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This textbook presents a rigorous approach to multivariable calculus in the context of model building and optimization problems. This comprehensive overview is based on lectures given at five SERC Schools from 2008 to 2012 and covers a broad range of topics that will enable readers to understand and create deterministic and nondeterministic models.  Researchers, advanced undergraduate, and graduate students in mathematics, statistics, physics, engineering, and biological sciences will find this book to be a valuable resource for finding appropriate models to describe real-life situations.

The first chapter begins with an introduction to fractional calculus moving on to discuss fractional integrals, fractional derivatives, fractional differential equations and their solutions.  Multivariable calculus is covered in the second chapter and introduces the fundamentals of multivariable calculus (multivariable functions, limits and continuity, differentiability, directional derivatives and expansions of multivariable functions). Illustrative examples, input-output process, optimal recovery of functions and approximations are given; each section lists an ample number of exercises to heighten understanding of the material. Chapter three  discusses deterministic/mathematical and optimization models evolving from differential equations, difference equations, algebraic models, power function models, input-output models and pathway models. Fractional integral and derivative models are examined.  Chapter four covers non-deterministic/stochastic models. The random walk model, branching process model, birth and death process model, time series models, and regression type models are examined. The fifth chapter covers optimal design. General linear models from a statistical point of view are introduced; the Gauss?Markov theorem, quadratic forms, and generalized inverses of matrices are covered. Pathway, symmetric, and asymmetric models are covered in chapter six, the concepts are illustrated with graphs.

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This textbook presents a rigorous approach to multivariable calculus in the context of model building and optimization problems. This comprehensive overview is based on lectures given at five SERC Schools from 2008 to 2012 and covers a broad range of topics that will enable readers to understand and create deterministic and nondeterministic models.  Researchers, advanced undergraduate, and graduate students in mathematics, statistics, physics, engineering, and biological sciences will find this book to be a valuable resource for finding appropriate models to describe real-life situations.

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CONTENTS
1 Essentials of Fractional Calculus = 1
 1.1 Introduction = 1
 1.2 Notions and Notations = 2
 1.3 The Mittag-Leffler Functions = 5
 1.4 The Wright Functions = 8
 1.5 Essentials of Fractional Calculus = 15
 1.6 Fractional Differential Equations of Relaxation and Oscillation Type = 19
 1.7 Fractional Differential Equations of Diffusion-Wave Type = 23
  1.7.1 Derivation of the Fundamental Solutions = 25
 References = 29
2 Multivariable Calculus = 39
 2.1 Review of Euclidean n-Space Rⁿ= 39
 2.2 Geometric Approach : Vectors in 2-Space and 3-Space = 39
  2.2.1 Components and the Norm of a Vector = 40
  2.2.2 Position Vector = 40
  2.2.3 Vectors as Ordered Triplets of Real Numbers = 40
 2.3 Analytic Approach = 41
  2.3.1 Properties of Addition and Scalar Multiplication = 41
 2.4 The Dot Product or the Inner Product = 41
  2.4.1 The Properties of Dot Product = 42
  2.4.2 Properties of the Norm = 43
  2.4.3 Geometric Interpretation of the Dot Product = 43
 2.5 Multivariable Functions = 44
  2.5.1 Scalar and Vector Functions and Fields = 45
  2.5.2 Visualization of Scalar-Valued Functions = 47
 2.6 Limits and Continuity = 48
  2.6.1 Limits = 48
  2.6.2 Continuity = 50
 2.7 Partial Derivatives and Differentiability = 52
  2.7.1 Partial Derivatives = 52
  2.7.2 Differentiability = 53
  2.7.3 Chain Rule = 55
  2.7.4 Gradient and Directional Derivatives = 56
  2.7.5 Tangent Plane and Normal Line = 58
  2.7.6 Differentiability of Vector-Valued Functions, Chain Rule = 59
  2.7.7 Particular Cases = 60
  2.7.8 Differentiation Rules = 62
  2.7.9 Taylor''''s Expansion = 63
 2.8 Introduction to Optimization = 64
  2.8.1 Unconstrained and Constrained Extremizers : Conditions for Local Extremizers = 66
 2.9 Classical Approximation Problems : A Relook = 73
  2.9.1 Input-Output Process = 73
  2.9.2 Approximation by Algebraic Polynomials = 73
 2.10 Introduction to Optimal Recovery of Functions = 74
  2.10.1 Some Motivating Examples = 76
  2.10.2 General Theory = 81
  2.10.3 Central Algorithms = 84
  2.10.4 Notes = 87
 References = 88
3 Deterministic Models and Optimization = 89
 3.1 Introduction = 89
 3.2 Deterministic Situations = 89
 3.3 Differential Equations = 90
 3.4 Algebraic Models = 92
 3.5 Computer Models = 93
 3.6 Power Function Models = 94
 3.7 Input-Output Models = 94
 3.8 Pathway Model = 95
  3.8.1 Optimization of Entropy and Exponential Model = 96
  3.8.2 Optimization of Mathai''''s Entropy = 97
 3.9 Fibonacci Sequence Model = 98
 3.10 Fractional Order Integral and Derivative Model = 101
 References = 106
4 Non-deterministic Models and Optimization = 107
 4.1 Introduction = 107
  4.1.1 Random Walk Model = 108
  4.1.2 Branching Process Model = 109
  4.1.3 Birth-and-Death Process Model = 109
  4.1.4 Time Series Models = 110
  4.1.5 Regression Type Models = 110
 4.2 Some Preliminaries of Statistical Distributions = 111
  4.2.1 Minimum Mean Square Prediction = 126
 4.3 Regression on Several Variables = 130
 4.4 Linear Regression = 133
  4.4.1 Correlation Between x₁and Its Best Linear Predictor = 140
 4.5 Multiple Correlation Coefficientρ₁.(₂...[TEX]$$His_{k}$$[/TEX]) = 142
  4.5.1 Some Properties of the Multiple Correlation Coefficient = 143
 4.6 Regression Analysis Versus Correlation Analysis = 150
  4.6.1 Multiple Correlation Ratio = 150
  4.6.2 Multiple Correlation as a Function of the Number of Regressed Variables = 152
 4.7 Residual Effect = 155
 4.8 Canonical Correlations = 157
  4.8.1 First Pair of Canonical Variables = 164
  4.8.2 Second an Subsequent Pair of Canonical Variables = 167
 4.9 Estimation of the Regression Function = 170
  4.9.1 Linear Regression of y on x = 171
  4.9.2 Linear Regression of y on x₁,..., x[TEX]$$His_{k}$$[/TEX] = 177
5 Optimal Regression Designs = 183
 5.1 Introduction = 183
 5.2 The General Linear Model = 183
 5.3 Estimable Linear Functions = 184
  5.3.1 The Gauss-Markov Theorem = 186
 5.4 Correlated Errors = 186
  5.4.1 Estimation of Some of the Parameters = 187
 5.5 Other Distance Measures = 188
 5.6 Means and Covariances of Random Vectors = 188
  5.6.1 Quadratic Forms and Generalized Inverses = 189
  5.6.2 Distribution of Quadratic Forms = 189
  5.6.3 Normal Errors = 191
  5.6.4 Rank and Nullity of a Matrix = 191
 5.7 Exact or Discrete Optimal Designs = 195
  5.7.1 Optimality Criteria = 198
  5.7.2 Two-Level Factorial Designs of Resolution Ⅲ = 199
  5.7.3 Optimal Block Designs = 201
  5.7.4 Approximate or Continuous Optimal Designs = 202
  5.7.5 Algorithms = 204
 References = 207
6 Pathway Models = 209
 6.1 Introduction = 209
 6.2 Power Transformation and Exponentiation of a Type 1 Beta Density = 209
  6.2.1 Asymmetric Models = 212
 6.3 Power Transformation and Exponentiation on a Type 2 Beta Model = 213
  6.3.1 Asymmetric Version of the Models = 216
 6.4 Laplace Transforms and Moment Generating Functions = 217
 6.5 Models with Appended Bessel Series = 219
 6.6 Models with Appended Mittag-Leffler Series = 220
 6.7 Bayesian Outlook = 221
 6.8 Pathway Model in the Scalar Case = 222
 6.9 Pathway Model in the Matrix-Variate Case = 224
  6.9.1 Arbitrary Moments = 227
  6.9.2 Some Special Cases = 228
 References = 228
Index = 231
Author Index = 233