Contents

Topology and Geometry

Quantum Mechanics

General Relativity, Black Holes and Cosmology

Superstrings

Diccionario de Topologia Lacaniana, PsiKolibro

Sumário

Math

Topology and Geometry

A comprehensive Introduction to differential Geometry IV

Finite Difference Methods for Ordinary and Partial Differential Equations

Riemmannian Geometry

Numerical Methods for Unconstrained Optimization and Nonlinear Equations

FOUNDATIONS of DIFFERENTIABLE MANIFOLDS and LIE GROUPS

DIFFERENTIAL TOPOLOGY

PHYSICS FOR MATHEMATICIANS - MECHANICS I

Solutions of Exercises of Introduction to Differential Geometry of Space Curves and Surfaces Taha Sochi

Modern Fisics

Exploring the Invisible Universe From Black Holes to Superstrings Belal E Baaquie, Frederick H Willeboordse

Gravitational Waves, Volume 2 Astrophysics and Cosmology Maggiore, Michele

How the Universe Works Introduction to Modern Cosmology Serge Parnovsky, Aleksei Parnowski

Introduction to General Relativity, Black Holes and Cosmology Yvonne Choquet-Bruhat

Introduction to Quantum Mechanics David J. Griffiths, Darrell F. Schroeter

Introduction to Superstrings and M-Theory 2nd Edition Michio Kaku

Lectures on Astrophysics Steven Weinberg

One Hundred Years of General Relativity From Genesis and Empirical Foundations to Gravitational Waves, Cosmology and Quantum… Wei-Tou Ni

Quantum field theory a modern introduction Michio Kaku

The Future of Humanity Terraforming Mars, Interstellar Travel, Immortality, and Our Destiny Beyond Earth Michio Kaku

Diccionario de Topologia Lacaniana, PsiKolibro

Math

Topology and Geometry

Glen E Bredon

Contents

PrefacAcknowledgments

I

General Topology

Metric Spaces

Topological Spaces

Subspaces

Connectivity and Components

Separation Axioms

Nets Moore-Smith Convergence

Compactness

Products

Metric Spaces Again

Existence of Real Valued Functions

Locally Compact Spaces

Paracompact Spaces

Quotient Spaces

Homotopy

Topological Groups

Convex Bodies

The Baire Category Theorem

II

Differentiable Manifolds

The Implicit Function Theorem

Differentiable Manifolds

Local Coordinates

Induced Structures and Examples

Tangent Vectors and Differentials

Sard’s Theorem and Regular Values

Local Properties of Immersions and Submersions

Vector Fields and Flows

Tangent Bundles

Embedding in Euclidean Space

Tubular Neighborhoods and Approximations

Classical Lie Groups

Fiber Bundles

Induced Bundles and Whitney Sums

Transversality

Thom~Pontryagin Theory

III

Fundamental Group

Homotopy Groups

The Fundamental Group

Covering Spaces

The Lifting Theorem

The Action of [ on the Fiber

Deck Transformations

Properly Discontinuous Actions

Classification of Covering Spaces

The Seifert~Van Kampen Theorem

Remarks on SO

IV

Homology Theory

Homology Groups

The Zeroth Homology Group

The First Homology Group

Functorial Properties

Homological Algebra

Axioms for Homology

Computation of Degrees

CW-Complexes

Conventions for CW-Complexes

Cellular Homology

Cellular Maps

Products of CW-Complexes

Euler’s Formula

Homology of Real Projective Space

Singular Homology

The Cross Product

Subdivision

The Mayer~ Vietoris Sequence

The Generalized Jordan Curve Theorem

The Borsuk~Ulam Theorem

Simplicial Complexes

Contents

Simplicial Maps

The Lefschetz-Hopf Fixed Point Theorem

V

Cohomology

Multilinear Algebra

Differential Forms

Integration of Forms

Stokes’ Theorem

Relationship to Singular Homology

More Homological Algebra

Universal Coefficient Theorems

Excision and Homotopy

de Rham’s Theorem

The de Rham Theory of cpn

Hopf’s Theorem on Maps to Spheres

Differential Forms on Compact Lie Groups;

VI

Products and Duality

The Cross Product and the Kiinneth Theorem

A Sign Convention

The Cohomology Cross Product

The Cup Product

The Cap Product

Classical Outlook on Duality

The Orientation Bundle

Duality Theorems

Duality on Compact Manifolds with Boundary

Applications of Duality

Intersection Theory;

The Euler Class, Lefschetz Numbers, and Vector Fields

The Gysin Sequence

Lefschetz Coincidence Theory

Steenrod Operations

Construction of the Steenrod Squares

Stiefel-Whitney Classes

Plumbing

VII

Homotopy Theory

Colibrations

The Compact-Open Topology

H-Spaces, H-Groups, and H-Cogroups

Homotopy Groups

The Homotopy Sequence of a Pair

Fiber Spaces

Free Homotopy

Classical Groups and Associated Manifolds

The Homotopy Addition Theorem

The Hurewicz Theorem

The Whitehead Theorem

Eilenberg-Mac Lane Spaces

Obstruction Theory

Obstruction Cochains and Vector Bundles

Appendices

App A The Additivity Axiom

App B Background in Set Theory

App C Critical Values

App D Direct Limits

App E Euclidean Neighborhood Retracts

Bibliography

Index of Symbols

Index

A comprehensive Introduction to differential Geometry IV

Michael Spivak

HIGHER DIMENSIONS AND CODIMENSIONS

A THE GEOMETRY OF CONSTANT CURVATURE MANIFOLDS

The standard models of S”Ko and H” Ko in R+ Stereographic projection and the conformal model of H” Conformal maps of R” and the isometries of H” Totally geodesic submanifolds and geodesic spheres of H” Horospheres and equidistant hypersurfaces Geodesic mappings; the projective model of H”; Beltrami’s theorem

B CURVES IN A RIEMANNIAN MANIFOLFrenet frames and curvatures

Curves whose jth curvature vanish

C THE FUNDAMENTAL EQUATIONS FOR SUBMANIFOLDS

The normal connection and the Weingarten equations Second fundamental forms and normal fundamental forms; the Codazzi-Mainardi equations

The Ricci equations

The fundamental theorem for submanifolds of Euclidean space The fundamental theorem for submanifolds of

constant curvature manifolds

D FIRST CONSEQUENCES

The curvatures of a hypersurface; Theorema Egregium; formula for the Gaussian curvaturThe mean curvature normal; umbilics;

all-umbilic submanifolds of Euclidean spacAll-umbilic submanifolds of constant curvature manifolds Positive curvature and convexity

E FURTHER RESULTS

Flat ruled surfaces in Flat ruled surfaces in constant curvature manifolds

Curves on hypersurfaces

F COMPLETE SURFACES OF CONSTANT CURVATURModifications of results for surfaces in R³

Surfaces of constant curvature in S³ surfaces with constant curvature the Hopf map

Surfaces of constant curvature in H³ Jörgens theorem; surfaces of constant curvature surfaces of constant curvature - rotation surfaces of constant curvature between and

G HYPERSURFACES OF CONSTANT CURVATURE IN

HIGHER DIMENSIONS

Hypersurfaces of constant curvature in dimensions >

The Ricci tensor; Einstein spaces, hypersurfaces which are Einstein spaces Hypersurfaces of the same constant curvature as the ambient manifolAddendum The Laplacian

Addendum The operator and the Laplacian on forms; Hodge’s Theorem

Addendum When are two Riemannian manifolds isometric?

Addendum Better imbedding invariants Problems

THE SECOND VARIATION

Two-parameter variations; the second variation formula

Jacobi fields; conjugate points

Minimizing and non-minimizing geodesics

The Hadamard-Cartan Theorem

The Sturm Comparison Theorem; Bonnet’s Theorem Generalizations to higher dimensions:

the Morse-Schoenberg Comparison Theorem;

Meyer’s Theorem; the Rauch Comparison Theorem

Synge’s lemma; Synge’s Theorem

Cut points; Klingenberg’s theorem

Problems A VARIATIONS OF LENGTH, AREA, AND VOLUMVariation of area for normal variations of surfaces in R³;

minimal surfaces

Isothermal coordinates on minimal surfaces; Bernstein’s Theorem

Weierstrass-Enneper representation

Associated minimal surfaces; Schwarz’s Theorem

Change of orientation; Henneberg’s minimal surfacClassical calculus of variations in a dimensions

Variation of volume formula

Isoperimetric problems

Addendum Isothermal coordinates

Addendum Immersed spheres with constant mean curvaturAddendum Imbedded surfaces with constant mean curvaturAddendum The second variation of volum

Finite Difference Methods for Ordinary and Partial Differential Equations

Preface

Boundary Value Problems and Iterative Methods

Finite Difference Approximations

Truncation errors

Deriving finite difference approximations

Second order derivatives

Higher order derivatives

A general approach to deriving the coefficients

Steady

States and Boundary Value Problems

The heat equation

Boundary conditions

The steady-state problem

A simple finite difference method

Local truncation error

Global error

Stability

Consistency

Convergence

Stability in the -norm

Green’s functions and max-norm stability

Neumann boundary conditions

Existence and uniqueness

Ordering the unknowns and equations

A general linear second order equation

Nonlinear equations

Discretization of the nonlinear boundary value problem

Nonuniqueness

Accuracy on nonlinear equations

Singular perturbations and boundary layers

Interior layers

ElliptiIterativNonuniform grids

Adaptive mesh selection

Continuation methods

Higher order methods

Fourth order differencing

Extrapolation methods

Deferred corrections

Spectral methods

Equations

Steady-state heat conduction

The -point stencil for the Laplacian

Ordering the unknowns and equations

Accuracy and stability

The -point Laplacian

Other elliptic equations

Solving the linear system

Sparse storage in MATLAB

Methods for Sparse Linear Systems

Jacobi and Gauss–Seidel

Analysis of matrix splitting methods

Rate of convergence

Successive overrelaxation

Descent methods and conjugate gradients

The method of steepest descent

The A-conjugate search direction

The conjugate-gradient algorithm

Convergence of conjugate gradient

Preconditioners

Incomplete Cholesky and ILU preconditioners

The Arnoldi process and GMRES algorithm

Krylov methods based on three term recurrences

Other applications of Arnoldi

Newton–Krylov methods for nonlinear problems

Multigrid methods

Slow convergence of Jacobi

The multigrid approach

II

Initial Value Problems

The Initial Value Problem for Ordinary Differential Equations

Linear ordinary differential equations

Duhamel’s principle

Lipschitz continuity

Existence and uniqueness of solutions

Systems of equations

Significance of the Lipschitz constant

Limitations

Some basic numerical methods

Truncation errors

One-step errors

Taylor series methods

Runge–Kutta methods

Embedded methods and error estimation

One-step versus multistep methods

Linear multistep methods

Local truncation error

Characteristic polynomials

Starting values

Predictor-corrector methods

Zero-Stability and Convergence for Initial Value Problems

Convergence

The test problem

One-step methods

Euler’s method on linear problems

Relation to stability for boundary value problems

Euler’s method on nonlinear problems

General one-step methods

Zero-stability of linear multistep methods

Solving linear difference equations

Absolute Stability for Ordinary Differential Equations

Unstable computations with a zero-stable method

Absolute stability

Stability regions for linear multistep methods

Systems of ordinary differential equations

Chemical kinetics

Linear systems

Nonlinear systems

Practical choice of step size

Plotting stability regions

The boundary locus method for linear multistep methods

Plotting stability regions of one-step methods

Relative stability regions and order stars

Stiff Ordinary Differential Equations

Numerical difficulties

Characterizations of stiffness

Numerical methods for stiff problems

A-stability and A ̨-stability

L-stability

BDF methods

The TR-BDF method

Runge–Kutta–Chebyshev explicit methods

Diffusion Equations and Parabolic Problems

Local truncation errors and order of accuracy

Method of lines discretizations

Stability theory

Stiffness of the heat equation

Convergence

PDE versus ODE stability theory

Von Neumann analysis

Multidimensional problems

The locally one-dimensional method

Boundary conditions

The alternating direction implicit method

Other discretizations

Advection Equations and Hyperbolic Systems

Advection

Method of lines discretization

Forward Euler time discretization

Leapfrog

Lax–Friedrichs

The Lax–Wendroff method

Stability analysis

Upwind methods

Stability analysis

The Beam–Warming method

Von Neumann analysis

Characteristic tracing and interpolation

The Courant–Friedrichs–Lewy condition

Some numerical results

Modified equations

Hyperbolic systems

Characteristic variables

Numerical methods for hyperbolic systems

Initial boundary value problems

Analysis of upwind on the initial boundary value problem

Outflow boundary conditions

Other discretizations

Mixed Equations

Some examples

Fully coupled method of lines

Fully coupled Taylor series methods

Fractional step methods

Implicit-explicit methods

Exponential time differencing methods

Implementing exponential time differencing methods

Appendices

Measuring Errors

Errors in a scalar value

Absolute error

Relative error

“Big-oh” and “little-oh” notation

Errors in vectors

Norm equivalence

Matrix norms

Errors in functions

Errors in grid functions

Norm equivalence

Estimating errors in numerical solutions

Estimates from the true solution

Estimates from a fine-grid solution

Estimates from coarser solutions

Polynomial Interpolation and Orthogonal Polynomials

The general interpolation problem

Polynomial interpolation

Monomial basis

Lagrange basis

Newton form

Error in polynomial interpolation

Orthogonal polynomials

Legendre polynomials

Chebyshev polynomials

Eigenvalues and Inner-Product Norms

Similarity transformations

Diagonalizable matrices

The Jordan canonical form

Symmetric and Hermitian matrices

Skew-symmetric and skew-Hermitian matrices

Normal matrices

Toeplitz and circulant matrices

The Gershgorin theorem

Inner-product norms

Other inner-product norms

Matrix Powers and Exponentials

The resolvent

Powers of matrices

Solving linear difference equations

Resolvent estimates

Matrix exponentials

Solving linear differential equations

Nonnormal matrices

Matrix powers

Matrix exponentials

Pseudospectra

Nonnormality of a Jordan block

Stable families of matrices and the Kreiss matrix theorem

Variable coefficient problems

Bibliography

Partial Differential Equations

Classification of differential equations

Second order equations

Elliptic equations

Parabolic equations

Hyperbolic equations

Derivation of partial differential equations from conservation principles

Advection

Diffusion

Source terms

Reaction-diffusion equations

Fourier analysis of linear partial differential equations

Fourier transforms

The advection equation

The heat equation

The backward heat equation

More general parabolic equations

Dispersive waves

Even- versus odd-order derivatives

The Schrödinger equation

The dispersion relation

Wave packets

Index

Riemmannian Geometry

Manfredo P do Carmo

Preface to the first edition

Preface to the second edition

Preface to the English edition

How to use this book

-DIFFERENTIABLE MANIFOLDS

Introduction

Differentiable manifolds; tangent spac Immersions and embeddings; examples

Other examples of manifolds Orientation

Vector fields; brackets Topology of manifolds

-RIEMANNIAN METRICS

Introduction

Riemannian Metrics

-AFFINE CONNECTIONS;

RIEMANNIAN CONNECTIONS

Introduction

Affine connections

Riemannian connections

-GEODESICS; CONVEX NEIGHBORHOODS

Introduction

The geodesic flow

Minimizing properties of geodesics

Convex neighborhoods

-CURVATUR Introduction

Curvatur Sectional curvatur Ricci curvature and scalar curvatur Tensors on Riemannian manifolds

-JACOBI FIELDS

Introduction

The Jacobi equation

Conjugate points

-ISOMETRIC IMMERSIONS

Introduction

The second fundamental form The fundamental equations

-COMPLETE MANIFOLDS; HOPF-RINOW AND HADAMARD THEOREMS

Introduction

Complete manifolds; Hopf-Rinow Theorem

The Theorem of Hadamar-SPACES OF CONSTANT CURVATUR Introduction

Theorem of Cartan on the determination of the metric by means of the curvatur Hyperbolic spac Space forms

Isometries of the hyperbolic space; Theorem of Liouvill-VARIATIONS OF ENERGY

Introduction

Formulas for the first and second variations of energy

The theorems of Bonnet-Myers and of Synge-Weinstein

-THE RAUCH COMPARISON THEOREM

Introduction

The Theorem of Rauch

Applications of the Index Lemma to immersions

Focal points and an extension of Rauch’s Theorem

-THE MORSE INDEX THEOREM

Introduction

The Index Theorem

-THE FUNDAMENTAL GROUP OF MANI- FOLDS OF NEGATIVE CURVATUR Introduction

Existence of closed geodesics

Preissman’s Theorem

-THE SPHERE THEOREM

Introduction

The cut locus

The estimate of the injectivity radius

The Sphere Theorem

Some further developments

References

Index

Numerical Methods for Unconstrained Optimization and Nonlinear Equations

J EDennis, Jr

Robert B Schnabel

PREFACE TO THE CLASSICS EDITION xi

PREFACE xiii

INTRODUCTION

Problems to be considered

Characteristics of “real-world” problems

Finite-precision arithmetic and measurement of error

Exercises

NONLINEAR PROBLEMS

IN ONE VARIABLE

What is not possible

Newton’s method for solving one equation in one unknown

Convergence of sequences of real numbers

A Convergence of Newton’s method

Globally convergent methods for solving one equation in one unknown

Methods when derivatives are unavailable

Minimization of a function of one variable

Exercises

VIIVIII

Contents

NUMERICAL LINEAR

ALGEBRA BACKGROUND

Vector and matrix norms and orthogonality

Solving systems of linear equations—matrix factorizations

Errors in solving linear systems

Updating matrix factorizations

Eigenvalues and positive definiteness

Linear least squares

Exercises

MULTIVARIABLE CALCULUS BACKGROUND

Derivatives and multivariate models

Multivariate finite-difference derivatives

Necessary and sufficient conditions for unconstrained minimization

Exercises

NEWTON’S METHOFOR NONLINEAR EQUATIONS

AND UNCONSTRAINED MINIMIZATION

Newton’s method for systems of nonlinear equations

Local convergence of Newton’s method

The Kantorovich and contractive mapping theorems

Finite-difference derivative methods for systems of nonlinear equations

Newton’s method for unconstrained minimization

Finite-difference derivative methods for unconstrained minimization

Exercises

GLOBALLY CONVERGENT MODIFICATIONS

OF NEWTON’S METHOD

The quasi-Newton framework

Descent directions

Line searches

Convergence results for properly chosen steps

Step selection by backtracking

The model-trust region approach

The locally constrained optimal “hook” step

The double dogleg step

Updating the trust region

Global methods for systems of nonlinear equations

Exercises Contents ix

STOPPING, SCALING, AND TESTING

Scaling

Stopping criteria

Testing

Exercises

SECANT METHODS FOR SYSTEMS

OF NONLINEAR EQUATIONS

Broyden’s method

Local convergence analysis of Broyden’s method

Implementation of quasi-Newton algorithms using Broyden’s update

Other secant updates for nonlinear equations

Exercises

SECANT METHODS

FOR UNCONSTRAINED MINIMIZATION

The symmetric secant update of Powell

Symmetric positive definite secant updates

Local convergence of positive definite secant methods

Implementation of quasi-Newton algorithms using the positive definite secant

update

Another convergence result for the positive definite secant method

Other secant updates for unconstrained minimization

Exercises

NONLINEAR LEAST SQUARES

The nonlinear least-squares problem

Gauss-Newton-type methods

Full Newton-type methods

Other considerations in solving nonlinear least-squares problems

Exercises

METHODS FOR PROBLEMS

WITH SPECIAL STRUCTURE

The sparse finite-difference Newton method

Sparse secant methods

Deriving least-change secant updates

Analyzing least-change secant methods

Exercises x Contents

APPENDIX: A MODULAR SYSTEM

OF ALGORITHMS

FOR UNCONSTRAINED MINIMIZATION

AND NONLINEAR EQUATIONS

by Robert Schnabel

|B|| APPENDIX: TEST PROBLEMS

by Robert Schnabel

FOUNDATIONS of DIFFERENTIABLE MANIFOLDS and LIE GROUPS

Frank W Warner

MANIFOLDS

Preliminaries

Differentiable Manifolds

The Second Axiom of Countability

Tangent Vectors and Differentials

Submanifolds, Diffeomorphisms, and the Inverse Function Theorem

Implicit Function Theorems

Vector Fields

Distributions and the Frobenius Theorem

Exercises

TENSORS AND DIFFERENTIAL FORMS

Tensor and Exterior Algebras

Tensor Fields and Differential Forms

The Lie DerivativDifferential Ideals

Exercises

S LIE GROUPS

Lie Groups and Their Lie Algebras

Homomorphisms

Lie Subgroups

Coverings

Simply Connected Lie Groups

Exponential Map

Continuous Homomorphisms

Closed Subgroups

The Adjoint Representation

Automorphisms and Derivations of Bilinear Operations and Forms

Homogeneous Manifolds

Exercises INTEGRATION ON MANIFOLDS

Orientation

Integration on Manifolds

de Rham Cohomology

Exercises

SHEAVES, COHOMOLOGY, AND THE DE RHAM THEOREM

Sheaves and Presheaves

Cochain Complexes

Axiomatic Sheaf Cohomology

The Classical Cohomology Theories

Alexander-Spanier Cohomology

de Rham Cohomology

Singular Cohomology

Cech Cohomology

The de Rham Theorem

Multiplicative StructurSupports

Exercises

d THE HODGE THEOREM

The Laplace-Beltrami Operator

The Hodge Theorem

Some Calculus

Elliptic Operators

Reduction to the Periodic CasEllipticity of the Laplace-Beltrami Operator

Exercises

BIBLIOGRAPHY

SUPPLEMENT TO THE BIBLIOGRAPHY

INDEX OF NOTATION

INDEX

DIFFERENTIAL TOPOLOGY

Victor Guillemin

Alan Pollack

Massachusetts Institute of Technology

PrefacIX

Straight forward to th’e Student XIII

Table of Symbols xv

Manifolds and Smooth Maps

DEFINITIONS

DERIVATIVES AND TANGENTS

THE INVERSE FUNCTION THEOREM AND,IMMERSIONS

SUBMERSIONS

TRANSVERSALITY

HOMOTOPY AND STABILITY

SARD’S THEOREM AND MORSE FUNCTIONS

EMBEDDING MANIFOLDS IN EUCLIDEAN SPACE

Transversality and Intersection

MANIFOLDS WITH BOUNDARY

ONE-MANIFOLDS AND SOME CONSEQUENCES

TRANSVERSALITY

INTERSECTION THEORY MOD

WINDING NUMBERS AND THE JORDAN-BROUWER SEPARATION

THEOREM

THE BORSUK-ULAM THEOREM

Oriented Intersection Theory

MOTIVATION

ORIENTATION

ORIENTED INTERSECTION NUMBER

LEFSCHETZ FIXED-POINT THEORY

VECTOR FIELDS AND THE POINCARE-HoPF THEOREM

THE HOPF DEGREE THEOREM

THE EULER CHARACTERISTIC AND TRIANGULATIONS

Integration on Manifolds

INTRODUCTION

EXTERIOR ALGEBRA

DIFFERENTIAL FORMS

INTEGRATION ON MANIFOLDS

EXTERIOR DERIVATIVE

COHOMOLOGY WITH FORMS

STOKES THEOREM

INTEGRATION AND MAPPINGS

THE GAuss-BoNNET THEOREM Contents

APPENDIX

Measure Zero and Sard’s Theorem

APPENDIX

Classification of Compact One-Manifolds

Bibliography

Index

PHYSICS FOR MATHEMATICIANS - MECHANICS I

MICHAEL SPIVAK

CONTENTS

Preface vii

PART I THE FOUNDATIONS OF MECHANICS

Prologue

Newtonian Mechanics

Mass and force

The first law

The second law

Mass and weight are different

yet not so different

The third law

The lures of symmetry

Composition offerees

Addendum A It Isn’t Rocket SciencWhy Easy Physics is So Hard: I

Addendum IB Weight Versus Mass

Problems

Newton’s Analysis of Central Forces

Problems

Conservation Laws

Conservation of momentum

Conservation of angular momentum

Conservation of energy: kinetic and potential energy

Conservation of energy in collisions

Conservation of energy in general

Addendum A Whips and Chains

Why Easy Physics is So Hard: II

Addendum B Follow the Bouncing Ball

Why Easy Physics is So Hard: III

Problems Ill

The One-Body and Two-Body Problems

The one-body problem

“The motion of bodies in mobile orbits,

and the motion of the absides”

The two-body problem

“The attractive forces of spherical bodies”

Addendum A A la Principia

Addendum B Reduction to a One-Dimensional Problem

Addendum C Rutherford Scattering

Addendum D Bertrand’s Theorem

Addendum E Power Force Laws and Duality

Problems

Rigid Bodies

Equilibrium

Virtual infinitesimal displacements

Configuration space

The principle of virtual work

d’Alembert’s principle

The inertia tensor

Calculating the inertia tensor

Rotation about an axis

Kinetic energy

Continuous bodies

Elementary examples

Addendum A The Strong Form of the Third Law

Problems

Constraints

Rigid bodies in contact

The pendulum

The compound physical pendulum

Equilibrium and Stability

Sliding

Rolling

Some subsidiary topics time-dependent constraints and hinges

Holonomic and differential constraints

Finding the constraint forces

The rolling sphere

Give a physics student enough rope problems

Addendum A The Bouncing SuperBall

Addendum B Statically Indeterminate Problems

Problems

Philosophical and Historical Questions

Early notions of conservation of momentum

Huygens and Galilean Invariance Contents xi

Newton’s proof of the third law

The parallelogram law

Newton at the hands of the scholars

PART II BUILDING ON THE FOUNDATIONS

Oscillations

Huygens cycloidal pendulum

The spherical pendulum

Springs

Harmonic oscillations

Damped oscillations

Forced oscillations

Damped forced oscillations

Coupled oscillators

The double pendulum

The vibrating string

Addendum A Abel’s Integral Equation

Addendum B Envelopes

Addendum C Stability of Solutions of Differential Equations

Problems

Rigid Body Motion

Rotating coordinate systems

The Euler equations

Poinsot’s geometric description

The free symmetric top, in body coordinates

The free symmetric top, in inertial coordinates

Euler angles

The heavy symmetrical top

The cuspidal case; fast tops

Precessing tops

Sleeping tops

The rising top

The polar cuspidal top

Gyroscopes

The gyrocompass

Precession of the equinoxes

Addendum A The Euler Equations for Rotating Principal Vectors

The Rolling Disc

Addendum B Secrets of the Herpolhode

Problems Xll

Contents

Non-Inertial Systems and Fictitious Forces

The basic equations

The translational or acceleration force

The centrifugal force

The deflection of a hanging body

The azimuthal or Euler force

The Coriolis force

The deflection of falling body

The southward deflection

Stupid experimenter tricks

Foucault’s pendulum

Hurricanes and bath-tubs

Mach’s Principle

Addendum A The Trojan Asteroids

The restricted three-body problem

Stability

Stability calculations

The collinear Lagrange points

Addendum B The Southward Deflection

Problems

Friction, Friend and Foe

The laws of friction

The Painleve paradoxes

The noble game of billiards

The Jellett invariant

Tippe Tops and hard boiled eggs

Problems

PART HI LAGRANGIAN MECHANICS

Analytical Mechanics

The mathematical arena for analytical mechanics

Specialized considerations for analytical mechanics

Lagrange’s equations

Using Lagrange’s equations

Constraint problems

Conservation of energy; action

Time-dependent Lagrangians

Lagrange multipliers

Addendum A Lagrange’s Rolling Disc

Problems Contents xiii

Variational Principles

The Euler equations

Hamilton’s principle

Maupertuis and the Principle of Least Action

Jacobi’s form of the principle of least action

Noether’s Theorem

The lures of symmetry, advanced version

Addendum A Lagrange Multipliers for Conditional Critical Points

Problem

Small Oscillations

Problems

INTERLUDE

Light

Optics in antiquity

Islamic scholars

Kepler and Galileo

Descartes

Fermat

Huygens

Newton

Maupertuis

Malus

Addendum A Battling to a Draw

Addendum B Huygens Principle

PART IV HAMILTONIAN MECHANICS

From Aragonite to the Schrodinger Wave Equation

The Cotangent Bundle

Special features of the cotangent bundle

The Legendre transform

Addendum A The Clairaut Equation

Problems

The Interplay of Mechanics and Optics

Optics emulates mechanics

Malus Theorem

Fermat’s Principle and Huygens Construction

Conical Refraction in Aragonite

Mechanics returns the compliment XIV

Contents

The equations on T - M

The partial derivatives of S

A partial differential equation for S

Invariant definitions; the interplay of TM and T - M

The extended Hamilton’s principle

Addendum A Liouville’s Volume Theorem

Problems

Hamilton-Jacobi Theory

The complete integral

Optional Envelopes of solutions

Optional Inverting the process; contact curves

JACOBFS THEOREM

Jacobi’s theorem and mechanics

Hamilton’s characteristic function

HAMILTON^JACOBI THEORY ANTHE SCHRODINGER WAVE EQUATION

Addendum A Motion in the Field of Two Fixed Masses

Geodesies on Ellipsoids

Addendum B Huygens’ Construction for Hyperbolic Equations

Problem

Canonical Transformations

Canonical transformations

Hamiltonian flows and integral invariants

Hamiltonian flows and canonical transformations

Generating functions

Time-dependent canonical transformations

Using generating functions to simplify Hamilton’s equations

Generating functions in the time-independent case

Other types of generating functions

Addendum A Time-InDependent Hamiltonians

Addendum B Generalized Canonical Transformations

Problems

Symplectic Manifolds

Symplectic vector spaces

Isotropic subspaces

Symplectic manifolds

Poisson brackets

Poisson brackets bis

Problems Contents xv

Liouville Integrability

Functions in involution

Conditional periodicity and the invariant tori

Action-angle variables

Action-angle variables on symplectic manifolds

Background

Problems

Epilogue

Adiabatic invariants

The averaging principle

An averaging theorem for one-dimensional systems

Adiabatic invariance of J

The Hannay angle

The Hannay hoop

Foucault’s pendulum revisited

Problems

Supplement A PDE Primer

Bibliography

Unabbreviated Journal Titles

Index

Solutions of Exercises of Introduction to Differential Geometry of Space Curves and Surfaces Taha Sochi

Preface

Nomenclature

Chapter 1 Preliminaries

Chapter 2 Curves in Space

Chapter 3 Surfaces in Space

Chapter 4 Curvature

Chapter 5 Special Curves

Chapter 6 Special Surfaces

Chapter 7 Tensor Differentiation over Surfaces

8 Footnotes

Computer Science

Matemática

1. Análise Matemática

   • Análise Funcional
   • Análise Numérica
   • Equações Diferenciais Parciais
   • Cálculo Variacional

2. Álgebra

   • Teoria dos Grupos
   • Álgebra Linear Avançada
   • Teoria dos Anéis e Corpos

3. Geometria e Topologia

   • Topologia Algébrica
   • Geometria Diferencial
   • Teoria dos Números

4. Probabilidade e Estatística

   • Probabilidade Avançada
   • Processos Estocásticos
   • Inferência Estatística

5. Matemática Aplicada

   • Otimização
   • Modelagem Matemática
   • Análise de Dados

Engenharia de Computação

1. Arquitetura de Computadores

   • Arquitetura de Sistemas Embarcados
   • Processadores de Alto Desempenho
   • Sistemas Operacionais

2. Inteligência Artificial e Aprendizado de Máquina

   • Aprendizado de Máquina
   • Processamento de Linguagem Natural
   • Redes Neurais e Deep Learning

3. Redes e Telecomunicações

   • Redes de Computadores
   • Segurança de Redes
   • Protocolos de Comunicação

4. Ciência de Dados

   • Big Data
   • Mineração de Dados
   • Algoritmos de Análise de Dados

5. Teoria da Computação

   • Algoritmos e Complexidade
   • Teoria da Informação
   • Criptografia

6. Engenharia de Software

   • Desenvolvimento de Software
   • Engenharia de Sistemas
   • Métodos Formais e Verificação

7. Robótica e Sistemas Autônomos

   • Controle de Robôs
   • Percepção e Navegação
   • Sistemas Autônomos e Inteligentes

Modern Fisics

Exploring the Invisible Universe From Black Holes to Superstrings Belal E Baaquie, Frederick H Willeboordse

The Question

What is a Field?

Pressure Field

Propagating pressure field

Electric and Magnetic Fields

Magnetic field

The Electromagnetic Field

Electromagnetic waves

Maxwell’s field equations

Do Fields Need a Medium?

Action at a distance

Lorentz Transformations

Gravitational Field

Quantum Fields

Feynman diagrams

Quantum Vacuum

Casimir effect and Lamb shift

Unification of Particles and Interactions

The Answer

The Geometry of Space

The Question

Is Space Curved?

Exploring the Invisible Universe

Description of Curved Space

Parallel Transport

Geodesics

Constructing a geodesic

Geodesics on a sphere S

Distances in Curved Spaces

Special Theory of Relativity

Spacetime Intervals

Null, timelike and spacelike

Curvature

The Answer

Gravity

The Question

Newton’s Gravity and Special Relativity

Equivalence Principle: Accelerating Frame

Gravity: Slowing Down Time

Gravity: Bending Spacetime

Geodesics and Freely Falling Frames

Geodesics and Cosmological Time

The Pattern of Spacetime and Gravity

Curvature and Matter

Gravitational Radiation

The Answer

Black

Holes

The Question

Brief History of Black Holes

Laplace’s Dark Star

The Schwarzschild Geometry

Astrophysical Black Holes

Black Holes: Dilating Time

Black Holes: Bending Space

Event Horizon: Black Hole’s Boundary

Stationary observer

Permanent Trapping of Light

Light cones: Stationary observer

Light cones: In-falling observer

Spinning Black Holes

Kerr black hole

Extremal Kerr Black Hole and Ergosphere

Energy from a Kerr Black Hole

Reissner–Nordstrom Charged Black Hole

Black Hole Entropy

Bekenstein–Hawking entropy

Black Hole Temperature

Black Hole Thermodynamics

Hawking Radiation

The Answer

Cosmology

The Question

Introduction

Expanding Universe: Newtonian Cosmology

Friedmann Equation

Hubble’s law revisited

The Curvature Parameter k

Critical universe

Closed universe

Open universe

The Cosmological Constant Λ

Age of the Universe

Critical density

Energy of the Universe

Mass-energy and radiation density

The Very Early Universe

Planck scale

Grand Unification scale

Electroweak scale

Condensation of quarks

Formation of light nuclei

Formation of atoms

Formation of galaxies

Present

Big Bang Nucleosynthesis

Inflationary Universe

Inflaton field

Flatness problem

Horizon problem

Exotic relics

Cosmic Background Radiation

Primordial Microscopic Black Holes

Exploring the Invisible Universe

The Dark Age of the Universe

Black Holes: Entropy of the Universe

The Answer

Dark

Universe

The Question

Dark Sky

Origin of Redshift

Dark Matter

Evidence : Galaxy rotation curve

Evidence : Cluster mass

Cluster mass — continued

Evidence : Gravitational lensing

Possible explanations for dark matter

Dark Energy

Evidence : Accelerating expansion

Evidence : Cosmic microwave background

Evidence : Flatness

Possible explanations for dark energy

The Answer

Galaxies, Stars and Planets

The Question

Primordial Gas Cloud

Formation of Galaxies

Formation of Stars

Stars: Hydrostatic Equilibrium

Classification of Stars

Spectral classification

Hertzsprung–Russell classification

End Points of Stellar Evolution

Normal and Active Galaxies

Active galaxies

Age of stars in the Milky Way

Supermassive Black Holes

Observing supermassive black holes

Active Galactic Nuclei AGN

Quasars

Formation of the Solar System

Solar Nebular Theory

Formation of the Terrestrial Planets

Early Earth

Formation of the Jovian Planets

Large Scale Structure of Our Solar System

The Answer

The Life of Stars

Nuclear Fusion: Star Burning

Stellar Thermonuclear Fusion

Binding energy of a nucleus

Nuclear Binding Energy: Fusion and Fission

Stellar Evolution: Formation of Red Giants

Helium Flash

Formation of a White Dwarf

Red Supergiant Stars

Evolution of High Mass Stars

Type II Supernovae

Type Ia Supernovae

Interstellar medium

Neutron Stars and Pulsars

Astrophysical Black Holes

Stellar size black holes

The Answer

The Origin of the Elements

Composition of the Universe

Elements: Stellar Nucleosynthesis

Main processes for nucleosynthesis

The pp-Process: Three-Step Hydrogen Burning

How much hydrogen does the Sun burn?

The CNO-Cycle for Hydrogen Burning

Helium Burning: Triple-Alpha Process

Alpha Capture and Other Processes

Silicon melting: Photodisintegration

Neutron Capture: s-Process and r-Process

Synthesis of Gold Au

Abundance of Elements in the Universe

The Answer

Elementary Particles

The Question

Elementary Building Blocks

Exploring the Invisible Universe

Particle Accelerators and Detectors

What is an Elementary Particle?

Symmetry

Symmetry and Conservation Law

Gauge invariance and gauge field

Baryon and Lepton Quantum Numbers

Antiparticles

Antiparticles and Causality

The Yukawa Interaction

Antiparticles and Quantum Field Theory

Energy Conservation and Quantum Numbers

Antiparticles: Baryons and Leptons

Hadrons: Strangeness Quantum Number

Quark Model

The Eight Fold Way

The Omega Minus Ω−

Experimental Evidence for Quarks

Quark jets

The Answer: Three Generations of Particles

Fundamental Interactions

The Question

Interactions in Nature

Strength and Duration of Interactions

Quantum Electrodynamics

Photons and electrons

Renormalization

The hydrogen atom revisited

Photons in a Plasma

Electroweak Interactions

Electroweak bosons: W± and Z

Electroweak Coupling Constants

Coupling of Weak Bosons to Fermions

Lepton–lepton couplings

Quark–quark couplings

Strangeness Changing Processes

Quantum Chromodynamics

Charmonium: Linear Potential

Gluonic Strings: Mesons and Baryons

Permanent Confinement of Quarks

The Answer

The Standard Model

The Question

The Standard Model of Particle Physics

β-Decay: Parity Violation in Nature

Fermions and Parity

Parity: Electron and neutrino

Fermions and Weak Bosons: Parity Violating Couplings

Pairing of Fermions: Chiral Anomaly Cancellation

Unification of the Weak and Electromagnetic Interactions

The Higgs Field and Phase Transition

Phase transition

Higgs condensation

The Higgs mechanism

Higgs interactions

The Masses of Electroweak Particles

Masses for the weak bosons

Masses for the fermions

Superconductivity and Higgs Mechanism

Analogy with the Higgs mechanism

Masses of Quarks and Leptons

Large Hadron Collider

The Atlas Experiment

Detection of the Higgs Boson

The Answer

Superstring

Unification

The Question

On the Road to Unification

Supersymmetry

Grand Unified Theories GUTs

Gravity and unification

Superstrings

Higher Spacetime Dimensions

Dimensional reduction and compactification

Topology and geometry

Superstrings: Observed Forces and Particles

Closed Superstrings

Self-interactions and quantum evolution

Superstring Interactions: Geometry versus Topology

Point-like versus topological interactions

Closed Superstrings: Type IIA and Type IIB

xvi

Exploring the Invisible Universe

Closed Heterotic Superstring

Spectrum of the heterotic string

Type I Open Superstrings

D-Branes

D-branes in various superstring theories

D-Brane: Our Universe

Two separated D-branes

Three coincident D-branes

Particles and forces in four dimensions

M and F Superstring Theories

Superconductor, Vortices and Duality

Superstring Theories: Connected by Dualities

The Answer

Superstring Gravity

The Question

Introduction

Quantum Gravity

Spacetime foam

Superstrings and Gravity

Brane Worlds and Gravity

Closed strings and gravity

Companion D-brane

Black Hole Entropy and Superstrings

Reissner–Nordstrom black hole

Colliding Branes, Cyclic Universes and the Big Bang

The Answer

Epilogue

Appendix

Laws

Equations

Maxwell’s Equations

Spacetime Metrics

Units

Constants

Periodic Table

Gravitational Waves, Volume 2 Astrophysics and Cosmology Maggiore, Michele

Stellar collapse

Neutron stars

Black-hole perturbation theory

Properties of dynamical space-times

GWs from compact binaries. Theory

GWs from compact binaries. Observations

Supermassive black holes

Basics of FRW cosmology

Helicity decomposition of metric perturbations

Evolution of cosmological perturbations

The imprint of GWs on the CMB

Inflation and primordial perturbations

Stochastic backgrounds of cosmological origin

Stochastic backgrounds and pulsar timing arrays

Abelian Higgs model

Absolute magnitude

Adaptive optics

Adiabatic index

Adiabatic perturbations

Adiabatic theorem

ADM formalism

ADM mass

ADM momentum

angular momentum

AEI formula

Affine parameter

Angular diameter distance

Anisotropic stress tensor

Anomalous X-ray pulsars

Apparent magnitude

Asiago Supernova Catalog

Asymptotic flatness

Axial perturbations of Schwarzschild

black holes

Baade’s window

Bar-mode instabilities

Bardeen variables

in flat space

Barotropic fluids

Baryon acoustic oscillations

Baryon asymmetry

Baryonic mass

Bayes factor

BBGKY hierarchy

BBKS transfer function

BICEPBig Bang Observer BBO

Big-bang nucleosynthesis

Bigravity

Black hole–black hole coalescence

observation

rate

theory

Black holes

extremal

irreducible mass

Kerr

perturbation theory

quasi-normal modes

spin measurement

supermassive

Blue supergiants

Bogoliubov transformation

Boltzmann equation

collision term

collisionless

Bondi–Sachs mass

Bounce solution

Boyer–Lindquist coordinates

Braking index

BSSN form of Einstein equations

Bulk viscosity

Bunch–Davies vacuum

Burke–Thorne potential

Cassini spacecraft

CFS instability

Chandra X-ray observatory

Chandrasekhar mass

Chandrasekhar–Schönberg limit

Chaotic inflation

Chapman–Jouguet detonation

Characteristic amplitude

Characteristic strain

Chirp mass

Christodoulou’s formula

Christoffel symbols

FRW

perturbed FRW

Close limit of colliding BHs

CMB temperature

Coherent search

Collapsar model

Common envelope evolution

Comoving coordinates

Comoving distance

Comoving horizon

in ΛCDM

816 Index

Compton scattering

Conformal gauge

Conformal Newtonian gauge

Conformal time

Continuity equation

Convective derivative

Cosmic censorship

Cosmic strings

bounds on tension

Cosmic variance

Cosmological constant

Covariant derivative of tensors

Cowling aproximation

Crab pulsar

braking index

characteristic age

period

spindown

Crab supernova

Critical density of the Universe

Curvature power spectrum

Dark energy

492–phantom

DECIGO

Deleptonization

Density contrast

gauge invariant

Deuterium bottleneck

de Sitter space

Distance modulus

Distance to the Galactic center

Dominant energy condition

Double-well potential

dRGT theory

Dynamical friction time-scale

e+ e− annihilation

Effective number of neutrinos

Effective number of relativistic species

Effective one-body description of

BH coalescence

inclusion of radiation reaction

waveform

Effective spin parameter χeff

Einstein luminosity

Einstein Telescope

Einstein tensor in FRW

Ekpyrotic model

Electron-capture SNe

eLISA

EMRIs

Energy function

Energy–momentum tensor

conservation

perfect fluid

perturbed

point-like particle

scalar field

Envelope approximation

Ergosphere

Escape velocities

EUCLID mission

Eulerian derivative

European Pulsar Timing Array EPTA

Extrinsic curvature

f R gravity

Fermi-GBM

Fitchett formula

Flatness problem

Flux function

Fourier transform

Friedmann equation

FRW cosmology

helicity decomposition

metric

scalar perturbations

tensor perturbations

Fundamental mode of neutron stars

92–ΓΓ Lagrangian

Gamma-ray bursts

afterglow

GRB 170817A

long and short

Generalized harmonic coordinate

Gravitational self-force

Gravitational slingshot

Graviton mass

Green’s function for black hole per-

turbations

Growth function

GstLAL analysis

GWastrophysical implications

parameters

properties

quasi-normal modes

search for electromagnetic coun-

terpart

statistical significance

tests of GR

GWastrophysical implications

parameters

tests of GR

GWGWGWHamiltonian formulation of GR

Hankel function

Harrison–Zeldovich spectrum

Hawking temperature

Hellings–Downs curve

Higgs mass

Hill-top inflation

Horizon problem

Hubble parameter

Hybrid inflation

Hyper-massive neutron star

Hypernovae

IMR-Phenom waveforms

Induced metric

Inflation

hill-top potential

natural

Starobinsky model

vacuum fluctuations

Instantons

INTEGRAL

International Pulsar Timing Array

IPTA

ISCO

Isocurvature perturbations

ISW effect

early

late

Index Jeffreys’ scale

K-correction

Kasner metric

Keck Array

Keck Observatory

Kerr geometry

ergosphere

static limit

Kerr metric

in quasi-isotropic coordinates

Kerr parameter

dimensionless form

Kilonova

Kinnersley tetrad

Klein–Gordon equation

Lagrangian derivative

ΛCDM

comoving distance

growth function

parameters

Landau–Lifshitz pseudotensor

Laplace transform

Lapse function

Last-scattering surface

thickness

Legendre polynomials

recursion relation

Lepton asymmetry

Lichnerowicz operator

Light ring

LIGO/Virgo collaboration

first binary neutron star de-

tection

first detection of gravitational

waves

first triple detection of a black

hole binary

limits on stochastic backgrounds

LISA space interferometer

orbit

sources

standard sirens

Longitudinal gauge

Loss cone

Love numbers

Luminosity

bolometric

818 Index

UBV filters

Luminosity distance

GWs

Lyth bound

M –σ relation for central BHs

Mészáros effect

MacLaurin spheroids

Magnetars

Magneto-elastic modes

Magneto-rotational core collapse

Massive gravity

Massless particle action

Master equation for Φ

Mathieu equation

Matter energy density

Matter–radiation equilibrium

redshift

temperature

Metallicity

Millennium Simulation

MiSaTaQuWa equation

Mode functions

Modified gravity

Mukhanov–Sasaki equation

Nambu–Goto action

NANOGrav

Neutrino decoupling

Neutrino free-streaming

damping of gravitational waves

486–Neutrinosphere

Neutron stars

compactness

spin

Neutron-star normal modes

f-modes

g-modes

glitches

interface modes

p-modes

r-modes

trapped modes

w-modes

Newman–Penrose formalism

Newtonian gauge

Nielsen–Olesen vortex

Noether charges

Nonlocal gravity

Novae

Novikov–Thorne accretion-disk model

Nucleosynthesis

primordial

r-process

s-process

Null tetrads

Numerical relativity

Optical depth

Padé approximants

Palatini formalism

Parametric resonance

Parkes Pulsar Timing Array PPTA

Particle horizon

Pauli matrices

Peculiar velocity

Perfect fluid

Phase-space distribution

Phillips relation

Physical coordinates in FRW

Pivot scale

Planck luminosity

Planck mass

reduced

Planck mission

Poisson equation

Poisson gauge

Polar perturbations of Schwarzschild

black holes

Polyakov action

Polytropic equation of state

Polytropic index

Positive-energy theorem

Power spectrum

definition

primordial

Power-law tails

Pre-big-bang model

Precursor

Preheating

Primordial scalar spectrum

amplitude

tilt

Proper time

Proto-neutron stars

normal modes

Pulsar timing arrays PTAs

723–Pulsars

Crab

discovery

fastest spinning

in globular clusters

kinetic age

millisecond

PSR B1919+PSR B1937+PSR J0348+PSR J1614-PSR J1734-PSR J1748-2446ad

PSR J1807−2500B

PSR J2144-recycled

spindown luminosity

Vela

PyCBC analysis

Quasi-normal modes

and Laplace transform

asymptotic spectrum

excitation

excitation factors

frequencies

GWQuasi-periodic oscillations QPOs

Radial infall into a black hole

Radiation energy density

Recoil of final BH after merger

due to mass asymmetry

spinning binaries

Recombination

Red supergiants

Reduced wavelength

Rees–Sciama effect

Regge–Wheeler equation

Regge–Wheeler gauge

Regge–Wheeler potential

Reheating

Reionization bump

Ricci scalar in FRW

Ricci tensor in FRW

Ringdown phase

Running index

Index RXTE

Sachs–Wolfe effect

Sachs–Wolfe plateau

Saha equation

Salpeter time-scale

Sasaki–Nakamura formalism

SASI standing accretion shock in-

stability

Scalar spectrum

amplitude

tilt

Schwarzschild metric

radial geodesics

SDSS

photometric scheme

Sgr A -

Shear viscosity

Shift vector

Silk damping

Slow-roll conditions

Slow-roll parameters

SNu unit

Soft gamma repeaters SGRs

Sound speed

adiabatic

gauge invariant

Spectral tilt

Spherical Bessel functions

Spherical harmonics

addition theorem

Spin weight

Spin–orbit Hamiltonian

Spin–spin Hamiltonian

Spin-weighted spherical harmonics

Spin-weighted spheroidal harmon-

ics

Square Kilometer Array SKA

Squeezed state

Standard sirens

Starobinsky inflationary model

Stars

color index

metallicity

Population I

Stefan–Boltzmann constant

Stokes parameters

820 Index

Sun

luminosity

mass

metallicity

radius

Superkicks

Supermassive BH binaries

Supermassive BHs

M –σ relation

formation

spin

Supernovae

Cas A

classification

electron capture

failed

galactic remnants

historical

pair instability

rates

SN SN 1054 Crab

SN SN 1572 Tycho’s

SN 1604 Kepler’s

SN SN 1987A

SN 2005ap

SN 2006gy

stripped-envelope

type Ia

type Ib

type Ic-BL

type II-L

type II-P

type IIb

type IIn

ultraluminous

Synchronous gauge

Tachyonic preheating

Tails

Tensor spherical harmonics

on the sphere

Tensor-to-scalar ratio r

Teukolsky equation

Thin-wall solution

Tight-coupling limit

Tolman–Oppenheimer–Volkov equa-

tions

Tortoise coordinate

Trace-K action

Transfer function

BBKS

scalar modes

tensor modes

Triple-alpha process

Tycho Brahe

Ultra-luminous X-ray sources

URCA process

Variation

√

of −g

of Ricci scalar

Vela pulsar

braking index

characteristic age

period

spindown

Velocity dispersion and M –σ rela-

tion

Velocity divergence

Velocity potential

Very Large Telescope VLT

Virgo interferometer

collaboration

Visibility function

wCDM model

Weak energy condition

Weyl scalars

Weyl tensor

White dwarfs

CO

He

NeO

Wick’s theorem

Wigner’s d-matrix

WMAP

Wolf–Rayet stars

Wronskian

Zerilli equation

derivation

Zerilli function

Zerilli potential

Zerilli spherical harmonics

How the Universe Works Introduction to Modern Cosmology Serge Parnovsky, Aleksei Parnowski

Preface

List of Table

List of Figures

Chapter The Laws of the Unerse

Roots of Cosmology

Principles of General Relatity

Perihelion precession

Deation of light

Gratational redshift

Other effects and tests

Chosen frame

Graty, inertia, and tidal forces

Lunar tides

Space, time, and space-time

Curved space-time

How Much Does Light Weigh?

Baryonic matter

Radiation

Dark energy and antigraty

How the Unerse Works: Introduction to Modern Cosmology

Chapter The Expanding Unerse

Einstein’s Static Unerse

Expansion and Redshift

Other galaxies and their recession

Expansion

Redshift

Hubble’s Law∗

Friedmann Models

Arrow of time

Geometry of the Unerse

Scale Factor

Deceleration parameter

Non-Relatistic Friedmann Solutions∗

Cosmological evolution without cosmological

constant∗

Study of solutions∗

Deceleration parameter∗

Matter with nonzero pressure in the expanding

Unerse∗

Modern Modification of the Model

Cosmological constant strikes back

Standard cosmological model

Distances in Astronomy

Chapter

Early Unerse

The Big Bang

Cosmic Microwave Background: an Echo of the Big Bang

CMB discovery

CMB anisotropy

Bringing cosmology to space

Ground studies of the CMB

CMB fluctuations spectrum

Conservation of energy

Speculations

Evolution of the Early Unerse∗

Cosmological Horizon

Distance to the Cosmological Horizon∗

Inflation of the Unerse

Inflation models

Multerse and the Anthropic Principle

Pulsating Unerse

The Matter in Making

Big Bang nucleosynthesis

Stellar nucleosynthesis

The antimatter problem

Chapter

Dark Matter

Revolution Comes

Edence for Dark Matter

rial mass

Galactic rotation curves

Mass-to-luminosity ratio

Galactic mergers

Cosmic flows

Growth rate of density fluctuations

Gratational lensing

Alternate models

What Makes up Dark Matter?

Chapter

Dark Energy

Cosmological Edence for Dark Matter and Dark Energy

Type Ia supernovae

Baryonic acoustic oscillations

CMB spectrum

Dark Energy

Time to Big Rip∗

Other Kinds of Matter

Chapter

Black Holes and Other Exotics

Black Holes

Schwarzschild black holes

Reissner–Nordström black hole

Kerr black hole

How the Unerse Works: Introduction to Modern Cosmology

Naked Singularities

Wormholes

Summary

Append A Cosmological Evolution with a

Cosmological Constant∗

De Sitter Solution∗

A CDM Model∗

A Flat CDM Model∗

Bibliography

Further Reading

Introduction to General Relativity, Black Holes and Cosmology Yvonne Choquet-Bruhat

Fundamentals

Riemannian and Lorentzian geometry

Introduction

Differentiable manifolds and mappings

Differentiable manifolds

Differentiable mappings

Submanifolds

Tangent and cotangent spaces

Vector fields and -forms

Mong frames

Tensors and tensor fields

Tensors, products and contraction

Tensor fields Pullback and Lie derate

Exterior forms

Structure coefficients of mong frames

Pseudo-Riemannian metrics

General properties

Riemannian metrics

Lorentzian metrics

Causality

Causal and null cones

Future and past

Spacelike submanifolds

Length and geodesics

Connections

Linear connection

Riemannian connection

Geodesics, another definition

Pseudo-Riemannian manifolds

Riemannian manifolds

Lorentzian manifolds

Curvature

Definitions

Symmetries and antisymmetries

Differential Bianchidentity and contractions

Geodesic deation

Linearized Ricctensor

Linearized Bianchidentities

Physical comment

Solutions of selected exercises

Problems

Lioulle theorem

Codifferential δ and Laplacian

of an exterior form

Geodesic normal coordinates

Cases d = , , and

Wave equation satisfied by the

Riemann tensor

The Bel–Robinson tensor

The Weyl tensor

The Cotton–York tensor

Linearization of the Riemann tensor

Second derate of the Ricctensor

Special relatity

Introduction

Newtonian mechanics

IThe Galileo–Newton Spacetime

INewtonian dynamics Galileo group

IPhysical comment

IThe Maxwell equations in

Galileo–Newton spacetime

The Lorentz and Poincaré groups

Lorentz contraction and dilation

Electromagnetic field and Maxwell equations

in Minkowskspacetime M

Maxwell equations in arbitrary dimensions

Special Relatity

IProper time

IProper frame and relate velocities

Some physical comments

Dynamics of a pointlike mass

INewtonian law

IRelatistic law

INewtonian approximation of the

relatistic equation

IEqualence of mass and energy

IParticles with zero rest mass

Continuous matter

ICase of dust incoherent matter,

masse particles

IPerfect fluids

IYang–Mills fields

Problems

ILorentz transformation of the

Maxwell equations

The relatistic Doppler–Fizeau effect

General Relatity

Introduction

Principle of general covariance

The Galileo–Newton equalence principle

General Relatity

Einstein equalence principles

Conclusion

Constants and units of measurement

Classical fields in General Relatity

Perfect fluid

Electromagnetic field

Charged fluid

Gratation and curvature

Observations and experiments

The Einstein equalence principle

Deation of light rays

Proper time, gratational time delay

Conclusion

Problems

Newtonian gratation theory in

absolute space and time E n × R

Mass in length units case n =

Planck units

TheEinsteinequations

Introduction

The Einstein equations

The Einstein equations in vacuum

Equations with sources

Matter sources

Field sources

The cosmological constant

General Einsteinian spacetimes

Regularity

Boundary conditions

Physical comment

Newtonian approximation

Determination of GE

Equations of motion

Post-Newtonian approximation

Minkowskian approximation

Linearized equations at η

Plane gratational waves

Further results on gratational waves

Tidal force

Gratational radiation

Strong high-frequency waves

Introduction

Phase and polarization

Propagation and backreaction

Observable displacements

Stationary spacetimes

Definition

Equations

Non-existence of gratational solitons

Gauss’s law

Lagrangians

Einstein–Hilbert Lagrangian in vacuo

Lagrangians for Einstein equations

with sources

Observations and experiments

Problems

The Einstein cylinder

de Sitter spacetime

Anti-de Sitter spacetime

Taub–NUT spacetime

The quadrupole formula

Gratational waves

Landau–Lifshitz pseudotensor

High-frequency waves from a

spherically symmetric star

Static solutions with compact

spacelike sections

Mass of an asymptotically

Euclidean spacetime

Taub Lagrangian

The Schwarzschildspacetime

Introduction

Spherically symmetric spacetimes

Schwarzschild metric

Other coordinates

Isotropic coordinates

Wave also called harmonic coordinates

Painlevé–Gullstrand-like coordinates

Regge–Wheeler coordinates

Schwarzschild spacetime and event horizon

The motion of the planets and perihelion precession

Equations

Results of observations

Escape velocity

Stability of circular orbits

Deflection of light rays

Theoretical prediction

Fermat’s principle and light travel

parameter time

Results of observation

VVVRedshift and time dilation

Redshift

Time dilation

Spherically symmetric interior solutions

Static solutions Upper limit on mass

Matching with an exterior solution

Non-static interior solutions

Spherically symmetric gratational collapse

Tolman, Gu, Hu, and Claudel–

Newman metrics

Monotonically decreasing density

Problems

Relatistic and Newtonian

gratational masses

The Reissner–Nordström solution

Schwarzschild spacetime in

dimension n +

Schwarzschild metric in isotropic

coordinates, n =

Wave coordinates for the

Schwarzschild metric in dimension

n +

Blackholes

Introduction

The Schwarzschild black hole

Eddington–Finkelstein extensions

VEddington–Finkelstein white hole

VKruskal spacetime

Stationary black holes

VAxisymmetric and stationary spacetimes

The Kerr spacetime and black hole

Boyer–Lindquist coordinates

The Kerr–Schild spacetime

Essential singularity

Horizons

Limit of stationarity The ergosphere

Extended Kerr spacetime

Absence of realistic interior solutions

or models of collapse

Uniqueness theorems for stationary black holes

The Israel uniqueness theorem

Uniqueness of the Kerr black hole

Stability of the Kerr black hole

General black holes

Definitions

Weak cosmic censorship conjecture

Thermodynamics of black holes

Conclusions

Observations

The interiors of black holes

Solution of Exercise

Problems

Lemaı̂tre coordinates

Reissner–Nordström black hole

Kerr–Newman metric

Irreducible mass Christodoulou–Ruffini

The Riemannian Penrose inequality

Introduction to cosmology

Introduction

The first cosmological models

Einstein static unerse

de Sitter spacetime

General models

Cosmological principle

Assumptions

Observational support

Robertson–Walker spacetimes

Robertson–Walker unerses, metric

at gen t

Robertson–Walker cosmologies

General properties of Robertson–Walker spacetimes

Cosmological redshift

The Hubble law

Deceleration parameter

Age and future of the unerse

Friedmann–Lemaı̂tre unerses

Equations

Density parameter

Einstein–de Sitter unerse

General models with p =

ΛCDM cosmological model

Primordial cosmology

Solution of Exercises and

Problems

Isotropic and homogeneous

Riemannian manifolds

Age of the unerse

Classical Friedmann–Lemaı̂tre unerses

Milne unerse

Part B

Advanced topics

General Einsteinian spacetimes The

Cauchy problem

Introduction

Wave coordinates

Generalized wave coordinates

Damped wave coordinates

Evolution in wave gauge

Solution of the reduced

equations in vacuum

Equations with sources

Preservation of the wave gauges

Wave gauge constraints

Local existence and uniqueness

Solution of the wave gauge constraints

Asymptotically Euclidean manifolds

Compact manifolds

Geometric n + splitting

Adapted frame and coframe

Dynamical system with

constraints for ḡ and K

Geometric Cauchy problem

Regularity assumptions

Solution of the constraints

by the conformal method

Conformally formulated CF

constraints

Elliptic system

Physical comment

Motion of a system of compact bodies

Effecte one-body EOB method

Numerical Relatity

Global properties

Global hyperbolicity and

global uniqueness

Global existence

Singularities and cosmic censorship conjectures

Strong cosmic censorship conjecture

Weak cosmic censorship conjecture

Problems

Symmetric hyperbolic systems

The wave equation as a

symmetric hyperbolic system

The evolution set of Maxwell

equations as a first-order

symmetric hyperbolic system

Conformal transformation of

the CF constraints

Einstein equations in dimension

Electrovac Einsteinian

spacetimes, constraints

Electrovac Einsteinian

spacetimes, Lorenz gauge

Wave equation for F

Wave equation for the Riemann tensor

First-order symmetric

hyperbolic system for the

Riemann tensor, Bel–Robinson energy

Schwarzschild trapped surface

Relatistic fluids

Introduction

Case of dust

Charged dust

Perfect fluid

Stress–energy tensor

Euler equations

Thermodynamics

Conservation of rest mass

Definitions Conservation of entropy

Equations of state n =

Wave fronts and propagation speeds

Characteristic determinant

Wave front propagation speed

Case of perfect fluids

Cauchy problem for the Euler and entropy system

The Euler and entropy equations

as a Leray hyperbolic system

First-order symmetric hyperbolic systems

Coupled Einstein–Euler–entropy system

Initial data

Evolution

Dynamical velocity

Fluid indeand Euler equations

Vorticity tensor and Helmholtz equations

General perfect fluid enthalpy h

Irrotational flows

Definition and properties

Coupling with the Einstein equations

Equations in a flow-adapted frame

Shocks

Charged fluids

Equations

Fluids with zero conductity

Fluids with finite conductity

Magnetohydrodynamics

Equations

Wave fronts

Yang–Mills fluids quark–gluon plasmas

scous fluids

Generalized Naer–Stokes equations

A Leray–Ohya hyperbolic system

for scous fluids

The heat equation

Conclusion

Solution of Exercise

Problems

Specific volume

Motion of isolated bodies

Euler equations for the dynamic velocity

Hyperbolic Leray system for the

dynamical velocity

Geodesics of conformal metric

Cosmological equation of state

p = γ − μ

Relatistic kinetic theory

Introduction

Distribution function

Definition

Interpretation

Moments of the distribution function

Particles of a gen rest mass

Vlasoequations

General relatistic GR–Vlasoequation

EM–GR–Vlasoequation

Yang–Mills plasmas

Solution of a Vlasoequation

Construction

Global existence theorem

Stress–energy tensor

The Einstein–Vlasosystem

Equations

Conservation law

The Cauchy problem

Cauchy data and constraints

Evolution

Local existence and uniqueness theorem

Global theorems

The Maxwell–Einstein–Vlasosystem

Particles with gen rest mass and charge

Particles with random masses and charges

Boltzmann equation Definitions

xMoments and conservation laws

Einstein–Boltzmann system

Thermodynamics

Entropy and the H theorem

Maxwell–Jüttner equilibrium distribution

Dissipate fluids

Extended thermodynamics

Solutions of selected exercises

Problems

Lioulle’s theorem and generalization

Vlasoequation for particles with

random charges

Distribution function on a

Robertson–Walker spacetime with

Vlasosource

References

Index

Introduction to Quantum Mechanics David J. Griffiths, Darrell F. Schroeter

Preface

Theory

The Wave Function

The Schrödinger Equation

The Statistical Interpretation

Probability

Discrete Variables

Continuous Variables

Normalization

Momentum

The Uncertainty Principle

Further Problems on Chapter

Time-Independent Schrödinger Equation

Stationary States

The Infinite Square Well

The Harmonic Oscillator

Algebraic Method

Analytic Method

The Free Particle

The Delta-Function Potential

Bound States and Scattering States

The Delta-Function Well

The Finite Square Well

Further Problems on Chapter

Formalism

Hilbert Space

Observables

Hermitian Operators

Determinate States

Eigenfunctions of a Hermitian Operator

Discrete Spectra

Continuous Spectra

Generalized Statistical Interpretation

The Uncertainty Principle

Proof of the Generalized Uncertainty Principle

The Minimum-Uncertainty Wave Packet

The Energy-Time Uncertainty Principle

Vectors and Operators

Bases in Hilbert Space

Dirac Notation

Changing Bases in Dirac Notation

Further Problems on Chapter

Quantum Mechanics in Three Dimensions

The Schröger Equation

Spherical Coordinates

The Angular Equation

The Radial Equation

The Hydrogen Atom

The Radial Wave Function

The Spectrum of Hydrogen

Angular Momentum

Eigenvalues

Eigenfunctions

Spin

Spin /

Electron in a Magnetic Field

Addition of Angular Momenta

Electromagnetic Interactions

Minimal Coupling

The Aharonov–Bohm Effect

Further Problems on Chapter

Identical Particles

Two-Particle Systems

Bosons and Fermions

Exchange Forces

Spin

Generalized Symmetrization Principle

Atoms

Helium

The Periodic Table

Solids

The Free Electron Gas

Band Structure

Further Problems on Chapter

Symmetries Conservation Laws

Introduction

Transformations in Space

The Translation Operator

How Operators Transform

Translational Symmetry

Conservation Laws

Parity

Parity in One Dimension

Parity in Three Dimensions

Parity Selection Rules

Rotational Symmetry

Rotations About the z Axis

Rotations in Three Dimensions

Degeneracy

Rotational Selection Rules

Selection Rules for Scalar Operators

Selection Rules for Vector Operators

Translations in Time

The Heisenberg Picture

Time-Translation Invariance

Further Problems on Chapter

IApplications

Time-Independent Perturbation Theory

Nondegenerate Perturbation Theory

General Formulation

First-Order Theory

Second-Order Energies

Degenerate Perturbation Theory

Two-Fold Degeneracy

“Good” States

Higher-Order Degeneracy

The Fine Structure of Hydrogen

The Relatistic Correction

Spin-Orbit Coupling

The Zeeman Effect

Weak-Field Zeeman Effect

Strong-Field Zeeman Effect

Intermediate-Field Zeeman Effect

Hyperfine Splitting in Hydrogen

Further Problems on Chapter

The Varitional Principle

Theory

The Ground State of Helium

The Hydrogen Molecule Ion

The Hydrogen Molecule

Further Problems on Chapter

The WKB Approximation

The “Classical” Region

Tunneling

The Connection Formulas

Further Problems on Chapter

Scattering

Introduction

Classical Scattering Theory

Quantum Scattering Theory

Partial Wave Analysis

Formalism

Strategy

Phase Shifts

The Born Approximation

Integral Form of the Schrödinger Equation

The First Born Approximation

The Born Series

Further Problems on Chapter

Quantum Dynamics

Two-Level Systems

The Perturbed System

Time-Dependent Perturbation Theory

Sinusoidal Perturbations

Emission and Absorption of Radiation

Electromagnetic Waves

Absorption, Stimulated Emission, and Spontaneous Emission

Incoherent Perturbations

Spontaneous Emission

Einstein’s A and B Coefficients

The Lifetime of an Excited State

Selection Rules

Fermi’s Golden Rule

The Adiabatic Approximation

Adiabatic Processes

The Adiabatic Theorem

Further Problems on Chapter

Afterword

The EPR ParadoBell’s Theorem

Med States and the Density Matr

Pure States

Med States

Subsystems

The No-Clone Theorem

Schrödinger’s Cat

Append Linear Algebra

A Vectors

A Inner Products

A Matrices

A Changing Bases

A Eigenvectors and Eigenvalues

A Hermitian Transformations

Introduction to Superstrings and M-Theory 2nd Edition Michio Kaku

Graduate Texts in Contemporary Physics

R N Mohapatra: Unification and Supersymmetry: The Frontiers of

Quark-Lepton Physics

R E Prange and S M Girn eds: The Quantum Hajj Effect, nd ed

M Kak:u: Introduction to Superstrings and M- Theory, nd ed

J W Lynn ed: High Temperature Superconductity

H Klapdor ed: Neutrinos

J H Hinken: Superconductor Electronics: Fundamentals and Microwave

Application

First Quantization and Path Integrals

IPatb Integrals and Point Particle!

Why Strings?

Historical Rcew orOauge Theory

Path Integrals and Point Particles

Relatislic Point Particles

First and Second Quantization

Faddeev- PopoQuantization

Scwnd Quantization

Harmonic Oscillators

Currents and Second Quanrition

Summary

References

Bosonic Slrngs

Oupta-Bleuler Quantization

Light Cone Quantization

BRST Quantization

Trees

From Path Intcgrals to Operators

Projecre lnwriance and Twists

Closed Strings

Ghost Elimination

Summary

References

Superstrings

Supersymmetric Point Particles

Two-Dimensional Supersymmetry

Trees

Local Two-Dimensional Supersymmetry

Quantization

GSO Projection

Superstrings

Light Cone Quantization of the GS Action

Vertices and Trees

Summary

References

Conformal Field Tbeory and Kac-Moody Algebras

Conformal Field Theory ’

Superconformal Field Theory

Spin Fields

Superconformal Ghosts

Fermion Verte

Spinors and Trees

Kao-Moody Algebras

Supersymmetry

Summary

References

Multiloops and Teichm ller Spaces

U nitarity

Single-Loop Amplitude

Harmonic Oscillators

Single-Loop Superstring Amplitudes

Closed Loops

Multiloop Amplitudes

Riemann Sufi’ aces and Teichmillier Spaces

Conformal Anomaly

Superstrings

Determinants and Singularities

ModulSpace and Grassmannians

Summary

References

S~ond Quantization and the Search for Geometry

Ught Cone fJ eld Theory

Why String Field Theory?

Dering Point Particlc Field Theory

Light Cone Field Theory

Interactions

Neumann Function Method

Equalence of the Scauering Amplitudes

four-String Intemction

Superstring Summary

Field Theory

References

BRST Field Theory

Covariant Siring Field Theory

eRST Field Theory

Gauge Fing

interactions

Winenfs String Field Theory

Proof of qualence

Closed Strings and Superstrings

Summary

References

Phenomclloloi:’ and Model Building

Anomlllies lind the Atiya h-Slngn Theorem

Beyond GUT Phenomenology

Anomalies and Fcynnum Diagrams

Anomalies in the Functional Fonnll sm

Anomalies and Characteristic Classes

Dirac index

G ratational a nd Gauge Anomalies

Anomaly Cancellation in Strings

Helerotic Strlni:s and Compactifiutioo

Compactificarion

The Hcterotic Smog

Spectrum

Covariant and Fennionic Fonnulations

Trees

Single-Loop Amplitude

E and Kac-Moody Algebras

Lorentzian Lattices

Summary

References

CaJabi-Yau Spaces and Orbifolds

Calabi-Yau Spaces

Reew of de Rahm Cohomology

Cohomology and Homology, Kahler Manifolds

Embedding the Spin Connection

Fermion Generations

Wilson Lines

O r b f o l d s

Four-Dimensional Superstrings

Summary

References

M-Theory

M-Theory and Duality

Introduction

Duality in Physics

Why Fe String Theories?

T -Duality

S-Duality

Type A Theory

Type B Theory

M-Theory and Type lIB Theory

, E ® E Heterotic String

Type Strings

Summary

References

Compactifications and BPS States

BPS States

Supersymmetry and P-Branes

Compactification

Example: D =

D = , N = , Theory

D=,N=,ITheories

M-Theory in D =

Example: D = , N = and D = , N =

Symmetry Enhancement and Tensionless Strings

F -Theory

FXl lple: D =

Summary

RefcrencC$

Solitons, D-Brane!, and Black Boles

Solitons

Supen:Ill:mbranc Actions

Fe-BnlDe Action

D-Branes

D-Bnmc:: Actions

Matr Models and Membranes

Black Holes

Summary

Conclusion

References

Lectures on Astrophysics Steven Weinberg

STARS

Hydrostatic Equilibrium

Equilibrium equation Central pressure Gratational binding energy

rial theorem Stability Initial contraction Keln time scale

Radiate Energy Transport

Differential energy density Transport term Absorption term

Scattering term Emission term Equilibrium Fludergence

Momentum tensor dergence Opacity Rosseland mean

Radiate transport equations

Radiate Models

Differential equations Conditions at center Conditions at

nominal surface True surface Vogt–Russell theorem

Effecte temperature Color temperature Hertzsprung–Russell relation

Eddington bound

Opacity

Contributions to opacity Stimulated emission Thomson scattering

Free–free absorption Kramers opacity Bound–free absorption

Bound–bound absorption Append: Calculation of free–free absorption

Nuclear Energy Generation

Proton–proton chain CNO cycle Suppression factors

Coulomb barrier Application to proton–proton chain Solar neutrinos

Application to CNO cycle Crossover Beyond hydrogen burning

Carbon synthesis Append: Calculation of suppression by

Coulomb barriers

Relations among Observables: The Main Sequence

Temperature and density dependence of energy generation and opacity

Dimensional analysis Gas pressure dominance: radius–mass relation,

luminosity–mass relation, central temperature versus effecte surface temperature,

Hertzsprung–Russell slope Hydrogen burning time Radiation pressure

dominance: radius–mass relation, luminosity–mass relation,

Hertzsprung–Russell slope

Convection

Stability against convection Eddington discriminant

Ming length theory Efficient convection Isentropic stars

The Sun Variational principle

Polytropes

Examples of polytropic stars The Lane–Emden differential equation

Exact solutions Numerical solutions

Instability

Onset of instability: general theorem, with exceptions Stars close to = /

Expansion in /c Append: Deration of relatistic energy correction

White Dwarfs and Neutron Stars

Equation of state for cold electrons High-mass and low-mass white dwarfs

Neutronization Relatistic instability Equation of state for

cold neutrons Low-mass neutron stars Landau–Oppenheimer–Volkoff limit

Neutron star spin Pulsars

Supermasse Stars

Gas/radiation pressure ratio Equation of state Mass

Stability Evolution

Bibliography for Chapter

BINARIES

Orbits

General orbits Spectroscopic binaries Energy and angular momentum

Relatistic corrections Append: Calculation of time dilation in binary stars

Close Binaries

Roche limit Sirius A and B Equipotential surfaces Roche lobes

Mass transfer Type a supernovae Roche lobe volumes

Gratational Wave Emission: Binary Pulsars

The Hulse–Taylor pulsar Quadrupole approximation for emitted power

Decrease in period Decrease in eccentricity Time to coalescence

Gamma ray bursts and kilonovae Total radiated energy

More binary pulsars Append: Reew of gratational radiation

Gratational Wave Detection: Coalescing Binaries

Weber bars Interferometers Sources Black holes versus neutron stars

Chirps Description of LIGO Transformation to

transverse-traceless gauge Response of LIGO to gratational waves

Shot noise and seismic noise Sensitity detection of

gratational wave Diagnosis of source: chirp mass, relatistic corrections

Estimate of distance More coalescing black-hole binaries

A coalescing neutron star binary Blind spots

Bibliography for Chapter

THE INTERSTELLAR MEDIUM

Spectral Lines

General transport equation Optical depth Solution for homogeneous

emission/absorption ratio Doppler broadening Einstein A and B

coefficients Emission lines from clouds in thermal equilibrium

Emission lines from non-equilibrium regions Absorption lines

cm lines CN absorption lines

HIRegions

Strömgren spheres Differential equation for ionization Interior of

the sphere Surface of the sphere Recombination lines Heating

Cooling

Cooling function Prompt radiation case Excitation by electrons

Hydrogen atoms Russell–Saunders classification of atom and ion states

CI O OI Cooling in HIregions Delayed radiation case

H and CO molecules Bremsstrahlung cooling

Star Formation

rial estimates Jeans radius and mass Molecular clouds

Dispersion relation for gratational perturbations

Transition to instability Collapse time

Accretion Disks

Exceeding the Eddington limit Role of scosity Differential equations

for surface density Mass and angular-momentum flow Steady disks

scous heating Spectral distribution Thickness of disk

Decaying disks Bessel function solution for constant scosity

Expansion of disk Accretion disks in binaries Cataclysmic variables

Accretion Spheres

Bondaccretion Conservation laws The wind equation

Transonic solutions Mass accretion rate M∗

Soft Bremsstrahlung

Emissity and Gaunt factor Born approximation A misleading formula

Low-energy theorem Debye screening

Bibliography for Chapter

GALAXIES

Collisionless Dynamics

Collisionless Boltzmann equation Surface density from velocity dispersion

Moment equations Solutions to Boltzmann equation

Eddington theorem

Polytropes and Isothermals

Polytrope solutions of Boltzmann equation Isothermal solutions of

Boltzmann equation Galaxy clusters Dark matter Missing baryons?

NFW distribution

Galactic Disks

Rotation curves Bulge dominance Disk dominance Halo dominance

Append A: Gratational potential of a disk Append B: Minimum

energy configuration for fed angular momentum

Spiral Arms

Trailing and leading spirals Differential equations for surface density

Lin–Shu density waves Winding from differential rotation

Pitch angle and winding problem Epicyclic frequency

Pattern frequency Crowding Lindblad resonances

Quasars

Quasi-stellar objects and sources Accretion on black holes Heating of

accretion disks Append: Orbits of minimum radius about black holes

Bibliography for Chapter

ASSORTED PROBLEMS

AUTHOR INDESUBJECT INDEX

One Hundred Years of General Relativity From Genesis and Empirical Foundations to Gravitational Waves, Cosmology and Quantum… Wei-Tou Ni

Part Genesis, Solutions and Energy

A genesis of special relatity

Valérie Messager and Christophe Letellier

IJMPD

Introduction

The Ether: From Celestial Body Motion to Light

Propagation

Its origin

The luminiferous ether

Galileo’s Composition Law for Velocities

Questioning the Nature of Light: Waves

or Corpuscles?

From Electrodynamics to Light

Ampère’s law

Maxwell’s electromagnetic waves as light

Helmholtz’s theory

Hertzs experiments for validating Maxwell’s

theory

Invariance of the Field Equations from a Frame

to Another One

Hertz’s electrodynamic theory

Voigt’s wave equation

Lorentz’s electrodynamical theory

Larmor’s theory

Poincaré’s Contribution

Einstein’s Contribution

Conclusion

Appendices

A Fizeau’s experiments

A Michelson and Morley’s experiments

Genesis of general relatity — A concise exposition

Wei-Tou Ni

IJMPD

Prelude — Before

The Period of Searching for Directions and New

Ingredients: –

The Period of Various Trial Theories: –

The Synthesis and Consolidation: –

Epilogue

Schwarzschild and Kerr solutions of Einstein’s field

equation: An Introduction

Christian Heinicke and Friederich W Hehl

IJMPD

Prelude

Newtonian graty

Minkowskspace

Null coordinates

Penrose diagram

Einstein’s field equation

The Schwarzschild Metric

Historical remarks

Approaching the Schwarzschild metric

S classical representations of the

Schwarzschild metric

The concept of a Schwarzschild black hole

Event horizon

Killing horizon

Surface graty

Infinite redshift

Using light rays as coordinate lines

Eddington–Finkelstein coordinates

Kruskal–Szekeres coordinates

Penrose–Kruskal diagram

Adding electric charge and the cosmological

constant: Reissner–Nordström

The interior Schwarzschild solution and the

TOequation

The Kerr Metric

Historical remarks

Approaching the Kerr metric

Papapetrou line element and vacuum

field equation

Ernst equation

From Ernst back to Kerr

Three classical representations of the

Kerr metric

The concept of a Kerr black hole

Depicting Kerr geometry

The ergoregion

Constrained rotation

Rotation of the event horizon

Penrose process and black hole

thermodynamics

Beyond the horizons

Using light rays as coordinate lines

Penrose–Carter diagram and Cauchy horizon

Gratoelectromagnetism, multipole moments

Gratoelectromagnetic field strength

Quadratic invariants

Gratomagnetic clock effect of

Mashhoon, Cohen et al

Multipole moments: Gratoelectric

and gratomagnetic ones

Adding electric charge and the cosmological

constant: Kerr–Newman metric

On the uniqueness of the Kerr black hole

On interior solutions with material sources

Kerr Beyond Einstein

Kerr metric accompanied by a propagating

linear connection

Kerr metric in higher dimensions and

in string theory

Append

A Exterior calculus and computer algebra

Gratational energy for GR and Poincaré

gauge theories: A covariant Hamiltonian approach

Chiang-MeChen, James Nester and Roh-Suan Tung

IJMPD

Introduction

Background

Some brief early history

From Einstein’s correspondence

Noether’s contribution

Noether’s result

The Noether Energy–Momentum Current

Ambiguity

Pseudotensors

Einstein, Klein and superpotentials

Other GR pseudotensors

Pseudotensors and the Hamiltonian

The Quasi-Local ew

Currents as Generators

Gauge and Geometry

Dynamical Spacetime Geometry and the

Hamiltonian

Pseudotensors and the Hamiltonian

Some comments

Differential Forms

Variational Principle for Form Fields

Hamiltons principle

Compact representation

Some Simple Examples of the Noether Theorems

Noether’s first theorem: Energy–momentum

Noether’s second theorem: Gauge fields

Field equations with local gauge theory

First-Order Formulation

The Hamiltonian and the + Spacetime Split

Canonical Hamiltonian formalism

The differential form of the spacetime

decomposition

Spacetime decomposition of the variational

formalism

The Hamiltonian and Its Boundary Term

The translational Noether current

The Hamiltonian formulation

Boundary terms: The boundary condition

and reference

Covariant-symplectic Hamiltonian

boundary terms

Standard Asymptotics

Spatial infinity

Null infinity

Energy fluApplication to Electromagnetism

Geometry: Covariant Differential Formulation

Metric and connection

Riemann–Cartan geometry

Regarding geometry and gauge

On the affine connection and gauge theory

Variational Principles for Dynamic Spacetime

Geometry

The Lagrangian and its variation

Local gauge symmetries, Noether currents

and differential identities

Interpretation of the differential identities

First-Order Form and the Hamiltonian

First-order Lagrangian and local gauge

symmetries

Generalized Hamiltonian and differential

identities

General geometric Hamiltonian boundary

terms

Quasi-local boundary terms

A preferred choice

Einstein’s GR

Preferred boundary term for GR

A “Best Matched” Reference

The choice of reference

Isometric matching of the -surface

Complete D isometric matching

xPart Complete D isometric matching

Concluding Discussion

EmpiricalFoundations

Equalence principles, spacetime structure

and the cosmic connection

Wei-Tou Ni

IJMPD

Introduction

Meaning of Various Equalence Principles

Ancient concepts of inequalence

Macroscopic equalence principles

Equalence principles for photons

wave packets of light

Microscopic equalence principles

Equalence principles including graty

Strong equalence principles

Inequalence and interrelations of various

equalence principles

Gratational Coupling to Electromagnetism and

the Structure of Spacetime

Premetric electrodynamics as a framework

to study gratational coupling

to electromagnetism

Wave propagation and the dispersion relation

The condition of vanishing of

B and B for all directions of

wave propagation

The condition of

Sk

B = P B = and A = A

for all directions of wave propagation

Nonbirefringence condition for the

skewonless case

Wave propagation and the dispersion

relation in dilaton field and axion field

No amplification/no attenuation and

no polarization rotation constraints

on cosmic dilaton field and cosmic axion field

Spacetime constitute relation including

skewons

Constitute tensor from asymmetric metric

and Fresnel equation

Empirical foundation of the closure relation

for skewonless case

From Galileo Equalence Principle to Einstein

Equalence Principle

EEP and Unersal Metrology

Gyrogratational Ratio

An Update of Search for Long Range/Intermediate

Range Spin–Spin, Spin–Monopole and

Spin–Cosmos Interactions

Prospects

Cosmic polarization rotation: An astrophysical test

of fundamental physics

Sperello dSerego Alighieri

IJMPD

Introduction

Impact of CPR on Fundamental Physics

Constraints from the Radio Polarization of RGs

Constraints from the UPolarization of RGs

Constraints from the Polarization of the

CMB Radiation

Other Constraints

Discussion

Outlook

Clock comparison based on laser ranging technologies

Étienne Samain

IJMPD

Introduction

Scientific Objectes

Time and frequency metrology

Fundamental physics

Solar System science

Solar System nagation based on clock

comparison

Time Transfer by Laser Link: TL on Jason-

Principle

Laser station ground segment

Space instrument

Time equation

Error budget

Link budget

Exploitation

One-Way Lunar Laser Link on LRO Spacecraft

Prospecte

Conclusion and Outlook

Solar-system tests of relatistic graty

Wei-Tou Ni

IJMPD

Introduction and Summary

Post-Newtonian Approximation, PPN Framework,

Shapiro Time Delay and Light Deflection

Post-Newtonian approximation

PPN framework

Shapiro time delay

Light deflection

Solar System Ephemerides

Solar System Tests

Outlook — On Going and Next-Generation Tests

Pulsars and graty

R N Manchester

IJMPD

Introduction

Pulsar timing

Tests of Relatistic Graty

Tests of general relatity with

double-neutron-star systems

The Hulse–Taylor binary, PSR

B +

PSR B +

The double pulsar, PSR

J − A/B

Measured post-Keplerian parameters

Tests of equalence principles and

alternate theories of gratation

Limits on PPN parameters

Part Gratational Dipolar gratational waves and the

constancy of G

General scalar–tensor and

scalar–vector–tensor theories

Future prospects

The Quest for Gratational-Wave Detection

Pulsar timing arrays

Nanohertz gratational-wave sources

Masse black-hole binary systems

Cosmic strings and the early unerse

Transient or burst GW sources

Pulsar timing arrays and current results

Existing PTAs

Limits on the nanohertz GW

background

Limits on GW emission from

indidual black-hole binary systems

Future prospects

Summary and Conclusion

Waves

Gratational waves: Classification, methods

of detection, sensitities, and sources

KazuakKuroda, Wei-Tou Nand Wei-Ping Pan

IJMPD

Introduction and Classification

GWs in GR

Methods of GW Detection, and Their Sensitities

Sensitities

Very high frequency band

kHz– THz and ultrahigh

frequency band above THz

High frequency band Hz– kHz

Doppler tracking of spacecraft μHz– mHz

in the low-frequency band

Space interferometers low-frequency band,

nHz– mHz; middle-frequency band,

mHz– Hz

Very-low-frequency band pHz– nHz

Ultra-low-frequency band fHz– pHz

xExtremely-low Hubble-frequency band

aHz– fHz

Sources of GWs

GWs from compact binaries

GWs from supernovae

GWs from masse black holes and their

coevolution with galaxies

GWs from extreme mass ratio inspirals EMRIs

Primordial/inflationary/relic GWs

Very-high-frequency and ultra-high-frequency

GW sources

Other possible sources

Discussion and Outlook

Ground-based gratational-wave detectors

KazuakKuroda

IJMPD

Introduction to Ground-Based Gratational-Wave

Detectors

Gratational-wave sources

Achieved sensitities of large projects

Coalescences of binary neutron stars

Coalescences of binary black holes

Supernova explosion

Quasi-normal mode oscillation at the

birth of black hole

Unstable fast rotating neutron star

Acceleration due to a gratational wave

Response of a resonant antenna

Response of a resonant antenna

Directity

Positioning

Comparison of a resonant antenna and

an interferometer

Resonant Antennae

Development of resonant antennae

Dynamical model of a resonant antenna with

two modes

Signal-to-noise ratio and noise temperature

Comparison of fe resonant antennae

Interferometers

First stage against technical noises

in prototype interferometers

m-Garching interferometer

m-Garching interferometer

Glasgow m-Fabry–Perot Michelson

interferometer

Caltech m-Fabry–Perot Michelson

interferometer

ISAS m and m delay-line

interferometer

Further RD efforts in the first-generation

detectors

Power recycling

Signal recycling and resonant

side-band extraction

Fighting with thermal noise of the second stage

Mirror and suspension thermal noise

Thermal noise of optical coating

Fighting against quantum noises and squeezing

Radiation pressure noise

Squeezing

Large Scale Projects

LIGO project

rgo project

GEO project

TAMA/CLIO/LCGTKAGRA project

TAMA

CLIO

LCGT KAGRA

Einstein telescope

Summary

Append A Thermal Noise

A Nyquist theorem

A Thermal noise of a harmonic oscillator

Append B Modulation

Append C Fabry–Perot Interferometer

xxi

xx

C Fabry–Perot caty

C Frequency response of a Fabry–Perot Michelson

interferometer

Append D Newtonian Noise

Gratational wave detection in space

Wei-Tou Ni

IJMPD

Introduction

Graty and Orbit Observations/Experiments

in the Solar System

Doppler Tracking of Spacecraft

Interferometric Space Missions

Frequency Sensitity Spectrum

Scientific Goals

Masse black holes and their co-evolution

with galaxies

Extreme mass ratio inspirals

Testing relatistic graty

Dark energy and cosmology

Compact binaries

Relic GWs

Basic Orbit Configuration, Angular Resolution

and Multi-Formation Configurations

Basic LISA-like orbit configuration

Basic ASTROD orbit configuration

Angular resolution

S/twelve spacecraft formation

Orbit Design and Orbit Optimization Using

Ephemerides

CGC ephemeris

Numerical orbit design and orbit

optimization for eLISA/NGO

Orbit optimization for ASTROD-GW

CGC ephemeris

Initial choice of spacecraft initial

conditions

Method of optimization

Deployment of Formation in Earthlike Solar Orbit

Time Delay Interferometry

Payload Concept

Outlook

Subject IndeAuthor IndeVolume

Foreword

Color plates

Part Cosmology

General Relatity and Cosmology

Martin Bucher and Wei-Tou Ni

IJMPD

Cosmic Structure

Marc Das

IJMPD

History of Cosmic Discovery

Measurement of the Galaxy Correlation Function

Before

After

Remarkable large-scale structure in simulations

Measurement of the BAO effect

Further measurements of the power spectrum

Lyman-α clouds

Large Scale Flows

Dwarf Galaxies as a Probe of Dark Matter

Gratational Lensing

Double images

Bullet cluster

Substructure of gratational lenses

Conclusion

Physics of the cosmic microwave background anisotropy

Martin Bucher

IJMPD

Obserng the Microwave Sky: A Short History

and Observational Overew

Brief Thermal History of the Unerse

I

-CP

Cosmological Perturbation Theory: Describing

a Nearly Perfect Unerse Using General Relatity

Characterizing the Primordial Power Spectrum

Recombination, Blackbody Spectrum, and

Spectral Distortions

Sachs–Wolfe Formula and More Exact Anisotropy

Calculations

What Can We Learn From the CMB Temperature

and Polarization Anisotropies?

Character of primordial perturbations:

Adiabatic growing mode versus field ordering

Boltzmann hierarchy evolution

Angular diameter distance

Integrated Sachs–Wolfe effect

Reionization

What we have not mentioned

Gratational Lensing of the CMB

CMB Statistics

Gaussianity, non-Gaussianity, and all that

Non-Gaussian alternates

Bispectral Non-Gaussianity

B Modes: A New Probe of Inflation

Suborbital searches for primordial B modes

Space based searches for primordial B modes

CMB Anomalies

Sunyaev–Zeldoch Effects

Experimental Aspects of CMB Observations

Intrinsic photon counting noise: Ideal

detector behaor

CMB detector technology

Special techniques for polarization

CMB Statistics Resited: Dealing with Realistic

Observations

Galactic Synchrotron Emission

Free–Free Emission

Thermal Dust Emission

Dust Polarization and Grain Alignment

Why do dust grains spin?

About which axis do dust grains spin?

A stochastic differential equation for Lt

Suprathermal rotation

Dust grain dynamics and the galactic

magnetic field

Origin of a magnetic moment along L

Magnetic precession

Barnett dissipation

Das–Greenstein magnetic dissipation

Alignment along B without

Das–Greenstein dissipation

Radiate torques

Small dust grains and anomalous

microwave emission AME

Compact Sources

Radio galaxies

Infrared galaxies

Other Effects

Patchy reionization

Molecular lines

Zodiacal emission

Extracting the Primordial CMB Anisotropies

Concluding Remarks

SNe Ia as a cosmological probe

Xiangcun Meng, Yan Gao and Zhanwen Han

IJMPD

Introduction

SNe Ia as a Standardizable Distance Candle

Progenitors of SNe Ia

Effect of SN Ia Populations on Their Brightness

SN Ia’s Role in Cosmology

Issues and Prospects

Gratational Lensing in Cosmology

ToshifumFutamase

IJMPD

Introduction and History

Basic Properties for Lens Equation

Deration of the cosmological lens equation

Properties of lens mapping

Caustic and critical curves

Circular lenses

The Einstein radius and radial arcs

Non-circular lenses

Strong Lensing

Methods of solng the lens equation:

LTM and non-LTM

Image magnification

Time delays

Comparison of lens model software

Non-light traces mass software

Light traces mass software

Lens statistics

Weak Lensing

Basic method

Shape measurements

E/B decomposition

Magnification bias

Simulation test

Higher-order weak lensing-flexion

and HOLICs

Cluster mass reconstruction

Density profile

Dark matter subhalos in the coma

cluster

Cosmic shear

How to measure the cosmic density field

Conclusion and Future

Inflationary cosmology: First + years

Katsuhiko Sato and Jun’ichYokoyama

IJMPD

Introduction

Developments in Japan

Developments in Russia

Inflation paradigm

Resolution of Fundamental Problems

Realization of Inflation

Three mechanisms

Inflation scenario

Slow-Roll Inflation Models

Large-field models

Small-field model

Hybrid inflation

Reheating

Generation of Quantum Fluctuations that

Eventually Behave Classically

Cosmological Perturbation

Generation of Curvature Fluctuations in

Inflationary Cosmology

Tensor Perturbation

The Most General Single-Field Inflation

Homogeneous background equations

Kinetically dren G-inflation

Potential-dren slow-roll G-inflation

Power Spectrum of Perturbations in Generalized

G-inflation

Tensor perturbations

Scalar perturbations

Inflationary Cosmology and Observations

Large-field models

Small-field model

Hybrid inflation model

Noncanonical models and multi-field models

Conclusion

Inflation, string theory and cosmic strings

Dad F Chernoff and S-H Henry Tye

IJMPD

Introduction

The Inflationary Unerse

String Theory and Inflation

String theory and flucompactification

Inflation in string theory

Small r Scenarios

Brane inflation

D-D̄-brane inflation

Quantum Inflection point inflation

DBmodel

D-D-brane inflation

Kähler modulinflation

Large r Scenarios

The Kim–Nilles–Peloso mechanism

Natural inflation

N-flation

Helical inflation

Axion monodromy

Discussions

Relics: Low Tension Cosmic Strings

Strings in brane world cosmology

Current bounds on string tension Gμ and

probability of intercommutation p

Scaling, Slowing, Clustering and Evaporating

Large-scale string distribution

Local string distribution

Detection

Detection a Microlensing

WFIRST microlensing rates

Gratational waves

Summary

Graty

Quantum graty: A brief history of ideas

and some outlooks

Steven Carlip, Dah-WeChiou, Wei-Tou Ni

and Richard Woodard

IJMPD

Prelude

Perturbate Quantum Graty

String Theory

Loop Quantum Graty

Black Hole Thermodynamics

Quantum Graty Phenomenology

Perturbate quantum graty comes of age

R P Woodard

IJMPD

Introduction

Why Quantum Gratational Effects from

Primordial Inflation are Observable

The background geometry

Inflationary particle production

Tree Order Power Spectra

The background for single-scalar inflation

Gauge-fed, constrained action

Tree order power spectra

The controversy over adiabatic regularization

Why these are quantum gratational effects

Loop Corrections to the Power Spectra

How to make computations

-Suppression and late-time growth

Nonlinear extensions

The promise of cm radiation

Other Quantum Gratational Effects

Linearized effecte field equations

Propagators and tensor Pfunctions

Results and open problems

Back-Reaction

Conclusions

Black hole thermodynamics

S Carlip

IJMPD

Introduction

Prehistory: Black Hole Mechanics and Wheeler’s

Cup of Tea

Hawking Radiation

Quantum field theory in curved spacetime

Hawking’s calculation

Back-of-the-Envelope Estimates

Entropy

Temperature

The Many Derations of Black Hole

Thermodynamics

Other settings

Unruh radiation

Particle detectors

Tunneling

Hawking radiation from anomalies

Periodic Greens functions

Periodic Gratational partition function

Periodic Pair production of black holes

Periodic Quantum field theory and the

eternal black hole

Periodic Quantized graty and classical

matter

Periodic Other approaches

Thermodynamic Properties of Black Holes

Periodic Black hole evaporation

Periodic Heat capacity

Periodic Phase transitions

Periodic Thermodynamic volume

Periodic Lorentz olation and perpetual

motion machines

Approaches to Black Hole Statistical Mechanics

Periodic “Phenomenology”

Periodic Entanglement entropy

Periodic String theory

Weakly coupled strings and branes

Fuzzballs

The AdS/CFT correspondence

Loop quantum graty

Microcanonical approach

Microcanonical approach

Other ensembles

Induced graty

Logarithmic corrections

The Holographic Conjecture

The Problem of Unersality

State-counting in conformal field theory

Application to black holes

Effecte descriptions

The Information Loss Problem

Nonunitary evolution

No black holes

Remnants and baby unerses

Hawking radiation as a pure state

Conclusion

Append A Classical Black Holes

Loop quantum graty

Dah-WeChiou

IJMPD

Introduction

Motations

Why quantum graty?

Difficulties of quantum graty

Background-independent approach

Connection Theories of General Relatity

Connection dynamics

Canonical Hamiltonian formulation

Remarks on connection theories

Quantum Kinematics

Quantization scheme

Cylindrical functions

Spin networks

S-knots

Operators and Quantum Geometry

Holonomy operator

Area operator

Volume operator

Quantum geometry

Scalar Constraint and Quantum Dynamics

Regulated classical scalar constraint

Quantum scalar constraint

Solutions to the scalar constraint

Quantum dynamics

Inclusion of Matter Fields

Yang–Mills fields

Fermions

Scalar fields

S-knots of geometry and matter

SubjectAuthorIndeIndeLow-Energy Physics

Weave states

Loop states versus Fock states

Holomorphic coherent states

Spin Foam Theory

From s-knots to spin foams

Spin foam formalism

Black Hole Thermodynamics

Statistical ensemble

Bekenstein–Hawking entropy

More on black hole entropy

Loop Quantum Cosmology

Symmetry reduction

Quantum kinematics

Quantum constraint operator

Physical Hilbert space

Quantum dynamics

Other models

Current Directions and Open Issues

The master constraint program

Algebraic quantum graty

Reduced phase space quantization

Off-shell closure of quantum constraints

Loop quantum graty versus spin foam theory

Covariant loop quantum graty

Spin foam cosmology

Quantum reduced loop graty

Cosmological perturbations in the Planck era

Spherically symmetric loop graty

Planck stars and black hole fireworks

Information loss problem

Quantum graty phenomenology

Supersymmetry and other dimensions

Quantum field theory a modern introduction Michio Kaku

Quantum Fields and Renormalization

Why Quantum Field Theory?

Historical Perspecte

Strong Interactions

Weak Interactions

Gratational Interaction

Gauge Revolution

Unification

Action Principle

From First to Second Quantization

Noether’s Theorem

Exercises

Symmetries and Group Theory

Elements of Group Theory

SOB

Representations of SOB and Ul

Representations of SOC and SUB

Representations of S О N

Spinors

Lorentz Group

Representations of the Poincare Group

Master Groups and Supersymmetry

Exercises

Spin- and \ Fields

Quantization Schemes

Klein-Gordon Scalar Field

Charged Scalar Field

Propagator Theory

Dirac Spinor Field

Quantizing the Spinor Field Weyl Neutrinos

Exercises

Quantum Electrodynamics

Maxwell’s Equations

Relatistic Quantum Mechanics

Quantizing the Maxwell Field

Gupta-Bleuler Quantization

C, P, and T Invariance

Parity

Charge Conjugation

Time Reversal

CPT Theorem

Exercises

Feynman Rules and LSZ Reduction

Cross Sections

Propagator Theory and Rutherford Scattering

LSZ Reduction Formulas

Reduction of Dirac Spinors

Time Evolution Operator

Wick’s Theorem

Feynman’s Rules

Exercises

Scattering Processes and the Matr

Compton Effect

Pair Annihilation

Mller Scattering

Bhabha Scattering

Bremsstrahlung

Radiate Corrections

Anomalous Magnetic Moment

Infrared Dergence

Lamb Shift

Dispersion Relations

Exercises

Renormalization of QED

The Renormalization Program

Renormalization Types

Nonrenormalizable Theories

Renormalizable Theories Contents xSuper-renormalizable Theories

Finite Theories

Overew of Renormalization in фл Theory

Overew of Renormalization in QED

Types of Regularization

Ward-TakahashIdentities

Overlapping Dergences

Renormalization of QED

Step One

Step Two

Step Three

Step Four

Exercises

П Gauge Theory and the Standard Model

Path Integrals

Postulates of Quantum Mechanics

Postulate

Postulate I

Deration of the Schrodinger Equation

From First to Second Quantization

Generator of Connected Graphs

Loop Expansion

Integration over Grassmann Variables

Schwinger-Dyson Equations

Exercises

Gauge Theory

Local Symmetry

Faddeev-PopoGauge Fing

Feynman Rules for Gauge Theory

Coulomb Gauge

The GriboAmbiguity

Equalence of the Coulomb and Landau Gauge

Exercises

The Weinberg-Salam Model

Broken Symmetry in Nature

The Higgs Mechanism

Weak Interactions

Weinberg-Salam Model

Lepton Decay Щ Gauge

‘tHooft Gauge

Coleman-Weinberg Mechanism

Exercises

The Standard Model

The Quark Model

QCD

Spin-Statistics Problem

Pair Annihilation

Jets

Absence of Exotics

Pion Decay

Asymptotic Freedom

Confinement

Chiral Symmetry

No Anomalies

Jets

Current Algebra

PC AC and the Adler-Weisberger Relation

CVC

PC AC

Adler-Weisberger Relation

Ming Angle and Decay Processes

Purely Leptonic Decays

Semileptonic Decays

Nonleptonic Decays

GIM Mechanism and Kobayashi-Maskawa Matr

Exercises

Ward Identities, BRST, and Anomalies

Ward-TakahashIdentity

Slavnov-Taylor Identities

BRST Quantization

Anomalies

Non-Abelian Anomalies

QCD and Pion Decay into Gamma Rays

Fujikawa’s Method

Exercises

BPHZ Renormalization of Gauge Theories

Counterterms in Gauge Theory

Dimensional Regularization of Gauge Theory Contents BPHZ Renormalization

Forests and Skeletons

Does Quantum Field Theory Really Exist?

Exercises

QCD and the Renormalization Group

Deep Inelastic Scattering

Parton Model

Neutrino Sum Rules

Product Expansion at the Light-Cone

Renormalization Group

Asymptotic Freedom

Callan-Symanzik Relation

Minimal Subtraction

Scale olations

Renormalization Group Proof

Step One

Step Two

Step Three

Exercises

Ш Nonperturbate Methods and Unification

Lattice Gauge Theory

The Wilson Lattice

Scalars and Fermions on the Lattice

Confinement

Strong Coupling Approximation

Monte Carlo Simulations

Hamiltonian Formulation

Renormalization Group

Exercises

Solitons, Monopoles, and Instantons

Solitons

Example: ф

Example: Sine-Gordon Equation

Example: Nonlinear C Model

Monopole Solutions

‘tHooft-PolyakoMonopole

WKB, Tunneling, and Instantons

Yang-Mills Instantons

в Vacua and the Strong С P Problem

Exercises xPhase Transitions and Critical Phenomena

Critical Exponents

The Ising Model

XFZ Heisenberg Model

IRF and VerteModels

Yang-Baxter Relation

Mean-Field Approximation

Scaling and the Renormalization Group

Step One

Step Two

Step Three

Step Four

e Expansion

Exercises

Grand Unified Theories

Unification and Running Coupling Constants

SUE

Anomaly Cancellation

Fermion Representation

Spontaneous Breaking of SUE

Hierarchy Problem

SO

Beyond GUT

Technicolor

Preons or Subquarks

Supersymmetry and Superstrings

Exercises

Quantum Graty

Equalence Principle

Generally Covariant Action

erbeins and Spinors in General Relatity

GUTs and Cosmology

Inflation

Cosmological Constant Problem

Kaluza-Klein Theory

Generalization to Yang-Mills Theory

Quantizing Graty

Counterterms in Quantum Graty

Exercises Contents Supersymmetry and Supergraty

Supersymmetry

Supersymmetric Actions

Superspace

Supersymmetric Feynman Rules

Nonrenormalization Theorems

Finite Field Theories

Super Groups

Supergraty

Exercises

Superstrings

Why Strings?

Points versus Strings

Quantizing the String

Gupta-Bleuler Quantization

Light-Cone Gauge

BRST Quantization

Scattering Amplitudes

Superstrings

Types of Strings

Type

ТуреПА

ТуреПВ

Heterotic String

Higher Loops

Phenomenology

Light-Cone String

BRST Action

Exercises

Append

Notes

SUN

Tensor Products

SUC

Lorentz Group

Dirac Matrices

Field Theory

Infrared Dergences to All Orders

Dimensional Regularization

References

Index

The Future of Humanity Terraforming Mars, Interstellar Travel, Immortality, and Our Destiny Beyond Earth Michio Kaku

Also by Michio Kaku

Title Page

Copyright

Dedication

Contents

Acknowledgments

Prologue

Introduction: Toward a Multiplanet Species

Part I: Leaving the Earth

1. Preparing for Liftoff

2. New Golden Age of Space Travel

3. Mining the Heavens

4. Mars or Bust

5. Mars: The Garden Planet

6. Gas Giants, Comets, and Beyond

Part II: Voyage to the Stars

7. Robots in Space

8. Building a Starship

9. Kepler and a Universe of Planets

Part III: Life in the Universe

10. Immortality

11. Transhumanism and Technology

12. Search for Extraterrestrial Life

13. Advanced Civilizations

14. Leaving the Universe

Notes

Suggested Reading

Illustration Credits

About the Author

Diccionario de Topologia Lacaniana, PsiKolibro

Abierto

Abierto básico

Acotado

Adherencia

Aplanamiento

Arcoconexo

Asíntota

Banda de Möbius

Bola n–dimensional

Botella de Klein

Cerrado

Circunferencia

Clausura

Compacidad

Compacto

Conexo

Continuidad

Convergencia

Cortar

Crosscap

Cubrimiento

Curva

Curva cerrada

Esfera

Espacio cociente

Espacio n–dimensional

Espacio topológico

Función continua

Geometría proyectiva

Geometrías no euclidianas

Grafo

Grupo fundamental

Hipérbola

Homeomorfismo

Homotopía

Identificar

Interior

Intervalo

Invariante

Lazo

Lazo reducible

Límite

Nudo

Nudo aplanado

Nudo borromeo

Nudo trivial

Ocho interior

Orientable

Pegar

Plano euclidiano

Plano proyectivo

Poliedro topológico

Polígono topológico

Problema de los cuatro colores

Problema de los puentes de Königsberg

Proyección

Punto

Punto impropio

Recta

Retracción

Subcubrimiento

Subsucesión

Sucesión

Sucesión convergente

Sucesión divergente

Sumergir

Superficie

Superficie cerrada

Superficie de revolución

Teoremas de Punto fijo

Topología

Topología combinatoria

Topología inducida

Topología usual

Toro

Álgebra

Abstract Algebra Groups, Rings and Fields, Advanced Group Theory, Modules and Noetherian Rings, Field Theory [Lecture notes (Yotsanan Meemark)

Forewordi

Contentsiii

Groups

Integers

Groups

Definitions and Examples

Subgroups

Homomorphisms

Group Actions

Quotient Groups and Cyclic Groups

Quotient Groups

Cyclic Groups

The Symmetric Group

Sylow Theorems

Sylow p-subgroups

Applications of Sylow Theorems

Finite Abelian Groups

Rings and Fields

Basic Concepts

Rings

Quaternions

Characteristic

Ring Homomorphisms and Group Rings

Ideals, Quotient Rings and the Field of Fractions

Maximal Ideals and Prime Ideals

Factorizations

Irreducible Elements and Prime Elements

Unique Factorization Domains

iii

Polynomial Rings

Polynomials and Their Roots

Factorizations in Polynomial Rings

Field Extensions

Algebraic and Transcendental Extensions

More on Roots of Polynomials

Advanced Group Theory

Jordan-Hölder Theorem

Solvable Groups

Nilpotent Groups

Linear Groups

Free Groups and Presentations

Modules and Noetherian Rings

Modules

Free Modules and Matrices

Projective and Injective Modules

Modules over a PID

Noetherian Rings

Artinian Rings

Symplectic Geometry

Symplectic Spaces

Symplectic Graphs

Field Theory

Splitting Fields

Algebraic Closure of a Field

Multiple Roots and Separability

Automorphisms of Fields and Galois Theory

Some Consequences of Galois Theory

Finite Fields

iv Cyclotomic Extensions

Normal Bases

Transcendental Extensions

A first course in noncommutative ring theory (TYLam, Tsit-Yuen Lam, 林節玄)

Forewordi

Contentsiii

Groups

Integers

Groups

Definitions and Examples

Subgroups

Homomorphisms

Group Actions

Quotient Groups and Cyclic Groups

Quotient Groups

Cyclic Groups

The Symmetric Group

Sylow Theorems

Sylow p-subgroups

Applications of Sylow Theorems

Finite Abelian Groups

Rings and Fields

Basic Concepts

Rings

Quaternions

Characteristic

Ring Homomorphisms and Group Rings

Ideals, Quotient Rings and the Field of Fractions

Maximal Ideals and Prime Ideals

Factorizations

Irreducible Elements and Prime Elements

Unique Factorization Domains

iii

Polynomial Rings

Polynomials and Their Roots

Factorizations in Polynomial Rings

Field Extensions

Algebraic and Transcendental Extensions

More on Roots of Polynomials

Advanced Group Theory

Jordan-Hölder Theorem

Solvable Groups

Nilpotent Groups

Linear Groups

Free Groups and Presentations

Modules and Noetherian Rings

Modules

Free Modules and Matrices

Projective and Injective Modules

Modules over a PID

Noetherian Rings

Artinian Rings

Symplectic Geometry

Symplectic Spaces

Symplectic Graphs

Field Theory

Splitting Fields

Algebraic Closure of a Field

Multiple Roots and Separability

Automorphisms of Fields and Galois Theory

Some Consequences of Galois Theory

Finite Fields

iv Cyclotomic Extensions

Normal Bases

Transcendental Extensions

An Introduction to Group Rings (César Polcino Milies, Sudarshan KSehgal)

Preface ix

I Groups

Basic Concepts

Homomorphisms and Factor Groups

Abelion Groups

Group Actions, p-groups and Sylow Subgroups

Solvable and Nilpotent Groups

FC Groups

Free Groups and Free Products

ttamiltonian Groups

The Hirsch Number

Rings, Modules and Algebras

Rings and Ideals

Modules and Algebras

Free Modales and Direct Sums

Finitehess Conditions

Semisimplicity

The Wedderburn-Artin Theorem

The Jacobson Radical

Rings of Algebraic Integers

Orders

Tensor Products

Group Rings

A Brief History

Basic Facts

Augmentation Ideals

vvi CONTBNTS

Semisimplicity

Abelian Group Algebras

Some Commutative Subalgebras

A Glance at Group Representations

Definition and Examples

Representations and Modules

Group Characters

Basic Facts

Characters and Isomorphism Questions

Ideals in Group Rings

Ring Theoretic Formulas

Nilpotent Ideals

Nilpotent Augmentation Meals

Semiprime Group Rings

Prime Group Rings

Chain Conditions in KG

Algebraic Elements

Introduction

Idempotent Elements

Torsion Units

Nilpotent Elements

Units of Group Rings

Introduction

Trivial Units

Finite Groups

Units of ZS

Infinite Groups

Finite Generation of L/(ZG)

Central Units

The

Isomorphism Problem

Introduction

The Normal Subgroup Correspondence

Mctabelian Groups

Circle Groups

Further ResultsCONTENTS vii

The Modular Isomorphism Problem

Free Groups of Units

Free Groups

Free Groups of Units

Explicit Free Groups

Explicit Free Groups in H

Properties of the Unit Group

Integral Group Rings

Group Algebras

Bibliography

Index

Groups, Rings and Group Rings (Antonio Giambruno, Cesar Polcino Milies etc)

On fine gradings on central simple algebras

Eli Aljadeff, Darrell Haile, and MichaelNatapov

On observable module categories

Nicolas Andruskiewitsch and Walter RFerrer Santos

Group gradings on integral group rings

Yuri ABahturin and Michael MParmenter

Profinite graphs — comparing notions

Gunther Bergauer and Wolfgang Herfort

Lie identities in symmetric elements in group rings: A survey

Osnel Broche Cristo and Manuel Ruiz Mann

Irreducible morphisms in subcategories

Gladys Chalom and Hector Merklen

Bol loops with a unique nonidentity commutator/associator

Orin Ghein and Edgar GGoodaire

Weil representations of symplectic groups

Gerald Cliff and David McNeilly

Gradings and graded identities for the upper triangular matrices

over an infinite field

Onofrio MDi Vincenzo, Plamen Koshlukov, and Angela Valenti

Structure of some classes of repeated-root constacyclic codes

over integers modulo m

Hai QDinh

Units in noncommutative orders

Ann Dooms and Eric Jespers

Idempotents in group algebras and coding theory

Raul AFerraz, Valeria OLuchetta, and Cesar Polcino Milies

Finitely generated constants of free algebras

Vitor OFerreira and Lucia SIMurakami

Partial actions of groups on semiprime rings

Miguel Ferrero

Representations of affine Lie superalgebras

Vyacheslav Futorny

On algebras and superalgebras with linear codimension growth

Antonio Giambruno, Daniela La Mattina, and Paola Misso

On spectra of group rings of finite abelian groups

Andre Gimenez Bueno and Michael Dokuchaev

© by Taylor & Francis Group, LLCWedderburn decomposition of small rational group algebras

Carmen Rosa Giraldo Vergara

Some questions on skewfields

Jairo ZGongalves

On the role of rings and modules in algebraic coding theory

Marcus Greferath and Sergio RLopez-Per mouth

Semiperfect rings with T-nilpotent prime radical

Nadiya MGubareni and Vladimir VKirichenko

The structure of the baric algebras

Henrique Guzzo, Jr

On torsion units of integral group rings of groups of small

order

Christian Hofert and Wolfgang Kimmerle

On a conjecture of Zassenhaus for metacyclic groups

Stanley OJuriaans and Sudarshan KSehgal

Nilpotent blocks revisited

Burkhard Kiilshammer

fiDecomposition of central units of integral group rings

Yuanlin Li and Michael MParmenter

Generic units in Z(

Zbigniew Marciniak and Sudarshan KSehgal

On quasi-Frobenius semigroup algebras

Boris VNovikov

Twisted loop algebras and Galois cohomology

Arturo Pianzola

Presentation of the group of units of ZZ?^

Antonio Pita and Angel del Rio

Engel theorem for Jordan superalgebras

Ivan Shestakov and Konstantin Okunev

A characterization of centre-by-finite subgroups of division

algebras

Mazi Shirvani

Isomorphic rational group algebras

Ana Cristina Vieira and Guilherme Leal

GTM Algebra (Graduate Texts in Mathematics) (Thomas WHungerford)

Preface

Acknowledgments

Suggestions on the Use of This Book

Introduction: Prerequisites and Preliminaries

Logic

Sets and Classes

Functions

Relations and Partitions

Products

The Integers

The Axiom of Choice, Order and Zorn’s Lemma

Cardinal Numbers

Chapter : Groups

Semigroups, Monoids and Groups

Homomorphisms and Subgroups

Cyclic Groups

Cosets and Counting

Normality, Quotient Groups, and Homomorphisms

Symmetric, Alternating, and Dihedral Groups

Categories: Products, Coproducts, and Free Objects

Direct Products and Direct Sums

Free Groups, Free Products, Generators & Relations

Chapter II: The Structure of Groups

Free Abelian Groups

Finitely Generated Abelian Groups

The Krull-Schmidt Theorem

The Action of a Group on a Set

The Sylow Theorems

Classification of Finite Groups

Nilpotent and Solvable Groups

Normal and Subnormal Series

Chapter III: Rings

Rings and Homomorphisms

Ideals

Factorization in Commutative Rings

Rings of Quotients and Localization

Rings of Polynomials and Formal Power Series

Factorization in Polynomial Rings

Chapter IV: Modules

Modules, Homomorphisms and Exact Sequences

Free Modules and Vector Spaces

Projective and Injective Modules

Hom and Duality

Tensor Products

Modules over a Principal Ideal Domain

Algebras

Chapter V: Fields and Galois Theory

Field Extensions

Appendix: Ruler and Compass Constructions

The Fundamental Theorem

Appendix: Symmetric Rational Functions

Splitting Fields, Algebraic Closure and Normality

Appendix: The Fundamental Theorem of Algebra

The Galois Group of a Polynomial

Finite Fields

Separability

Cyclic Extensions

Cyclotomic Extensions

Radical Extensions

Appendix: The General Equation of Degree n

Chapter VI: The Structure of Fields

Transcendence Bases

Linear Disjointness and Separability

Chapter VII: Linear Algebra

Matrices and Maps

Rank and Equivalence

Appendix: Abelian Groups Defined by

Generators and Relations

Determinants

Decomposition of a Single Linear Transformation and Similarity

The Characteristic Polynomial, Eigenvectors and Eigenvalues

Chapter VIII: Commutative Rings and Modules

Chain Conditions

Prime and Primary Ideals

Primary Decomposition

Noetherian Rings and Modules

Ring Extensions

Dedekind Domains

The Hilbert Nullstellensatz

Chapter IX: The Structure of Rings

Simple and Primitive Rings

The Jacobson Radical

Semisimple Rings

The Prime Radical; Prime and Semiprime Rings

Algebras

Division Algebras

Chapter X: Categories

Functors and Natural Transformations

Adjoint Functors

Morphisms

List of Symbols

Bibliography

Index

Lectures on Rings and Modules (Karl HHofmann)

PMCOHN

Skew Fields of Fractions, and the Prime Spectrum of a General Ring

The Category of R-Fields

The Universal I!-Inverting Ring

The Z-Rational Closure of a Mapping

The Description of R-Fields by Localizing Sets

Sufficient Conditions for the Existence of Universal Fields

of Fractions

Firs: A Class of Rings with a Universal Field of Fractions

Ideal Sets

The Prime Spectrum of a Ring

Appendix: All Primes are Good

VLASTIMIL DLAB and CLAUS MICHAEL RINGEL

Balanced Rings

IPreliminaries

Notation and Terminology

Generators and Cogenerators

• Uniserial Rings are Balanced

IILocal Rings

A Necessary Length Condition

Rings with W =

Exceptional Rings

Exceptional Rings are Balanced

Structure of Local Balanced Rings

IIIGeneral Theorems

Morita Equivalence

Left Balanced Rings are Left Artinian

The Structure of Balanced Rings

Rings Finitely Generated over their Centres

IIIThe Module Category of a Balanced Ring

Centralizers of Indecomposable Modules

Existence of Exceptional Rings VIII

CARL FAITH

Modules Finite over Endomorphism Ring

Abstract

Acknowledgements

Introduction

Annihilators

Modules Finite over Endomorphism Ring

Regular Rings

QI Rings

V-Rings

PF-Rings

LFUCHS

The Cancellation Property for Modules

Quasi-Injective Modules

Projective Modules

Further Necessary Conditions

Equipollence of Pullbacks

The Lifting Property

Consequences of the Necessary Conditions

Modules whose Endomorphisms are Monic

Torsion-Free Modules of Rank

Final Comments

ALFRED ffGOLDIE

The Structure of Noetherian Rings

Semi-Prime Rings

Quotient Rings

Semi-Prime Rings

The Quotient Problem

Non-Commutative Local Rings

The Structure of Orders

Dedekind Prime Rings IX

KWANGIL KOH

Quasisimple Modules and other Topics in Ring Theory

GMICHLER

Blocks and Centers of Group Algebras

Introduction

Notation

Block Ideals

Linear Characters

Defect Groups

First Main Theorem on Blocks

Osima’s Theorem

Blocks and Normal Subgroups

Blocks with Normal Defect Groups

Brauer’s Main Theorem on Blocks with Normal Defect Groups

Conjugacy Classes and Blocks

Conjugacy Classes and Simple Modules

Probenius Reciprocity Theorem and Clifford’s Theorem

The Blocks of a p-Nilpotent Group

Group Algebras with Central Radicals

RSPIERCE

Closure Spaces with Applications to Ring Theory

Introduction

Closure Operators

Closure Spaces

Additive Morphisms ,

Bounded Morphisms

Inductive Families

Applications to Modules

An Application to Projective Dimension

HANS HSTRRER

On Goldman’s Primary Decomposition

Rational Extensions of Modules

Atoms X

The Primary Decomposition

A Characterization of Certain Artinian Rings

Goldman’s Theory

The Tertiary Decomposition

Compressible and Quasi-Simple Modules

Método de regularização

EMENTA: Introdução: exemplos clássicos e modelagem; Definição de Método de regularização; Métodos de regularização contínuos; Regularização de Tikhonov: operadores lineares e não lineares.

OBJETIVO: Introduzir o analisante/analistas à teoria de regularização de problemas Inversos e a técnicas de obtenção de soluções estáveis para os mesmos.

PROGRAMA DETALHADO:

Unidade 1: Problemas inversos e sua modelagem

• Exemplos clássicos
• Equações integrais de 1a espécie

Referencia: [1] §1.1 a §1.7 [5] $1.1 [7] $1.1 a §1.2

Unidade 2: Equações de Operadores mal postas

• Inversa Generalizada
• Operadores compactos e svd
• Teoria espectral e calculo funcional

Referencia: [1] §2.1 a §2.3 [5] $1.2 a §1.3

Unidade 3: Regularização de operadores

• Definições e conceitos básicos
• Ordem ótima
• Regularização por projeção

Referencia: [1] §3.1 a §3.3 [5] $2.1 a §2.4

Unidade 4: Métodos de regularização contínuos

• Escolha de parâmetros a-priori
• Saturação e Principio da discrepância
• Escolha de parâmetros heurística
• Métodos tipo mollifier

Referencia: [1] §4.1 a §4.6

4: Regularização de Tikhonov

• Teoria clássica
• Regularização por projeção
• Método da máxima entropia
• Restrições convexas

Referencia: [1] §5.1 a §4.4

5: Regularização de problemas não-lineares

• Tikhonov não linear, analise de convergência
• Escolha de parâmetros a-posteriori
• Escalas de Hilbert

Transposição da teoria de regularização matemática para a clínica psicanalítica. A ideia é que o sujeito em análise é um “problema inverso mal posto”: não sabemos diretamente a origem do sintoma (efeito observado), mas buscamos reconstruir o que no inconsciente o produziu. Assim, o método de regularização fornece metáforas e critérios para manter a análise estável.

Vou organizar o material em ementa, objetivos e programa clínico-psicanalítico — paralelamente ao modelo matemático.

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📘 Análise Clínica como Teoria da Regularização

EMENTA

Introdução: exemplos clássicos da clínica (Freud: fobia do pequeno Hans, homem dos ratos, homem dos lobos) como “problemas inversos”. Definição de método clínico de regularização: como o analista sustenta o setting e evita explosão caótica da transferência. Métodos contínuos de estabilização: intervalos, cortes, remarcações, esquecimento, interpretação mínima. Regularização de Tikhonov como paradigma da clínica: manejo de operadores lineares (fala literal, sintoma manifesto) e não-lineares (lapsos, atos falhos, formações do inconsciente).

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OBJETIVO

Introduzir o analisante/analista à teoria de regularização do inconsciente:

• Manejo do excesso de gozo (mal-posed problem).
• Técnicas de obtenção de soluções estáveis para os impasses clínicos (interpretação, corte, silêncio, escrita).
• Critérios de escolha de parâmetros (quantidade de fala, frequência de sessões, intensidade da intervenção).

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PROGRAMA DETALHADO

Unidade 1: Problemas Inversos e sua Modelagem Clínica

• Exemplos clássicos da clínica freudiana como problemas inversos (efeito = sintoma, causa = inconsciente).
• Sintoma = equação integral de 1ª espécie: não aparece diretamente, precisa ser reconstruído. 📖 Freud, “O Homem dos Ratos”, “O Pequeno Hans”.

Unidade 2: Operadores Mal Postos

• Inversa generalizada = o sujeito tenta dar sentido imediato ao sintoma → fracassa.
• Operadores compactos: traumas, pontos de condensação significante.
• Teoria espectral: leitura das formações inconscientes como decomposição em modos (S1, S2, objeto a). 📖 Lacan, Seminário XI (conceito de repetição e trauma).

Unidade 3: Regularização de Operadores

• Setting como operador de regularização.
• Ordem ótima: quanto interpretar, quanto silenciar.
• Regularização por projeção: identificar “núcleo fantasmático” e projetar no discurso do sujeito. 📖 Freud, “Recordar, repetir e elaborar”.

Unidade 4: Métodos Contínuos de Regularização

• Escolha de parâmetros a-priori: regra da frequência (ex. 3 sessões/semana).
• Saturação e princípio da discrepância: ponto em que a fala gira em torno do mesmo e pede corte.
• Escolha de parâmetros heurísticos: quando o analista intervém sem cálculo exato (ato interpretativo).
• Métodos tipo mollifier: pequenas intervenções amortecedoras que suavizam o excesso. 📖 Lacan, Seminário XX (sutileza interpretativa e furo).

Unidade 5: Regularização de Tikhonov (Clínica da Suposição)

• Teoria clássica: aposta na suposição de saber.
• Regularização por projeção: reescrever o sintoma no campo da metáfora paterna.
• Método da máxima entropia: deixar o discurso proliferar até mostrar a falta estrutural.
• Restrições convexas: limites do setting (tempo, dinheiro, ausência de invasão sexual/física). 📖 Lacan, “Subversão do Sujeito e Dialética do Desejo”.

Unidade 6: Regularização de Problemas Não-Lineares

• Sintomas modernos, fora da estrutura clássica (toxicomanias, transexualidades, psicopatologias do excesso).
• Tikhonov não-linear: aposta em uma regularização singular, não universal.
• Escolha de parâmetros a-posteriori: retroativamente o analista entende se interveio bem ou não.
• Escalas de Hilbert: pluralidade de espaços psíquicos, RSI como diferentes métricas de estabilização. 📖 Lacan, Seminário XXIII (“O Sinthoma”).

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Equação-Metáfora Clínica (Tikhonov adaptada)

No problema inverso clássico:

• : sintoma manifesto.
• : operador de linguagem (discurso).
• : causa inconsciente buscada.
• : custo de transformação psíquica (resistência, recalque).
• : parâmetro de regularização = manejo do analista (quanto cortar, quanto interpretar).

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👉 A clínica psicanalítica pode ser pensada como a arte de escolher :

• Muito alto = silêncio absoluto (excesso de estabilidade → inibição).
• Muito baixo = excesso de interpretações (explosão → angústia).
• Ótimo = análise se sustenta, permite deslocar o sintoma e dar consistência ao sujeito.

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