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Astronomy

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Astronomical Distance Units .
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Celestial Coordinates .
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Celestial Navigation .
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Location of North and South Celestial Poles .

Chemistry

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Avogadro's Number
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Balancing Chemical Equations
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Stochiometry
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The Periodic Table .

Classical Physics

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Archimedes Principle
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Bernoulli Principle
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Blackbody (Cavity) Radiation and Planck's Hypothesis
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Center of Mass Frame
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Comparison Between Gravitation and Electrostatics
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Compton Effect .
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Coriolis Effect
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Cyclotron Resonance
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Dispersion
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Doppler Effect
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Double Slit Experiment
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Elastic and Inelastic Collisions .
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Electric Fields
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Error Analysis
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Fick's Law
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Fluid Pressure
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Gauss's Law of Universal Gravity .
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Gravity - Force and Acceleration
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Hooke's law
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Ideal and Non-Ideal Gas Laws (van der Waal)
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Impulse Force
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Inclined Plane
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Inertia
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Kepler's Laws
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Kinematics
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Kinetic Theory of Gases .
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Kirchoff's Laws
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Laplace's and Poisson's Equations
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Lorentz Force Law
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Maxwell's Equations
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Moments and Torque
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Nuclear Spin
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One Dimensional Wave Equation .
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Pascal's Principle
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Phase and Group Velocity
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Planck Radiation Law .
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Poiseuille's Law
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Radioactive Decay
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Refractive Index
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Rotational Dynamics
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Simple Harmonic Motion
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Specific Heat, Latent Heat and Calorimetry
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Stefan-Boltzmann Law
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The Gas Laws
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The Laws of Thermodynamics
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The Zeeman Effect .
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Wien's Displacement Law
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Young's Modulus

Climate Change

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Keeling Curve .

Cosmology

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Baryogenesis
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Cosmic Background Radiation and Decoupling
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CPT Symmetries
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Dark Matter
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Friedmann-Robertson-Walker Equations
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Geometries of the Universe
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Hubble's Law
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Inflation Theory
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Introduction to Black Holes .
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Olbers' Paradox
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Penrose Diagrams
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Planck Units
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Stephen Hawking's Last Paper .
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Stephen Hawking's PhD Thesis .
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The Big Bang Model

Finance and Accounting

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Amortization
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Annuities
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Brownian Model of Financial Markets
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Capital Structure
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Dividend Discount Formula
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Lecture Notes on International Financial Management
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NPV and IRR
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Periodically and Continuously Compounded Interest
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Repurchase versus Dividend Analysis

Game Theory

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The Truel .

General Relativity

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Accelerated Reference Frames - Rindler Coordinates
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Catalog of Spacetimes .
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Curvature and Parallel Transport
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Dirac Equation in Curved Spacetime
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Einstein's Field Equations
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Geodesics
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Gravitational Time Dilation
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Gravitational Waves
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One-forms
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Quantum Gravity
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Relativistic, Cosmological and Gravitational Redshift
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Ricci Decomposition
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Ricci Flow
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Stress-Energy Tensor
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Stress-Energy-Momentum Tensor
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Tensors
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The Area Metric
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The Equivalence Principal
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The Essential Mathematics of General Relativity
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The Induced Metric
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The Metric Tensor
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Vierbein (Frame) Fields
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World Lines Refresher

Lagrangian and Hamiltonian Mechanics

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Classical Field Theory .
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Euler-Lagrange Equation
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Ex: Newtonian, Lagrangian and Hamiltonian Mechanics
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Hamiltonian Formulation .
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Liouville's Theorem
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Symmetry and Conservation Laws - Noether's Theorem

Macroeconomics

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Lecture Notes on International Economics
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Lecture Notes on Macroeconomics
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Macroeconomic Policy

Mathematics

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Amplitude, Period and Phase
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Arithmetic and Geometric Sequences and Series
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Asymptotes
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Augmented Matrices and Cramer's Rule
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Basic Group Theory
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Basic Representation Theory
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Binomial Theorem (Pascal's Triangle)
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Building Groups From Other Groups
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Completing the Square
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Complex Numbers
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Composite Functions
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Conformal Transformations .
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Conjugate Pair Theorem
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Contravariant and Covariant Components of a Vector
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Derivatives of Inverse Functions
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Double Angle Formulas
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Eigenvectors and Eigenvalues
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Euler Formula for Polyhedrons
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Factoring of a3 +/- b3
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Fourier Series and Transforms .
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Fractals
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Gauss's Divergence Theorem
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Grassmann and Clifford Algebras .
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Heron's Formula
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Index Notation (Tensors and Matrices)
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Inequalities
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Integration By Parts
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Introduction to Conformal Field Theory .
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Inverse of a Function
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Law of Sines and Cosines
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Line Integrals, ∮
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Logarithms and Logarithmic Equations
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Matrices and Determinants
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Matrix Exponential
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Mean Value and Rolle's Theorem
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Modulus Equations
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Orthogonal Curvilinear Coordinates .
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Parabolas, Ellipses and Hyperbolas
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Piecewise Functions
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Polar Coordinates
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Polynomial Division
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Quaternions 1
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Quaternions 2
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Regular Polygons
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Related Rates
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Sets, Groups, Modules, Rings and Vector Spaces
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Similar Matrices and Diagonalization .
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Spherical Trigonometry
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Stirling's Approximation
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Sum and Differences of Squares and Cubes
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Symbolic Logic
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Symmetric Groups
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Tangent and Normal Line
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Taylor and Maclaurin Series .
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The Essential Mathematics of Lie Groups
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The Integers Modulo n Under + and x
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The Limit Definition of the Exponential Function
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Tic-Tac-Toe Factoring
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Trapezoidal Rule
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Unit Vectors
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Vector Calculus
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Volume Integrals

Microeconomics

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Marginal Revenue and Cost

Particle Physics

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Feynman Diagrams and Loops
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Field Dimensions
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Helicity and Chirality
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Klein-Gordon and Dirac Equations
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Regularization and Renormalization
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Scattering - Mandelstam Variables
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Spin 1 Eigenvectors .
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The Vacuum Catastrophe

Probability and Statistics

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Box and Whisker Plots
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Categorical Data - Crosstabs
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Chebyshev's Theorem
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Chi Squared Goodness of Fit
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Conditional Probability
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Confidence Intervals
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Data Types
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Expected Value
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Factor Analysis
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Hypothesis Testing
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Linear Regression
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Monte Carlo Methods
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Non Parametric Tests
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One-Way ANOVA
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Pearson Correlation
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Permutations and Combinations
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Pooled Variance and Standard Error
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Probability Distributions
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Probability Rules
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Sample Size Determination
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Sampling Distributions
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Set Theory - Venn Diagrams
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Stacked and Unstacked Data
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Stem Plots, Histograms and Ogives
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Survey Data - Likert Item and Scale
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Tukey's Test
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Two-Way ANOVA

Programming and Computer Science

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Hashing
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How this site works ...
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More Programming Topics
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MVC Architecture
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Open Systems Interconnection (OSI) Standard - TCP/IP Protocol
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Public Key Encryption

Quantum Computing

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The Qubit .

Quantum Field Theory

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Creation and Annihilation Operators
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Field Operators for Bosons and Fermions
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Lagrangians in Quantum Field Theory
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Path Integral Formulation
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Relativistic Quantum Field Theory

Quantum Mechanics

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Basic Relationships
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Bell's Theorem
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Bohr Atom
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Clebsch-Gordan Coefficients .
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Commutators
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Dyson Series
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Electron Orbital Angular Momentum and Spin
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Entangled States
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Heisenberg Uncertainty Principle
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Ladder Operators .
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Multi Electron Wavefunctions
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Pauli Exclusion Principle
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Pauli Spin Matrices
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Photoelectric Effect
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Position and Momentum States
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Probability Current
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Schrodinger Equation for Hydrogen Atom
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Schrodinger Wave Equation
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Schrodinger Wave Equation (continued)
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Spin 1/2 Eigenvectors
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The Differential Operator
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The Essential Mathematics of Quantum Mechanics
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The Observer Effect
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The Quantum Harmonic Oscillator .
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The Schrodinger, Heisenberg and Dirac Pictures
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The WKB Approximation
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Time Dependent Perturbation Theory
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Time Evolution and Symmetry Operations
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Time Independent Perturbation Theory
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Wavepackets

Semiconductor Reliability

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The Weibull Distribution

Solid State Electronics

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Band Theory of Solids .
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Fermi-Dirac Statistics .
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Intrinsic and Extrinsic Semiconductors
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The MOSFET
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The P-N Junction

Special Relativity

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4-vectors .
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Electromagnetic 4 - Potential
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Energy and Momentum, E = mc2
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Lorentz Invariance
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Lorentz Transform
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Lorentz Transformation of the EM Field
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Newton versus Einstein
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Spinors - Part 1 .
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Spinors - Part 2 .
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The Lorentz Group
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Velocity Addition

Statistical Mechanics

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Black Body Radiation
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Entropy and the Partition Function
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The Harmonic Oscillator
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The Ideal Gas

String Theory

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Bosonic Strings
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Extra Dimensions
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Introduction to String Theory
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Kaluza-Klein Compactification of Closed Strings
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Strings in Curved Spacetime
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Toroidal Compactification

Superconductivity

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BCS Theory
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Introduction to Superconductors
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Superconductivity (Lectures 1 - 10)
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Superconductivity (Lectures 11 - 20)

Supersymmetry (SUSY) and Grand Unified Theory (GUT)

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Chiral Superfields
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Generators of a Supergroup
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Grassmann Numbers
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Introduction to Supersymmetry
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The Gauge Hierarchy Problem

The Standard Model

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Electroweak Unification (Glashow-Weinberg-Salam)
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Gauge Theories (Yang-Mills)
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Gravitational Force and the Planck Scale
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Introduction to the Standard Model
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Isospin, Hypercharge, Weak Isospin and Weak Hypercharge
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Quantum Flavordynamics and Quantum Chromodynamics
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Special Unitary Groups and the Standard Model - Part 1 .
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Special Unitary Groups and the Standard Model - Part 2
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Special Unitary Groups and the Standard Model - Part 3 .
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Standard Model Lagrangian
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The Higgs Mechanism
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The Nature of the Weak Interaction

Topology

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Units, Constants and Useful Formulas

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Constants
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Formulas
Last modified: November 20, 2021 ✓

Introduction to Black Holes --------------------------- The basic definition of a black hole is a mass where the escape velocity is greater than the speed of light. From classical physics we get the escape velocity for earth to be: v = √(2GM/R) where R is the radius of the earth. Replacing v with c we get and rearranging we get: Rs = 2GM/c2 ... A. Very roughly, if the diameter of earth were .08 m it would be a black hole. If we took a 1 kg mass, the force on the surface of this compressed earth calculated using Newton's law would be roughly 1019 N which is a force greater than any other force known in physics. The radius, Rs, at which light can begin to escape is called the SCHWARZCHILD RADIUS and the surface of the sphere formed with that radius is called the EVENT HORIZON. Schwarzchild Metric ------------------- In a similar manner to flat spacetime, we can construct an invariant interval in curved spacetime called the SCHWARZCHILD METRIC that describes the space time associated with a black hole. The metric in spherical coordinates is derived from Einstein's field equations (a complicated process not attempted here!). It is (assuming c = 1): dτ2 = (1 - Rs/R)dt2 - dR2/(1 - Rs/R) - R22 Or, dτ2 = (1 - 2MG/R)dt2 - dR2/(1 - 2MG/R) - R22 Where R is the distance from center of the black hole to a particular point in space. Note when R is very large this equation reduces to the form of the flat space time metric. dτ2 = dt2 - dR2 - R22 The metric represents the relationship between the proper time measured by the person, τ, and the time measured by an observer. There are 2 interesting cases. As R approaches Rs, the observed time, t, becomes longer and longer to maintain the balance of the equation. At Rs = R the metric blows up and to all intents and purposes, the observed time becomes infinite. Therefore, a person accelerating towards a black hole sees their velocity growing at an increasing rate as they approach it. However, due to time dilation, an observer would see the person actually travelling slower and slower and never actually reaching the event horizon. This can be seen in more detail in the following diagram. Consider 2 people - Bob and Alice - travelling in a spaceship. The path the rocket takes in spacetime is a hyperbola. At point A, Bob 'ejects' Alice from the spaceship while it continues on its hyperbolic path. Between A and B, Alice and Bob can see each other with no issue. Once Alice passes through B, however, things change. Alice will continue to be able to see Bob, albeit at an ever increasing distance, but Bob will no longer be able to see Alice. Alice can see Bob because light from Bob is travelling at a 45 degree angle indicated by the dotted lines. However, light leaving Alice at C is not traveling at a 45 degree angle and would have to exceed c to reach Bob at D. What actually happens is that if Bob looks back at Alice as she zooms away, and before she arrives at B, he will perceive her to be moving at a slower and slower rate as she approaches point B. In the limit it appears to take an infinite amount of time for Alice to reach B. The other interesting situation is when R = 0. This is more complex to explain and will not be discussed here. It is sufficient to say that the point R = 0 represents a SINGULARITY where descriptions of space and time are physically not well understood. Entropy ------- Assume we want to add 1 photon (equivalent to a 'bit' of information) to a black hole. For such a photon to interact with a black hole, its wavelength has to be roughly equal to Rs. Thus, E = hf = hc/Rs. From Einstein, we can say that: ΔE = Δmc2 ∴ Δm = ΔE/c2 = hc/Rsc2 = h/Rsc Using equation A., the radius, R, changes with mass as follows: ΔR = 2ΔmG/c2 = 2Gh/c3Rs ∴ ΔRRs = 2Gh/c3 which has the dimensions of AREA! The surface area of the Schwarzchild sphere is: A = 4πRs2 ∴ ΔA = 8πRsΔR = 8π2Gh/c3 after substitution. If we consider that one photon constitutes one unit of entropy then we can say that for ΔS photons: ΔA = 8π2GhΔS/c3 Or ΔS = ΔAc3/16πGh Which can also be written in terms of the PLANCK LENGTH, lP = √(hG/2πc3) = 1.616 x 10-35 m, as: ΔS = ΔA/8lP2 This is the BEKENSTEIN FORMULA for a 'simple' black hole. It basically says that the black hole entropy is proportional to the AREA of its event horizon. Classically, entropy is associated with volumes, so this is very different! The Unruh Effect ---------------- The Unruh effect is a prediction of Quantum Field Theory. It says that the vacuum when being accelerated is not the empty state, but instead is filled with real particles. An observer at rest sees the vacuum as a state with no real particles present (on average there are fluctuations, but they fluctuate around an average of zero particles present). However, a uniformly accelerated observer has his own vacuum which is different to the vacuum of the observer at rest. From his point of view, the accelerating observer sees the vacuum as containing a bath of real particles at a temperature proportional to the acceleration. As a consequence, the accelerating observer will observe blackbody radiation. The Unruh effect also applies to detectors. For example, all things being equal, a thermometer waved around in empty space, will indicate a slightly higher temperature relative to a its stationary value. The Unruh temperature is given by: T = ha/4π2cKB Hawking Radiation ----------------- Gravity and acceleration are related via the equivalence principle. Close to the event horizon of a black hole, virtual particle/antiparticle pairs resulting from vacuum fluctuations are boosted by the intense curvature of spacetime and become real particles. This process is consistent with the Unruh effect. An observer near the event horizon must accelerate to keep from falling in and will see a bath of real particles that pop out of the horizon. In contrast, an observer that free falls through the horizon will see a quantum field state that looks like a vacuum state. Thus, when quantum field theory is combined with curved spacetime, the concept of a particle being present becomes relative, not absolute. Now the bath of particles that the accelerated observer sees contains energy and the only place that this energy can come from is the black hole itself. Normally, these particle/antiparticles annihilate each other after a very short period of time and the total energy is conserved. However, if some of these particles escape before they can be annihilated, their antiparticles will fall back through the event horizon. Since, the escaped particles must have positive energy, the implication is that the particles falling back in must have negative energy if the total energy is to be conserved. This causes the black hole to lose mass. The escaped particles constitute HAWKING RADIATION. Hawking radiation is black body radiation. Therefore, to someone observing from a distance, it would appear that the black hole is radiating, and therefore will have a temperature. This temperature can be found from the laws of thermodynamics. Temperature is defined in terms of entropy and energy as: kBTΔS = ΔE For a single photon ΔS = 1 and ΔE = hc/λ = hc/Rs = hc3/2mG Thus, T = hc3/2mGkB = constant/m A more rigorous proof leads to: T = hg/4π2cKB This, has the same form as the Unruh temperature. It is worth noting that the formulas for entropy and temperature both contain Planck's constant and therefore demonstrate that both are quantum-gravitational effects. Thus, the temperature of a black hole is finite and inversely proportional to its mass. In other words, a black hole has negative specific heat. A black hole of about 5 solar masses would have a temperature of about 12×10-9 K. Since this is much less than the background radiation temperature of about 2.7 K it would absorb more radiation than it emits. The net influx of heat energy adds to the mass of the black hole. As the mass increases, the temperature falls further and the influx of energy becomes greater and greater and so on. For a 'small' black hole it is possible that the rate of growth for the hole is less than the amount of emitted Hawking radiation. This is saying that the temperature of the emitted radiation is greater than the background radiation temperature. Under these circumstances the energy flow is in the opposite direction and the hole 'evaporates' away. This may explain why we never see small black holes in reality. They form but evaporate so quickly that we don't have time to observe them. For example, a black hole with the mass of a car would evaporate away in about 10-9 seconds. Near the horizon there is a lot of 'quantum activity'. There is chaos (entropy) and agitated motion. As a result the temperature near (just outside) the event horizon is extremely high. The black body temperature discussed above is much lower than the horizon temperature because the emitted radiation has been gravitationally red shifted. Bob and Alice Revisited ----------------------- Previously we had said that Alice sailed through the event horizon unscathed. Well, now we a have a conflict. It would appear that Alice is not unscathed but gets thermalized at the event horizon and is radiated out. Which alternative is it? The answer is both. Bob sees Alice thermalized near the horizon and and sees her 'bits' flying outward in the Hawking radiation. Alice in her own frame of reference just falls through the horizon unscathed. Since Alice and Bob can't communicate there is no apparent contradiction. Bob thinks she's dead while Alice can't communicate otherwise. Still, something seems odd about this. How can we reconcile these two scenarios. Let's consider the following. Before Alice crosses the horizon, Bob decides to take a look at her to see if she is being thermalized. In order to do this, however, he has to hit her with short enough wavelength radiation to resolve her position near the horizon. Unfortunately, the energy associated with these waves is enough on its own to thermalize her. In essence it is Bob looking at Alice that thermalizes her!. Therefore, we can now reconcile Bob and Alice's situation by treating as a quantum effect, namely, the act of observing a system disturbs the it! Therefore, it seems there are 2 distinct versions of the same reality. The 3 dimensions that Alice sees inside the black hole and the 2 dimensions associated with the thin surface of hot material which thermalizes her and radiates her back out. So the question becomes, "is there a way of representing the 3 dimensions inside the black hole with the 2 dimensions associated with surface of the event horizon. This is exactly what the HOLOGRAPHIC PRINCIPLE is. Information stored on the surface of the event horizon can be visualized in terms of a volume in the same way that an optical holograms can represent 2D images in 3D. Black Hole Information Paradox ------------------------------ Information comes in bits. Bits are indestructible. They can be ejected but this results in the addition of heat to the environment that causes a corresponding increase in entropy. In fact, for a computer LANDAUER's PRINCIPLE states that: E = kTln(2) Where E is the energy to erase 1 bit and T is the temperature of the circuitry. This is one reason why computers need to be cooled. Information consumed by a black hole would seem to violate the idea of information conservation because once the information is inside the black hole it can never escape. The only way it could escape is via Hawking radiation but this would require that a duplicate copy of the information inside the event horizon exists outside of it. However, having two copies of the information would violate quantum theory. Another issue concerns the 'monogamy of entanglement'. In the Hawking process the ingoing information is necessarily entangled with the outgoing information. However, the outgoing information also needs to be entangled independently with all Hawking radiation emitted in the past. However, entanglement involving both is not allowed. The Holographic principle is an attractive solution to both problems since information just outside the 2D event horizon can be regarded as a representation of infomation in the 3D interior. AdS/CFT Correspondence ---------------------- The Holographic principle got a large boost in 1997 with the formulation of anti-de Sitter/Conformal Field Theory correspondence (AdS/CFT) by Maldacena. Maldacena's original example of AdS/CFT showed the correspondence between a particular type of gravity in 5 dimensionsal anti-de Sitter space and a 4 dimensional quantum field theory. de Sitter space is the vacuum solution of Einstein's field equations with a positive cosmological constant (expanding universe). It represents a universe with zero to slightly positive scalar curvature (curved spherically). anti-de Sitter space is the vacuum solution of Einstein's field equations with a negative cosmological constant (contracting universe). It represents a universe with negative scalar curvature (hyperbolic space). Let's look at this qualitatively in terms of the following model. Hyperbolic space can be viewed as a disk as illustrated below. One can define the distance between points of this disk in such a way that all the triangles and squares are the same size and the circular outer boundary are infinitely far from any point in the interior. Now imagine a stack of hyperbolic disks where each disk represents the state of the universe at a given time. The result is a solid cylinder in which any cross section is a copy of the hyperbolic disk. This construction describes a hypothetical universe with only 2 space and 1 time dimension. However, hyperbolic space can have more than 2 dimensions and one can "stack up" copies of hyperbolic space to get higher-dimensional models of anti-de Sitter space. The key feature of the anti-de Sitter space is the boundary of the cylinder. One property of this boundary is that, locally around any point, it looks just like flat spacetime. AdS/CFT conjectures that this boundary can be regarded as the "spacetime" for a conformal field theory. A conformal field theory (CFT) is a quantum field theory which is invariant under conformal transformations. This means that the physics of the theory looks the same at all length scales. Conformal field theories care about angles, but not about distances. Notice that the boundary of the cylinder has fewer dimensions than anti-de Sitter space itself. For example, in the 3 dimensional example illustrated above, the boundary is a 2 dimensional surface. The upshot of all this is that inside the cylinder we have a gravitational theory in D dimensions and on the boundary we have a QFT in D - 1 dimensions. These theories are conjectured to be exactly equivalent, despite living in different numbers of dimensions. Therefore, one can perform calculations in one theory and translate those calculations into calculations in the other theory. Following Maldacena's discovery, physicists have discovered many different realizations of the AdS/CFT correspondence. The theories involved are generally not viable models of the real world, but they have certain features which make them useful for solving problems in quantum field theory and quantum gravity. Regardless, AdS/CFT correspondence surely has a bearing on the black hole information paradox. Caveats ------- Despite these advances in understanding there are still problems. Crudely speaking, when a black hole evaporates it will eventually reach a point where so information has departed the black hole that not enough remains at the event horizon for holography to represent the interior. In this scenario, an in falling observer would crash into the 'firewall' just aboves the horizon and would be fried to a crisp! This would indicate that the picture of black holes painted by General Relativity is dramatically wrong.