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Alexey Stakhov. Mathematics of Harmony as a New Interdisciplinary Direction and “Golden” Paradigm of Modern Science
Mathematics of Harmony as a New Interdisciplinary Direction and “Golden” Paradigm of Modern Science
Mathematics of Harmony as a New Interdisciplinary Direction and “Golden” Paradigm of Modern Science
Chapter 1. Foundations of the Constructive (Algorithmic) Measurement Theory. 1.1. The Evolution of the Concept of “Measurement” in Mathematics
1.2. Axioms of Eudoxus–Archimedes and Cantor
1.3. The Problem of Infinity in Mathematics
1.4. Criticism of the Cantor Theory of Infinite Sets. 1.4.1. Infinitum Actu Non Datur
1.4.2. Criticism of Cantor’s theory of sets in 19th and early 20th centuries
1.4.3. Research by Alexander Zenkin
1.5. Constructive Approach to the Creation of the Mathematical Measurement Theory
1.6. The “Indicatory” Model of Measurement. 1.6.1. The conceptions of the “indicatory” element (IE) and the “indicatory” model of measurement
1.7. The Concept of the Optimal Measurement Algorithm
1.8. Classical Measurement Algorithms
1.8.1. Counting algorithm
1.8.2. “Binary” algorithm
1.8.3. Readout algorithm
1.8.4. Restrictions S
1.9. Optimal (n, k, 0)-Algorithms. 1.9.1. Recursion method
1.9.2. Synthesis of the optimal (n, k, 0)-algorithm
1.9.3. Special cases of the optimal (n, k, 0)-algorithm
1.9.4. Optimal (n, k, 0)-algorithms and positional numeral systems
1.10. Optimal (n, k, 1)-Algorithms Based on Arithmetic Square. 1.10.1. Synthesis of the optimal (n, k, 1)-algorithm
1.10.2. Arithmetic square
1.10.3. Optimal (n, k, 1)-algorithm
1.10.4. An example of the optimal (n, k, 1)-algorithm
1.10.5. The extreme particular cases of the optimal (n, k, 1)-algorithm
1.10.6. The importance of the binomial algorithms for mathematics and computer science
Chapter 2. Principle of Asymmetry of Measurement and Fibonacci Algorithms of Measurement. 2.1. Bachet–Mendeleev Problem
2.2. Asymmetry Principle of Measurement
2.3. A New Formulation of Bachet–Mendeleev Problem
2.4. Synthesis of the Optimal Fibonacci’s Algorithm of Measurement. 2.4.1. Fibonacci measurement algorithms
2.4.2. Special cases
2.4.3. The system of the standard weights for Fibonacci measurement algorithms
2.5. Example of the Fibonacci Algorithm for the Case p = 1
2.6. The Main Result of the Algorithmic Measurement Theory
2.7. Isomorphism Between “Lever Balances” and “Rabbits Reproduction” 2.7.1. What is isomorphism?
2.7.2. The generalized “principle of asymmetry of reproduction” in wildlife
2.8. Isomorphism Between the Algorithmic Theory of Measurement and the Theory of Positional Numeral Systems
2.9. International Recognition of the Algorithmic Measurement Theory. 2.9.1. All-union conference “information-measuring systems 1973”
2.9.2. Scientific trip to Austria
2.10. Letter of the USSR Ambassador to Austria, Mikhail Efremov, on Patenting Fibonacci Inventions
Chapter 3. Evolution of Numeral Systems. 3.1. Brief History of the Most Known Numeral Systems. 3.1.1. The main stages in the development of numeral systems
3.1.2. Babylonian numeral system with the base 60
3.1.3. Alphabetical and Roman numbering systems
3.1.4. Ancient Egyptian decimal non-positional numeral system
3.1.5. Mayan numeral system
3.1.6.Decimal system
3.1.7. The decimal system as the greatest mathematical discovery in the history of mathematics
3.1.8.Binary system
3.2. Ternary Symmetric Numeral System and Computer “Setun” 3.2.1.The concept of canonical positional numeral system
3.2.2.The concept of the “symmetric” numeral system
3.2.3.Ternary symmetric numeral system
3.2.4.Representation of the negative numbers
3.2.5.Ternary “symmetric” arithmetic
3.2.6. Ternary computer “Setun”
Chapter 4. Bergman’s System and “Golden” Number Theory. 4.1. George Bergman and Bergman’s System. 4.1.1.Definition of Bergman’s system
4.1.2.George Bergman
4.1.3.The main distinction between Bergman’s system and binary system
4.2. The “Golden” Number Theory and the New Properties of Natural Numbers. 4.2.1.Euclid’s definition of natural numbers
4.2.2.Constructive definition of the real numbers
4.2.3.Newton’s definition of the real number
4.2.4.The “extended” Fibonacci and Lucas numbers
4.2.5.Bergman’s system as a new definition of the real number
4.2.6.The “golden” representation of natural numbers
4.2.7. Multiplicity and MINIMAL FORM of the “golden” representations
4.3. Z-and D-properties of Natural Numbers
4.4. F-and L-codes
4.5. The Left and Right Shifts of the Φ- F-, and L-codes
4.5.1.The shifts of the Φ-code to the left and to the right
4.5.2.The shifts of the F-code to the left and to the right
4.5.3.The shifts to the left and to the right of the L-code
4.5.4.Numerical examples
4.5.5.Algebraic summation of integers
4.6. Evaluation of Bergman’s System
4.6.1.The fundamental significance of Bergman’s system for the development of modern mathematics and computer science
4.6.2.Evaluation of the applications of Bergman’s system
Chapter 5. The “Golden” Ternary Mirror-Symmetrical Arithmetic. 5.1. The Ternary Mirror-Symmetrical Representation. 5.1.1. Brief history
5.1.2. The conversion of the binary MINIMAL FORM into the ternary “golden” representation of the same integer N
5.2. The “Golden” Ternary F-and L-representations
5.2.1. The representation of negative numbers
5.3. Mirror-Symmetrical Property: The Base and Comparison of the “Golden” Ternary Representations and Their Redundancy. 5.3.1. Mirror-symmetrical property
5.3.2. The base of the “golden” ternary mirror-symmetrical numeral system
5.3.3. Comparison of the “golden” ternary mirror-symmetrical numbers
5.3.4. The range of the ternary mirror-symmetrical representation of numbers
5.3.5. Code redundancy of the ternary mirror-symmetrical numeral system
5.4. The Ternary Mirror-Symmetrical Summation and Subtraction. 5.4.1. Mirror-symmetrical summation-subtraction
5.4.2. Examples of the ternary mirror-symmetrical summation (subtraction)
5.5.. Ternary Mirror-Symmetrical Multi-Digit Summator (Subtractor)
5.5.1. “Swing”-phenomenon
5.6. The Ternary Mirror-Symmetrical Multiplication and Division. 5.6.1. Mirror-symmetrical multiplication
5.6.2. Mirror-symmetrical division
5.6.3. The main arithmetical advantages of the ternary mirror-symmetrical arithmetic
5.7. Technical Realizations of the Ternary Mirror-Symmetrical Arithmetical Devices. 5.7.1. General information
5.7.2. The ternary mirror-symmetrical accumulator
5.8. Matrix and Pipeline Mirror-Symmetrical Arithmetical Unit. 5.8.1. Matrix mirror-symmetrical summator (subtractor)
5.8.2. The pipeline mirror-symmetrical summator (subtractor)
5.8.3. The pipeline ternary mirror-symmetrical multiplier
5.9. Evaluation of the Ternary Mirror-Symmetrical Arithmetic. 5.9.1. The ternary mirror-symmetrical arithmetic as the synthesis of Bergman’s system and Brousentsov’s ternary principle
5.9.2. Support of Prof. Donald Knuth
5.9.3. Support of Nikolay Brusentsov
Chapter 6. Fibonacci p-Codes and Fibonacci Arithmetic for Mission-Critical Applications. 6.1. “Trojan Horse” of Modern Computers for Mission-Critical Applications. 6.1.1.The main disadvantage of the binary system
6.1.2.The first Stakhov publications on the redundant numeral systems
6.1.3.Stakhov’s scientific trip to Austria, Stakhov’s speech on the joint meeting of Austrian Cybernetics and Computer Societies and the Fibonacci patents by Alexey Stakhov
6.2. Fibonacci p-Codes and Their Partial Cases. 6.2.1.Definition of the Fibonacci p-codes
6.2.2.The partial cases of the Fibonacci p-codes
6.3. Unusual Peculiarities of the Fibonacci Representations. 6.3.1.A range of number representation in the Fibonacci p-codes
6.3.2.Multiplicity of number representation
6.3.3.“Convolution” and “devolution” of the Fibonacci digits
6.3.4.The p-convolution and p-devolution for the case p = 1
6.3.5. MINIMAL p-FORM for the Fibonacci p-code
6.3.6.The base of the Fibonacci p-code
6.3.7.Code redundancy of the Fibonacci p-codes
6.4. The Simplest Fibonacci Arithmetical Operations. 6.4.1.Comparison of numbers in the Fibonacci p-codes
6.4.2.The Fibonacci summing and subtracting counters
6.4.3.Fibonacci summation of the binary numerals
6.4.4. Fibonacci summation of the multi-bit numbers (for the case p = 1)
6.4.5.Fibonacci subtraction
6.5. Fibonacci Multiplication and Division. 6.5.1.Fibonacci multiplication
6.5.2.Example of the Fibonacci multiplication
6.5.3.Example of Fibonacci division
6.6. Harvard Realization of Some Fibonacci Devices. 6.6.1.The device for the reduction of the Fibonacci representations to the MINIMAL FORM
6.6.2.The convolution register as a self-checking device
6.6.3.Device for checking the MINIMAL FORM
6.7. Fibonacci Counter for the MINIMAL FORM and Zeckendorf’s Theorem. 6.7.1.The simplest mathematical identities for the Fibonacci numbers
6.7.2.The Fibonacci code for the case p = 1
6.7.3.Fibonacci representation
6.7.4.Zeckendorf’s theorem
6.8. Algorithm of the Fibonacci Counter for the MINIMAL FORM. 6.8.1.An example of the Fibonacci counting for the MINIMAL FORM
6.9. General Operational Rules of the Fibonacci Counter for the MINIMAL FORM
6.10. Boolean Realization of the Fibonacci Counter for the MINIMAL FORM. 6.10.1.A general description
6.10.2. The Register
6.10.3.Block for Analysis, Block for Error Checking, Disposition Block, and Block for Zero Installation
6.11.Functioning of the Fibonacci Counter for the MINIMAL FORM. 6.11.1. The initial state
6.11.2.Increasing the Fibonacci counter capacity
6.11.3.Functioning of the Block for Error Checking
6.12. The Main Peculiarities of the Fibonacci Counter for the MINIMAL FORM. 6.12.1.A range of number representation for the MINIMAL FORM
6.12.2.A redundancy of the Fibonacci code for the MINIMAL FORM
6.12.3.Error-detecting ability of the Fibonacci code
6.12.4. The main advantages of the Fibonacci counter for the MINIMAL FORM
6.12.5.Significance of Zeckendorf’s theorem for future development of computing and micro-electronic technology
6.13. The Main Principles of the Fibonacci Computer
6.13.1.“Soft” and “hard” errors
6.13.2.Formula for error-detecting ability of the soft errors
6.13.3.Fibonacci Parity Code
6.13.4.Energy consumption and power dissipation in ROM
6.14. Fibonacci Arithmetic Based on the Basic Micro-Operations. 6.14.1.The basic micro-operations
6.14.2.Logical operations
6.15. Fibonacci Summation and Subtraction by Using “Basic Micro-Operations” 6.15.1.Fibonacci summation
6.15.2.Fibonacci subtraction
6.15.3.The example of the Fibonacci subtraction
6.16. A Conception of the Fibonacci Arithmetical Processor for the Noise-Immune Computations. 6.16.1.Noise-immune computations
6.16.2.Checking the basic micro-operations
6.16.3. The hardware realization of the noise-immune Fibonacci processor
6.17. Evaluation of the Results Obtained in Chapter 6
Chapter 7
7.1. Definition of the Codes of the Golden p-Proportions and Their Partial Cases. 7.1.1 The classic binary system and codes of the golden p-proportions
7.1.2 The partial cases of the codes of the golden p-proportions
7.2. Conversion of Numbers from Traditional Numeral Systems to Bergman’s System. 7.2.1 The table method
7.2.2 Conversion of the fractional numbers
7.2.3 Conversion of integers
7.3. The “Golden” Arithmetic. 7.3.1 The “golden” multiplicative and additive properties
7.3.2 The “golden” summation and subtraction
7.4. Representation of Numbers with the Floating Point
7.5. The “Golden” Multiplication
7.6. The “Golden” Division
7.7. Digital Metrology and a New Theory of Resistive Dividers Based on Golden p-Proportion Codes. 7.7.1 What is metrology?
7.7.2 A structure of the resistive dividers
7.7.3 The “binary” resistive divider
7.8. The “Golden” Self-Correcting Digital-to-Analog and Analog-to-Digital Converters. 7.8.1 The electrical properties of the “golden” resistive dividers
7.8.2 The “golden” digital-to-analog converters
7.8.3 Checking the “golden” DAC
7.8.4 The “golden” self-correcting ADC
Conclusions. General Conclusions
List of the Most Important Modern Scientific Results Chapters 1 and 2
Chapters 3–7
Chapter 3. Evolution of Numeral Systems
Chapter 4. Bergman’s System and “Golden” Number Theory
Chapter 5. The “Golden” Ternary Mirror-Symmetrical Arithmetic
Chapter 6. Fibonacci p-Codes and Fibonacci Arithmetic for Mission-Critical Applications
Chapter 7. Codes of the Golden p-Proportions (p = 0, 1, 2, 3, …)
Bibliography
