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Format: 1 online resource (361 pages)

Edition: First edition.

ISBN: 9783031603945

Series Statement: Lecture Notes in Physics Series ; Volume 1030

Note: Intro -- Preface -- Contents -- 1 Topology -- 1.1 Introduction -- 1.2 Sets and Functions -- 1.3 Countable and Uncountable Sets -- 1.4 Continuous Functions -- 1.4.1 Continuity in Metric Spaces -- 1.4.2 Sequential Continuity -- 1.4.3 Open Spheres and Open Sets -- 1.5 Topological Spaces -- 1.5.1 Closed Sets and Limit Points -- 1.5.2 Hausdorff and Second Countable Spaces -- 1.6 Homeomorphism -- 1.7 Connectedness and Arcwise Connectedness -- 1.7.1 Connectedness -- 1.7.2 Arcwise Connected Spaces and Homotopy -- 1.8 Compactness -- Exercises -- References -- 2 Hilbert Spaces -- 2.1 Introduction -- 2.2 Vector Spaces -- 2.3 Inner Product and Inner Product Spaces -- 2.4 l2 Space: The Space of Fourier Coefficients -- 2.5 The Space L2 (-∞, ∞) -- 2.5.1 Intuitive Lebesgue Measure Theory on R -- 2.5.2 The Lebesgue Integral -- 2.5.3 The Function Space L2 (-∞, ∞) -- 2.6 Completeness -- 2.7 Parallelogram Law -- 2.8 Complete Orthonormal Sets -- 2.8.1 Orthonormal Sets -- 2.8.2 Gram-Schmidt Orthonormalisation Procedure -- 2.8.3 Complete Orthonormal Sets -- Exercises -- References -- 3 Fourier Analysis -- 3.1 The Cycles and Epicycles of Ptolemy -- 3.2 Fourier Series -- 3.3 Fourier Transform -- 3.4 Dirac-Delta Distribution and Its Fourier Transform -- 3.5 The Uncertainty Principle -- 3.6 The Discrete Fourier Transform -- 3.7 The Fast Fourier Transform (FFT) -- 3.7.1 Divide and Conquer -- 3.7.2 Number of Operations -- 3.8 At What Rate Should One Sample the Data? -- 3.8.1 The Sampling Theorem -- 3.8.2 Aliasing -- Exercises -- References -- 4 Complex Analysis: Hands On -- 4.1 Preliminaries -- 4.1.1 The Argand Plane -- 4.1.2 Functions -- 4.2 Analytic Functions -- 4.2.1 The Cauchy-Riemann Conditions -- 4.2.2 Orthogonal Families of Curves -- 4.3 Understanding Integrals -- 4.3.1 The Riemann Integral -- 4.3.2 The Contour Integral -- 4.4 Cauchy's Theorem. , 4.5 Cauchy Integral Formula -- 4.6 Singularities and Power Series Expansions of AnalyticFunctions -- 4.7 Branch Points, Branch Cuts, and Riemann Sheets -- 4.8 Residue Theorem -- 4.9 Analytic Continuation -- 4.10 The Stationary Phase Approximation -- Exercises -- References -- 5 Understanding Differential Equations -- 5.1 Introduction -- 5.2 Various Equations Occurring in Physics -- 5.3 First Order Ordinary Differential Equations -- 5.3.1 Equations that Yield to Standard Methods -- General Discussion and Geometrical Insights -- The Numerical Approach -- Taylor Series Solution -- 5.3.2 Equations that Do Not Yield to Standard Methods -- Orbit of a Photon Falling into a Black Hole -- Motion of a Heavy Symmetrical Top -- 5.4 Second Order Ordinary Differential Equations -- 5.4.1 The Simple Harmonic Oscillator -- 5.4.2 The Anharmonic Oscillator: A Simple Pendulum with Large Amplitude -- 5.5 Linear Partial Differential Equations in Two Independent Variables -- 5.6 Second Order Partial Differential Equations -- 5.6.1 General Discussion -- 5.6.2 Classification -- 5.7 Solving the One-Dimensional Wave Equation -- Exercises -- References -- 6 Solving Differential Equations -- 6.1 Introduction -- 6.1.1 The Method of Separation of Variables -- 6.1.2 Green's Functions -- 6.2 Separation of Variables: Three Examples -- 6.2.1 The Laplace Equation: Rectangular Box -- 6.2.2 The Diffusion Equation: Semi-infinite CylindricalRod -- Derivation of the Heat Equation -- Semi-infinite Cylindrical Rod -- The Steady State Solution Ω0 -- 6.2.3 The Wave Equation: Spherical Antenna -- 6.3 The Sturm-Louiville Problem -- 6.3.1 Geometrical Insight into Sturm-Louiville Problem: Simple Examples -- The Classical Harmonic Oscillator -- The Quantum Harmonic Oscillator -- 6.3.2 The Sturm-Louiville Problem: General Discussion -- 6.4 Forced Oscillations of a String Fixed at Two Ends. , 6.4.1 The Eigenfunction Expansion Method -- 6.4.2 Green's Function Method -- 6.5 Green's Functions in Electrostatics -- 6.5.1 Poisson's Equation with Homogeneous Boundary Condition at Infinity -- 6.5.2 Volume and Surface Greens Functions -- 6.5.3 Dirichlet's Green's Function for a Sphere -- 6.6 Eigenfunction Expansion of the Greens Function -- Exercises -- References -- 7 Differential Geometry and Tensors -- 7.1 Introduction -- 7.2 Differentiable Manifolds -- 7.2.1 Example of a 2-Sphere as Differentiable Manifold -- 7.2.2 Smooth Functions on Manifolds -- 7.3 Tangent Vectors and 1-Forms -- 7.3.1 Transformation Law for Tangent Vectors -- 7.3.2 Covariant Vectors or 1-Forms -- 7.3.3 Cotangent Space -- 7.3.4 Non-coordinate Bases -- 7.4 Tensors of Higher Rank -- 7.5 The Covariant Derivative and the Affine Connection -- 7.5.1 Smooth Tensor Fields -- 7.5.2 Differentiating Tensors -- 7.5.3 Curvature -- 7.6 Riemannian Manifolds and the Metric -- 7.6.1 The Metric -- 7.6.2 Raising and Lowering Indices -- 7.6.3 The Metric Affine Connection -- 7.7 Length of a Curve and Geodesics -- 7.7.1 Geodesic Deviation -- 7.8 Product Manifolds and Vector Bundles -- 7.8.1 Products of Manifolds -- 7.8.2 Vector Bundles -- Exercises -- References -- 8 The Rotation Group, Lorentz Group and Lie Groups -- 8.1 Introduction -- 8.2 Rotations as Orthogonal Linear Transformations with Determinant Unity -- 8.3 The Rotation Matrix Parametrised by Euler Angles -- 8.4 The Axis and Angle Description of Rotations -- 8.5 Representations of the Rotation Group -- 8.6 Tensor Representations -- 8.7 Reducible and Irreducible Representations -- 8.8 Infinite Dimensional Representations -- 8.9 Lie Groups and Lie Algebras -- 8.9.1 Lie Groups -- 8.9.2 The Lie Bracket and Derivative -- 8.10 Lie Algebra of a Lie Group -- 8.10.1 General Discussion -- 8.10.2 The Lie Algebra of SO(3) -- 8.11 The Group SU(2). , 8.12 Topological Aspects -- 8.12.1 The Homomorphism Between SU(2) and SO(3) -- 8.12.2 The Topological Structure of SU(2) and SO(3) -- 8.13 The Lorentz Group -- 8.13.1 O(3, 1) and its Subgroups -- 8.13.2 The Generators of SO+(3, 1) -- 8.14 Spinors and the SL(2, C) Group -- 8.14.1 The Concept of a 2-Spinor -- 8.14.2 Spin Dyad and the Null Tetrad -- 8.14.3 SL(2, C): The Universal Covering Groupof SO+ (3, 1) -- 8.15 Other Lie Groups and Concluding Remarks -- Exercises -- References -- 9 Probability and Random Variables -- 9.1 Introduction -- 9.2 Probability -- 9.2.1 Axiomatic Definition of Probability -- 9.2.2 Conditional Probability, Bayes Theorem and Independent Events -- 9.3 Random Variables and Their Distributions -- 9.4 Probability Distributions on RN: Multivariate Distributions -- 9.5 Independence of Random Variables -- 9.6 Functions of Random Variables -- 9.7 Moment Generating and Characteristic Functions of an r. v. -- Exercises -- Reference -- 10 Probability Distributions in Physics -- 10.1 Introduction -- 10.2 The Binomial Distribution and Random Walk -- 10.3 The Poisson Distribution -- 10.4 Probability Space on the Real Line R -- 10.5 The Gaussian (Normal) Distribution and the Central Limit Theorem -- 10.6 The χ2 Distribution -- 10.6.1 The Mean and Variance of χ2 -- 10.6.2 χ2 for Large n -- 10.7 The Student's t: Distribution -- 10.8 The Reduced χ2 Distribution -- Exercises -- References -- 11 The Statistical Detection of Signals in Noisy Data -- 11.1 Introduction -- 11.2 Characterisation of Noise -- 11.3 The Matched Filter -- 11.4 Binary Hypothesis Testing -- 11.4.1 The One-Dimensional Case -- 11.4.2 The Two-Dimensional Case -- 11.4.3 Neyman-Pearson Criterion -- 11.5 Composite Hypothesis and Maximum Likelihood Detection -- 11.5.1 The Signal Manifold -- 11.5.2 Maximum Likelihood -- 11.6 Ambiguity Function and the Metric on the Signal Manifold. , 11.7 Errors in MLE: Fisher Information Matrix and the Rao-Cramer Bound -- 11.8 χ2 Discriminators -- 11.8.1 What Is a χ2 Discriminator? -- 11.8.2 Constructing a Generic χ2 Discriminator -- 11.8.3 The Traditional χ2 Discriminator -- 11.9 Concluding Remarks -- Exercises -- References -- A Independence of X and S -- Reference -- Index.

Additional Edition: ISBN 9783031603938

Language: English