Kooperativer Bibliotheksverbund

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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

UID:

Format: 1 online resource (361 pages)

Edition: 1st ed. 2024.

ISBN: 9783031603945

Series Statement: Lecture Notes in Physics, 1030

Content: Modern mathematics has become an essential part of today’s physicist’s arsenal and this book covers several relevant such topics. The primary aim of this book is to present key mathematical concepts in an intuitive way with the help of geometrical and numerical methods - understanding is the key. Not all differential equations can be solved with standard techniques. Examples illustrate how geometrical insights and numerical methods are useful in understanding differential equations in general but are indispensable when extracting relevant information from equations that do not yield to standard methods. Adopting a numerical approach to complex analysis it is shown that Cauchy’s theorem, the Cauchy integral formula, the residue theorem, etc. can be verified by performing hands-on computations with Python codes. Figures elucidate the concept of poles and essential singularities. Further the book covers topology, Hilbert spaces, Fourier transforms (discussing how fast Fourier transform works), modern differential geometry, Lie groups and Lie algebras, probability and useful probability distributions, and statistical detection of signals. Novel features include: (i) Topology is introduced via the notion of continuity on the real line which then naturally leads to topological spaces. (ii) Data analysis in a differential geometric framework and a general description of χ2 discriminators in terms of vector bundles. This book is targeted at physics graduate students and at theoretical (and possibly experimental) physicists. Apart from research students, this book is also useful to active physicists in their research and teaching.

Note: Dedication -- Preface -- Topology -- Hilbert Spaces -- Fourier Analysis -- Complex analysis: hands on -- Understanding differential equations: geometrical insights and general analysis -- Solving Differential Equations -- Differential Geometry and Tensors -- Representations of the rotation group and Lie groups -- Probability and Random Variables -- Probability distributions in physics -- The statistical detection of signals in noisy data -- Bibliography.

Additional Edition: ISBN 9783031603938

Language: English

DOI: 10.1007/978-3-031-60394-5

URL: lizenzpflichtig

UID:

Format: XVI, 351 p. 57 illus., 42 illus. in color. , online resource.

Edition: 1st ed. 2024.

ISBN: 9783031603945

Series Statement: Lecture Notes in Physics, 1030

Content: Modern mathematics has become an essential part of today's physicist's arsenal and this book covers several relevant such topics. The primary aim of this book is to present key mathematical concepts in an intuitive way with the help of geometrical and numerical methods - understanding is the key. Not all differential equations can be solved with standard techniques. Examples illustrate how geometrical insights and numerical methods are useful in understanding differential equations in general but are indispensable when extracting relevant information from equations that do not yield to standard methods. Adopting a numerical approach to complex analysis it is shown that Cauchy's theorem, the Cauchy integral formula, the residue theorem, etc. can be verified by performing hands-on computations with Python codes. Figures elucidate the concept of poles and essential singularities. Further the book covers topology, Hilbert spaces, Fourier transforms (discussing how fast Fourier transform works), modern differential geometry, Lie groups and Lie algebras, probability and useful probability distributions, and statistical detection of signals. Novel features include: (i) Topology is introduced via the notion of continuity on the real line which then naturally leads to topological spaces. (ii) Data analysis in a differential geometric framework and a general description of χ2 discriminators in terms of vector bundles. This book is targeted at physics graduate students and at theoretical (and possibly experimental) physicists. Apart from research students, this book is also useful to active physicists in their research and teaching.

Note: Dedication -- Preface -- Topology -- Hilbert Spaces -- Fourier Analysis -- Complex analysis: hands on -- Understanding differential equations: geometrical insights and general analysis -- Solving Differential Equations -- Differential Geometry and Tensors -- Representations of the rotation group and Lie groups -- Probability and Random Variables -- Probability distributions in physics -- The statistical detection of signals in noisy data -- Bibliography.

In: Springer Nature eBook

Additional Edition: Printed edition: ISBN 9783031603938

Additional Edition: Printed edition: ISBN 9783031603952

Additional Edition: Printed edition: ISBN 9783031603969

Language: English

DOI: 10.1007/978-3-031-60394-5

URL: https://doi.org/10.1007/978-3-031-60394-5