Tabla de Contenidos: “…Chapter 2: Dispelling Ghosts from the Past: A Review of Pre-Calculus and Calculus IForgotten but Not Gone: A Review of Pre-Calculus; Recent Memories: A Review of Calculus I; Finding Limits Using L'Hopital's Rule; Chapter 3: From Definite to Indefinite: The Indefinite Integral; Approximate Integration; Knowing Sum-Thing about Summation Formulas; As Bad as It Gets: Calculating Definite Integrals Using the Riemann Sum Formula; Light at the End of the Tunnel: The Fundamental Theorem of Calculus; Understanding the Fundamental Theorem of Calculus; Your New Best Friend: The Indefinite Integral…”

Tabla de Contenidos: “…Gauss, príncipe de los matemáticos / José Ferreirós -- 4. Bernhard Riemann : la potencia de un genio / María de Paz -- 5. …”

Tabla de Contenidos: “…Problemas resueltos; Capıtulo 3 Calculo Integral con una Variable; 3.1. La integral de Riemann; 3.2. Algunos teoremas sobre integrales; 3.3. …”

Tabla de Contenidos: “…; 1.11 Potential and Kinetic Energies; 1.12 Riemann Zeta Function and Prime Numbers; 1.13 How to Solve It; 1.13.1 Understanding the Problem…”

Tabla de Contenidos: “…Concepto de superficie de Riemann…”

Tabla de Contenidos: “…11 Using Derivatives to Graph 173 -- Relative Extrema 174 -- Finding Critical Numbers 175 -- Classifying Extrema 176 -- The Wiggle Graph 178 -- The Extreme Value Theorem 180 -- Determining Concavity 182 -- Another Wiggle Graph 183 -- The Second Derivative Test 184 -- 12 Derivatives and Motion 187 -- The Position Equation 188 -- Velocity 190 -- Acceleration 191 -- Vertical Projectile Motion 193 -- 13 Common Derivative Applications 195 -- Newton's Method 196 -- Evaluating Limits: L'Hôpital's Rule 199 -- More Existence Theorems 200 -- The Mean Value Theorem 201 -- Rolle's Theorem 203 -- Related Rates 204 -- Optimization 208 -- Part 4: The Integral 215 -- 14 Approximating Area 217 -- Riemann Sums 218 -- Right and Left Sums 219 -- Midpoint Sums 221 -- The Trapezoidal Rule 222 -- Simpson's Rule 225 -- 15 Antiderivatives 227 -- The Power Rule for Integration 228 -- Integrating Trigonometric Functions 230 -- Separation 232 -- The Fundamental Theorem of Calculus 233 -- Part One: Areas and Integrals Are Related 233 -- Part Two: Derivatives and Integrals Are Opposites 235 -- u-Substitution 236 -- Tricky u-Substitution and Long Division 237 -- Technology Focus: Definite and Indefinite Integrals 239 -- 16 Applications of the Fundamental Theorem 245 -- Calculating Area Between Two Curves 246 -- The Mean Value Theorem for Integration 249 -- A Geometric Interpretation 249 -- The Average Value Theorem 251 -- Finding Distance Traveled 253 -- Accumulation Functions 255 -- Arc Length 256 -- Rectangular Equations 256 -- Parametric Equations 257 -- Part 5: Differential Equations and More 259 -- 17 Differential Equations 261 -- Separation of Variables 262 -- Types of Solutions 263 -- Family of Solutions 264 -- Specific Solutions 266 -- Exponential Growth and Decay 267 -- 18 Visualizing Differential Equations 275 -- Linear Approximation 276 -- Slope Fields 277.…”

Tabla de Contenidos: “…Antiderivatives and Indefinite Integration. Area. Riemann Sums and Definite Integrals. The Fundamental Theorem of Calculus. …”

Tabla de Contenidos: “…Intro -- Examples and Problems in Mathematical Statistics -- Contents -- Preface -- List of Random Variables -- List of Abbreviations -- 1 Basic Probability Theory -- PART I: THEORY -- 1.1 OPERATIONS ON SETS -- 1.2 ALGEBRA AND σ-FIELDS -- 1.3 PROBABILITY SPACES -- 1.4 CONDITIONAL PROBABILITIES AND INDEPENDENCE -- 1.5 RANDOM VARIABLES AND THEIR DISTRIBUTIONS -- 1.6 THE LEBESGUE AND STIELTJES INTEGRALS -- 1.6.1 General Definition of Expected Value: The Lebesgue Integral -- 1.6.2 The Stieltjes-Riemann Integral -- 1.6.3 Mixtures of Discrete and Absolutely Continuous Distributions -- 1.6.4 Quantiles of Distributions -- 1.6.5 Transformations -- 1.7 JOINT DISTRIBUTIONS, CONDITIONAL DISTRIBUTIONS AND INDEPENDENCE -- 1.7.1 Joint Distributions -- 1.7.2 Conditional Expectations: General Definition -- 1.7.3 Independence -- 1.8 MOMENTS AND RELATED FUNCTIONALS -- 1.9 MODES OF CONVERGENCE -- 1.10 WEAK CONVERGENCE -- 1.11 LAWS OF LARGE NUMBERS -- 1.11.1 The Weak Law of Large Numbers (WLLN) -- 1.11.2 The Strong Law of Large Numbers (SLLN) -- 1.12 CENTRAL LIMIT THEOREM -- 1.13 MISCELLANEOUS RESULTS -- 1.13.1 Law of the Iterated Logarithm -- 1.13.2 Uniform Integrability -- 1.13.3 Inequalities -- 1.13.4 The Delta Method -- 1.13.5 The Symbols op and Op -- 1.13.6 The Empirical Distribution and Sample Quantiles -- PART II: EXAMPLES -- PART III: PROBLEMS -- PART IV: SOLUTIONS TO SELECTED PROBLEMS -- 2 Statistical Distributions -- PART I: THEORY -- 2.1 INTRODUCTORY REMARKS -- 2.2 FAMILIES OF DISCRETE DISTRIBUTIONS -- 2.2.1 Binomial Distributions -- 2.2.2 Hypergeometric Distributions -- 2.2.3 Poisson Distributions -- 2.2.4 Geometric, Pascal, and Negative Binomial Distributions -- 2.3 SOME FAMILIES OF CONTINUOUS DISTRIBUTIONS -- 2.3.1 Rectangular Distributions -- 2.3.2 Beta Distributions -- 2.3.3 Gamma Distributions -- 2.3.4 Weibull and Extreme Value Distributions…”

Tabla de Contenidos: “…PROPIEDADES -- 6.2 SUMA DE RIEMANN -- 6.3 ÁREA BAJO UNA CURVA -- 6.4 INTEGRAL DEFINIDA -- 6.4.1 DEFINICIÓN -- 6.4.2 PROPIEDADES DE LA INTEGRAL DEFINIDA -- 6.4.3 TEOREMAS FUNDAMENTALES DEL CÁLCULO INTEGRAL -- 6.4.4 CAMBIO DE VARIABLE PARA LA INTEGRAL DEFINIDA -- 6.5 INTEGRACIÓN NUMÉRICA -- 6.5.1 REGLA TRAPEZOIDAL -- 6.5.2 REGLA DE SIMPSON…”

Tabla de Contenidos: “…. -- How it works... -- The Riemann zeta function -- Getting ready -- How to do it... -- How it works... -- Airy and Bairy functions -- Getting ready... -- How to do it... -- The Bessel and Struve functions -- Getting ready... -- How to do it... -- How it works... -- There's more -- Chapter 8: Calculus, Interpolation, and Differential Equations -- Introduction -- Integration -- Getting ready -- How to do it... -- How it works... -- Computing integrals using the Newton-Cotes method -- Computing integrals using a Gaussian quadrature -- Getting ready -- How to do it…”

Tabla de Contenidos: “…Machine generated contents note: Preface List of Abbreviations 1 Introduction 1.1 Forward Problem 1.2 Inverse Problem 1.3 Issues in Inverse Problem Solving 1.4 Linear, Nonlinear and Linearized Problems 2 Signal and System as Vectors 2.1 Vector Space 2.1.1 Vector Space and Subspace 2.1.2 Basis, Norm and Inner Product 2.1.3 Hilbert Space 2.2 Vector Calculus 2.2.1 Gradient 2.2.2 Divergence 2.2.3 Curl 2.2.4 Curve 2.2.5 Curvature 2.3 Taylor's Expansion 2.4 Linear System of Equations 2.4.1 Linear System and Transform 2.4.2 Vector Space of Matrix 2.4.3 Least Square Solution 2.4.4 Singular Value Decomposition (SVD) 2.4.5 Pseudo-inverse 2.5 Fourier Transform 2.5.1 Series Expansion 2.5.2 Fourier Transform 2.5.3 Discrete Fourier Transform (DFT) 2.5.4 Fast Fourier Transform (FFT) 2.5.5 Two-dimensional Fourier Transform References 3 Basics for Forward Problem 3.1 Understanding PDE using Images as Examples 3.2 Heat Equation 3.2.1 Formulation of Heat Equation 3.2.2 One-dimensional Heat Equation 3.2.3 Two-dimensional Heat Equation and Isotropic Diffusion 3.2.4 Boundary Conditions 3.3 Wave Equation 3.4 Laplace and Poisson Equations 3.4.1 Boundary Value Problem 3.4.2 Laplace Equation in a Circle 3.4.3 Laplace Equation in Three-dimensional Domain 3.4.4 Representation Formula for Poisson Equation References 4 Analysis for Inverse Problem 4.1 Examples of Inverse Problems in Medical Imaging 4.1.1 Electrical Property Imaging 4.1.2 Mechanical Property Imaging 4.1.3 Image Restoration 4.2 Basic Analysis 4.2.1 Sobolev Space 4.2.2 Some Important Estimates 4.2.3 Helmholtz Decomposition 4.3 Variational Problems 4.3.1 Lax-Milgram Theorem 4.3.2 Ritz Approach 4.3.3 Euler-Lagrange Equations 4.3.4 Regularity Theory and Asymptotic Analysis 4.4 Tikhonov Regularization and Spectral Analysis 4.4.1 Overview of Tikhonov Regularization 4.4.2 Bounded Linear Operators in Banach Space 4.4.3 Regularization in Hilbert Space or Banach Space 4.5 Basics of Real Analysis 4.5.1 Riemann Integrable 4.5.2 Measure Space 4.5.3 Lebesgue Measurable Function 4.5.4 Pointwise, Uniform, Norm Convergence and Convergence in Measure 4.5.5 Differentiation Theory References 5 Numerical Methods 5.1 Iterative Method for Nonlinear Problem 5.2 Numerical Computation of One-dimensional Heat equation 5.2.1 Explicit Scheme 5.2.2 Implicit Scheme 5.2.3 Crank-Nicolson Method 5.3 Numerical Solution of Linear System of Equations 5.3.1 Direct Method using LU Factorization 5.3.2 Iterative Method using Matrix Splitting 5.3.3 Iterative Method using Steepest Descent Minimization 5.3.4 Conjugate Gradient (CG) Method 5.4 Finite Difference Method (FDM) 5.4.1 Poisson Equation 5.4.2 Elliptic Equation 5.5 Finite Element Method (FEM) 5.5.1 One-dimensional Model 5.5.2 Two-dimensional Model 5.5.3 Numerical Examples References 6 CT, MRI and Image Processing Problems 6.1 X-ray CT 6.1.1 Inverse Problem 6.1.2 Basic Principle and Nonlinear Effects 6.1.3 Inverse Radon Transform 6.1.4 Artifacts in CT 6.2 MRI 6.2.1 Basic Principle 6.2.2 K-space Data 6.2.3 Image Reconstruction 6.3 Image Restoration 6.3.1 Role of p in (6.35) 6.3.2 Total Variation Restoration 6.3.3 Anisotropic Edge-preserving Diffusion 6.3.4 Sparse Sensing 6.4 Segmentation 6.4.1 Active Contour Method 6.4.2 Level Set Method 6.4.3 Motion Tracking for Echocardiography References 7 Electrical Impedance Tomography 7.1 Introduction 7.2 Measurement Method and Data 7.2.1 Conductivity and Resistance 7.2.2 Permittivity and Capacitance 7.2.3 Phasor and Impedance 7.2.4 Admittivity and Trans-impedance 7.2.5 Electrode Contact Impedance 7.2.6 EIT System 7.2.7 Data Collection Protocol and Data Set 7.2.8 Linearity between Current and Voltage 7.3 Representation of Physical Phenomena 7.3.1 Derivation of Elliptic PDE 7.3.2 Elliptic PDE for Four-electrode Method 7.3.3 Elliptic PDE for Two-electrode Method 7.3.4 Min-max Property of Complex Potential 7.4 Forward Problem and Model 7.4.1 Continuous Neumann-to-Dirichlet Data 7.4.2 Discrete Neumann-to-Dirichlet Data 7.4.3 Nonlinearity between Admittivity and Voltage 7.5 Uniqueness Theory and Direct Reconstruction Method 7.5.1 Calderon's Approach 7.5.2 Uniqueness and Three-dimensional Reconstruction: Infinite Measurements 7.5.3 Nachmann's D-bar Method in Two Dimension 7.6 Backprojection Algorithm 7.7 Sensitivity and Sensitivity Matrix 7.7.1 Perturbation and Sensitivity 7.7.2 Sensitivity Matrix 7.7.3 Linearization 7.7.4 Quality of Sensitivity Matrix 7.8 Inverse Problem of EIT 7.8.1 Inverse Problem of RC Circuit 7.8.2 Formulation of EIT Inverse Problem 7.8.3 Ill-posedness of EIT Inverse Problem 7.9 Static Imaging 7.9.1 Iterative Data Fitting Method 7.9.2 Static Imaging using 4-channel EIT System 7.9.3 Regularization 7.9.4 Technical Difficulty of Static Imaging 7.10 Time-difference Imaging 7.10.1 Data Sets for Time-difference Imaging 7.10.2 Equivalent Homogeneous Admittivity 7.10.3 Linear Time-difference Algorithm using Sensitivity Matrix 7.10.4 Interpretation of Time-difference Image 7.11 Frequency-difference Imaging 7.11.1 Data Sets for Frequency-difference Imaging 7.11.2 Simple Difference Ft,ω2− Ft,ω1 7.11.3 Weighted Difference Ft,ω2− [alpha] Ft,ω1 7.11.4 Linear Frequency-difference Algorithm using Sensitivity Matrix 7.11.5 Interpretation of Frequency-difference Image References 8 Anomaly Estimation and Layer Potential Techniques 8.1 Harmonic Analysis and Potential Theory 8.1.1 Layer Potentials and Boundary Value Problems for Laplace Equation 8.1.2 Regularity for Solution of Elliptic Equation along Boundary of Inhomogeneity 8.2 Anomaly Estimation using EIT 8.2.1 Size Estimation Method 8.2.2 Location Search Method 8.3 Anomaly Estimation using Planar Probe 8.3.1 Mathematical Formulation 8.3.2 Representation Formula References 9 Magnetic Resonance Electrical Impedance Tomography 9.1 Data Collection using MRI 9.1.1 Measurement of Bz 9.1.2 Noise in Measured Bz Data 9.1.3 Measurement of B = (Bx,By,Bz) 9.2 Forward Problem and Model Construction 9.2.1 Relation between J , Bz and σ 9.2.2 Three Key Observations 9.2.3 Data Bz Traces σ∇u © e z-directional Change of σ 9.2.4 Mathematical Analysis toward MREIT Model 9.3 Inverse Problem Formulation using B or J 9.4 Inverse Problem Formulation using Bz 9.4.1 Model with Two Linearly Independent Currents 9.4.2 Uniqueness 9.4.3 Defected Bz Data in a Local Region 9.5 Image Reconstruction Algorithm 9.5.1 J-substitution Algorithm 9.5.2 Harmonic Bz Algorithm 9.5.3 Gradient Bz Decomposition and Variational Bz Algorithm 9.5.4 Local Harmonic Bz Algorithm 9.5.5 Sensitivity Matrix Based Algorithm 9.5.6 Anisotropic Conductivity Reconstruction Algorithm 9.5.7 Other Algorithms 9.6 Validation and Interpretation 9.6.1 Image Reconstruction Procedure using Harmonic Bz Algorithm 9.6.2 Conductivity Phantom Imaging 9.6.3 Animal Imaging 9.6.4 Human Imaging 9.7 Applications References 10 Magnetic Resonance Elastography 10.1 Representation of Physical Phenomena 10.1.1 Overview of Hooke's Law 10.1.2 Strain Tensor in Lagrangian Coordinates 10.2 Forward Problem and Model 10.3 Inverse Problem in MRE 10.4 Reconstruction Algorithms 10.4.1 Reconstruction of [mu] with the Assumption of Local Homogeneity 10.4.2 Reconstruction of [mu] without the Assumption of Local Homogeneity 10.4.3 Anisotropic Elastic Moduli Reconstruction 10.5 Technical Issues in MRE References…”

Tabla de Contenidos: “…9.2.3 Explicitly filtered LES and the grid convergence issue -- 9.3 Implicit coupling between numerics and explicit subgrid models -- 9.3.1 Statement of the problem -- 9.3.2 A first analysis via Taylor expansions -- 9.3.3 Static and dynamic analysis -- 9.3.4 Observations in nonhomogeneous flows -- 9.3.4.1 Plane channel flow case -- 9.3.4.2 Circular cylinder case -- 9.4 LES numerics: beyond order of accuracy -- 9.4.1 Statement of the problem -- 9.4.2 Structure-preserving schemes for LES -- 9.4.3 Extension to LES of compressible flows -- 9.5 Concluding remarks -- References -- 10 Numerical approximations formulated as LES models -- 10.1 Coarse grained simulations -- 10.1.1 Introduction -- 10.1.2 Low-pass filtered and finite-volume discretized Navier-Stokes equations -- 10.1.3 Modified equation analysis of subgrid scale modeling -- 10.1.3.1 Finite scale Navier-Stokes -- 10.1.3.2 Spatial filtering and ensemble averaging -- 10.2 Turbulence Reynolds number and mixing transition -- 10.2.1 Effective kinematic viscosity -- 10.3 Compressible numerical hydrodynamics -- 10.3.1 Riemann solvers -- 10.3.2 Low-Ma correction -- 10.3.3 Modified equation analysis -- 10.4 Case studies -- 10.4.1 Taylor-Green vortex -- 10.4.1.1 Impact of numerical scheme on quantities of interest -- 10.4.1.2 Numerical Reynolds number -- 10.4.2 Accelerated interface Rayleigh-Taylor driven mixing -- 10.4.2.1 Impact of numerical scheme on quantities of interest -- 10.4.2.2 Numerical Reynolds number -- 10.5 Summary and conclusions -- Acknowledgments -- References -- 11 Numerical treatment of incompressible turbulent flow -- 11.1 Introduction -- 11.1.1 Numerical challenge -- 11.1.2 Short history of energy-preserving discretization -- 11.1.3 Turbulence modeling -- 11.2 Flow equations -- 11.3 Energy-preserving discretization -- 11.3.1 Finite-volume -- 11.3.2 Evolution of energy…”

“…He is the author of a number of Springer books, including Dynamical Systems (2005), Postmodern Analysis (3rd ed. 2005, also translated into Japanese), Compact Riemann Surfaces (3rd ed. 2006) and Riemannian Geometry and Geometric Analysis (4th ed., 2005). …”