Sums of Exponential Functions and their New Fundamental Properties,

with Applications to Natural Phenomena

Table of Contents

Preface  7

Communication 11

Acknowledgements. 12

1. Exponential functions in Nature phenomena 14

1.1. Cellsí replication and growth as an example of exponential and logarithmic processes   14

1.2. Relationship between some universal quantitative characteristics that define Natureís evolvement and existence  16

1.3. Exponential functions in electrical processes 18

1.4. Inverse proportional function and the synthesis and destruction processes   19

1.5. Logarithmic functions  23

1.6. Natureís evolvement as a multifaceted phenomenon governed by laws   25

1.7. Fermatís Last Theorem and its relationship  37

with the physical world  37

1.7.1. A property of rarefaction of a discrete space  37

1.7.2. Notes about the proof of Fermatís Last Theorem  44

1.7.3. Inseparability and Singularity of the Universe  45

2. Properties of exponential functions and introduction of auxiliary concepts    50

2.1. Overview of the research and its applications  50

2.2. The Theorem and its Corollaries  54

2.3. Pair functions and synchronization concept   56

2.4. Lemma 1. Properties of pair functions  59

2.4.1. Uniqueness of characteristic points 60

2.4.2. The order of characteristic points 62

2.4.3. Properties of an HPF  63

2.4.4. Properties of an LPF  64

2.5. Lemma 2. Synchronization of pair functions  65

2.5.1. Synchronizing HPFs  66

2.5.2. Synchronizing LPFs  69

2.5.3. Synchronizing pair functions at an extremum point   70

2.5.4. Synchronization at an inflection point and other characteristic points    72

2.5.5. Corollary 1 of Lemma 2   73

2.5.6. Corollary 2 of Lemma 2   73

2.5.7. Corollary 3 of Lemma 2   74

2.5.8. A numerical example for Lemma 2 and its Corollaries  76

2.5.9. An example of a practical computation of the adjusted coefficients   79

2.6. Lemma 3. The sum of an HPF and pi-functions   80

2.6.1. Corollary 1 of Lemma 3. (Monotonic increase of the total function to the left of an HPF maximum)   86

2.6.2. Corollary 2 of Lemma 3  87

2.7. Lemma 4. Adding strong pi-functions to a sum of synchronized HPFs   88

2.7.1. Corollary 1 of Lemma 4  98

2.7.2. Corollary 2 of Lemma 4  99

2.8. Lemma 5. Adding weak positive functions to an LPF  99

2.8.1. Corollary 1 of Lemma 5  102

2.8.2. Corollary 2 of Lemma 5  102

2.9. Lemma 6. Equivalency of the sums of exponential functions and their derivatives  180  104

3. The Theorem  107

3.1. A proof of the Theorem  107

3.2. Changes in the functionís and its derivativesí form with the addition of more pi-functions    127

3.3. Considering the sum of exponential functions in a general form    130

3.4. Proof of Corollaries  132

3.4.1. Corollary 1. (One-time oscillation property)  132

3.4.2. Corollary 2. (Confluence of dialectical opposites)  134

3.4.3. Corollary 3. (Three solutions property)  135

3.4.4. Corollary 4. (Shift invariance of the properties of sums of exponential functions)  135

3.4.5. Corollary 5. (Two solutions property)  136

3.4.6. Corollary 6. (Two intersections of two exponential curves)  137

3.4.7. Corollary 7. (Properties of exponential series)  137

3.4.8. Corollary 8. (The number of solutions of the IRR equation)  138

3.4.9. Series Conjecture  145

4. Superposition of exponential electrical signals  146

5. Conclusion  149

References  150

 

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Last modified: 04/25/15