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A First Course in Applied Mathematics

Book 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.

Table of Contents

  1. Cover Page
  2. Title Page
  3. Copyright
  4. Dedication
  5. Contents
  6. PREFACE
  7. CHAPTER 1: BASICS OF LINEAR ALGEBRA
    1. 1.1 NOTATION AND TERMINOLOGY
    2. 1.2 VECTOR AND MATRIX NORMS
    3. 1.3 DOT PRODUCT AND ORTHOGONALITY
    4. 1.4 SPECIAL MATRICES
    5. 1.5 VECTOR SPACES
    6. 1.6 LINEAR INDEPENDENCE AND BASIS
    7. 1.7 ORTHOGONALIZATION AND DIRECT SUMS
    8. 1.8 COLUMN SPACE, ROW SPACE, AND NULL SPACE
    9. 1.9 ORTHOGONAL PROJECTIONS
    10. 1.10 EIGENVALUES AND EIGENVECTORS
    11. 1.11 SIMILARITY
    12. 1.12 BEZIER CURVES AND POSTSCRIPT FONTS
    13. 1.13 FINAL REMARKS AND FURTHER READING
  8. CHAPTER 2: RANKING WEB PAGES
    1. 2.1 THE POWER METHOD
    2. 2.2 STOCHASTIC, IRREDUCIBLE, AND PRIMITIVE MATRICES
    3. 2.3 GOOGLE'S PAGERANK ALGORITHM
    4. 2.4 ALTERNATIVES TO THE POWER METHOD
    5. 2.5 FINAL REMARKS AND FURTHER READING
  9. CHAPTER 3: MATRIX FACTORIZATIONS
    1. 3.1 LU FACTORIZATION
    2. 3.2 QR FACTORIZATION
    3. 3.3 SINGULAR VALUE DECOMPOSITION (SVD)
    4. 3.4 SCHUR FACTORIZATION
    5. 3.5 INFORMATION RETRIEVAL
    6. 3.6 PARTITION OF SIMPLE SUBSTITUTION CRYPTOGRAMS
    7. 3.7 FINAL REMARKS AND FURTHER READING
  10. CHAPTER 4: LEAST SQUARES
    1. 4.1 PROJECTIONS AND NORMAL EQUATIONS
    2. 4.2 LEAST SQUARES AND QR FACTORIZATION
    3. 4.3 LAGRANGE MULTIPLIERS
    4. 4.4 FINAL REMARKS AND FURTHER READING
  11. CHAPTER 5: IMAGE COMPRESSION
    1. 5.1 COMPRESSING WITH DISCRETE COSINE TRANSFORM
    2. 5.2 HUFFMAN CODING
    3. 5.3 COMPRESSION WITH SVD
    4. 5.4 FINAL REMARKS AND FURTHER READING
  12. CHAPTER 6: ORDINARY DIFFERENTIAL EQUATIONS
    1. 6.1 ONE-DIMENSIONAL DIFFERENTIAL EQUATIONS
    2. 6.2 LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS
    3. 6.3 SOLUTIONS VIA EIGENVALUES AND EIGENVECTORS
    4. 6.4 FUNDAMENTAL MATRIX SOLUTION
    5. 6.5 FINAL REMARKS AND FURTHER READING
  13. CHAPTER 7: DYNAMICAL SYSTEMS
    1. 7.1 LINEAR DYNAMICAL SYSTEMS
    2. 7.2 NONLINEAR DYNAMICAL SYSTEMS
    3. 7.3 PREDATOR–PREY MODELS WITH HARVESTING
    4. 7.4 FINAL REMARKS AND FURTHER READING
  14. CHAPTER 8: MATHEMATICAL MODELS
    1. 8.1 OPTIMIZATION OF A WASTE MANAGEMENT SYSTEM
    2. 8.2 GROUPING PROBLEM IN NETWORKS
    3. 8.3 AMERICAN CUTANEOUS LEISHMANIASIS
    4. 8.4 VARIABLE POPULATION INTERACTIONS
  15. REFERENCES
  16. INDEX