Table of Contents

ABOUT THE AUTHOR 10
PREFACE 11
CHAPTERSí CONTENT AT A GLANCE 13
ACKNOWLEDGEMENTS 15
CHAPTER 1 16
EXPONENTIAL, LOGARITHMIC AND POWER FUNCTIONS 16
1.1. Polynomial functions 16
1.1.1. Definition of polynomial functions 16
1.1.2. Properties of polynomial functions 17
1.2. Exponential functions 22
1.2.1. Exponential functions as Nature functions 22
1.2.2. Properties of exponential functions 23
1.3. Power functions and equations 29
1.3.1. Power functions in natural applications 29
1.3.2. Properties of power functions 30
1.4. Logarithmic and exponential functions and equations 33
1.4.1. General properties and relationship of logarithmic and exponential functions 33
1.4.2. Logarithmic and exponential functions and natural phenomena 35
1.4.3. Synthesis and destruction processes 39
CHAPTER 2 42
POLYNOMIAL, POWER, EXPONENTIAL AND LOGARITHMIC FUNCTIONS AND THEIR RELATIONSHIP 42
2.1. Introducing equivalent equation composed of the sum of exponential functions 42
2.2. Number of solutions of a polynomial, and the number of sign changes of sums of exponential
functions 45
2.3. Generalization of Lemma for real powers 50
2.4. Number of solutions of the power equation and its correspondence with the sum of exponential
functions 58
2.5. The application of the theorem about the correspondence of power and exponential equations 62
CHAPTER 3 64
MATHEMATICAL MODELING OF NATURAL PHENOMENA 64
3.1. Cellsí replication and growth 64
3.2. Relationship between some universal characteristics 66
3.3. Example of exponential functions in electrical processes 72
3.3.1. Adding exponential electrical signals 73
3.4. Inverse proportional function and the synthesis and destruction processes 74
3.4.1. Changing the fundamental laws. Electrical processes 76
3.4.2. Boundary function and natural selection 79
3.5. Modeling natural processes. Unity of interrelated factors 80
3.5.1. Dialectical laws and their unity 80
3.5.2. Dialectics and Natureís evolvement 87
3.5.3. Dialectics from the historical perspective 92
3.5.4. Determinism and randomness 94
3.6. Fermatís Last Theorem and its relationship with the physical world 98
3.6.1. Rarefaction properties of the discrete space 98
3.6.2. Notes on the proof of Fermatís Last Theorem 108
3.6.3. Inseparability and Uniqueness of the Universe 108
CHAPTER 4 115
PROPERTIES OF EXPONENTIAL FUNCTIONS AND INTRODUCTION OF AUXILIARY CONCEPTS 115
4.1. Overview of the research and its applications 115
4.2. Pair functions and the synchronization concept 120
4.2.1. The axial symmetry of the high and low pair functions 122
4.2.2. Lemma 1. Properties of pair functions 123
4.2.3. Uniqueness of characteristic points 124
4.2.4. The order of characteristic points 126
4.2.5. Properties of an HPF 127
4.2.6. Properties of an LPF 129
4.3. Lemma 2. Synchronization of pair functions 130
4.3.1. Synchronizing HPFs 130
4.3.2. Synchronizing LPFs 134
4.3.3. Synchronizing pair functions at the extremum point 135
4.3.4. The role of exponential functions in the evolvement of Nature 137
4.3.5. Synchronization at an inflection point and other characteristic points 137
4.3.6. Corollary 1 of Lemma 2 138
4.3.7. Corollary 2 of Lemma 2 139
4.3.8. Corollary 3 of Lemma 2. 139
4.3.9. A numerical example for Lemma 2 and its Corollaries 141
4.3.10. An example of a practical computation of adjusted coefficients 144
4.4. Lemma 3. The sum of an HPF and pi-functions 147
4.4.1. Corollary 1 of Lemma 3. (Monotonic increase of the total function to the left of the HPFís
maximum) 152
4.4.2. Corollary 2 of Lemma 3 154
4.5. Lemma 4. Adding strong pi-functions to the sum of synchronized HPFs 156
4.5.1. Corollary 1 of Lemma 4 167
4.5.2. Corollary 2 of Lemma 4 168
4.6. Lemma 5. Adding weak positive functions to LPF 169
4.6.1. Corollary 1 of Lemma 5 171
4.6.2. Corollary 2 of Lemma 5 172
4.7. Lemma 6. Equivalency of sums of exponential functions and their derivatives 174
CHAPTER 5 177
SOME PROPERTIES OF NATURAL PHENOMENA AND THEIR MATHEMATICAL REPRESENTATION 177
5.1. Properties of natural phenomena 177
5.1.1. Oscillation property 177
5.1.2. Confluence of dialectical opposites 179
5.1.3. Shift invariance of sums of exponential functions 181
5.1.4. Intersections of exponential curves 182
AFTERWORD 183
REFERENCES 184
INDEX 186

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Last modified: 04/25/15