Description
This book details how applied mathematics involves predictions, interpretations, analysis, and mathematical modeling to solve real-world problems. Due to the broad range of applications, mathematical concepts and techniques and reviewed throughout, especially those in linear algebra, matrix analysis, and differential equations. Some classical definitions and results from analysis are also discussed and used. Some applications (postscript fonts, information retrieval, etc.) are presented at the end of a chapter as an immediate application of the theory just covered, while those applications that are discussed in more detail (ranking web pages, compression, etc.) are presented in dedicated chapters. Acollection of mathematical models of a slightly different nature, such as basic discrete mathematics and optimization, is also provided. Clear proofs of the main theorems ultimately help to make the statements of the theorems more understandable, and a multitude of examples follow important theorems and concepts. In addition, the author builds material from scratch and thoroughly covers the theory needed to explain the applications in full detail, while not overwhelming readers with unneccessary topics or discussions. In terms of exercises, the author continuously refers to the real numbers and results in calculus when introducing a new topic so readers can grasp the concept of the otherwise intimidating expressions. By doing this, the author is able to focus on the concepts rather than the rigor. The quality, quantity, and varying level of difficulty of the exercises provides instructors more classroom flexibility. Topical coverage includes linear algebra; ranking web pages; matrix factorizations; least squares; image compression; ordinary differential equations; dynamical systems; and mathematical models. JORGE REBAZA, PhD, is Associate Professor in the Department of Mathematics at Missouri State University. Dr. Rebaza has published numerous journal articles in his areas of research interest, which include numerical analysis, dynamical systems, matrix computations, and applied mathematics. Preface xi 1. Basics of Linear Algebra 1 1.1 Notation and Terminology 1 1.2 Vector and Matrix Norms 4 1.3 Dot Product and Orthogonality 8 1.4 Special Matrices 9 1.5 Vector Spaces 21 1.6 Linear Independence and Basis 24 1.7 Orthogonalization and Direct Sums 30 1.8 Column Space, Row Space and Null Space 34 1.9 Orthogonal Projections 43 1.10 Eigenvalues and Eigenvectors 47 1.11 Similarity 56 1.12 Bezier Curves Postscripts Fonts 59 1.13 Final Remarks and Further Reading 68 2. Ranking Web Pages 79 2.1 The Power Method 80 2.2 Stochastic, Irreducible and Primitive Matrices 84 2.3 Google’s PageRank Algorithm 92 2.4 Alternatives to Power Method 106 2.5 Final Remarks and Further Reading 120 3. Matrix Factorizations 131 3.1 LU Factorization 132 3.2 QR Factorization 142 3.3 Singular Value Decomposition (SVD) 155 3.4 Schur Factorization 166 3.5 Information Retrieval 186 3.6 Partition of Simple Substitution Cryptograms 194 3.7 Final Remarks and Further Reading 203 4. Least Squares 215 4.1 Projections and Normal Equations 215 4.2 Least Squares and QR Factorization 224 4.3 Lagrange Multipliers 228 4.4 Final Remarks and Further Reading 231 5. Image Compression 235 5.1 Compressing with Discrete Cosine Transform 236 5.2 Huffman Coding 260 5.3 Compression with SVD 267 5.4 Final Remarks and Further Reading 271 6. Ordinary Differential Equations 277 6.1 One-Dimensional Differential Equations 278 6.2 Linear Systems of Differential Equations 307 6.3 Solutions via Eigenvalues and Eigenvectors 308 6.4 Fundamentals Matrix Solution 312 6.5 Final Remarks and Further Reading 316 7. Dynamical Systems 325 7.1 Linear Dynamical Systems 326 7.2 Nonlinear Dynamical Systems 340 7.3 Predator-Prey Models with Harvesting 374 7.4 Final Remarks and Further Reading 385 8. Mathematical Models 395 8.1 Optimization of a Waste Management System 396 8.2 Grouping Problem in Networks 404 8.3 American Cutaneous Leishmaniasis 410 8.4 Variable Population Interactions 420 References 431 Index 435
