Mathematics That Power Our World, The: How Is It Made?

by Joseph Khoury, Gilles Lamothe

Overview

The Mathematics That Power Our World: How Is It Made? is an attempt to unveil the hidden mathematics behind the functioning of many of the devices we use on a daily basis. For the past years, discussions on the best approach in teaching and learning mathematics have shown how much the world is divided on this issue. The one reality we seem to agree on globally is the fact that our new generation is lacking interest and passion for the subject. One has the impression that the vast majority of young students finishing high school or in their early post-secondary studies are more and more divided into two main groups when it comes to the perception of mathematics. The first group looks at mathematics as a pure academic subject with little connection to the real world. The second group considers mathematics as a set of tools that a computer can be programmed to use and thus, a basic knowledge of the subject is sufficient. This book serves as a middle ground between these two views. Many of the elegant and seemingly theoretical concepts of mathematics are linked to state-of-the-art technologies. The topics of the book are selected carefully to make that link more relevant. They include: digital calculators, basics of data compression and the Huffman coding, the JPEG standard for data compression, the GPS system studied both from the receiver and the satellite ends, image processing and face recognition.This book is a great resource for mathematics educators in high schools, colleges and universities who want to engage their students in advanced readings that go beyond the classroom discussions. It is also a solid foundation for anyone thinking of pursuing a career in science or engineering. All efforts were made so that the exposition of each topic is as clear and self-contained as possible and thus, appealing to anyone trying to broaden his mathematical horizons.

Product Details

ISBN-13:	9789813144088
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	06/24/2016
Pages:	204
Product dimensions:	5.90(w) x 8.90(h) x 0.60(d)

Table of Contents

Preface vii

1 What makes a calculator calculate? 1

1.1 Introduction 1

1.1.1 A view from the inside 2

1.1.2 Before you go further 2

1.2 Number systems 2

1.2.1 Why O's and 1's? 3

1.2.2 The binary system 4

1.2.3 Binary Coded Decimal representation (BCD) 6

1.2.4 Signed versus unsigned binary numbers 7

1.3 Binary arithmetics 11

1.3.1 Binary addition of unsigned integers 12

1.3.2 Binary addition of signed integers 13

1.3.3 Two's complement subtraction 14

1.4 Logic 15

1.4.1 Logic gates 16

1.5 Boolean Algebra 19

1.5.1 Sum of products - Product of sums 21

1.5.2 Sum of products 23

1.5.3 Product of sums 24

1.6 Digital adders 25

1.6.1 Half-adder 25

1.6.2 Full-adder 26

1.6.3 Lookahead adder 28

1.6.4 Two's complement implementation 30

1.6.5 Adder-subtractor combo 31

1.7 BCD to seven-segment decoder 32

1.8 So, how does the magic happen? 36

1.9 What next? 36

1.10 References 37

2 Basics of data compression, prefix-free codes and Huffman codes 39

2.1 Introduction 39

2.1.1 What is data compression and do we really need it? 39

2.1.2 Before you go further 40

2.2 Storage inside computers 40

2.2.1 Measuring units 41

2.3 Lossy and lossless data compression 41

2.4 Binary codes 43

2.4.1 Binary trees 43

2.4.2 Fixed length and variable length codes 44

2.5 Prefix-free code 45

2.5.1 Decoding a message using a prefix-free code 45

2.5.2 How to decide if a code is prefix-free? 46

2.5.3 The Kraft inequality for prefix-free binary codes 47

2.6 Optimal codes 49

2.7 The source entropy 54

2.8 The Huffman code 55

2.8.1 The construction 57

2.8.2 The Huffman algorithm 58

2.8.3 An example 59

2.9 Some remarks 63

2.10 References 64

3 The JPEG standard 65

3.1 Introduction 65

3.1.1 Before you go further 66

3.2 The Discrete Cosine Transform (DCT) 66

3.2.1 The one-dimensional DCT 67

3.2.2 The two-dimensional DCT 70

3.3 DCT as a tool for image compression 72

3.3.1 Image pre-processing 72

3.3.2 Level shifting 73

3.3.3 Applying the DCT 73

3.3.4 Quantization 74

3.3.5 Encoding 75

3.4 JPEG decompression 82

3.5 The mathematics of DCT 84

3.5.1 Two-dimensional DCT as a linear transformation 84

3.5.2 What is the deal with orthogonal bases anyway? 87

3.5.3 Proof of the orthogonality of the DCT matrix 89

3.5.4 Proof of Theorem 4.1 92

3.6 References 94

4 Global Positioning System (GPS) 95

4.1 Introduction 95

4.1.1 Before you go further 96

4.2 Latitude, longitude and altitude 96

4.3 About the GPS system 98

4.3.1 The GPS constellation 98

4.3.2 The GPS signal 98

4.4 Pinpointing your location 99

4.4.1 Where am I on the map? 99

4.4.2 Measuring the distance to a satellite 100

4.4.3 Where am I on the surface of the planet? 102

4.4.4 Is it really that simple? 103

4.4.5 The fix 104

4.4.6 Finding the coordinates of the receiver 105

4.4.7 Conversion from cartesian to (latitude, longitude, altitude) coordinates 108

4.5 The mathematics of the GPS signal 109

4.5.1 Terminology 109

4.5.2 Linear Feedback Shift Registers 110

4.5.3 Some modular arithmetic 113

4.5.4 Groups 115

4.5.5 Fields - An introduction and basic results 120

4.5.6 The field Z_p 121

4.5.7 Polynomials over a field 123

4.5.8 The field Z_pr - A first approach 127

4.5.9 The field F_pr - A second approach 129

4.5.10 The lead function 132

4.6 Key properties of GPS signals: Correlation and maximal period 133

4.6.1 Correlation 133

4.6.2 The LFSR sequence revisited 134

4.6.3 Proof of Theorem 4.1 135

4.6.4 More about the signal 138

4.7 A bit of history 140

4.8 References 141

5 Image processing and face recognition 143

5.1 Introduction 143

5.1.1 Before you go further 144

5.2 Paster image 144

5.3 Invertible linear transformations 145

5.4 Gray level for the new image 149

5.5 Bilinear interpolation 150

5.6 The centroid of the face 152

5.7 Optimal transformation for the average face 153

5.8 Convex sets and extremal points 157

5.9 Least squares method 159

5.9.1 Dot product - Inner product 159

5.10 Face recognition 164

5.10.1 Descriptive statistics 165

5.10.2 The principle components from the covariance matrix 172

5.10.3 Comparison of the principle features 179

5.10.4 Visualizing the features 180

5.11 References 182

Index 183