 is an accumulation point of 𝐴 if 𝑥 ∈ (𝐴⧵{𝑥})− A point 𝑥 in 𝐴− is a point of adherence in 𝐴 or is adherent to 𝐴 if 𝑥 ∈ 𝐴− . Proposition D.2. 61 Let (𝑋, 𝑻 ) be a TOPOLOGICAL SPACE (Definition D.16 page 170). Let 𝐴− be the CLOSURE, 𝐴∘ the INTERIOR, and 𝜕𝐴 the BOUNDARY of a set 𝐴. Let 𝟚𝑋 be the POWER SET of 𝑋. − 1. 𝐴 is CLOSED ∀𝐴∈𝟚𝑋 . ∘ 2. 𝐴 is OPEN ∀𝐴∈𝟚𝑋 . Lemma D.5. 62 Let 𝐴− be the CLOSURE, 𝐴∘ the INTERIOR, and 𝜕𝐴 the BOUNDARY of a set 𝐴 in a topological space (𝑋, 𝑻 ). Let 𝟚𝑋 be the POWER SET of 𝑋. ∘ − 1. 𝐴 ⊆ 𝐴 ⊆ 𝐴 ∀𝐴∈𝟚𝑋 . ∘ 2. 𝐴 = 𝐴 ⟺ 𝐴 is OPEN ∀𝐴∈𝟚𝑋 . − 3. 𝐴 = 𝐴 ⟺ 𝐴 is CLOSED ∀𝐴∈𝟚𝑋 . Definition D.19. 63 Let (𝑋, 𝑻𝑥 ) and (𝑌 , 𝑻𝑦 ) be TOPOLOGICAL SPACEs (Definition D.16 page 170). Let 𝖿 be a −1 function in 𝑌 𝑋 . A function 𝖿 ∈ 𝑌 𝑋 is continuous if 𝑈⏟ ∈ 𝑻𝑦 ⟹ 𝖿⏟⏟⏟⏟⏟⏟⏟⏟⏟ (𝑈 ) ∈ 𝑻𝑥 . OPEN in (𝑌 , 𝑻𝑦 )

OPEN in (𝑋, 𝑻𝑥 )

(

(

(

)

)

)

A function is discontinuous in (𝑋, 𝑻𝑦 )(𝑋,𝑻𝑥 ) if it is not CONTINUOUS in (𝑋, 𝑻𝑦 )(𝑋,𝑻𝑥 ) .

(

)

continuous

( continuous

)

(

]

discontinuous

Figure D.3: continuous/discontinuous functions (Example D.15 page 171) Example D.15. Some continuous/discontinuous functions are illustrated in Figure D.3 (page 171). Definition D.19 (previous definition) defines continuity using open sets. Continuity can alternatively be defined using closed sets or closure (next theorem). 58

📘 Davis (2005) page 46 ⟨Example 4.16⟩ 📘 Munkres (2000) page 81, 📘 Davis (2005) page 46 ⟨Example 4.16⟩ 60 📘 Gemignani (1972) pages 55–56 ⟨Definition 3.5.7⟩, 📘 McCarty (1967) page 90, 📘 Munkres (2000) page 95 ⟨§Closure and Interior of a Set⟩, 📘 Thron (1966), pages 21–22 ⟨definition 4.8, defintion 4.9⟩, 📘 Kelley (1955) page 42, 📘 Kubrusly (2001) pages 115–116 61 📘 McCarty (1967) page 90 ⟨IV.1 THEOREM⟩ 62 📘 McCarty (1967) pages 90–91 ⟨IV.1 THEOREM⟩, 📘 ALIPRANTIS AND BURKINSHAW (1998) PAGE 59 63 📘 Davis (2005) page 34 59

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D.4. TOPOLOGICAL SPACES

Theorem D.18. 64 Let (𝑋, 𝑻 ) and (𝑌 , 𝑺) be topological spaces. Let 𝖿 be a function in 𝑌 𝑋 . The following are equivalent: 1. 𝖿 is CONTINUOUS ⟺ −1 𝑌 2. 𝐵 is closed in (𝑌 , 𝑺) ⟹ 𝖿 (𝐵) is closed in (𝑋, 𝑻 ) ∀𝐵∈𝟚 ⟺ 3. 𝖿(𝐴− ) ⊆ 𝖿(𝐴)− ∀𝐴∈𝟚𝑋 ⟺ − −1 −1 − 𝑌 4. 𝖿 (𝐵) ⊆ 𝖿 (𝐵 ) ∀𝐵∈𝟚 Remark D.1. A word of warning about defining continuity in terms of topological spaces—continuity is defined in terms of a pair of topological spaces, and whether function is continuous or discontinuous in general depends very heavily on the selection of these spaces. This is illustrated in Proposition D.3 (next). The ramification of this is that when declaring a function to be continuous or discontinuous, one must make clear the assumed topological spaces. Proposition D.3. 1. 2.

65

Let (𝑋, 𝑻 ) and (𝑌 , 𝑺) be TOPOLOGICAL SPACEs. Let 𝖿 be a FUNCTION in (𝑌 , 𝑺)(𝑋,𝑻 ) .

𝑻 is the DISCRETE TOPOLOGY 𝑺 is the INDISCRETE TOPOLOGY

⟹ 𝖿 is CONTINUOUS ⟹ 𝖿 is CONTINUOUS

∀𝖿∈(𝑌 ,𝑺)(𝑋,𝑻 ) ∀𝖿∈(𝑌 ,𝑺)(𝑋,𝑻 )

Definition D.20. 66 Let (𝑋, 𝑻 ) be a TOPOLOGICAL SPACE (Definition D.16 page 170). A sequence ⦅𝑥𝑛 ⦆𝑛∈ℤ converges in (𝑋, 𝑻 ) to a point 𝑥 if for each OPEN SET (Definition D.16 page 170) 𝑈 ∈ 𝑻 that contains 𝑥 there exists 𝘕 ∈ ℕ such that 𝑥𝑛 ∈ 𝑈 for all 𝑛 > 𝘕 . This condition can be expressed in any of the following forms: 1. The limit of the sequence ⦅𝑥𝑛 ⦆ is 𝑥. 3. lim ⦅𝑥𝑛 ⦆ = 𝑥. 𝑛→∞ 2. The sequence ⦅𝑥𝑛 ⦆ is convergent with limit 𝑥. 4. ⦅𝑥𝑛 ⦆ → 𝑥. A sequence that converges is convergent. A sequence that does not converge is said to diverge, or is divergent. An element 𝑥 ∈ 𝐴 is a limit point of 𝐴 if it is the limit of some 𝐴-valued sequence ⦅𝑥𝑛 ∈ 𝐴⦆. Example D.16. 67 Let (𝑋, 𝑻31 ) be a topological space where 𝑋 ≜ {𝑥, 𝑦, 𝑧} and 𝑻31 ≜ {∅, {𝑥}, {𝑥, 𝑦}, {𝑥, 𝑧}, {𝑥, 𝑦, 𝑧}}. In this space, the sequence ⦅𝑥, 𝑥, 𝑥, …⦆ converges to 𝑥. But this sequence also converges to both 𝑦 and 𝑧 because 𝑥 is in every open set (Definition D.16 page 170) that contains 𝑦 and 𝑥 is in every open set that contains 𝑧. So, the limit (Definition D.20 page 172) of the sequence is not unique. Example D.17. In contrast to the low resolution topological space of Example D.16, the limit of the sequence ⦅𝑥, 𝑥, 𝑥, …⦆ is unique in a topological space with sufficiently high resolution with respect to 𝑦 and 𝑧 such as the following: Define a topological space (𝑋, 𝑻56 ) where 𝑋 ≜ {𝑥, 𝑦, 𝑧} and 𝑻56 ≜ {∅, {𝑦}, {𝑧}, {𝑥, 𝑦}, {𝑦, 𝑧}, {𝑥, 𝑦, 𝑧}}. In this space, the sequence ⦅𝑥, 𝑥, 𝑥, …⦆ converges to 𝑥 only. The sequence does not converge to 𝑦 or 𝑧 because there are open sets (Definition D.16 page 170) containing 𝑦 or 𝑧 that do not contain 𝑥 (the open sets {𝑦}, {𝑧}, and {𝑦, 𝑧}). Theorem D.19 (The Closed Set Theorem). 68 Let (𝑋, 𝑻 ) be a TOPOLOGICAL SPACE. Let 𝐴 be a subset of 𝑋 (𝐴 ⊆ 𝑋). Let 𝐴− be the CLOSURE (Definition D.18 page 171) of 𝐴 in (𝑋, 𝑻 ). Every 𝐴-valued sequence ⦅𝑥𝑛 ∈ 𝐴⦆𝑛∈ℤ 𝐴 is CLOSED in (𝑋, 𝑻 ) ⟺ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ { that CONVERGES in (𝑋, 𝑻 ) has its LIMIT in 𝐴 } (𝐴 = 𝐴− ) 64

📘 McCarty (1967) pages 91–92 ⟨IV.2 THEOREM⟩, 📘 SEARCóID (2006) PAGE 130 ⟨“THEOREM 8.3.1 (CRITERIA FOR CONTINUITY)”, SET IN metric spaceS⟩ 65 📘 Crossley (2006) page 18 ⟨Proposition 3.9⟩, 📘 Ponnusamy (2002) page 98 ⟨2.64. Theorem.⟩ 66 📘 Joshi (1983) page 83 ⟨(3.1) Definition⟩, 📘 Leathem (1905), page 13 ⟨“→” symbol, section III.11⟩ 67 📘 Munkres (2000) page 98 ⟨Hausdorff Spaces⟩ 68 📘 Kubrusly (2001) page 118 ⟨Theorem 3.30⟩, 📘 Haaser and Sullivan (1991) page 75 ⟨6⋅9 Proposition⟩, 📘 Rosenlicht (1968) pages 47–48

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Theorem D.20. 69 Let (𝑋, 𝑻 ) and (𝑌 , 𝑺) be a TOPOLOGICAL SPACEs. Let 𝖿 be a function in (𝑌 , 𝑺)(𝑋,𝑻 ) . ⦅𝑥𝑛 ⦆ → 𝑥 ⟹ 𝖿(⦅𝑥𝑛 ⦆) → 𝖿(𝑥) 𝖿 is CONTINUOUS in (𝑌 , 𝑺)(𝑋,𝑻 ) ⟺ { } { } (Definition D.20 page 172) (Definition D.19 page 171) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ SEQUENTIAL CHARACTERIZATION OF CONTINUITY

INVERSE IMAGE CHARACTERIZATION OF CONTINUITY

✎PROOF: 1. Proof for the ⟹ case (proof by contradiction): (a) Let 𝑈 be an open set in (𝑌 , 𝑻 ) that contains 𝖿(𝑥) but for which there exists no 𝘕 such that 𝖿(𝑥𝑛 ) ∈ 𝑈 for all 𝑛 > 𝘕 . (b) Note that the set 𝖿 −1 (𝑈 ) is also open by the continuity hypothesis. (c) If ⦅𝑥𝑛 ⦆ → 𝑥, then 𝖿(⦅𝑥𝑛 ⦆) ↛ 𝖿 (𝑥) ⟹ there exists no 𝘕 such that 𝖿(𝑥𝑛 ) ∈ 𝑈 for all 𝑛 > 𝘕 ⟹ there exists no 𝘔 such that 𝑥𝑛 ∈ 𝖿 ⟹ ⦅𝑥𝑛 ⦆ ↛ 𝑥

−1

(𝑈 ) for all 𝑛 > 𝘔

by Definition D.20 by definition of 𝖿 −1

by continuity hyp. and def. of convergence (Definition D.20 page 172)

⟹ contradiction of ⦅𝑥𝑛 ⦆ → 𝑥 hypothesis ⟹ 𝖿 (⦅𝑥𝑛 ⦆) → 𝖿 (𝑥) 2. Proof for the ⟸ case (proof by contradiction): (a) Let 𝐷 be a closed set in (𝑌 , 𝑺). (b) Suppose 𝖿 −1 (𝐷) is not closed… (c) then by the closed set theorem (Theorem D.19 page 172), there must exist a convergent sequence ⦅𝑥𝑛 ⦆ in (𝑋, 𝑻 ), but with limit 𝑥 not in 𝖿 −1 (𝐷). (d) Note that 𝖿(𝑥) must be in 𝐷. Proof: i. by definition of 𝐷 and 𝖿, 𝖿(⦅𝑥𝑛 ⦆) is in 𝐷 ii. by left hypothesis, the sequence 𝖿(⦅𝑥𝑛 ⦆) is convergent with limit 𝖿(𝑥) iii. by closed set theorem (Theorem D.19 page 172), 𝖿(𝑥) must be in 𝐷. (e) Because 𝖿 (𝑥) ∈ 𝐷, it must be true that 𝑥 ∈ 𝖿 −1 (𝐷). (f) But this is a contradiction to item (2c) (page 173), and so item (2b) (page 173) must be wrong, and 𝖿 −1 (𝐷) must be closed. (g) And so by Theorem D.18 (page 172), 𝖿 is continuous.

✏

D.5 Polynomial interpolation D.5.1 Lagrange interpolation Definition D.21. 70 The Lagrange polynomial 𝐿𝑃 ,𝑛 (𝑥) with respect to the 𝑛 + 1 points 𝑃 = {(𝑥𝑘 , 𝑦𝑘 )|𝑘 = 0, 1, 2, … , 𝑛} is defined as 𝑛 𝑥 − 𝑥𝑚 𝐿𝑃 ,𝑛 (𝑥) ≜ 𝑦𝑘 ∑ ∏ 𝑥 𝑘 − 𝑥𝑚 𝑘=0 𝑚≠𝑛 69

📘 Ponnusamy (2002) pages 94–96 ⟨“2.59. Proposition.”; in the context of metric spaces; includes the “inverse image and “sequential characterization of continuity ” terminology; this terminology does not seem to be widely used in the literature in general, but has been adopted for use in this text⟩ 70 📃 Waring (1779) page 60, 📃 Euler (1783) page 165 ⟨§. 10. Problema 2. Corollarium 3.⟩, 📃 Gauss (1866), 📃 Lagrange (1877), 📘 Matthews and Fink (1992), page 206, 📃 Meijering (2002) ⟨historical background⟩ characterization of continuity ”

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Proposition D.4. Let 𝐿𝑃 ,𝑛 (𝑥) be the Lagrange polynomial with respect to the points 𝑃 = {(𝑥𝑘 , 𝑦𝑘 )|𝑘 = 0, 1, 2, … , 𝑛}. 1. 𝐿𝑃 ,𝑛 (𝑥) is an 𝑛th order polynomial. 2. 𝐿𝑃 ,𝑛 (𝑥) intersects all 𝑛 + 1 points in 𝑃 . Example D.18 (Lagrange interpolation). The Lagrange polynomial 𝐿𝑃 ,3 (𝑥) with respect to the 4 points 𝑃 = {(−2, 1), (−1, 3), (3, 2), (5, 4)} is 2970 79 3 −378 2 −7 𝑥 + 𝑥 + 𝑥+ 𝐿𝑃 ,3 (𝑥) = 840 840 840 840 ✎PROOF: 𝑛

𝐿𝑃 ,3 (𝑥) =

∑

𝑦𝑘

𝑘=0

𝑥 − 𝑥𝑚 ∏ 𝑥 − 𝑥𝑚 𝑚≠𝑛 𝑘

by Definition D.21

(𝑥 + 1)(𝑥 − 3)(𝑥 − 5) (𝑥 + 2)(𝑥 − 3)(𝑥 − 5) + 𝑦1 (𝑥0 − 𝑥1 )(𝑥0 − 𝑥2 )(𝑥0 − 𝑥3 ) (𝑥1 − 𝑥0 )(𝑥1 − 𝑥2 )(𝑥1 − 𝑥3 ) (𝑥 + 2)(𝑥 + 1)(𝑥 − 5) (𝑥 + 2)(𝑥 + 1)(𝑥 − 3) + 𝑦2 + 𝑦3 (𝑥2 − 𝑥0 )(𝑥2 − 𝑥1 )(𝑥2 − 𝑥3 ) (𝑥3 − 𝑥0 )(𝑥3 − 𝑥1 )(𝑥3 − 𝑥2 ) (𝑥 + 2)(𝑥 − 3)(𝑥 − 5) (𝑥 + 1)(𝑥 − 3)(𝑥 − 5) +3 =1 (−2 + 1)(−2 − 3)(−2 − 5) (−1 + 2)(−1 − 3)(−1 − 5) (𝑥 + 2)(𝑥 + 1)(𝑥 − 5) (𝑥 + 2)(𝑥 + 1)(𝑥 − 3) +2 +4 (3 + 2)(3 + 1)(3 − 5) (5 + 2)(5 + 1)(5 − 3) 3 2 3 𝑥 − 7𝑥 + 7𝑥 + 15 𝑥 − 6𝑥2 − 𝑥 + 30 𝑥3 − 2𝑥2 − 13𝑥 − 10 𝑥3 − 7𝑥 − 6 =1 +3 +2 +4 −35 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 24 −40 84 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟ = 𝑦0

roots=−2, 3, 5

roots=−1, 3, 5 3

2

3

2

roots=−2, −1, 3

roots=−2, −1, 5 3

2

3

𝑥 − 7𝑥 + 7𝑥 + 15 𝑥 − 6𝑥 − 𝑥 + 30 𝑥 − 2𝑥 − 13𝑥 − 10 𝑥 − 7𝑥 − 6 + − + 35 8 20 21 3 −8 ⋅ 20 ⋅ 21 + 35 ⋅ 20 ⋅ 21 − 35 ⋅ 8 ⋅ 21 + 35 ⋅ 8 ⋅ 20 =𝑥 ( ) 35 ⋅ 8 ⋅ 20 ⋅ 21 7 ⋅ 8 ⋅ 20 ⋅ 21 − 6 ⋅ 35 ⋅ 20 ⋅ 21 + 2 ⋅ 35 ⋅ 8 ⋅ 21 + 0 ⋅ 35 ⋅ 8 ⋅ 20 + 𝑥2 ( ) 35 ⋅ 8 ⋅ 20 ⋅ 21 −7 ⋅ 8 ⋅ 20 ⋅ 21 − 35 ⋅ 20 ⋅ 21 + 13 ⋅ 35 ⋅ 8 ⋅ 21 − 7 ⋅ 35 ⋅ 8 ⋅ 20 + 𝑥( ) 35 ⋅ 8 ⋅ 20 ⋅ 21 −15 ⋅ 8 ⋅ 20 ⋅ 21 + 30 ⋅ 35 ⋅ 20 ⋅ 21 + 10 ⋅ 35 ⋅ 8 ⋅ 21 − 6 ⋅ 35 ⋅ 8 ⋅ 20 +( ) 35 ⋅ 8 ⋅ 20 ⋅ 21 11060 3 −52920 2 −980 415800 = 𝑥 + 𝑥 + 𝑥+ 117600 117600 117600 117600 79 3 −378 2 −7 2970 = 𝑥 + 𝑥 + 𝑥+ 840 840 840 840 =−

✏

D.5.2 Newton interpolation Definition D.22. 71 The Newton polynomial 𝑁𝑃 ,𝑛 (𝑥) with respect to the 𝑛 + 1 points 𝑃 = {(𝑥𝑘 , 𝑦𝑘 )|𝑘 = 0, 1, 2, … , 𝑛} is defined as 𝑘

𝑛

𝑁𝑃 ,𝑛 (𝑥) ≜

∑ 𝑘=0

𝛼𝑘

∏

(𝑥 − 𝑥𝑚 )

𝑚=0

71

📓 Newton (1711), 📓 Fraser (1919), pages 9–17 ⟨Methodus differentialis: “A photographic reproduction of the original Latin text”⟩, 📓 Fraser (1919), pages 18–25 ⟨Methodus differentialis: English translation⟩, 📘 Fraser (1919), pages 1–8 ⟨historical background and notes⟩, 📃 Meijering (2002) ⟨historical background⟩, 📘 Matthews and Fink (1992), page 220

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Proposition D.5. Let 𝑁𝑃 ,𝑛 (𝑥) be the Newton polynomial with respect to the points 𝑃 = {(𝑥𝑘 , 𝑦𝑘 )|𝑘 = 0, 1, 2, … , 𝑛}. 1. 𝑁𝑃 ,𝑛 (𝑥) is an 𝑛th order polynomial. 2. 𝑁𝑃 ,𝑛 (𝑥) intersects all 𝑛 + 1 points in 𝑃 . Example D.19 (Newton polynomial interpolation). The Newton polynomial 𝑁𝑃 ,3 (𝑥) with respect to the 4 points 𝑃 = {(−2, 1), (−1, 3), (3, 2), (5, 4)} is 79 𝑥3 + −378 𝑥2 + −7 𝑥 + 2970 𝑁𝑃 ,3 (𝑥) = 840 840 840 840 ✎PROOF: 𝑘

𝑛

𝑁𝑃 ,3 (𝑥) =

∑

𝛼𝑘

𝑘=0

∏

(𝑥 − 𝑥𝑚 )

𝑚=1

= 𝛼0 + 𝛼1 (𝑥 − 𝑥0 ) + 𝛼2 (𝑥 − 𝑥0 )(𝑥 − 𝑥1 ) + 𝛼3 (𝑥 − 𝑥0 )(𝑥 − 𝑥1 )(𝑥 − 𝑥2 ) = 𝛼0 + 𝛼1 (𝑥 + 2) + 𝛼2 (𝑥 + 2)(𝑥 + 1) + 𝛼3 (𝑥 + 2)(𝑥 + 1)(𝑥 − 3) = 𝛼0 + 𝛼1 (𝑥 + 2) + 𝛼2 (𝑥2 + 3𝑥 + 2) + 𝛼3 (𝑥3 − 7𝑥 − 6) = 𝑥3 (𝛼3 ) + 𝑥2 (𝛼2 ) + 𝑥(−7𝛼3 + 3𝛼2 + 𝛼1 ) + (−6𝛼3 + 2𝛼2 + 2𝛼1 + 𝛼0 ) 0 ⎡ 1 ⎢ 2 1 = [𝛼0 𝛼1 𝛼2 𝛼3 ] ⎢ 2 3 ⎢ ⎣ −6 −7 ⎡ ⎢ ⎢ ⎢ ⎣

⎡ ⎢ ⎢ ⎢ ⎣

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⎤ ⎡ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎦ ⎣

𝑦0 𝑦1 𝑦2 𝑦3

⎡ ⎢ =⎢ ⎢ ⎣

1 1 1 1

⎤ ⎥ ⎥ ⎥ ⎦ 0 (𝑥1 − 𝑥0 ) (𝑥2 − 𝑥0 ) (𝑥3 − 𝑥0 )

⎡ ⎢ =⎢ ⎢ ⎣

1 1 1 1

0 (−1 + 2) (3 + 2) (5 + 2)

⎡ ⎢ =⎢ ⎢ ⎣

1 1 1 1

0 1 5 7

0 0 20 42

0 0 0 84

⎤⎡ ⎥⎢ ⎥⎢ ⎥⎢ ⎦⎣

0 0 0 1 0 0 5 20 0 7 42 84

1 0 0 0

0 1 0 0

0 0 1 0

0 0 1 0

0 0 0 1

⎤⎡ ⎥⎢ ⎥⎢ ⎥⎢ ⎦⎣

0 0 (𝑥2 − 𝑥0 )(𝑥2 − 𝑥1 ) (𝑥3 − 𝑥0 )(𝑥3 − 𝑥1 ) 0 0 (3 + 2)(3 + 1) (5 + 2)(5 + 1) 𝛼0 𝛼1 𝛼2 𝛼3 0 0 0 1

1 𝑥 𝑥2 𝑥3

⎤ ⎥ ⎥ ⎥ ⎦

0 0 0 (𝑥3 − 𝑥0 )(𝑥3 − 𝑥1 )(𝑥3 − 𝑥2 )

0 0 0 (5 + 2)(5 + 1)(5 − 3)

⎤⎡ ⎥⎢ ⎥⎢ ⎥⎢ ⎦⎣

𝛼0 𝛼1 𝛼2 𝛼3

⎤⎡ ⎥⎢ ⎥⎢ ⎥⎢ ⎦⎣

𝛼0 𝛼1 𝛼2 𝛼3

⎤ ⎥ ⎥ ⎥ ⎦

⎤ ⎥ ⎥ ⎥ ⎦

⎤ ⎥ ⎥ ⎥ ⎦

⎤ ⎡ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎦ ⎣

1 0 0 0

0 0 0 1 0 0 1 0 0 −1 1 0 5 20 0 −1 0 1 7 42 84 −1 0 0

⎡ ⎢ =⎢ ⎢ ⎣

1 0 0 0

0 0 0 1 0 0 1 0 0 −1 1 0 0 20 0 4 −5 1 0 42 84 6 −7 0

⎡ ⎢ =⎢ ⎢ ⎣

1 0 0 0

0 0 0 1 0 1 0 0 −1 1 1 −1 0 1 0 4 5 0 42 84 6 −7


0 0 0 1

0 0 1 20

0

⎤ ⎥ ⎥ ⎥ ⎦ 0 ⎤ 0 ⎥ 0 ⎥ ⎥ 1 ⎦ 0 ⎤ 0 ⎥ 0 ⎥ ⎥ 1 ⎦

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⎡ ⎢ ⎢ ⎢ ⎣

𝛼0 𝛼1 𝛼2 𝛼3

⎡ ⎢ =⎢ ⎢ ⎣

1 0 0 0

0 1 0 0

0 0 1 0 0 0 0 −1 1 0 1 1 1 1 0 − 20 4 5 42 − 42 0 84 6 − 42 −7 + 4 20 5

⎡ ⎢ =⎢ ⎢ ⎣

1 0 0 0

0 1 0 0

0 0 1 0 0 0 0 −1 1 0 1 1 1 1 0 −4 20 5 14 − 42 0 84 − 12 4 20 5

⎡ ⎢ =⎢ ⎢ ⎣

1 0 0 0

0 1 0 0

0 0 1 0 0 0 −1 1 4 5 1 0 − 20 20 35 0 84 − 24 10 10

⎡ ⎢ =⎢ ⎢ ⎣

1 0 0 0

0 1 0 0

0 0 1 0

1 0 0 ⎤ ⎡ 1 0 ⎥ ⎢ −1 1 4 5 ⎥=⎢ − 20 20 20 ⎥ ⎢ 24 35 21 − 840 ⎦ ⎣ − 840 840 ⎡ 1 ⎤ ⎢ 2 ⎥ =⎢ 9 ⎥ − 20 ⎢ 79 ⎥ ⎣ 840 ⎦

0 ⎤⎡ 1 0 ⎥⎢ 3 0 ⎥⎢ 2 ⎢ 10 ⎥ 4 840 ⎦ ⎣

0 ⎡ 1 1 ⎢ 2 𝑁𝑃 ,3 (𝑥) = [𝛼0 𝛼1 𝛼2 𝛼3 ] ⎢ 2 3 ⎢ ⎣ −6 −7 =[ 1 2

= [ 1+4−

=

9 − 20

9 10

−

79 840

79 140

0 1 0 0 −1 1 5 4 − 0 20 20 24 35 1 − 840 840

0 0 1 0

⎡ 1 ⎢ 2 ]⎢ 2 ⎢ ⎣ −6 2−

27 20

−

0 0 0 1

⎤ ⎥ ⎥ ⎥ ⎦

0 0 0 1

⎤ ⎥ ⎥ ⎥ ⎦ 0 0 ⎤ 0 0 ⎥ 1 0 ⎥ 20 ⎥ 21 10 − 10 10 ⎦ 0 0 ⎤ 0 0 ⎥ 1 0 ⎥ 20 21 10 ⎥ − 840 840 ⎦

⎤ ⎥ ⎥ ⎥ ⎦

⎤⎡ 1 ⎤ ⎥⎢ 𝑥 ⎥ ⎥ ⎢ 𝑥2 ⎥ ⎥⎢ 3 ⎥ ⎦⎣ 𝑥 ⎦ 0 0 0 ⎤⎡ 1 0 0 ⎥⎢ 3 1 0 ⎥⎢ ⎥⎢ −7 0 1 ⎦ ⎣ 0 0 0 1

79 120

9 − 20

79 840

1 𝑥 𝑥2 𝑥3

⎤ ⎥ ⎥ ⎥ ⎦

⎡ ⎢ ]⎢ ⎢ ⎣

1 𝑥 𝑥2 𝑥3

⎤ ⎥ ⎥ ⎥ ⎦

79 3 378 2 7 2970 𝑥 − 𝑥 − 𝑥+ 840 840 840 840

✏

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APPENDIX E C++ SOURCE CODE SUPPORT

This document seeks to conform to the principles of Reproducible Research as detailed at http://reproducibleresearch.net/. This section contains a partial C++ source code listing for ssp.exe, written by the author of this paper, which produced the TEX files used to generate the 128 or so data plot files presented in CHAPTER 3. The complete and downloadable source code for ssp.exe is to accompany any online version of this document. There are some who might hesitate to use computer simulation to demonstrate mathematical concepts. And arguably their concern is not unfounded. There is in fact evidence to suggest that while the invention of the printing press has greatly assisted the progress of mathematical discovery, the introduction of computers has harmed it. Evidence of this hypothesis is given in Figure E.1 (page 178).1 This graph shows the number of “notable” mathematicians alive during the last 3000 years; Here a “notable mathematician” is defined as one whose name appears in Saint Andrew's University's Who Was There website.2 Note the following: The number of mathematicians starts to exponentially increase at about the time of Gutenberg's invention of the printing press— that is, when information of discoveries and results could be widely and economically circulated.3 There is another increase after the invention of the slide rule in the early 1600s— that is, when computational power increased. There are huge increases around the time of the first and second industrial revolutions— that is, when there were many applications that called for mathematical solutions. After the invention of the pocket scientific calculator in 1972 and home IBM PC in 1981— machines that could often make hard-core mathmatical analysis unnecessary in real-world applications— there was a huge drop in the number of mathematicians.

E.1

Symbolic sequence routines

1 2

/*============================================================================ * Daniel J . Greenhoe 1

Data for Figure E.1 (page 178) extracted from http://www-history.mcs.st-andrews.ac.uk/Timelines/WhoWasThere.html 2 http://www-history.mcs.st-andrews.ac.uk/Timelines/WhoWasThere.html 3 This point is also made by Resnikoff and Wells in 📘 Resnikoff and Wells (1984), page 9.

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E.1. SYMBOLIC SEQUENCE ROUTINES IBM PC 653

calculator second industrial revolution industrial revolution

435

slide rule invented

302

Gutenberg printing press invented

144 0

12

1

14

-1000

-350

0

1000 Middle Ages / “Dark Ages”

59

21

1450 1600

1800

154 112 2000

Renaissance

Figure E.1: Number of notable mathematicians alive over time 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

* header f i l e f o r r o u t i n e s f o r char sequence functions *============================================================================*/ c l a s s symseq { private : long N; char * x ; public : symseq ( const long M) ; // c o n s t r u c t o r i n i t i a l i z i n g to ' . ' symseq ( const long M, const unsigned seed , const char * symbols ) ; // c o n s t r u c t o r i n i t i a l i z i n g using seed void c l e a r ( void ) ; // f i l l sequence with the value 'A ' char g e t ( const long n ) ; // g e t a value from x a t l o c a t i o n n char g e t ( const long n , char * symbols ) ; // g e t a value from x a t l o c a t i o n n but e x i t i f not in s t r i n g void put ( long n , char symbol ) ; void put ( const long s t a r t , const long end , const char symbol ) ; const long getN ( void ) { const long M=( const long )N; r e t u r n M; } // g e t N void downsample ( i n t M, symseq * y ) ; / / downsample by a f a c t o r o f and w r i t e to void l i s t ( const long s t a r t , const long end , const char * s t r 1 , const char * s t r 2 , const i n t d i s p l a y , FILE * f p t r ) ; void l i s t ( const long s t a r t , const long end , FILE * f p t r ) { l i s t ( s t a r t , end , ” ” , ” ” , 1 , f p t r ) ; } void l i s t ( const long s t a r t , const long end , const i n t d i s p l a y , FILE * f p t r ) { l i s t ( s t a r t , end , ”” , ”” ,1 , fptr ) ; } void l i s t ( const long s t a r t , const long end , const char * s t r 1 , const char * s t r 2 , FILE * f p t r ) { list ( start , end , str1 , str2 , 1 , fptr ) ; } void l i s t ( long s t a r t , long end ) { l i s t ( s t a r t , end , ” ” , ” ” ,NULL) ; } // l i s t contents o f sequence void l i s t ( void ) { l i s t (0 , N−1 , ” ” , ” ” ,NULL) ; } // l i s t contents o f sequence void l i s t ( long s t a r t ) { l i s t ( s t a r t , N−1 , ” ” , ” ” ,NULL) ; } // l i s t contents o f sequence void s h i f t L ( long n ) ; // s h i f t symseq n elements to the l e f t void s h i f t R ( long n ) ; // s h i f t symseq n elements to the r i g h t void prngseed ( unsigned seed ) { srand ( seed ) ; } // void randomize ( const char * symbols ) ; / / randomize using a l i s t o f symbols void randomize ( const unsigned seed , const char * symbols ) { prngseed ( seed ) ; randomize ( symbols ) ; } void operator =( symseq y ) ; // x=y void operator > >=( long n ) { s h i f t R ( n ) ; } // s h i f t symseq n elements to the r i g h t void operator < # include < s t d l i b . h> # include < s t r i n g . h> # include # include

13 14

/*−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

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APPENDIX E. C++ SOURCE CODE SUPPORT 15 16 17 18 19 20 21 22 23 24 25 26 27 28

* c o n s t r u c t o r i n i t i a l i z i n g symseq to ' . ' *−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*/ symseq : : symseq ( long M) { long n ; void *memptr ; N=M; memptr=malloc (N* s i z e o f ( char ) ) ; i f ( memptr==NULL) { f p r i n t f ( s t d e r r , ”symseq : : symseq memory a l l o c a t i o n f o r %l d elements f a i l e d \n” ,M) ; e x i t ( EXIT_FAILURE ) ; } x = ( char * ) memptr ; f o r ( n= 0 ; n=0;m−−)x [m]= x [m−n ] ; f o r (m= 0 ;m c l e a r ( ) ; i f ( end>=Nx ) { f p r i n t f ( s t d e r r , ”ERROR using copy ( s t a r t , end , seqR1 * x , seqR1 * y ) : =%l d i s too l a r g e . \ n” , end ) ; e x i t ( EXIT_FAILURE ) ; } i f ( ( end− s t a r t +1) ! =Ny) { f p r i n t f ( s t d e r r , ”ERROR using copy ( s t a r t , seqR1 * x , seqR1 * y ) : l e n g t h o f [ x _ s t a r t . . . x_end ] (% l d ) ! = l e n g t h o f y(% l d ) . \ n” ,Nx−s t a r t , Ny) ; e x i t ( EXIT_FAILURE ) ; } f o r ( n= s t a r t ,m= 0 ; ng e t ( n ) ; y−>put (m, xx ) ; } }

264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288

/*−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− * downsample sequence by a f a c t o r o f * and w r i t e to sequence pointed to by *−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*/ void downsample ( i n t M, symseq * x , symseq * y ) { const long Nx = x−>getN ( ) ; const long Ny = y−>getN ( ) ; char symbol ; long n ,m; i f (Mput (m, symbol ) ; } }

E.2

Die routines

1 2 3 4 5 6 7 8 9 10 11 12 13

/*============================================================================ * Daniel J . Greenhoe * header f i l e f o r r o u t i n e s f o r die r o u t i n e s * 'A '−−> die f a c e value 1 * 'B '−−> die f a c e value 2 * 'C '−−> die f a c e value 3 * 'D '−−> die f a c e value 4 * ' E '−−> die f a c e value 5 * ' F '−−> die f a c e value 6 *============================================================================*/ c l a s s dieseq : p u b l i c symseq { public : dieseq ( const long M) : symseq (M) { } ; / / c o n s t r u c t o r i n i t i a l i z i n g to ' . '

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dieseq ( const long M, const unsigned seed ) : symseq (M, seed , ”ABCDEF” ) { } ; / / c o n s t r u c t o r i n i t i a l i z i n g random v a l u e s void randomize ( void ) { symseq : : randomize ( ”ABCDEF” ) ; } // void randomize ( unsigned seed ) { srand ( seed ) ; randomize ( ) ; } int randomize ( long s t a r t , long end , i n t wA, i n t wB, i n t wC, i n t wD, i n t wE, i n t wF) ; int randomize ( unsigned seed , i n t wA, i n t wB, i n t wC, i n t wD, i n t wE, i n t wF) { srand ( seed ) ; r e t u r n randomize ( 0 , getN ( ) −1 ,wA, wB, wC,wD, wE, wF) ; } int randomize ( i n t wA, i n t wB, i n t wC, i n t wD, i n t wE, i n t wF) { r e t u r n randomize ( 0 , getN ( ) −1 ,wA, wB, wC,wD, wE, wF) ; } int randomize ( long s t a r t , long end , unsigned seed , i n t wA, i n t wB, i n t wC, i n t wD, i n t wE, i n t wF) { srand ( seed ) ; r e t u r n randomize ( s t a r t , end , wA, wB, wC,wD, wE, wF) ; } char g e t ( long n ) { r e t u r n symseq : : g e t ( n , ”ABCDEF” ) ; } // g e t a value from x a t l o c a t i o n n void put ( long n , char symbol ) { symseq : : put ( n , symbol ) ; } seqR1 dietoR1 ( void ) ; //map die f a c e v a l u e s to R^1 seqC1 dietoC1 ( void ) ; //map die f a c e v a l u e s to R^1 seqR1 dietoR1pam ( void ) ; //map die f a c e v a l u e s to R^1 using PAM scheme ( symmetric about zero ) seqR3 dietoR3 ( void ) ; //map die f a c e v a l u e s to R^3 seqR1 histogram ( const long s t a r t , const long end , i n t d i s p l a y , FILE * f p t r ) ; / / compute , d i s p l a y , and w r i t e histogram seqR1 histogram ( ) { r e t u r n histogram ( 0 , getN ( ) −1 ,0 ,NULL) ; } / / compute histogram seqR1 histogram ( const long s t a r t , const long end ) { r e t u r n histogram ( s t a r t , end , 0 ,NULL) ; } / / compute histogram seqR1 histogram ( i n t d i s p l a y , FILE * f p t r ) { r e t u r n histogram ( 0 , getN ( ) −1 ,1 , f p t r ) ; } / / p r i n t histogram to file seqR1 histogram ( FILE * f p t r ) { r e t u r n histogram ( 0 , getN ( ) −1 ,0 , f p t r ) ; } / / p r i n t histogram to f i l e void operator =( dieseq y ) ; // x=y

14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

};

33 34 35 36 37 38 39 40

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

extern extern extern extern extern extern

int double double vectR3 vectR6 complex

die_domain ( char c ) ; / / check i f value i s in the domain o f d ie die _dietoR 1 ( char c ) ; die_dietoR1pam ( char c ) ; die _di etoR3 ( char c ) ; die _di etoR6 ( char c ) ; die_dietoC1c ( char c ) ;

/*============================================================================ * Daniel J . Greenhoe * r o u t i n e s f o r Real Die d i e s e q s *============================================================================*/ /*===================================== * headers *=====================================*/ # include < s t d i o . h> # include < s t d l i b . h> # include < s t r i n g . h> # include # include # include # include < r1 . h> # include < r2 . h> # include < r3 . h> # include < r6 . h> # include # include < die . h>

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/*===================================== * prototyp es *=====================================*/ void phistogram ( seqR1 * data , const long s t a r t , const long end , FILE * p t r ) ;

25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

/*−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− * f i l l the dieseq with weighted pseudo−random die f a c e v a l u e s *−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*/ i n t dieseq : : randomize ( long s t a r t , long end , i n t wA, i n t wB, i n t wC, i n t wD, i n t wE, i n t wF) { int r , u ; long n ; char symbol ; i n t sum=wA+wB+wC+wD+wE+wF ; i f (sum! = 1 0 0 ) { f p r i n t f ( s t d e r r , ” dieseq : : randomize e r r o r : sum o f weight v a l u e s = %d ! = 100\n” ,sum) ; r e t u r n −1; } // p r i n t f ( ” s t a r t=%l d end=%l d weights =(%03d %03d %03d %03d %03d %03d ) \n” , s t a r t , end , wA, wB, wC,wD, wE, wF) ; f o r ( n= s t a r t ; n6 * a l l other v a l u e s −−> 0 *−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*/ seqR1 dieseq : : dietoR1 ( void ) { const long N=getN ( ) ; long n ; char sym ; double xR1 ; seqR1 y (N) ; f o r ( n= 0 ; n−2.5 B−−>−1.5 C−−>−0.5 D−−>0.5 E−−>1.5 F−−>2.5 * a l l other v a l u e s −−> 0 *−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*/ seqR1 dieseq : : dietoR1pam ( void ) { const long N=getN ( ) ; long n ; char sym ; double xR1 ; seqR1 y (N) ; f o r ( n= 0 ; nnumber o f dna 'C ' symbols , * y[4]−−>number o f dna 'D ' symbols , * y[5]−−>number o f dna 'D ' symbols , * y[6]−−>number o f dna 'D ' symbols , * y[0]−−>number o f a l l other v a l u e s * y[7]−−> t o t a l number o f symbols y [ 1 ] , y [ 2 ] , . . . , y [ 6 ] *−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*/ seqR1 dieseq : : histogram ( const long s t a r t , const long end , i n t d i s p l a y , FILE * f p t r ) { seqR1 data ( 8 ) ; long n ; long bin ; double p ; int i ; char symbol ; FILE * p t r ; data . c l e a r ( ) ; f o r ( n= s t a r t ; n|+1| C−−>| 0 | D−−>| 0 | E− − >| −1| F−−>| 0 | * | 0| | 0| |+1| | −1| | 0| | 0| *−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*/ vectR3 d ie_d ietoR3 ( char c ) { vectR3 xyz ; switch ( c ) { case 'A ' : xyz . put ( + 1 , 0 , 0 ) ; break ; case 'B ' : xyz . put ( 0 , + 1 , 0 ) ; break ; case 'C ' : xyz . put ( 0 , 0 , + 1 ) ; break ; case 'D ' : xyz . put ( 0 , 0 , −1) ; break ; case ' E ' : xyz . put ( 0 , −1 , 0 ) ; break ; case ' F ' : xyz . put ( −1 , 0 , 0 ) ; break ; default : f p r i n t f ( s t d e r r , ”ERROR using die_dieto R3 ( c ) : c=%c ( 0 x%x ) i s not in the v a l i d domain { A , B , C , D, E , F } \ n” , c , c ) ; e x i t ( EXIT_FAILURE ) ; } r e t u r n xyz ; }

303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326

/*−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− * map die f a c e v a l u e s to R^6 * A− − > ( 1 , 0 , 0 , 0 , 0 , 0 ) D− − > ( 0 , 0 , 0 , 1 , 0 , 0 ) * B− − > ( 0 , 1 , 0 , 0 , 0 , 0 ) E− − > ( 0 , 0 , 0 , 0 , 1 , 0 ) * C− − > ( 0 , 0 , 1 , 0 , 0 , 0 ) F− − >(0 ,0 ,0 ,0 ,0 ,1) * 0 − − >(0 ,0 ,0 ,0 ,0 ,0) * on ERROR r e t u r n ( 0 , 0 , 0 , 0 , 0 , 0 ) *−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*/ vectR6 d ie_d ietoR6 ( char c ) { vectR6 r s i x ; switch ( c ) { case 'A ' : r s i x . put ( 1 , 0 , 0 , 0 , 0 , 0 ) ; break ; case 'B ' : r s i x . put ( 0 , 1 , 0 , 0 , 0 , 0 ) ; break ; case 'C ' : r s i x . put ( 0 , 0 , 1 , 0 , 0 , 0 ) ; break ; case 'D ' : r s i x . put ( 0 , 0 , 0 , 1 , 0 , 0 ) ; break ; case ' E ' : r s i x . put ( 0 , 0 , 0 , 0 , 1 , 0 ) ; break ; case ' F ' : r s i x . put ( 0 , 0 , 0 , 0 , 0 , 1 ) ; break ; default : f p r i n t f ( s t d e r r , ”ERROR using dietoR6 ( char c ) : c=%c ( 0 x%x ) i s not in the v a l i d domain { 0 , A , B , C , D, E , F } . \ n” , c , c ) ; e x i t ( EXIT_FAILURE ) ; } return r s i x ; }

E.3

Real die routines

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/*============================================================================ * Daniel J . Greenhoe * header f i l e f o r r o u t i n e s f o r r e a l die r o u t i n e s * 'A '−−> die f a c e value 1 * 'B '−−> die f a c e value 2 * 'C '−−> die f a c e value 3 * 'D '−−> die f a c e value 4 * ' E '−−> die f a c e value 5 * ' F '−−> die f a c e value 6

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*============================================================================*/ c l a s s r d i e s e q : p u b l i c dieseq { public : r d i e s e q ( const long M) : dieseq (M) { } ; r d i e s e q ( const long M, const unsigned seed ) : dieseq (M, seed ) { } ; void operator =( dieseq y ) ; // x=y void operator > >=( long n ) { s h i f t R ( n ) ; } // s h i f t r d i e s e q n elements to the r i g h t void operator < # include < s t d l i b . h> # include # include # include # include < r1 . h> # include < r2 . h> # include < r3 . h> # include < r4 . h> # include < r6 . h> # include # include < e u c l i d . h> # include < l a r c . h> # include < die . h> # include < r e a l d i e . h>

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/*−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− * d i s p l a y r e a l die metric t a b l e *−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*/ i n t r d i e s e q : : m e t r i c t b l ( void ) { char a , b ; f o r ( a= 'A ' ; aput (m+N, rxxm ) ; i f ( showcount ) f p r i n t f ( s t d e r r , ” \b\b\b\b\b\b\b\b” ) ; } i f ( showcount ) f p r i n t f ( s t d e r r , ”%8l d . . . . done . \ n” ,m+N) ; return r v a l ; }

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/*−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− * a u t o c o r r e l a t i o n Rxx (m) *−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*/ double r d i e s e q : : Rxx ( const long m) { const long mm= l a b s (m) ; const long N=getN ( ) ; long n ,nmm; double d , sum ; char a , b ; f o r ( n=0 ,sum= 0 ; n2 C−−>3 D−−>4 E−−>5 F−−>6 *−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*/ i n t r d i e _ d i e t o R 1 ( char c ) { int n , rval ; char domain [ 6 ] = { 'A ' , 'B ' , 'C ' , 'D ' , ' E ' , ' F ' } ; char element ; f o r ( n=0 , r v a l = −1;n< 6 ;n++) i f ( c==domain [ n ] ) r v a l =n ; i f ( r v a l ==−1) { f p r i n t f ( s t d e r r , ”ERROR using r d i e _ d i e t o R 1 ( char c ) : c=%c ( 0 x%x ) i s not in the v a l i d domain { 0 , A , B , C , D, E , F } \ n” , c , c ) ; e x i t ( EXIT_FAILURE ) ; } return r v a l ; }

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E.3. REAL DIE ROUTINES

/*−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− * map die f a c e v a l u e s to R^3 * |+1| | 0| | 0| | 0| | 0| | −1| | 0| * A−−>| 0 | B− − >|+1| C−−>| 0 | D−−>| 0 | E− − >| −1| F−−>| 0 | 0−−>| 0 | * | 0| | 0| |+1| | −1| | 0| | 0| | 0| *−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*/ vectR3 r d i e _ d i e t o R 3 ( char c ) { vectR3 xyz ; switch ( c ) { case 'A ' : xyz . put ( + 1 , 0 , 0 ) ; break ; case 'B ' : xyz . put ( 0 , + 1 , 0 ) ; break ; case 'C ' : xyz . put ( 0 , 0 , + 1 ) ; break ; case 'D ' : xyz . put ( 0 , 0 , −1) ; break ; case ' E ' : xyz . put ( 0 , −1 , 0 ) ; break ; case ' F ' : xyz . put ( −1 , 0 , 0 ) ; break ; default : f p r i n t f ( s t d e r r , ”ERROR using r d i e _ d i e t o R 3 ( char c ) : c=%c ( 0 x%x ) i s not in the v a l i d domain { A , B , C , D, E , F } \ n” , c , c ) ; e x i t ( EXIT_FAILURE ) ; } r e t u r n xyz ; }

147 148 149 150 151 152 153 154 155 156 157 158 159

/*−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− * map R^3 v a l u e s to die f a c e v a l u e s using Lagrange Arc metric *−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*/ r d i e s e q r d i e _ R 3 t o d i e _ l a r c ( seqR3 xyz ) { long n ; i n t m; long N=xyz . getN ( ) ; double d [ 7 ] ; double s m a l l e s t d ; char c l o s e s t f a c e ; vectR3 p , q [ 7 ] ; r d i e s e q r d i e (N) ;

160 161 162 163 164 165 166 167

//q [ 0 ] . put ( 0 , 0 , 0 ) ; q[1]= rdie_dietoR3 ( q[2]= rdie_dietoR3 ( q[3]= rdie_dietoR3 ( q[4]= rdie_dietoR3 ( q[5]= rdie_dietoR3 ( q[6]= rdie_dietoR3 (

'A ' ) ; 'B ' ) ; 'C ' ) ; 'D ' ) ; 'E ' ) ; 'F ' ) ;

168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186

f o r ( n= 0 ; n6 *−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*/ seqR1 spinseq : : spintoR1 ( void ) { const long N=getN ( ) ; long n ; seqR1 y (N) ; f o r ( n= 0 ; n−1 G−−>−1 * a l l other v a l u e s −−> 0 *−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*/ seqR1 dnaseq : : dnatoR1bin ( void ) { const long N=getN ( ) ; long n ; char symbol ; seqR1 seqR1 (N) ; f o r ( n= 0 ; n(0 ,1 ,0 ,0) G− − >(0 ,0 ,0 ,1) * 0 − − >(0 ,0 ,0 ,0) * on ERROR r e t u r n ( 0 , 0 , 0 , 0 ) *−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*/ vectR4 dnatoR4c ( char c ) { vectR4 r f o u r ; switch ( c ) { case ' 0 ' : r f o u r . put ( 0 , 0 , 0 , 0 ) ; break ; case 'A ' : r f o u r . put ( 1 , 0 , 0 , 0 ) ; break ; case 'C ' : r f o u r . put ( 0 , 1 , 0 , 0 ) ; break ; case 'T ' : r f o u r . put ( 0 , 0 , 1 , 0 ) ; break ; case 'G ' : r f o u r . put ( 0 , 0 , 0 , 1 ) ; break ; d e f a u l t : r f o u r . put ( 0 , 0 , 0 , 0 ) ; f p r i n t f ( s t d e r r , ”ERROR using dnatoR4 ( char c ) : c=%c ( 0 x%x ) i s not in the v a l i d domain { A , C , T , G } . Returning ( 0 , 0 , 0 , 0 ) . \ n” , c , c ) ; } return rfour ; }

234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259

/*−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− * map gsp f a c e v a l u e s to R^1 * A−−>1 T−−>2 C−−>3 G−−>4 *−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*/ seqR1 dnaseq : : dnatoR1 ( void ) { const long N=getN ( ) ; long n ; char symbol ; seqR1 seqR1 (N) ; f o r ( n= 0 ; nnumber o f dna 'C ' symbols , * y[4]−−>number o f dna 'G ' symbols , * y[0]−−>number o f a l l other v a l u e s * y[5]−−> t o t a l number o f symbols y [ 1 ] , y [ 2 ] , . . . , y [ 5 ] *−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*/ seqR1 dnaseq : : histogram ( const long s t a r t , const long end , i n t d i s p l a y , FILE * f p t r ) { seqR1 data ( 6 ) ; long n ; long bin ; double p ; int i ; char symbol ; FILE * p t r ; data . c l e a r ( ) ; f o r ( n= s t a r t ; ng e t ( 1 ) +data−>g e t ( 2 ) , data−>g e t ( 3 ) +data−>g e t ( 4 ) , data−>g e t ( 0 ) ) ; f o r ( bin = 1 ; bin g e t ( bin ) / ( double )N* 1 0 0 . 0 ) ; f p r i n t f ( ptr , ” (%6.2 l f %%) (%6.2 l f %%) (%6.2 l f %%) | \ n” , ( data−>g e t ( 1 ) +data−>g e t ( 2 ) ) / ( double )N* 1 0 0 . 0 , ( data−>g e t ( 3 ) +data−>g e t ( 4 ) ) / ( double )N* 1 0 0 . 0 , data−>g e t ( 0 f p r i n t f ( ptr , ” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−\n” ) ; }

352 353 354 355 356 357 358 359 360 361

/*===================================== * operators *=====================================*/ /*−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− * operator dnaseq x = dnaseq y *−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*/ void dnaseq : : operator =( dnaseq y ) { const long N=getN ( ) ; const long M=y . getN ( ) ;

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long n ; char symbol ; i f (N! =M) { f p r i n t f ( s t d e r r , ” \nERROR using dnaseq : : operator = : l e n g t h o f x (% l d ) d i f f e r s from l e n g t h o f y (% l d ) . \ n” ,N,M) ; e x i t ( EXIT_FAILURE ) ; } f o r ( n= 0 ; n1 T−−>2 C−−>3 G−−>4 0−−>0 other −−>−1 *−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*/ double dna_dnatoR1 ( char symbol ) { double r ; switch ( symbol ) { case 'A ' : r = 1 . 0 ; break ; case 'T ' : r = 2 . 0 ; break ; case 'C ' : r = 3 . 0 ; break ; case 'G ' : r = 4 . 0 ; break ; default : f p r i n t f ( s t d e r r , ”ERROR using dna_dnatoR1 ( symbol ) : symbol='%c ' ( 0 x%x ) i s not in the v a l i d domain { A , T , C , G} \ n” , symbol , symbol ) ; e x i t ( EXIT_FAILURE ) ; } return r ; }

394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410

/*−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− * map dna symbols to R^2 *−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*/ vectR2 dna_dnatoR2 ( char symbol ) { vectR2 r ; switch ( symbol ) { case 'A ' : r . put ( 1 . 0 , 0 ) ; break ; case 'T ' : r . put ( − 1 . 0 , 0 ) ; break ; case 'C ' : r . put ( 0 , + 1 . 0 ) ; break ; case 'G ' : r . put ( 0 , −1.0) ; break ; default : f p r i n t f ( s t d e r r , ”ERROR using dna_dnatoR2 ( symbol ) : symbol='%c ' ( 0 x%x ) i s not in the v a l i d domain { A , T , C , G} \ n” , symbol , symbol ) ; e x i t ( EXIT_FAILURE ) ; } return r ; }

411 412 413 414 415 416 417 418 419 420 421 422 423 424

/*−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− * map R^2 v a l u e s to dna v a l u e s using Euclidean metric * 0 A B C D E F A+..+F *−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*/ dnaseq dna_R2todna_euclid ( seqR2 xy ) { long n ; i n t m; long N=xy . getN ( ) ; double d [ 5 ] ; double s m a l l e s t d ; char c l o s e s t f a c e ; vectR2 p , q [ 5 ] ; dnaseq rdna (N) ;

425 426 427 428 429 430

//q [ 0 ] . put ( 0 , 0 , 0 ) ; q [ 1 ] = dna_dnatoR2 ( 'A ' ) ; q [ 2 ] = dna_dnatoR2 ( 'T ' ) ; q [ 3 ] = dna_dnatoR2 ( 'C ' ) ; q [ 4 ] = dna_dnatoR2 ( 'G ' ) ;

431 432 433 434 435

f o r ( n= 0 ; n # include < r6 . h> # include < e u c l i d . h> # include < l a r c . h>

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E.6. LEGRANGE ARC DISTANCE ROUTINES

/*−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− * path l e n g t h s o f Lagrange arc from a point p a t polar coordinate ( rp , tp ) * to point q a t polar coordinate ( rq , tq ) . * __tq __tq * | | ( dr ) ^2 * s = | ds dtheta = | s q r t ( r ^2 + (−−−−−−−− ) ) dtheta * __ | tp __ | tp ( dtheta ) * * r e f e r e n c e : Paul Dawkins , * http : / / t u t o r i a l . math . lamar . edu/ C l a s s e s / C a l c I I / PolarArcLength . aspx * h t t p s : / / books . google . com/ books ? i d =b4ksCQAAQBAJ&pg=PA533 *−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*/ double l a r c _ a r c l e n g t h ( double rp , double rq , double t d i f f ) { double y ; const double phi= f a b s ( t d i f f ) ; const double rho=rq−rp ; const double sp= s q r t ( rp * rp * phi * phi+rho * rho ) ; const double sq= s q r t ( rq * rq * phi * phi+rho * rho ) ; const double up=rp * rho * phi+ f a b s ( rho ) * sp ; const double uq= rq * rho * phi+ f a b s ( rho ) * sq ; i f ( rp ==0) { f p r i n t f ( s t d e r r , ” \nERROR using l a r c _ a r c l e n g t h ( rp , rq , t d i f f ) : rp=% l f \n” , rp ) ; e x i t ( EXIT_FAILURE ) ; } i f ( rq ==0) { f p r i n t f ( s t d e r r , ” \nERROR using l a r c _ a r c l e n g t h ( rp , rq , t d i f f ) : rp=% l f \n” , rq ) ; e x i t ( EXIT_FAILURE ) ; } i f ( t d i f f PI ) { f p r i n t f ( s t d e r r , ” \nERROR using l a r c _ a r c l e n g t h ( rp , rq , t d i f f ) : t d i f f =%l f >PI \n” , t d i f f ) ; e x i t ( EXIT_FAILURE ) ; } // y = ( l a r c _ i n d e f i n t ( rp , rq , 0 , t d i f f , t d i f f )− l a r c _ i n d e f i n t ( rp , rq , 0 , t d i f f , 0 ) ) / t d i f f ; // y2 = ( rho >=0) ? ( f a b s ( rho ) / ( 2 * phi ) ) * ( log ( rq * phi+sq )−log ( rp * phi+sp ) ) // : ( f a b s ( rho ) / ( 2 * phi ) ) * ( log (− rq * phi+sq )−log (− rp * phi+sp ) ) ; i f ( f a b s ( rho ) %l f =maxerror b e s t r q=% l f t h e t a =%.2 l f PI \n” , s m a l l e s t e r r o r , maxerror , bestrq , t h e t a / PI ) ; e x i t ( EXIT_FAILURE ) ; } r e t u r n bestq ; }

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Back Matter ❝It

appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils.❞ Niels Henrik Abel (1802–1829), Norwegian mathematician

4

❝When

evening comes, I return home and go to my study. On the threshold I strip naked, taking off my muddy, sweaty workaday clothes, and put on the robes of court and palace, and in this graver dress I enter the courts of the ancients and am welcomed by them, and there I taste the food that alone is mine, and for which I was born. And there I make bold to speak to them and ask the motives of their actions, and they, in their humanity reply to me. And for the space of four hours I forget the world, remember no vexation, fear poverty no more, tremble no more at death; I pass indeed into their world.❞ 5

Niccolò Machiavelli (1469–1527), Italian political philosopher, in a 1513 letter to friend Francesco Vettori.

6

ancient library of Alexandria 4 5

The Book Worm by Carl Spitzweg, circa 1850

quote: 📘 Simmons (2007), page 187. image: http://en.wikipedia.org/wiki/Image:Niels_Henrik_Abel.jpg, public domain quote: 📘 Machiavelli (1961), page 139?. image: http://commons.wikimedia.org/wiki/File:Santi_di_Tito_-_Niccolo_Machiavelli%27s_portrait_headcrop.jpg,

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❝To sit alone in the lamplight with a book spread out before you, and hold intimate converse with men of unseen generations— such is a pleasure beyond compare.❞ Yoshida Kenko (Urabe Kaneyoshi) (1283? – 1350?), Japanese author and Buddhist monk

7

quote: image:

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📘 Kenko (circa 1330) http://en.wikipedia.org/wiki/Yoshida_Kenko A book concerning symbolic sequence processing

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BIBLIOGRAPHY

Colin Conrad Adams and Robert David Franzosa. Introduction to Topology: Pure and Applied. Featured Titles for Topology Series. Pearson Prentice Hall, 2008. ISBN 9780131848696. URL http://books.google.com/books?vid=ISBN0131848690. Charalambos D. Aliprantis and Owen Burkinshaw. Principles of Real Analysis. Acedemic Press, London, 3 edition, 1998. ISBN 9780120502578. URL http://www.amazon.com/dp/0120502577. Herbert Amann and Joachim Escher. Analysis II. Birkhäauser Verlag AG, Basel–Boston–Berlin, 2008. ISBN 978-3-7643-7472-3. URL http://books.google.com/books?vid=ISBN3764374721. Tom M. Apostol. Mathematical Analysis. Addison-Wesley series in mathematics. Addison-Wesley, Reading, 2 edition, 1975. ISBN 986-154-103-9. URL http://books.google.com/books?vid= ISBN0201002884. Aristotle. Metaphysics. University of Adelaide, Adelaide, 330BC? URL http://etext.library. adelaide.edu.au/a/aristotle/metaphysics/. Stefan Banach. Sur les opérations dans les ensembles abstraits et leur applications aux équations intégrales (on abstract operations and their applications to the integral equations). Fundamenta Mathematicae, 3:133–181, 1922. URL http://matwbn.icm.edu.pl/ksiazki/fm/fm3/ fm3120.pdf. Stefan Banach. Théorie des opérations linéaires. Monografje Matematyczne, Warsaw, Poland, 1932a. URL http://matwbn.icm.edu.pl/kstresc.php?tom=1&wyd=10. (Theory of linear operations). Stefan Banach. Theory of Linear Operations, volume 38 of North-Holland mathematical library. North-Holland, Amsterdam, 1932b. ISBN 0444701842. URL http://www.amazon.com/ dp/0444701842/. English translation of 1932 French edition, published in 1987. Alexander Barvinok. A Course in Convexity, volume 54 of Graduate studies in mathematics. American Mathematical Society, 2002. ISBN 9780821872314. URL http://books.google.com/books? vid=ISBN0821872311. Ladislav Beran. Orthomodular Lattices: Algebraic Approach. Mathematics and Its Applications (East European Series). D. Reidel Publishing Company, Dordrecht, 1985. ISBN 90-277-1715-X. URL http://books.google.com/books?vid=ISBN902771715X. Sterling Khazag Berberian. Introduction to Hilbert Space. Oxford University Press, New York, 1961. URL http://books.google.com/books?vid=ISBN0821819127. 217

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Mihaly Bessenyei and Zsolt Pales. A contraction principle in semimetric spaces. arXiv.org, January 8 2014. URL http://arxiv.org/abs/1401.1709. D. Besso. Teoremi elementari sui massimi i minimi. Annuari Ist. Tech. Roma, pages 7–24, 1879. see Bullen(2003) pages 453, 203. M. Jales Bienaymé. Société philomatique de paris—extraits des procès-verbaux. Scéance, pages 67–68, June 13 1840. URL http://www.archive.org/details/extraitsdesproc46183941soci. see Bullen(2003) pages 453, 203. Garrett Birkhoff. On the combination of subalgebras. Mathematical Proceedings of the Cambridge Philosophical Society, 29:441–464, October 1933. doi: 10.1017/S0305004100011464. URL http: //adsabs.harvard.edu/abs/1933MPCPS..29..441B. Garrett Birkhoff. Lattices and their applications. Bulletin of the American Mathematical Society, 44:1:793–800, 1938. doi: 10.1090/S0002-9904-1938-06866-8. URL http://www.ams.org/bull/ 1938-44-12/S0002-9904-1938-06866-8/. Garrett Birkhoff. Lattice Theory. American Mathematical Society, New York, 2 edition, 1948. URL http://books.google.com/books?vid=ISBN3540120440. R. B. Blackman and J. W. Tukey. The measurement of power spectra from the point of view of communications engineering—part ii. The Bell System Technical Journal, 37:485–569, March 1958. URL https://archive.org/download/bstj37-2-485. R. B. Blackman and J. W. Tukey. The Measurement of Power Spectra from the Point of View of Communications Engineering. Dover Publications, New York, 1959. ISBN 486-60507-8. URL https://archive.org/download/TheMeasurementOfPowerSpectra. “unabridged and corrected republication of the work originally published in January and March, 1958, in Volume XXXVII of the Bell System Technical Journal”. Leonard Mascot Blumenthal. Distance geometries: a study of the development of abstract metrics. The University of Missouri studies. A quarterly of research, 13(2):145, 1938. Leonard Mascot Blumenthal. Theory and Applications of Distance Geometry. Oxford at the Clarendon Press, 1 edition, 1953. ISBN 0-8284-0242-6. URL http://books.google.com/books?vid= ISBN0828402426. Béla Bollobás. Linear Analysis; an introductory course. Cambridge mathematical textbooks. Cambridge University Press, Cambridge, 2 edition, March 1 1999. ISBN 978-0521655774. URL http://books.google.com/books?vid=ISBN0521655773. Umberto Bottazzini. The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass. Springer-Verlag, New York, 1986. ISBN 0-387-96302-2. URL http://books.google. com/books?vid=ISBN0387963022. Bourbaki. Éléments de mathématique. Prewmieère partie: Les structures fondamentales de l'analyse. Livre I: Théorie des ensembles (Fascicule des résultats). Hermann & Cie, Paris, 1939. Bourbaki. Theory of Sets. Elements of Mathematics. Hermann, Paris, 1968. URL http://www. worldcat.org/oclc/281383. J. L. Brenner. Limits of means for large values of the variable. Pi Mu Epsilon Journal, 8(3):160–163, Fall 1985. URL http://www.pme-math.org/journal/issues.html. version 0.50


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Hermann Weyl. Emmy noether. Scripta Mathematica…, 3:201–220, April 26 1935. URL http:// link.springer.com/chapter/10.1007/978-1-4684-0535-4_6. memorial address at Bryn Mawr College. Norbert Wiener. A simplification of the logic of relations. Proceedings of the Cambridge Philosophical Society, 17:387–390, 1914. URL http://books.google.com/books?vid=ISBN0674324498&pg= PA224. reprinted in Jean Van Heijenoort's From Frege to G odel : A Source Book in Mathematical Logic, 1879-1931 page 224–227. Stephen Willard. General Topology. Addison-Wesley Series in Mathematics. Addison-Wesley, 1970. ISBN 9780486434797. URL http://books.google.com/books?vid=ISBN0486434796. a 2004 Dover edition has been published which “is an unabridged republication of the work originally published by the Addison-Wesley Publishing Company …1970”. W. A. Wilson. On quasi-metric spaces. American Journal of Mathematics, 53(3):675–684, July 1931a. URL http://www.jstor.org/stable/2371174. Wallace Alvin Wilson. On semi-metric spaces. American Journal of Mathematics, 53(2):361–373, April 1931b. URL http://www.jstor.org/stable/2370790. Robert S. Wolf. Proof, Logic, and Conjecture; The Mathematician's Toolbox. W.H. Freeman and Company, 1998. ISBN 0716730502. URL http://www.worldcat.org/isbn/0716730502. Qinglan Xia. The geodesic problem in quasimetric spaces. Journal of Geometric Analysis, 19(2):452– 479, April 2009. doi: 10.1007/s12220-008-9065-4. URL http://link.springer.com/article/10. 1007/s12220-008-9065-4. William Henry Young. On classes of summable functions and their fourier series. Proceedings of the Royal Society of London, 87(594):225–229, August 1912. URL http://www.archive.org/details/ philtrans02496252. Ernst Zermelo. Untersuchungen über die grundlagen der mengenlehre i. Mathematische Annalen, 65:261–281, 1908a. URL http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader? ht=VIEW&did=D38200. Ernst Zermelo. Investigations in the foundations of set theory. In Jean Van Heijenoort, editor, From Frege to G odel : A Source Book in Mathematical Logic, 1879-1931, pages 199–215. Harvard University Press, 1967, Cambridge, Massechusettes, 1908b. URL http://www.amazon.com/dp/ 0674324498. Vladimir A. Zorich. Mathematical Analysis I. Universitext Series. Springer Science & Business Media, 2004. ISBN 9783540403869. URL http://books.google.com/books?vid=ISBN3540403868.

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REFERENCE INDEX

Aliprantis and Burkinshaw (1998), 128, 141, 142, 151, 160, 171 Adams and Franzosa (2008), 21, 129 Amann and Escher (2008), 116 Apostol (1975), 6, 9, 24, 130, 143 Aristotle (330BC?), 77 Banach (1922), 149, 151 Banach (1932b), 151 Banach (1932a), 151 orgen Bang-Jensen and Gutin (2007), 8 Barvinok (2002), 9, 165 Beran (1985), 8 Berberian (1961), 139, 144, 164 Bessenyei and Pales (2014), 127, 137, 138 Besso (1879), 166 Bienaymé (1840), 166 Birkhoff (1933), 10, 11 Birkhoff (1938), 10 Birkhoff (1948), 10–12 Blackman and Tukey (1958), 25 Blackman and Tukey (1959), 25 Blumenthal (1938), 127 Blumenthal (1953), 127, 133– 136 Bollobás (1999), 149, 154, 165, 166, 169, 170 Bottazzini (1986), 6 Bourbaki (1939), 6 Bourbaki (1968), 5 Brenner (1985), 166 Brown and Watson (1996), 19, 129 Bruckner et al. (1997), 127, 150

Brunschwig et al. (2003), 136 Bryant (1985), 131 Bryc (2012), 22 Bullen (1990), 166 Bullen (2003), 166, 169 Burkill (2004), 23 Burris and Sankappanavar (1981), 10, 12 Burris and Sankappanavar (2000), 12–16 Burstein et al. (2005), 125 Busemann (1955), 155, 157 Cajori (1993), viii Carothers (2000), 166, 169 Cauchy (1821), 24, 169 Chatterji (1967), 19, 129 Cohn (2012), 5 Cohn (2002), 9 Comtet (1966), 19, 129 Comtet (1974), 6, 7, 19, 129 Copson (1968), 20, 127, 138, 155, 157 Corazza (1999), 158, 160 Cori and Lascar (2001), 21 Costa et al. (2004), 125 Crammer et al. (2007), 125 Crammer et al. (2008), 125 Cristianini and Hahn (2007), 104 Crossley (2006), 172 Curry and Feldman (2010), 22 Czerwik (1993), 138 Davey and Priestley (2002), 11 Davis and Hersh (1999), 137 Davis (2005), 5, 132, 157, 170, 171 Dedekind (1888b), 21 Dedekind (1888a), 21 Dedekind (1900), 8, 10, 12 Descartes (1637b), 1 Descartes (1637a), 1

237

Descartes (1684b), viii Descartes (1684a), viii Deza and Deza (2006), 107, 155, 159, 160 Deza and Deza (2009), 19, 158–160 Deza and Deza (2014), 107, 127, 138, 155 DiBenedetto (2002), 170 Dieudonné (1969), 128, 138, 139, 155–157 Doboš (1998), 160 Dobrowolski and Mogilski (1995), 150 Dominguez-Torres (2010), 24 Dominguez-Torres (2015), 24 Durbin (2000), 7 Duthie (1942), 9 Euclid (circa 300BC), 138, 155 Euler (1783), 173 Evans et al. (1967), 19, 129 Ewen (1950), vi Ewen (1961), vi Fagin et al. (2003b), 138 Fagin et al. (2003a), 138 Feldman and Valdez-Flores (2010), 22, 27 Fourier (1820), 164 Fraenkel (1922), 5 Fraser (1919), 174 Fréchet (1906), 138, 155 Fréchet (1928), 138, 155 Friedman (1970), 150 Fuhrmann (2012), 7 Galleani and Garello (2010), vii, 83, 100 Galvin and Shore (1984), 127, 128 Gauss (1866), 173 Gemignani (1972), 171

page 238 GenBank (2014), 2, 45, 63 Gibbons et al. (1977), 138 Giles (1987), 19, 130, 131, 142, 143, 151, 152, 155, 157, 159 Giles (2000), 151, 152 Grabiec et al. (2006), 155 Gradshteyn and Ryzhik (2007), 111, 113 Grattan-Guinness (1990), 107 Grätzer (2003), 11 Greenhoe (2013), 101 Greenhoe (2016a), xvi Greenhoe (2016b), xvi, 127 Greenhoe (2016c), xvi, 121 Greenhoe (2015), xvi, 22, 160 S. G. Gregory (2006), 2, 45, 63 Haaser and Sullivan (1991), 116, 131, 149, 172 Halmos (1948), 149 Halmos (1950), 7 Halmos (1960), 5, 6, 9, 21 Haray (1969), 8 Hardy (1929), 163 Hardy et al. (1952), 165, 166, 169 Hardy (1940), vi Hausdorff (1914), 170 Hausdorff (1937), 6, 9, 127, 132, 138, 155, 170 Runtao He (2004), 2, 45, 63 Heath (1961), 128 Heijenoort (1967), vi Heinonen (2001), 138 Hijab (2016), 116 Hocking and Young (1961), 21 Hoehn and Niven (1985), 138, 169 Housman (1936), vi Hull and Dobell (1962), 59 Isham (1989), 19 Isham (1999), 19, 160 Jeffrey (1995), 111, 112 Jeffrey and Dai (2008), 111, 112 Jensen (1906), 165, 169 Jiménez and Yukich (2006), 125 Joshi (1983), 21, 129, 170, 172 Jůza (1956), 160 Kelley (1955), 5, 6, 158, 171 Kelly (1963), 155 Kenko (circa 1330), 216 Jr. and Gentle (1980), 59 Khamsi and Kirk (2001), 19, 20, 127, 130, 157 Kirk and Shahzad (2014), 138 Klauder (2010), 19 Kolmogorov (1933), 22

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REFERENCE INDEX Korselt (1894), 8, 11 Krishnamurthy (1966), 19, 129 Kubrusly (2001), 19, 131, 149, 171, 172 Kubrusly (2011), 170 Kuratowski (1921), 5 Kuratowski (1961), 5 Lagrange (1795), 107 Lagrange (1877), 110, 173 Landau (1966), 21 Laos (1998), 127, 136 Leathem (1905), 130, 172 Lin (1998), 125 Loève (1977), 150 Machiavelli (1961), 215 MacLane and Birkhoff (1999), 8, 10 Maddox (1989), 150 Maddux (2006), 6 Maeda and Maeda (1970), 10 Maligranda and Orlicz (1987), 125 Maligranda (1995), 166, 169 Matthews and Fink (1992), 173, 174 McCarty (1967), 171, 172 Meijering (2002), 108, 110, 173, 174 Mendel (1853), 2, 45, 63 Menger (1928), 127 Michel and Herget (1993), 7, 127, 139, 151, 155, 156 Miller (2006), 22 Milovanović and Milovanović (1979), 166 Minkowski (1910), 166, 169 Mitrinović et al. (2010), 149, 165 Molchanov (2005), 127 Mulholland (1950), 166 Müller-Olm (1997), 11 Munkres (2000), 5, 6, 170– 172 Murdeshwar (1990), 21, 129 GenBank-AF086833.2 (2013), 81 GenBank-NZ_CM003360.1 (2015), 81 GenBank-DS982815.1 (2015), 81 GenBank-NC_004718.3 (2011), 80 Newton (1711), 110, 174 Norfolk (1991), 150 Oikhberg and Rosenthal (2007), 153 Oppenheim and Schafer (1999), 25 Ore (1935), 8–10 Pap (1995), 7 A book concerning symbolic sequence processing

Papoulis (1991), 22 Patty (1967), 155 Peano (1888b), 149 Peano (1888a), 149 Peano (1889b), 21 Peano (1889a), 21 Pečarić et al. (1992), 165 Peirce (1880), 9 Peirce (1883), 6 Poincaré (1902b), 27 Poincaré (1902a), 27 Pommerville (2013), 2, 45, 63 Ponnusamy (2002), 172, 173 Prabhu (2013), 25 Lycaeus (circa 450), 125 Rana (2002), 5 Ratcliffe (2013), 107 Resnikoff and Wells (1984), 177 Ribeiro (1943), 155 Riesz (1909), 170 Rolewicz (1985), 150 Rosenlicht (1968), 128, 130– 132, 142–144, 157, 172 Rota (1964), 10 Rota (1997), 10 Rudeanu (2001), 15 Rudin (1976), 168, 169 Salzmann et al. (2007), 21, 129 Schechter (1996), 116 Schubring (2005), 149 Schweizer and Sklar (1983), 137 Searcóid (2006), 172 Severance (2001), 59 Shen and Vereshchagin (2002), 9 Sherstnev (1962), 137 Silver and Stokes (2007), 107 Simmons (2007), xvi, 215 Simmons (2016), 23 Simon (2011), 165 Singh et al. (1997), 102 Snell (1997), 22 Snell (2005), 22 Steen and Seebach (1978), 21, 160, 170 Steiner (1966), 19 Stewart (2012), 110 Stoltenberg (1969), 155 Stopple (2003), 85 Suppes (1972), 5, 6 Sutherland (1975), 131 Szirmai (2007), 125 Thomson et al. (2008), 144 Thron (1966), 19, 127, 171 Thurston (1956), 21 Tietze (1923), 170 Tolsted (1964), 166, 169 Topkis (1978), 15

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REFERENCE INDEX Vallin (1999), 158, 160 van de Vel (1993), 149 Veltkamp and Hagedoorn (2000), 125 Veltkamp (2001), 125 Vitányi (2011), 125 Voss (1992), 83 Walmsley (1920), 136

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Waring (1779), 110, 173 Watson and Crick (1953b), 2, 45, 63 Watson and Crick (1953a), 2, 45, 63 Weyl (1935), 170 Wiener (1914), 5 Willard (1970), 21


Wilson (1931b), 127 Wilson (1931a), 155 Wolf (1998), 5 Xia (2009), 138 Young (1912), 169 Zermelo (1908b), 5 Zermelo (1908a), 5 Zorich (2004), 141

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SUBJECT INDEX

2𝘕 -offset autocorrelation, 78, 81, 86 ℝ1 die distance linear space, 83, 85 ℝ1 spinner random variable, 83, 87 ℝ2 spinner distance linear space, 84, 84, 85, 88 ℝ3 die distance linear space, 84, 84, 85 ℝ3 die random variable, 83, 84, 86, 91 ℝ3 distance linear space, 86, 91 ℝ4 DNA random variable, 83, 100, 101, 103, 104 ℝ6 die distance linear space, 84, 89 ℝ6 die random variable, 83, 84, 89, 96–98, 102 ℝ6 fair die distance linear space, 84 ℝ𝘕 ordered distance linear space, xi 𝛼-scaled, 159 𝛼-scaled metric, 159 𝛼-truncated, 159 𝛼-truncated metric, 159 (𝑋, 𝖽) is not translation invariant, 120 𝗀-transform metric, 19, 20 𝜙-mean, x, 165 1 𝜋 -scaled Euclidean metric, 122 𝘔 -offset autocorrelation, 78 𝘕 -tuple, 157 𝑛th-moment, x, 27 (strictly monotonically increasing in 𝜙), 116 LATEX, ii TEX-Gyre Project, ii XƎLATEX, ii GLB, 10

LUB, 10 σ-algebra, x, 22 σ-inframetric inequality, 126, 138, 138 σ-inframetric space, 126, 138 3-tuple, 6 6 element discrete metric, 21, 22 6 element ring, 21, 22 Abel, Niels Henrik, 215 absolute value, ix, x, 9, 9, 110, 111, 133, 139, 151, 156, 157 absorptive, 10, 12 accumulation point, 126, 127, 171 addition operator, xi, 7, 7, 24, 26 additive, 22 additive identity element, 31, 32, 37, 46 additive property, 23 adherent, 171 Adobe Systems Incorporated, ii affine, 139, 165 algebra of set, 19 algebra of sets, x alphabetic order relation, 9 AM-GM inequality, 169 AND, ix anti-symmetric, 8 antitone, 15, 15 arcs, 8 Aristotle, 77 arithmatic operator, 110, 111 arithmetic center, x, 28, 28, 33 arithmetic mean, 126, 139 arithmetic mean geometric mean inequality, 169 associates, 149 associative, 10, 149, 164

241

asymmetric metric, 155 auto-correlation, x autocorrelation, 77, 78, 78, 85 Avant-Garde, ii b-metric, 138 Bacterium DNA sequence, 81 ball, 121 base, 33, 126–130, 133–135, 141, 170, 170, 171 bijection, 48 bijective, 7, 7, 13, 16, 51 board game spinner outcome subspace, 35 Bohr, Harald, 163 bound greatest lower bound, 10 infimum, 10 least upper bound, 10 supremum, 10 boundary, 171 bounded, 126, 127, 131, 143, 159 bounded metric, 160 Bourbaki notation, 5 Cantor Intersection Theorem, 126 Cantor intersection theorem, 132, 132 cardinality, 76 cardinality of 𝐴, 7 carica papaya, 81 Carl Spitzweg, 215 Cartesian coordinate system, 108 Cartesian product, 6 Cartesian product of 𝑋 and 𝑌 , viii Cauchy, 126, 128, 130, 131, 134–136, 143, 146

page 242 Cauchy condition, 130 Cauchy sequence, 130 Cauchy sequences, 130 ceiling, x ceiling function, 10, 10 center, x, 3, 8, 27, 45, 74, 76 center measures, 37 chain, 8 chain rule, 111, 112 characterization, 155 closed, 128, 128, 131, 132, 170, 171–173 closed ball, x, 128 closed interval, ix, 9 closed set, 126, 127, 128 closed set theorem, 173 closure, x, 126, 127, 131, 171, 171, 172 closure point, 171 commutative, 10, 62, 140, 144, 149 commutative ring, 9 comparable, 8, 8 complement, ix complete, 130, 131, 132 complex numbers, viii complex plane, 21, 21, 49, 52, 64 concave, 149, 165, 165, 166 constant-sequence addition, xi constant-sequence multiplication, xi continuity, 115, 117, 118, 126, 127, 133, 136, 139, 172, 173 continuous, 119, 126, 128, 133, 133, 136–139, 144, 146, 147, 151, 165, 166, 171, 171– 173 contradiction, 145 converge, 130 convergence, 19, 126, 127, 134, 143, 173 metric space, 142 convergent, 126, 128, 130, 130, 131, 133–136, 143, 144, 146, 172, 173 convergent sequence, 126 converges, 172, 172 converse, 6 Convex, 10 convex, 9, 23, 120, 149, 150, 152, 153, 165, 165, 166 functional, 165 strictly, 165 convex subset, 24, 77 convolution, xi, 24, 24, 25 Coordinatewise order relation, 9 coordinatewise order relation, 9, 21

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SUBJECT INDEX cross-correlation, x, 78 decreasing, 165 definitions ℝ1 die distance linear space, 83 ℝ2 spinner distance linear space, 84 ℝ3 die distance linear space, 84 ℝ6 die distance linear space, 84 𝑛th-moment, 27 σ-inframetric space, 138 6 element discrete metric, 22 6 element ring, 22 accumulation point, 171 alphabetic order relation, 9 arcs, 8 arithmetic center, 28 base, 33, 170 bijective, 7 Cartesian product, 6 Cauchy sequence, 130 center, 8 chain, 8 closed ball, 128 closed interval, 9 closed set, 128 closure point, 171 complex plane, 21 coordinatewise order relation, 9 dictionary order relation, 9 directed graph, 8 distance space, 127 DNA outcome subspace, 29 DNA scaffold outcome subspace, 29 empty set, 5 extended probability space, 27 extended real numbers, 5 fair die outcome subspace, 28 fully ordered set, 8 function, 6 genome outcome subspace, 29 genome scaffold outcome subspace, 29 geometric center, 28 graph, 8 half-open interval, 9 harmonic center, 28 inframetric space, 138 A book concerning symbolic sequence processing

injective, 7 integer line, 21 interior, 171 interior point, 171 isometry, 19 Lagrange polynomial, 173 lattice, 10 lexicographical order relation, 9 linear order relation, 8 linear space, 149 linearly ordered set, 8 maxmin center, 28 metric, 155 metric geometry, 32 metric linear space, 150 metric space, 138, 155 minimal center, 28 moment, 27 n-tuple, 6 near metric space, 138 Newton polynomial, 174 normed linear space, 151 one-to-one, 7 onto, 7 open ball, 32, 128 open interval, 9 open set, 128 ordered metric space, 20 ordered pair, 5 ordered quasi-metric space, 20 ordered set, 8 ordered triple, 6 outcome center, 28 outcome subspace, 27 outcome variance, 28 partially ordered set, 8 point of adherence, 171 poset, 8 power distance space, 138 probability space, 22 quasi-metric space, 155 real die outcome subspace, 29 real line, 21 relation, 6 sequence, 23 set of all functions, 6 set of all relations, 6 set of complex numbers, 6 set of integers, 5 set of natural numbers, 5 set of non-negative real numbers, 5

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SUBJECT INDEX set of outcomes, 22 set of postive real numbers, 5 set of real numbers, 5 set of whole numbers, 5 spinner outcome subspace, 29 summation, 164 surjective, 7 topological space, 32 topological space induced by (𝑋, 𝖽), 130 topology, 33, 170 topology induced by (𝑋, 𝖽) on 𝑋, 130 totally ordered set, 8 triple, 6 undirected graph, 8 unordered metric space, 20 unordered quasi-metric space, 20 unordered set, 8 vertices, 8 weighted die outcome subspace, 28 weighted graph, 8 weighted real die outcome subspace, 29 Descartes, René, viii, 1 diameter, x, 127 dicrete metric preserving function, 75 dictionary order relation, 9 die outcome subspace, 78 die sequence, 85, 86, 89, 91 difference, ix dilated, 159 dilated Euclidean, 135, 146 dilated metric, 159 directed, 8, 8 directed graph, 8, 43 directed metric, 155 directed PRNG state machine, 21 directed weighted graph, 3 discontinuity, 117–119 discontinuous, 118, 119, 133, 133–136, 171, 171, 172 discrete, 134, 145, 159 Discrete Fourier Transform, xi, 96–98 discrete Fourier transform, 26, 26 discrete metric, 2, 22, 28–30, 45, 46, 49, 51, 63, 65, 77, 157, 157 discrete metric geometry, 3 discrete metric preserving function, 75, 160 discrete metric transform on 2016 Jul 4

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outcome subspaces, 75 discrete topology, 32, 33, 170, 172 distance, x, 4, 19, 78, 82, 110, 127, 127, 128, 133–136 distance function, 2, 19, 115, 119, 133, 135, 136 Distance space, 4 distance space, x, 3, 4, 19, 120, 125, 126, 127, 127–139, 143–147 distance spaces, 126 distributes, 72, 149 distributive, 11 distributive inequalities, 11 diverge, 135, 172 divergent, 172 DNA, 29, 45 DNA mapping with extended range, 65 DNA outcome subspace, 29 DNA scaffold outcome subspace, 29 DNA to linear structures, 62 domain, ix, 6, 24, 25, 107 dot product, 109 down sample, 91, 94 down sampled by a factor of 𝘔 , 24 dual, 9, 21 Ebola virus DNA sequence, 81 edge weight, 8 empty set, viii, 5, 128 equality by definition, ix equality relation, ix Euclidean, 133–135, 145, 146 Euclidean metric, 83–85, 87– 95, 107, 110, 120–122, 157 Euclidean metric space, 146 events, 22 examples 𝛼-scaled, 159 𝛼-scaled metric, 159 𝛼-truncated, 159 𝛼-truncated metric, 159 Bacterium DNA sequence, 81 board game spinner outcome subspace, 35 bounded, 159 bounded metric, 160 Coordinatewise order relation, 9 dilated, 159 dilated metric, 159 discrete, 159 discrete metric preserving function, 160 discrete metric transform on outcome subspaces, A book concerning symbolic sequence processing

75 DNA, 45 DNA mapping with extended range, 65 DNA to linear structures, 62 Ebola virus DNA sequence, 81 fair die, 56, 57 fair die mapping to isomorphic structure, 51 fair die mapping with extended range, 52 fair die mappings to real line and integer line, 49 fair die sequence, 78 Fourier analysis of Ebola DNA sequence, 100 GSP to complex plane, 64 GSP with Markov model, 65 high pass filtering of weighted die sequence, 94 high pass filtering of weighted real die sequence, 91 high pass filtering of weighted spinner sequence, 92 Inverse tangent metric, 20 Lagrange arc distance balls in ℝ2 , 123 Lagrange arc distance balls in ℝ3 , 123 Lagrange arc distance in ℝ2 , 122 Lagrange arc distance in ℝ3 , 122 Lagrange arc distance versus Euclidean metric, 121 LCG mappings standard ordering, 61 LCG mappings, sequential directed graph, 62 LCG mappings, sequential ordering, 61 LCG mappings, standard ordering, 59 length 1200 nonstationary die sequence with 10Hz oscillating mean, 95 length 12000 nonstationary artificial DNA sequence with 10Hz oscillating mean, 98 length 12000 nonstationary die sequence with 100Hz oscillating mean, 97 length 12000 non-

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page 244 stationary die sequence with 10Hz oscillating mean, 97 Lexicographical order relation, 9 linear addition, 72 linear addition with metric transform, 76 low pass filtering of fair die sequence, 89 low pass filtering of real die sequence, 85 low pass filtering of spinner sequence, 87 pair of dice and hypothesis testing, 68 pair of dice outcome subspace, 66 pair of spinner and hypothesis testing, 71 pair of spinners, 70 Papaya DNA sequence, 81 power transform, 76, 159 power transform metric, 159 radar screen, 159 radar screen metric, 159 Random DNA sequence, 80 real die mappings, 53 real die sequence, 78 ring multiplication, 73 SARS coronavirus DNA sequence, 80 snowflake, 159 snowflake transform, 76 snowflake transform metric, 159 spinner mappings, 56 spinner outcome subspace, 57 spinner sequence, 78 statistical edge detection using Haar wavelet on non-stationary artificial DNA sequence, 102 statistical edge detection using Haar wavelet on non-stationary die sequence, 101 the standard topology on the real line, 171 The usual norm, 151 Wavelet analysis of Phage Lambda DNA sequence, 104 weighted die mappings, 54 weighted die sequence, 79, 80

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SUBJECT INDEX weighted real die sequence, 79 weighted ring, 39 weighted ring outcome subspace, 60 weighted spinner mappings, 57 weighted spinner outcome subspace, 37, 58 weighted spinner sequence, 80, 80 exclusive OR, ix existential quantifier, ix expectation, 3 expected value, 29 extended probability space, x, 27, 27 extended real numbers, viii, 5, 137 extended set of integers, viii, 5 extension, 82, 107, 114 fair die, 2, 29, 51, 56, 57, 75, 78 fair die mapping to isomorphic structure, 51 fair die mapping with extended range, 52 fair die mappings to real line and integer line, 49 fair die outcome space, 90 fair die outcome subspace, 28, 29, 32, 49, 51, 52, 75, 84 fair die sequence, 78 Fast Wavelet Transform, 101 field, iii, 3, 24, 25, 78, 149 field of real numbers, 7 Filter, 85–89, 91 filter, 25, 85, 87, 89, 91, 92, 94, 102 filter bank, 101 filtered, 25 finite, 7 FIR filtering, 3 First Fundamental Theorem of Calculus, 116, 118 floor, x floor function, 10, 10 FontLab Studio, ii for each, ix Fourier analysis, 3 Fourier analysis of Ebola DNA sequence, 100 Free Software Foundation, ii Fréchet product metric, 77 full period, 59 fully ordered set, 8 function, iii, 6, 7, 8, 15, 16, 19, 22, 23, 84, 172 functions ℝ3 die random variable, A book concerning symbolic sequence processing

84

ℝ4 DNA random variable, 100, 101, 103, 104 ℝ6 die random variable, 84, 96–98, 102 1 𝜋 -scaled Euclidean metric, 122 absolute value, x, 9, 9, 110, 111, 133, 139, 151, 156, 157 arithmetic center, 33 arithmetic mean, 139 asymmetric metric, 155 auto-correlation, x autocorrelation, 78 cardinality, 76 cardinality of 𝐴, 7 ceiling function, 10, 10 center, 3 center measures, 37 constant-sequence addition, xi constant-sequence multiplication, xi cross-correlation, x, 78 diameter, x dicrete metric preserving function, 75 die sequence, 85, 86, 89, 91 directed metric, 155 discrete metric, 2, 22, 28, 29, 45, 46, 49, 51, 63, 65, 77, 157 discrete metric preserving function, 75 distance, x, 4, 19, 78, 82, 110, 127, 127, 128, 133–136 distance function, 19, 115, 119, 133, 135, 136 distance space, 120 edge weight, 8 Euclidean metric, 83–85, 87–95, 107, 110, 120–122, 157 expectation, 3 floor function, 10, 10 geometric mean, 139 great circle metric, 107 harmonic mean, 139 isometry, 84, 85 isomorphism, 13 Lagrange arc distance, 4, 84, 86–88, 90–93, 95, 110, 119–123 Lagrange arc distance space, 120 Langrange arc distance, 108, 110, 114, 115, 118, 120 length 𝘔 Haar scaling sequence, 26 length 𝘔 Haar wavelet

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SUBJECT INDEX sequence, 26 length 𝘔 high pass Hanning sequence, 26 length 𝘔 high pass rectangular sequence, 25 length 𝘔 low pass Hanning sequence, 25 length 𝘔 low pass rectangular sequence, 25 length 16 Hanning low pass sequence, 86, 88, 90 length 16 high pass rectangular sequence, 25 length 16 low pass rectangular sequence, 25 length 16 Rectangular low pass sequence, 88, 89 length 16 rectangular low pass sequence, 85–87, 89 length 1600 Haar scaling sequence, 104 length 200 Haar wavelet sequence, 102, 103 length 50 Hanning low pass sequence, 86, 89, 90 length 50 high pass Hanning sequence, 26 length 50 high pass rectangular sequence, 92 length 50 low pass Hanning sequence, 25 length 8 Haar scaling sequence, 26 length 8 Haar wavelet sequence, 26 length 9 Haar wavelet sequence, 26 limit, x low pass rectangular sequence, 87 low pass sequence, 85, 89 maximum, 139 metric, 4, 19, 20, 29, 43, 77, 110, 120, 135, 157, 163 metric function, 126, 133 metric induced by the norm, 151 metric preserving function, 74–76, 158, 158–160 metric transform, 74 minimum, 139 modulus, 9 natural log, 110, 111 Newton polynomial, 108 norm, 120, 151 ordered triple, ix outcome center, 33, 46 outcome expected value, xi, 46, 47, 47, 49, 55 2016 Jul 4

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outcome random variable, 46, 46, 52, 68 outcome random variables, 59 outcome subspace sequence metric, 77, 77, 78 outcome subspace variance, 47 outcome variance, xi, 28, 37, 46, 47, 49 PAM die random variable, 96–98, 102 PAM DNA random variable, 99, 100, 103, 104 polar angle, 120 power distance, 138 power distance function, 138, 139, 147 Power mean, 126, 137 power mean, x, 126, 137, 166, 166 power triangle function, x, 126, 137, 137–139, 143, 144 power triangle inequality, 4 probability function, 3, 22, 29 pullback, 48 pullback function, 75 pushforward, 48 QPSK die random variable, 96–98, 102 QPSK DNA random variable, 99, 100, 103, 104 QPSK spinner random variable, 84 quadratic mean, 139 quasi-metric, 20, 43, 54, 62, 155, 155 random variable, 46, 47, 49, 53, 54, 71, 82 real die sequence, 4 real-valued random variable, iii, 1, 22 RMS, 100–102 root mean square, 100 scaling function, 101 sequence, 3, 78, 88, 92, 127, 130–132 set function, 7 slope-intercept, 108 spherical metric, 4, 107, 108, 110, 114 spinner sequence, 4, 87, 92 square root, 110, 111 sum, x traditional die random variable, 94 traditional expected value, x, 47, 49, 51, 55 A book concerning symbolic sequence processing

traditional random variable, 1, 2, 22, 49 traditional variance, x, 47, 49 triangle function, 137, 146 usual metric, iii, 1, 21, 49, 51, 65, 157 variance, 28 vertice weight, 8 wavelet, 101 weighted die sequence, 94 Gauss, Karl Friedrich, xvi GenBank, 80 Generalized AM-GM inequality, 169 generalized arithmetic mean geometric mean inequality, 169 Generalized associative property, 164 genome, 2, 45, 62, 63 genome outcome subspace, 29 genome scaffold outcome subspace, 29 Genomic Signal Processing, 2, 45, 62 geometric center, x, 28 geometric inequality, 138 geometric mean, 126, 139 GNU Octave clear, 178, 179, 182, 185, 204 cos, 209, 212 data, 185, 204 nolig , 183, 185, 200, 204 exit, 178–182, 186, 187, 189, 190, 193, 195, 196, 200– 205, 207, 208, 210, 212, 213 function, 210, 211 functions, 178, 181 help, 207, 208 if, 79, 80, 178–185, 188– 201, 203–213 int, 178–183, 185, 188– 201, 203–209, 211–213 log, 210 mapping, 202 output, 182 polar, 209, 210, 212, 213 print, 183, 185, 200, 204 rand, 78–80, 181, 183, 201 real, 186–188, 192, 193, 202, 206 samples, 190, 191, 196, 197, 206 sin, 209, 212 sqrt, 196, 210

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page 246 test, 178 using, 178, 183, 184, 190–192, 196–199, 201, 205, 206, 209, 211–213 with, 178–180, 183, 185, 188, 194, 200, 201, 204, 207, 209, 212, 213 zero, 183, 199 Golden Hind, ii graph, x, 8, 8, 27, 45 great circle metric, 107 greatest lower bound, ix, 10, 10 GSP, 2, 45, 62 GSP to complex plane, 64 GSP with Markov model, 65 Gutenberg Press, ii half-open interval, ix, 9, 133 Hardy, G.H., 163 harmonic center, x, 28, 37 harmonic inequality, 138 harmonic mean, 126, 139 head, 8 Heuristica, ii high pass filtering, 108 high pass filtering of weighted die sequence, 94 high pass filtering of weighted real die sequence, 91 high pass filtering of weighted spinner sequence, 92 homogeneous, 9, 120, 121, 150, 151, 154 horizontal half-open interval, 133, 145 Housman, Alfred Edward, v idempotent, 10, 12–14 identity, 149 if, ix if and only if, ix IIR filtering, 3 image, ix image set, 6 imaginary part, 21 implied by, ix implies, ix implies and is implied by, ix inclusive OR, ix incomparable, 8, 64, 84 increasing, 165 indiscrete topological space, 130 indiscrete topology, 170, 172 inequalities AM-GM, 169 distributive, 11 Jensen's, 165 median, 12

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SUBJECT INDEX minimax, 11 minimax inequality, 28 Minkowski (sequences), 169 modular, 12 modularity inequality, 12 power triangle inequality, 144, 145 Young, 169 inequality triangle, 151 inframetric inequality, 138, 138 inframetric space, 138, 139 injective, 7, 7, 19, 20, 48 integer line, xi, 21, 21, 49, 50, 57, 60, 63, 64, 82 integers, viii interior, x, 126, 127, 171, 171 interior point, 171 Interpolation, 108 intersection, ix interval topology, 21 into, 7 inverse, viii, 6, 149 inverse image characterization, 133 inverse image characterization of continuity, 133, 173 Inverse tangent metric, 20 inverse tangent metric, 20 irreflexive ordering relation, ix isometric, iii, 3, 19, 69, 82, 84, 85 isometry, 19, 84, 85 isomorphic, iii, 3, 13, 13–15, 52, 56–59, 61, 62, 68, 71, 72, 74, 82, 85 isomorphism, 13 isotone, 15, 15, 18, 158 Jacobi, Carl Gustav Jacob, 149 Jacobi, Karl Gustav Jakob, 137 Jensen's Inequality, 165, 169 join, ix, 10 join super-distributive, 11 juxtaposition, 7 Kaneyoshi, Urabe, 216 Kenko, Yoshida, 216 Kuratowski, 5 L'Hôpital's rule, 118, 119 l'Hôpital's rule, 168 Lagrange arc distance, 4, 84, 86–88, 90–93, 95, 110, 119– 123 A book concerning symbolic sequence processing

Lagrange arc distance balls in ℝ2 , 123 Lagrange arc distance balls in ℝ3 , 123 Lagrange arc distance in ℝ2 , 122 Lagrange arc distance in ℝ3 , 122 Lagrange arc distance space, 120 Lagrange arc distance versus Euclidean metric, 121 Lagrange interpolation, 4, 108–110 Lagrange polynomial, 173 Lagrange, Joseph Louis, 85 Langrange arc distance, 108, 110, 114, 115, 118, 120 Langrange, Joseph-Louis, 107 lattice, 10, 10–13, 15–18 isomorphic, 13 LCG mappings standard ordering, 61 LCG mappings, sequential directed graph, 62 LCG mappings, sequential ordering, 61 LCG mappings, standard ordering, 59 least upper bound, ix, 10, 10 Leibniz, Gottfried, viii length 𝘔 Haar scaling sequence, 26 length 𝘔 Haar wavelet sequence, 26 length 𝘔 high pass Hanning sequence, 26 length 𝘔 high pass rectangular sequence, 25 length 𝘔 low pass Hanning sequence, 25 length 𝘔 low pass rectangular sequence, 25 length 1200 non-stationary die sequence with 10Hz oscillating mean, 95 length 12000 non-stationary artificial DNA sequence with 10Hz oscillating mean, 98 length 12000 non-stationary die sequence with 100Hz oscillating mean, 97 length 12000 non-stationary die sequence with 10Hz oscillating mean, 97 length 16 Hanning low pass sequence, 86, 88–90 length 16 Hanning sequence, 94, 95 length 16 high pass rectangu-

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SUBJECT INDEX lar sequence, 25, 94 length 16 low pass rectangular sequence, 25 length 16 Rectangular low pass sequence, 88, 89 length 16 rectangular low pass sequence, 85–89 length 16 rectangular sequence, 94 length 1600 Haar scaling sequence, 104 length 200 Haar wavelet sequence, 102, 103 length 50 Hanning high pass filter, 91, 93 length 50 Hanning low pass sequence, 86, 88–90 length 50 Hanning sequence, 95 length 50 high pass Hanning sequence, 26 length 50 high pass rectangular sequence, 91, 92 length 50 low pass Hanning sequence, 25 length 50 rectangular high pass filter, 91, 93 length 50 rectangular high pass sequence, 91 length 50 Rectangular low pass sequence, 89, 90 length 50 rectangular low pass sequence, 86 length 50 rectangular sequence, 95 length 8 Haar scaling sequence, 26 length 8 Haar wavelet sequence, 26 length 9 Haar wavelet sequence, 26 Lexicographical order relation, 9 lexicographical order relation, 9, 21 limit, x, 130, 131, 172, 172 limit point, 172 linear, 10, 16, 59, 62 linear addition, 72 linear addition with metric transform, 76 linear congruential, 59 linear congruential pseudorandom number generator, 2, 20, 22 linear interpolation, 108 linear order relation, 8, 21 linear space, xi, 3, 85, 149, 149–151, 154 linearity, 111, 112, 121 linearly ordered, 1, 16–18, 49, 2016 Jul 4

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51, 129 linearly ordered set, 8, 10 Liquid Crystal, ii low pass filtering, 108 low pass filtering of fair die sequence, 89 low pass filtering of real die sequence, 85 low pass filtering of spinner sequence, 87 low pass rectangular sequence, 87 low pass sequence, 85, 89 lower bound, 10, 10 Machiavelli, Niccolò, 215 maps to, ix maximally likely, 59 maximin, 11 maximum, 126, 139 maxmin center, xi, 28 mean square, 126 measurable, iii measurable function, 22 median, 12 median inequality, 11 meet, ix, 10 meet sub-distributive, 11 Melissococcus plutonius strain 49.3 plasmid pMP19, 81 metric, 4, 19, 20, 28, 29, 32, 43, 77, 82, 110, 120, 135, 150, 155, 157, 158, 160, 161, 163 generated by norm, 151 induced by norm, 151 Metric ball, 19 metric function, 126, 133 metric geometry, 1, 2, 32, 49, 63 metric induced by the norm, 151 metric linear space, xi, 150, 150, 152, 153 metric preserving function, 74–76, 158, 158–160 Metric space, 125 metric space, 3, 4, 19, 32, 125–136, 138, 138, 139, 142– 147, 155, 156–160, 171–173 metric transform, 74 metrics, 160 𝛼-scaled metric, 159 𝛼-truncated metric, 159 bounded, 160 dilated metric, 159 discrete, 157 inverse tangent, 20 power transform metric, 159 radar screen, 159 usual, 157 A book concerning symbolic sequence processing

minimal center, xi, 28 minimal inequality, 138 minimax, 11 minimax inequality, 11, 11, 12, 28 minimum, 126, 139 Minkowski's Inequality, 165 Minkowski's inequality, 163 Minkowski's Inequality for sequences, 169 modular, 12 Modular inequality, 12 modular inequality, 12 modularity inequality, 12 modulus, 9 moment, x, 27 monontone, 126 monotone, 15, 141 monotonicity, 115, 116 multiplication operation, 8 multiplication operator, xi, 7, 24, 26 n-tuple, 6 natural log, 110, 111 natural numbers, viii near metric space, 126, 138, 138, 139 negative infinity, viii Newton interpolation, 108– 110 Newton polynomial, 108, 174 non-commutative, 62 non-linearly ordered, 49 non-monotone, 160 non-negative, 9, 119, 127, 140, 145, 155–157, 162, 163, 166 non-negative real numbers, viii non-symmetric, 20 non-uniform, 59 nondegenerate, 9, 19, 48, 54, 119, 127, 141, 145, 151, 154, 155, 159, 161–163 nonnegative, 22 norm, 120, 151, 151, 154 usual, 151 normalized, 22 normed linear space, 120, 150, 151, 151, 153 NOT, ix not Cauchy, 135 not closed, 173 not convex, 120, 121 not isomorphic, 14, 84 not open, 133, 134 not order preserving, 14, 51, 84 not translation invariant, 120 not unique, 134, 144, 172

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page 248 null space, ix, 6 one-to-one, 7, 7 one-to-one and onto, 7 only if, ix onto, 7, 7 open, 126, 127, 128, 128–130, 133–135, 141, 142, 170, 170, 171, 173 open ball, x, 32, 32, 33, 126, 127, 128, 128–130, 133–135, 140–143, 151, 152 open balls, 134, 141 open interval, x, 9, 133 open set, 128, 128–130, 133, 134, 142, 170, 172, 173 operations 2𝘕 -offset autocorrelation, 78, 81, 86 ℝ1 spinner random variable, 83, 87 ℝ3 die random variable, 83, 86, 91 ℝ4 DNA random variable, 83 ℝ6 die random variable, 83, 89 𝜙-mean, x, 165 𝘔 -offset autocorrelation, 78 addition operator, xi, 7, 7, 24, 26 arithmatic operator, 110, 111 arithmetic mean, 126 autocorrelation, 77, 78, 85 closure, x, 171 convolution, xi, 24, 24, 25 Discrete Fourier Transform, xi, 96–98 discrete Fourier transform, 26, 26 dot product, 109 down sample, 91, 94 down sampled by a factor of 𝘔 , 24 expected value, 29 Fast Wavelet Transform, 101 Filter, 85–89, 91 filter, 25, 85, 87, 89, 91, 92, 94, 102 filter bank, 101 filtered, 25 FIR filtering, 3 Fourier analysis, 3 geometric mean, 126 harmonic mean, 126 high pass filtering, 108 IIR filtering, 3

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SUBJECT INDEX Interpolation, 108 join, 10 juxtaposition, 7 Lagrange interpolation, 4, 108–110 length 16 Hanning low pass sequence, 86, 88, 89 length 16 Hanning sequence, 94, 95 length 16 high pass rectangular sequence, 94 length 16 rectangular low pass sequence, 86, 88, 89 length 16 rectangular sequence, 94 length 50 Hanning high pass filter, 91, 93 length 50 Hanning low pass sequence, 86, 88, 90 length 50 Hanning sequence, 95 length 50 high pass rectangular sequence, 91 length 50 rectangular high pass filter, 91, 93 length 50 rectangular high pass sequence, 91 length 50 Rectangular low pass sequence, 89, 90 length 50 rectangular low pass sequence, 86 length 50 rectangular sequence, 95 limit, 130, 172 linear interpolation, 108 low pass filtering, 108 maximum, 126 mean square, 126 meet, 10 minimum, 126 multiplication operation, 8 multiplication operator, xi, 7, 24, 26 Newton interpolation, 108–110 open set, 129 PAM die random variable, 82 PAM DNA random variable, 83 pulse amplitude modulation, 83 QPSK die random variable, 83 QPSK DNA random variable, 83 QPSK spinner random variable, 83, 88 quadrature phase shift keying, 83 A book concerning symbolic sequence processing

sampled, 101 set difference, 128 set intersection, 129 SMR, 138 square-mean-root, 138 traditional die random variable, 82, 85, 89 traditional stochastic processing, 1 Voss Spectrum, 83 wavelet analysis, 3 operator, 149 order, ix, 82 order preserving, 12, 13, 14, 15, 15, 51, 84, 85 order relation, 2, 8, 8, 21 order relations alphabetic, 9 coordinatewise, 9 dictionary, 9 lexicographical, 9 order topology, 21 ordered distance linear space, iii, 3, 82 ordered distance space, xi, 3 ordered metric space, 20, 20, 21, 48, 53, 71, 72, 138 ordered pair, viii, 5, 5, 133 ordered quasi-metric space, 20, 20, 21, 27, 46–48 ordered set, ix, 3, 8, 8–10, 15, 16, 18, 20 linearly, 8 totally, 8 ordered set of partitions of an integer, 10 ordered set of real numbers, 9 ordered triple, ix, 6 oriented triangle inequality, 155 origin, 118, 119 outcome center, x, 27, 28, 28, 33, 37, 39, 41, 43, 46, 75 outcome expected value, xi, 46, 47, 47, 49, 55 outcome random variable, 46, 46, 52, 68 outcome random variables, 59 outcome subspace, 27, 27– 29, 43, 45–48, 56, 59, 63, 65, 68, 71, 72, 74, 77, 78, 82 outcome subspace sequence, 78 outcome subspace sequence metric, 77, 77, 78 outcome subspace variance, 47 outcome variance, xi, 28, 28, 37, 46, 47, 49

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SUBJECT INDEX outcomes, 22 pair of dice, 68 pair of dice and hypothesis testing, 68 pair of dice outcome subspace, 66, 68 pair of real dice, 66, 67 pair of spinner, 71, 73 pair of spinner and hypothesis testing, 71 pair of spinners, 70, 71 PAM die random variable, 82, 96–98, 102 PAM DNA random variable, 83, 99, 100, 103, 104 Papaya DNA sequence, 81 partial order relation, 8 partially ordered set, 8 Peano's Axioms, 21 pi, viii Poincaré, Jules Henri, 27 point of adherence, 171 polar angle, 120 polar form, 109 polynomial Lagrange, 173 Newton, 174, 175 poset, 8 order preserving, 12 positive, 116 positive infinity, viii positive real numbers, viii, 137 positivity, 115, 116 power distance, 138 power distance function, 138, 139, 147 Power distance space, 4 power distance space, 4, 126, 127, 138, 139–147 power distance spaces, 126 Power mean, 126, 137 power mean, x, 126, 137, 166, 166, 169 Power mean metrics, 161 power mean metrics, 28 power set, x, 127, 170, 171 power set of 𝑋, 6 power transform, 76, 159 power transform metric, 159 power triangle function, x, 126, 137, 137–139, 143, 144 power triangle inequality, x, 4, 126, 137, 138, 138–140, 143–145, 147 power triangle triangle space, 139 preorder, 8 preserves joins, 13 preserves meets, 13, 14 probabilistic metric spaces, 2016 Jul 4

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137 probability function, x, 3, 22, 22, 29 probability measure, 22 probability space, iii, x, 1, 22, 22, 23, 27, 46 Proclus, 125 product rule, 111 proper subset, ix proper superset, ix properties (𝑋, 𝖽) is not translation invariant, 120 (strictly monotonically increasing in 𝜙), 116 absolute value, ix absorptive, 10, 12 additive, 22 additive property, 23 adherent, 171 affine, 139, 165 algebra of sets, x AND, ix anti-symmetric, 8 antitone, 15, 15 associates, 149 associative, 10, 149, 164 bijection, 48 bijective, 7, 13, 16, 51 bounded, 126, 127, 131, 143 Cauchy, 126, 128, 130, 131, 134–136, 143, 146 Cauchy condition, 130 closed, 128, 128, 131, 132, 170, 171–173 closed set, 127 commutative, 10, 62, 140, 144, 149 comparable, 8, 8 complement, ix complete, 130, 131, 132 concave, 149, 165, 165, 166 continuity, 115, 117, 118, 126, 127, 133, 136, 139, 172, 173 continuous, 119, 126, 128, 133, 133, 136–139, 144, 146, 147, 151, 165, 166, 171, 171–173 converge, 130 convergence, 19, 126, 127, 134, 143, 173 convergent, 126, 128, 130, 130, 131, 133–136, 143, 144, 146, 172, 173 converges, 172, 172 Convex, 10 convex, 9, 23, 120, 149, 150, 152, 153, 165, 165, 166 A book concerning symbolic sequence processing

decreasing, 165 difference, ix dilated Euclidean, 135, 146 directed, 8, 8 discontinuity, 117–119 discontinuous, 118, 119, 133, 133–136, 171, 171, 172 discrete, 134, 145 distance space, 132 distributes, 72, 149 distributive, 11 diverge, 135, 172 divergent, 172 equality by definition, ix equality relation, ix Euclidean, 133–135, 145, 146 exclusive OR, ix existential quantifier, ix extension, 82, 107, 114 finite, 7 for each, ix full period, 59 greatest lower bound, ix, 10 homogeneous, 9, 120, 121, 150, 151, 154 idempotent, 10, 12–14 identity, 149 if, ix if and only if, ix implied by, ix implies, ix implies and is implied by, ix inclusive OR, ix incomparable, 8, 64, 84 increasing, 165 injective, 7, 19, 20, 48 intersection, ix into, 7 irreflexive ordering relation, ix isometric, iii, 3, 19, 69, 82, 84, 85 isometry, 19 isomorphic, iii, 3, 13, 13–15, 52, 56–59, 61, 62, 68, 71, 72, 74, 82, 85 isotone, 15, 15, 18, 158 join, ix join super-distributive, 11 least upper bound, ix, 10 limit, 130 linear, 10, 16, 59, 62 linearity, 111, 112, 121 linearly ordered, 1, 16– 18, 49, 51, 129 maps to, ix

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page 250 maximally likely, 59 measurable, iii median inequality, 11 meet, ix meet sub-distributive, 11 metric, 82 modular, 12 monontone, 126 monotone, 15, 141 monotonicity, 115, 116 non-commutative, 62 non-linearly ordered, 49 non-monotone, 160 non-negative, 9, 119, 127, 140, 145, 155–157, 162, 163, 166 non-symmetric, 20 non-uniform, 59 nondegenerate, 9, 19, 48, 54, 119, 127, 141, 145, 151, 155, 159, 161–163 nonnegative, 22 normalized, 22 NOT, ix not Cauchy, 135 not closed, 173 not convex, 120, 121 not isomorphic, 14, 84 not open, 133, 134 not order preserving, 14, 51, 84 not translation invariant, 120 not unique, 134, 144, 172 one-to-one, 7 one-to-one and onto, 7 only if, ix onto, 7 open, 126, 128, 128–130, 133–135, 141, 142, 170, 170, 171 open set, 130 order, ix, 82 order preserving, 12, 13, 14, 15, 15, 51, 84, 85 oriented triangle inequality, 155 positive, 116 positivity, 115, 116 power set, x power triangle inequality, 139, 140, 147 preserves joins, 13 preserves meets, 13, 14 proper subset, ix proper superset, ix pseudo-distributes, 149 randomness, 22 real, 151

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SUBJECT INDEX reflexive, 8 reflexive ordering relation, ix ring of sets, x strictly antitone, 15, 15– 17, 32, 36, 38, 40, 42, 44 strictly convex, 165 strictly isotone, 15, 15– 18, 31, 35, 36, 39–44, 46, 48, 50, 52, 54, 57, 60, 62, 64, 65, 74 strictly monontone, 137 strictly monotone, 15, 131, 138–140, 143, 146, 166 strictly monotonic, 165 strictly monotonically increasing, 113, 116, 241, 249 strictly positive, 151 subadditive, 9, 20, 150, 151, 155–158, 161, 163 subaddtive, 162 submultiplicative, 9 subset, ix super set, ix surjective, 7, 59 symmetric, 23, 47, 57, 58, 64, 71, 115, 119, 127, 140, 144, 152, 155 symmetric difference, ix symmetry, 20, 115, 156, 161, 163 there exists, ix topology of sets, x transitive, 8 translation invariant, 120, 150, 153, 154 triangle inequality, 4, 9, 92, 120, 125–127, 133–136, 144, 147, 155–157, 162 triangle inequality fails, 120 triangle inquality, 151 uncorrelated, 78, 80, 91 undirected, 8, 8 uniform, 59 uniformly distributed, 78, 80 union, ix unique, 126, 128, 131– 133, 135, 136, 146, 172 universal quantifier, ix unordered, 21, 28, 29, 49, 84 vector norm, xi pseudo-distributes, 149 pseudo-random number generator, 59 pstricks, ii pullback, 48 pullback function, 75 Pullback metric, 19, 20 A book concerning symbolic sequence processing

pulse amplitude modulation, 83 pushforward, 48 QPSK die random variable, 83, 96–98, 102 QPSK DNA random variable, 83, 99, 100, 103, 104 QPSK spinner random variable, 83, 84, 88 quadratic inequality, 138 quadratic mean, 139 quadrature phase shift keying, 83 quasi-metric, 20, 43, 54, 62, 155, 155 quasi-metric space, 20, 155 quotes Abel, Niels Henrik, 215 Aristotle, 77 Bohr, Harald, 163 Descartes, René, viii, 1 Gauss, Karl Friedrich, xvi Hardy, G.H., 163 Housman, Alfred Edward, v Jacobi, Carl Gustav Jacob, 149 Jacobi, Karl Gustav Jakob, 137 Kaneyoshi, Urabe, 216 Kenko, Yoshida, 216 Lagrange, Joseph Louis, 85 Langrange, JosephLouis, 107 Leibniz, Gottfried, viii Machiavelli, Niccolò, 215 Poincaré, Jules Henri, 27 Proclus, 125 Russull, Bertrand, v Stravinsky, Igor, v Weyl, Hermann, 170 quotient structures, 9 radar screen, 159 radar screen metric, 159 Random DNA sequence, 80 random variable, x, 23, 46– 49, 53, 54, 71, 72, 82 randomness, 22 range, ix, 6, 80, 158 real, 151 real dice, 68 real die, 3, 4, 29, 67, 75, 78, 79, 82 real die mappings, 53 real die outcome subspace, 29, 31, 32, 53, 75, 82, 84 real die sequence, 4, 78, 108

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SUBJECT INDEX real line, xi, 1–3, 20, 21, 21, 46, 47, 49–51, 54, 57–59, 63, 64, 68, 71, 72, 82, 129 real line ordered metric space, 48 real numbers, viii, 151 real part, 21 real-valued random variable, iii, 1, 22 reflexive, 8 reflexive ordering relation, ix relation, viii, 6, 6, 8, 84, 138 inverse, 6 relations σ-inframetric inequality, 126, 138, 138 converse, 6 coordinatewise order relation, 21 distance function, 2 domain, 6 dual, 9, 21 geometric inequality, 138 harmonic inequality, 138 image set, 6 inframetric inequality, 138, 138 inverse, viii, 6 lexicographical order relation, 21 minimal inequality, 138 null space, 6 order relation, 2, 8, 21 partial order relation, 8 power triangle inequality, x, 126, 137, 138, 138, 143, 144, 147 preorder, 8 quadratic inequality, 138 range, 6 relaxed triangle inequality, 126, 138, 138 square mean root inequality, 138 standard ordering relation, 83 triangle inequality, 125, 137, 138, 138 triangle relation, 145, 146 relaxed triangle inequality, 126, 138, 138 Reproducible Research, 177 ring, 157 ring multiplication, 73 ring of sets, x RMS, 100–102 root mean square, 100 2016 Jul 4

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Russull, Bertrand, v Saint Andrew's University's, 177 sampled, 101 SARS coronavirus DNA sequence, 80 scaffold DNA, 29 scaling function, 101 Second Fundamental Theorem of Calculus, 116 sequence, xi, 3, 23, 24–26, 77, 78, 88, 92, 127, 130–132, 151 sequences Cauchy, 130 sequential characterization, 133 sequential characterization of continuity, 133, 173 set, 5–7, 9, 22, 128, 155 set difference, 128 set function, 7 set intersection, 129 set of all functions, 6 set of all functions in 𝑋 × 𝑌 , ix set of all relations, 6 set of all relations in 𝑋 × 𝑌 , ix set of complex numbers, 6, 21 set of integers, 5, 21 set of natural numbers, 5 set of non-negative real numbers, 5 set of outcomes, x, 22 set of postive real numbers, 5 set of real numbers, iii, 1, 5, 21 set of whole numbers, 5 sets, 6 open ball, 151 real numbers, 151 sigma-algebra, 19 slope-intercept, 108 SMR, 138 snowflake, 159 snowflake transform, 76 snowflake transform metric, 159 source code, 177 space, 110 linear, 149 metric, 128, 151, 155, 156 metric vector, 150 normed vector, 151 topological, 170 vector, 149 spherical metric, 4, 107, 108, 110, 114 spinner, 29, 69, 70, 75, 78 spinner mappings, 56 A book concerning symbolic sequence processing

spinner outcome subspace, 29, 35, 57, 75, 76, 84, 88 spinner sequence, 4, 78, 87, 92, 93, 108 spinners, 71 square mean root inequality, 138 square root, 110, 111 square-mean-root, 138 standard ordered set of real numbers, 10 standard ordering relation, 83 statistical edge detection using Haar wavelet on nonstationary artificial DNA sequence, 102 statistical edge detection using Haar wavelet on nonstationary die sequence, 101 stochastic process, 2, 49 Stravinsky, Igor, v strictly antitone, 15, 15–17, 32, 36, 38, 40, 42, 44 strictly convex, 165, 165 strictly isotone, 15, 15–18, 31, 35, 36, 39–44, 46, 48, 50, 52, 54, 57, 60, 62, 64, 65, 74 strictly monontone, 137 strictly monotone, 15, 131, 138–140, 143, 146, 166 strictly monotonic, 165 strictly monotonically increasing, 113, 116, 241, 249 strictly positive, 151, 154 structures ℝ1 die distance linear space, 83, 85 ℝ2 spinner distance linear space, 84, 84, 85, 88 ℝ3 die distance linear space, 84, 84, 85 ℝ3 distance linear space, 86, 91 ℝ6 die distance linear space, 84, 89 ℝ6 fair die distance linear space, 84 ℝ𝘕 ordered distance linear space, xi 𝗀-transform metric, 20 𝘕 -tuple, 157 𝑛th-moment, x, 27 σ-algebra, x, 22 σ-inframetric space, 126, 138 3-tuple, 6 6 element discrete metric, 21, 22 6 element ring, 21, 22 accumulation point,

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page 252 126, 127, 171 algebra of set, 19 arcs, 8 arithmetic center, x, 28, 28 b-metric, 138 ball, 121 base, 33, 126–130, 133– 135, 141, 170, 170, 171 boundary, 171 Cartesian coordinate system, 108 Cartesian product, 6 Cartesian product of 𝑋 and 𝑌 , viii Cauchy sequence, 130 ceiling, x center, x, 8, 27, 45, 74, 76 chain, 8 closed ball, x, 128 closed interval, ix, 9 closed set, 126, 128 closure, 126, 127, 131, 171, 172 closure point, 171 commutative ring, 9 complex numbers, viii complex plane, 21, 21, 49, 52, 64 convergent sequence, 126 convex subset, 24, 77 die outcome subspace, 78 die sequence, 91 directed graph, 8, 43 directed PRNG state machine, 21 directed weighted graph, 3 discrete metric, 30 discrete metric geometry, 3 discrete topology, 32, 33, 170, 172 distance function, 133 Distance space, 4 distance space, x, 3, 4, 19, 125, 126, 127, 127–139, 143–147 distance spaces, 126 DNA, 29 DNA outcome subspace, 29 DNA scaffold outcome subspace, 29 domain, ix, 24, 25, 107 empty set, viii, 5, 128 Euclidean metric space, 146 extended probability

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SUBJECT INDEX space, x, 27, 27 extended real numbers, viii, 5, 137 extended set of integers, viii, 5 extension, 107 fair die, 2, 29, 51, 75, 78 fair die outcome space, 90 fair die outcome subspace, 28, 29, 32, 49, 51, 52, 75, 84 field, iii, 3, 24, 25, 78, 149 field of real numbers, 7 floor, x fully ordered set, 8 function, iii, 6, 7, 8, 15, 16, 19, 22, 23, 84, 172 genome, 2, 45, 62, 63 genome outcome subspace, 29 genome scaffold outcome subspace, 29 Genomic Signal Processing, 2, 45, 62 geometric center, x, 28 graph, x, 8, 8, 27, 45 greatest lower bound, 10 GSP, 2, 45, 62 half-open interval, ix, 9, 133 harmonic center, x, 28, 37 horizontal half-open interval, 133, 145 identity, 149 image, ix indiscrete topological space, 130 indiscrete topology, 170, 172 inframetric space, 138, 139 integer line, xi, 21, 21, 49, 50, 57, 60, 63, 64, 82 integers, viii interior, x, 126, 127, 171, 171 interior point, 171 interval topology, 21 inverse, 149 Lagrange arc distance space, 120 lattice, 10, 10–13, 15–18 least upper bound, 10 limit, 172 linear congruential, 59 linear congruential pseudo-random number generator, 2, 20, 22 linear order relation, 21 A book concerning symbolic sequence processing

linear space, xi, 3, 85, 149–151, 154 linearly ordered set, 8, 10 maxmin center, xi, 28 measurable function, 22 metric, 19, 20, 28, 32, 150, 158, 160, 161 Metric ball, 19 metric geometry, 1, 2, 32, 49, 63 metric linear space, xi, 150, 150, 152, 153 metric preserving function, 76, 158 Metric space, 125 metric space, 3, 4, 19, 32, 125–136, 138, 138, 139, 142– 147, 156–160, 171–173 metrics, 160 minimal center, xi, 28 moment, x, 27 n-tuple, 6 natural numbers, viii near metric space, 126, 138, 138, 139 negative infinity, viii non-negative real numbers, viii norm, 151, 154 normed linear space, 120, 150, 151, 151, 153 null space, ix open, 127, 173 open ball, x, 32, 32, 33, 126, 127, 128, 128–130, 133– 135, 140–143, 152 open balls, 134, 141 open interval, x, 9, 133 open set, 128, 128, 129, 133, 134, 142, 170, 172, 173 operator, 149 order relation, 8 order topology, 21 ordered distance linear space, iii, 3, 82 ordered distance space, xi, 3 ordered metric space, 20, 20, 21, 48, 53, 71, 72, 138 ordered pair, viii, 5, 5, 133 ordered quasi-metric space, 20, 20, 21, 27, 46–48 ordered set, ix, 3, 8, 8–10, 15, 16, 18, 20 ordered set of partitions of an integer, 10 ordered set of real numbers, 9 ordered triple, 6

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SUBJECT INDEX origin, 118, 119 outcome center, x, 27, 28, 28, 37, 39, 41, 43, 75 outcome random variable, 46 outcome subspace, 27, 27–29, 43, 45–48, 56, 59, 63, 65, 68, 71, 72, 74, 77, 78, 82 outcome subspace sequence, 78 outcome variance, 28 pair of dice, 68 pair of dice outcome subspace, 68 pair of real dice, 66, 67 pair of spinner, 71, 73 pair of spinners, 71 partially ordered set, 8 Peano's Axioms, 21 pi, viii point of adherence, 171 polar form, 109 poset, 8 positive infinity, viii positive real numbers, viii, 137 Power distance space, 4 power distance space, 4, 126, 127, 138, 139–147 power distance spaces, 126 power mean metrics, 28 power set, 127, 170, 171 power set of 𝑋, 6 power triangle inequality, 126 power triangle triangle space, 139 probability function, x, 22 probability space, iii, x, 1, 22, 22, 23, 27, 46 pseudo-random number generator, 59 Pullback metric, 20 quasi-metric space, 20, 155 random variable, x, 23, 48, 72 range, ix, 80, 158 real dice, 68 real die, 3, 4, 29, 67, 75, 78, 79, 82 real die outcome subspace, 29, 31, 32, 53, 75, 82, 84 real die sequence, 108 real line, xi, 1–3, 20, 21, 21, 46, 47, 49–51, 54, 57–59, 63, 64, 68, 71, 72, 82, 129 real line ordered metric 2016 Jul 4

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space, 48 real numbers, viii relation, viii, 6, 6, 8, 84, 138 ring, 157 scaffold DNA, 29 sequence, xi, 3, 23, 24– 26, 77, 151 set, 5–7, 9, 22, 128, 155 set function, 7 set of all functions, 6 set of all functions in 𝑋 × 𝑌 , ix set of all relations, 6 set of all relations in 𝑋 × 𝑌 , ix set of complex numbers, 6, 21 set of integers, 5, 21 set of natural numbers, 5 set of non-negative real numbers, 5 set of outcomes, x, 22 set of postive real numbers, 5 set of real numbers, iii, 1, 5, 21 set of whole numbers, 5 sets, 6 sigma-algebra, 19 space, 110 spinner, 29, 69, 70, 75, 78 spinner outcome subspace, 29, 35, 75, 76, 84, 88 spinner sequence, 92, 93, 108 spinners, 71 standard ordered set of real numbers, 10 stochastic process, 2, 49 surface of a sphere with radius 𝑟, 107 surface of a sphere…, 107 topological space, x, 32, 126, 127, 129, 133, 144, 170– 173 topological space (𝑋, 𝑻 ) induced by (ℝ, 𝖽), 134 topological space induced by (ℝ, 𝖽), 133 topological space induced by (𝑋, 𝖽), 130, 130, 131, 142 topologogical space, 129 topology, 19, 33, 126, 127, 129, 130, 133–135, 170, 170 topology induced by (𝑋, 𝖽) on 𝑋, 130 A book concerning symbolic sequence processing

totally ordered set, 8 traditional random variable, 23 triangle inequality, 125 triangle relation, 138 triple, 4, 6 trivial topology, 170 undirected graph, 8, 43 unit ball, 121 unit Lagrange arc distance ball, 124 unordered metric space, 20, 20, 22, 51, 52 unordered quasi-metric space, 20, 20, 22 unordered set, 8 usual Borel σ-algebra, iii, 1, 22 usual metric space, 171 vertical half-open interval, 133, 145 vertices, 8 wagon wheel, 75 wagon wheel outcome subspace, 75 wagon wheel output subspace, 75 weighted die, 3, 29, 55, 75, 79, 80 weighted die outcome subspace, 28, 28, 54, 76 weighted graph, iii, x, 3, 8, 8, 29, 31, 82 weighted graphs, 82 weighted real die, 29, 79, 80 weighted real die outcome subspace, 29, 29, 33, 37 weighted real die sequence, 108 weighted spinner, 73 weighted spinner outcome subspace, 37, 57 weighted spinner sequence, 108 whole numbers, viii subadditive, 9, 20, 150, 151, 155–158, 161, 163 subaddtive, 162 submultiplicative, 9 subset, ix sum, x summation, 164 super set, ix surface of a sphere with radius 𝑟, 107 surface of a sphere…, 107 surjective, 7, 7, 59 symmetric, 23, 47, 57, 58, 64, 71, 115, 119, 127, 140, 144, 152, 155

Daniel J. Greenhoe

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page 254 symmetric difference, ix symmetry, 20, 115, 156, 161, 163 tail, 8 The Archimedean Property, 140, 141 The Book Worm, 215 The Closed Set Theorem, 172 The principle of Archimedes, 141 the standard topology on the real line, 171 The usual norm, 151 theorems 𝗀-transform metric, 19 Cantor intersection, 132 Cantor Intersection Theorem, 126 Cantor intersection theorem, 132 chain rule, 111, 112 closed set theorem, 173 distributive inequalities, 11 First Fundamental Theorem of Calculus, 116, 118 Fréchet product metric, 77 Generalized AM-GM inequality, 169 Generalized associative property, 164 inverse image characterization, 133 inverse image characterization of continuity, 133, 173 Jensen's Inequality, 165, 169 L'Hôpital's rule, 118, 119 l'Hôpital's rule, 168 minimax inequality, 11, 11, 12 Minkowski's Inequality, 165 Minkowski's inequality, 163 Minkowski's Inequality for sequences, 169 Modular inequality, 12 Power mean metrics, 161 product rule, 111 Pullback metric, 19 Second Fundamental Theorem of Calculus, 116 sequential characterization, 133 sequential characterization of continuity, 133, 173

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SUBJECT INDEX The Archimedean Property, 140, 141 The Closed Set Theorem, 172 The principle of Archimedes, 141 Uniqueness of limit, 144 Young's Inequality, 169 there exists, ix topological space, x, 32, 126, 127, 129, 133, 144, 170–173 topological space (𝑋, 𝑻 ) induced by (ℝ, 𝖽), 134 topological space induced by (ℝ, 𝖽), 133 topological space induced by (𝑋, 𝖽), 130, 130, 131, 142 topologies discrete, 170 indiscrete, 170 trivial, 170 topologogical space, 129 topology, 19, 33, 126, 127, 129, 130, 133–135, 170, 170 topology induced by (𝑋, 𝖽) on 𝑋, 130 topology of sets, x totally ordered set, 8 traditional die random variable, 82, 85, 89, 94 traditional expected value, x, 23, 23, 47, 49, 51, 55 traditional random variable, 1, 2, 22, 23, 49 traditional stochastic processing, 1 traditional variance, x, 23, 23, 47, 49 transitive, 8 translation invariant, 120, 150, 153, 154 triangle function, 137, 146 triangle inequality, 4, 9, 92, 120, 125–127, 133–137, 138, 138, 144, 147, 151, 155–157, 162 triangle inequality fails, 120 triangle inquality, 151 triangle relation, 138, 145, 146 triangle-inequality, 154 triple, 4, 6 trivial topology, 170 uncorrelated, 78, 80, 91 undirected, 8, 8 undirected graph, 8, 43 uniform, 59 uniformly distributed, 78, 80 union, ix unique, 126, 128, 131–133, 135, 136, 146, 172 A book concerning symbolic sequence processing

Uniqueness of limit, 144 unit ball, 121 unit Lagrange arc distance ball, 124 universal quantifier, ix unordered, 21, 28, 29, 49, 84 unordered metric space, 20, 20, 22, 51, 52 unordered quasi-metric space, 20, 20, 22 unordered set, 8 upper bound, 10, 10 usual Borel σ-algebra, iii, 1, 22 usual metric, iii, 1, 20, 21, 49, 51, 65, 157, 157 usual metric space, 171 usual norm, 151 Utopia, ii values GLB, 10 LUB, 10 additive identity element, 31, 32, 37, 46 contradiction, 145 diameter, 127 greatest lower bound, 10 head, 8 imaginary part, 21 least upper bound, 10 limit, 131 limit point, 172 lower bound, 10, 10 outcome expected value, 47 real part, 21 tail, 8 traditional expected value, 23, 23, 47 traditional variance, 23, 23 upper bound, 10, 10 variance, 28 vector norm, xi vertical half-open interval, 133, 145 vertice weight, 8 vertices, 8 Voss Spectrum, 83 wagon wheel, 75 wagon wheel outcome subspace, 75 wagon wheel output subspace, 75 wavelet, 101 wavelet analysis, 3 Wavelet analysis of Phage Lambda DNA sequence, 104 weighted, 166

Daniel J. Greenhoe

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page 255

SUBJECT INDEX weighted die, 3, 29, 55, 75, 79, 80 weighted die mappings, 54 weighted die outcome subspace, 28, 28, 54, 76 weighted die sequence, 79, 80, 94 weighted graph, iii, x, 3, 8, 8, 29, 31, 82 weighted graphs, 82 weighted real die, 29, 79, 80

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weighted real die outcome subspace, 29, 29, 33, 37 weighted real die sequence, 79, 108 weighted ring, 39 weighted ring outcome subspace, 60 weighted spinner, 73 weighted spinner mappings, 57 weighted spinner outcome


subspace, 37, 37, 57, 58 weighted spinner sequence, 80, 80, 108 Weyl, Hermann, 170 Who Was There, 177 whole numbers, viii Wáng Hàn Zōng Zhōng Mińg Tǐ Fán, ii

Young's Inequality, 169

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Daniel J. Greenhoe

2016 Jul 4

10:12am UTC

A book concerning symbolic sequence processing

A book concerning symbolic sequence processing

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