Redshift Academy

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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: January 26, 2018

Expansion of the Universe -------------------------- Mass Dominated Period ------------------------- . . . . . M . . . . +x = 0 .m . . . . . . . . <-- D --> Consider the case of a mass, m sitting on the surface of a sphere with mass, M, that is being ejected from M at velocity, v, we can use the conservation of energy and Newton's law that states: 1. A spherically symmetric body affects external objects gravitationally as though all of its mass were concentrated at a point at its centre. 2. If the body is a spherically symmetric shell (i.e., a hollow ball), no net gravitational force is exerted by the shell on any object inside, regardless of the object's location within the shell. We can write: mv2/2 - GMm/D = constant where mv2/2 is the KE of m and GMm/D is the gravitational PE due to M. Note when the RHS is equal to 0 we get the familiar formula for the escape velocity. mv2 - 2GMm/D = constant * 2 v2 - 2GM/D = constant * 2/m = constant Now D = ax where a is the scale factor and x is a distance. . . v = D = ax So, . a2x2 - 2GM/ax = constant Rewrite in terms of mass density, ρ . ∴ a2x2 - (2G/ax)(4/3)πρD3 = constant . ∴ a2x2 - (2G/ax)(4/3)πρa3x3 = constant . ∴ a2x2 - 2G(4/3)πρa2x2 = constant To make sense the constant on the RHS must be ∝ x2. Therefore we can divide LHS and RHS by x2 . ∴ a2 - 2G(4/3)πρa2 = constant By convention we write: . ∴ (a/a)2 = (8/3)πρG - k/a2 where k relates to the curvature of the universe. This is the FRIEDMANN-ROBERTSON-WALKER equation. ρ is a function of t (as volume expands density will decrease). We can get around this by replacing with M/a3. Thus, . (a/a)2 = (8/3)πMG/a3 - k/a2 This is the FRW EQUATION for the MASS DOMINATED PERIOD Consider k = 0 (just at the escape velocity). . (a/a)2 = (8/3)πMG/a3 Assume a solution a = βtp where p is a power. . ∴ (a/a)2 = p2/t2 ∴ p2/t2 = 8πMG/3β3t3p ∴ p = 2/3 and a = βt2/3 ∴ p2 = 8πMG/3β3 ∴ 4/9 = 8πMG/3β3 ∴ β3 = 6πMG ∴ a = (6πMG)1/3t2/3 . ∴ H = (a/a) = 2/(3t) Therefore, during the mass dominated period, the universe grew according to a power law of 2/3. Radiation Dominated Period --------------------------- This occurred before the mass dominated period. Universe has 109 photons for every proton. Photons are massless but have energy by virtue of their motion. Slow one down and it disappears. Ep = hν = c/λ (de Broglie) As universe expands λ gets stretched and so the energy (and temperature) diminish by a factor of 1/a. There will now be an extra a in the denominator of the FRW equation. Thus, . (a/a)2 = (8/3)πMG/a4 Solve this equation in the same way as before to get. p = 1/2. So, a = (6πMG)1/3t1/2 . ∴ H = (a/a) = 1/(2t) Therefore, during the radiation dominated period, the universe grew according to a power law of 1/2. Vacuum Dominated Period ----------------------- First, we need to talk about pressure and energy density for radiation dominated period. Consider many photons sloshing backwards and forwards inside a 1D box |---------------------|<- wall <------ L ------> F = dp/dt The change in photon momentum is 2p as it bounces off the wall. The bouncing creates a pressure on the wall. Time between collisions = 2L/c F = 2p/(2L/c) = pc/L = E/L = ρ in 1D. Move to a cube with side a: -->x ------ / /| -----/ |-> P (pressure) | |A| | | / | |/ ----- <- a -> F = E/L Therefore P = E/LA => P = ρ But this only one direction. To get the correct answer for 3D we need to divide by 3, Thus, P = ρR/3 Now consider the above box again. Let's move the face A of the cube by an amount dx. The work done is PAdx or PdV. The decrease in the internal energy of the box is -PdV. Now E inside box is: E = ρV Thus, dE = ρdV + Vdρ = -PdV Vdρ = -(P + ρ)dV Now introduce the parameter, w, as P = ρw Vdρ = -(w + 1)ρdV dρ = -(w + 1)ρdV/V dρ/ρ = -(w + 1)dV/V lnρ = -(w + 1)lnV after integration = ln{1/Vw + 1} + c' c' is the constant of integration ρ = c'/Vw + 1 = c'/a3(w + 1) since V = a3 Rewrite the FRW equation (a/a)2 + k/a2 = (8/3)πρG, as follows: . (a/a)2 + k/a2 = (8/3)πGc'/a3(1 + w) We can now use w to describe any combination of periods. w = 1/3 => (8/3)πGc'/a4 => (8/3)πGρR ... RADIATION DOMINATED w = 0 => (8/3)πGc'/a3 => (8/3)πGρM ... MASS DOMINATED w = -1 => c' => (8/3)πGρo ... VACUUM DOMINATED The quantity 8πGρo is the COSMOLOGICAL CONSTANT, Λ Summary ------- (a). ρm = mc2/V m is constant as V increases. Therefore, ρ decreases. (b). ργ = hf/V = hc/λV λ increases as V increases. Therefore, ρ rapidly decreases. (c). ρV = constant. Today, ρV is approximately 70% of the total energy density. ρm is approximately 30% (including Dark matter) and ρR is negligible. If we consider just the vacuum energy we get: . (a/a) = √(8/3)πGρo ∴ a = c'exp{(√(8πGρo/3)t} Period | Density | Expansion | Pressure | w ----------+----------+---------------------+----------+--------- Radiation | ∝ 1/a4 | ∝ t1/2    | P = ρR/3 | w = 1/3 | Mass | ∝ 1/a3 | ∝ t2/3    | P = 0  | w = 0 | Vacuum | constant | ∝ exp{(√(8πGρV/3)t} | P = -ρV | w = -1 | The Universe grew fairly rapidly during the radiation period, slowed down during the mass period and grows exponentially during the vacuum period. This is illustrated below. Image courtesy of Wikipedia. de SITTER SPACE --------------- A universe where the only source of energy comes from the vacuum (Λ ≠ = 0, ρM = ρR = 0) is referred to as de SITTER SPACE. Thus, . (a/a)2 = H2 = Λ/3 - k/a2 For the k = 0 case we get: . (a/a) = H = √8πGρ0 With the solution: ∴ a = Aexp(√(8πGρ0)t) Alternatively, a = Aexp(√H t) Thus, in the vacuum dominated phase the universe is growing exponentially. Recall that: v = HD = D√8πGρ0 ∴ DH = c/√8πGρ0 Since Λ is a constant, DH is fixed. This distance corresponds to the distance at which we can't see beyond because the universe is expanding at the speed of light. DH is referred to as the COSMIC HORIZON. The implication of this is that as the universe continues to expand, the objects that we see in the night sky today will eventually slip through the cosmic horizon and become invisible. Another way of saying this is that the light emitted at the horizon is effectively redshifted to infinite wavelength. Observers in the distant future will look up at the heavens and see a different universe than we see today. Ultimately, the night sky will become completely black! The radius of the cosmic horizon is approximately 46.6 billion light years. The radius of the surface of last scattering is about 45.7 billion light years. Eventually the SOLS will pass through the cosmic horizon and the CMB will disappear. Note: The age of the universe is estimated to be 13.8 billion years. However, the radius of the observable universe is not this number. In the real universe, spacetime is expanding and the distances obtained by multiplying the speed of light multiplied by this interval has no direct physical significance. It is worth noting that while the universe as a whole is expanding, it can be contracting locally due to gravity. A good example is a black hole.