Table of Contents
 

ABOUT THE AUTHOR       11

PREFACE      13

CONTENT AT A GLANCE 16

ACKNOWLEDGEMENTS   19

1. EXPONENTIAL, LOGARITHMIC, POWER AND POLYNOMIAL FUNCTIONS  21

1.1. Polynomial functions       21

1.1.1. Definition of polynomial functions      21

1.1.2. The presence of polynomial functions in Nature          23

1.1.3. Properties of polynomial functions       26

1.1.4. Characteristic points of a polynomial function 30

1.2. Exponential functions      33

1.2.1. Exponential functions as Nature functions       33

1.2.2. Properties of exponential functions      34

1.3. Power functions and equations    40

1.3.1. Power functions in natural applications            40

1.3.2. Properties of power functions   41

1.4. Logarithmic functions and equations      44

1.4.1. General properties and relationship of logarithmic and exponential functions            44

1.4.2. Logarithmic and exponential functions and natural phenomena          45

1.4.3. Synthesis and destruction processes     49

CHAPTER 2.  52

POLYNOMIAL, POWER, EXPONENTIAL AND LOGARITHMIC  FUNCTIONS AND THEIR RELATIONSHIP       52

2.1. Introducing equivalent equation composed of the sum of exponential functions           52

2.2. Number of solutions of a polynomial and the number of sign changes of sums of exponential functions         55

2.3. Generalization of Lemma for real powers           60

2.3. Number of solutions of the power equation and its correspondence with the sum of exponential functions     68

2.4. The application of the theorem about the corresponding power and exponential equations       72

CHAPTER 3.  74

MATHEMATICAL MODELING OF NATURAL PHENOMENA           74

3.1. Cellsí replication and growth as an example of exponential and logarithmic processes 74

3.2. Relationship between some universal quantitative characteristics that define Natureís evolvement      76

3.3. Example of exponential functions in electrical processes            82

3.3.1. Adding exponential electrical signals   83

3.4. Inverse proportional function and the synthesis and destruction processes        84

3.4.1. Changing the fundamental laws. Electrical processes  86

3.4.2. Boundary function and natural selection          89

3.5. Modeling natural processes as a unity of interrelated factors, whose development is governed by laws           90

3.5.1. Dialectical laws and cognition of the Whole    90

3.5.2. Dialectics and Natureís evolvement     96

3.5.3. Dialectics from the historical perspective         101

3.5.4. Dialectics and social teachings 103

3.5.5. Determinism and randomness of natural phenomena   110

3.6. Fermatís Last Theorem and its relationship with the physical world      114

3.6.1. Rarefaction properties of the discrete space     114

3.6.2. Notes on the proof of Fermatís Last Theorem 124

3.6.3. Inseparability and Uniqueness of the Universe            124

CHAPTER 4.  131

PROPERTIES OF EXPONENTIAL FUNCTIONS AND INTRODUCTION OF AUXILIARY CONCEPTS  131

4.1. Overview of the research and its applications     131

4.2. Pair functions and the synchronization concept  136

4.2.1. The axial symmetry of the high and low pair functions           138

4.2.2. Lemma 1. Properties of pair functions  139

4.2.3. Uniqueness of characteristic points      140

4.2.4. The order of characteristic points         142

4.2.5. Properties of an HPF    143

4.2.6. Properties of an LPF     145

4.3. Lemma 2. Synchronization of pair functions       146

4.3.1. Synchronizing HPFs     146

4.3.2. Synchronizing LPFs      149

4.3.3. Synchronizing pair functions at the extremum point    151

4.3.4. The role of exponential functions in the evolvement of Nature           152

4.3.5. Synchronization at an inflection point and other characteristic points 153

4.3.6. Corollary 1 of Lemma 2            154

4.3.7. Corollary 2 of Lemma 2            154

4.3.8. Corollary 3 of Lemma 2.           155

4.3.9. A numerical example for Lemma 2 and its Corollaries 157

4.3.10. An example of a practical computation of adjusted coefficients       160

4.4. Lemma 3. The sum of an HPF and pi-functions  163

4.4.1. Corollary 1 of Lemma 3. (Monotonic increase of the total function to the left of the HPFís maximum)       168

4.4.4. Corollary 2 of Lemma 3            170

4.5. Lemma 4. Adding strong pi-functions to a sum of synchronized HPFs 172

4.5.1. Corollary 1 of Lemma 4            183

4.5.2. Corollary 2 of Lemma 4            184

4.6. Lemma 5. Adding weak positive functions to an LPF    185

4.6.1. Corollary 1 of Lemma 5            187

4.6.2. Corollary 2 of Lemma 5            188

4.7. Lemma 6. Equivalency of the sums of exponential functions and their derivatives       190

5. GENERALIZATIONS AND SOME APPLICATIONS OF THEOREMS ABOUT THE CORRESPONDING FUNCTIONS 194

5.1. Properties of natural phenomena 194

5.1.1. Oscillation property      194

5.1.2. Confluence of dialectical opposites      196

5.1.3. Shift invariance of sums of exponential functions       198

5.1.4. Intersections of exponential curves      199

5.2. Solutions of IRR equation           200

5.2.1. Specifics of IRR equation        201

5.2.2. Transformation of IRR Equation          204

5.2.3. Examples           208

5.2.4. The property of intermediate market values and their relation to values of cash flows           209

5.2.5. Conclusion        214

6. EPILOGUE            216

REFERENCES          218

INDEX           221

 

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Last modified: 04/25/15