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

Geometries of the Universe -------------------------- Unit 1 Sphere in Spherical Coordinates --------------------------------------- The 1-sphere is just a circle swept out by angle φ dS2 = dr2 + r22 It is conventional to refer to dφ as dΩ1 - the metric for a circle. Thus: dS2 = dr2 + r212 In cartesian coordinates this is equivalent to writing: x2 + y2 = 1 Where, x = cosφ y = sinφ Unit 2 Sphere in Spherical Coordinates --------------------------------------- The 2-sphere can be formed by stacking 1-spheres on top of each other where the circle diameters continuously vary from 0 to 1 and then back to 0. θ ranges 0 -> π dS2 = dθ2 + sin2θdφ2   = dθ2 + sin2θdΩ12   = dΩ22 In cartesian coordinates this is equivalent to writing: x2 + y2 + z2 = 1 Where, x = sinθcosφ y = sinθsinφ z = cosθ Unit 3 Sphere in Spherical Coordinates --------------------------------------- The 3-sphere can be visualized as a 2-sphere that can be inflated/deflated just like a ballon with r ranging from 0 -> π dS2 = dr2 + sin2r[dθ2 + sin2θdφ2]   = dr2 + sin2r[dθ2 + sin2θdΩ12]   = dr2 + sin2rdΩ22   = dΩ32 In cartesian coordinates this is equivalent to writing: x2 + y2 + z2 + w2 = 1 Where, x = sinrsinθcosφ y = sinrsinθsinφ z = sinrcosθ w = cosr FRW Universes ------------- Positive spacial curvature: For a 3 sphere, we can incorporate the expansion of space by modifying the metric as follows: dS2 = a(t)2[dr2 + sin2rdΩ22] dτ2 = dt2 - dS22 = dt2 - a(t)2[dr2 + sin2rdΩ22] Negative spacial curvature (Hyperbolic plane): For the negative curvature case we have to use the ideas of hyperbolic geometry. It turns out that in this regime the metric is given by: dS2 = a(t)2[dr2 + sinh2rdΩ22]   = dH32 Note: sinhr = (er - e-r)/2 r: 0 -> ∞ dτ2 = dt2 - a(t)2H32 Flat spacial curvature: dS2 = a(t)2[dr2 + r222] dτ2 = dt2 - a(t)2[dr2 + r222] We can summarize all of the above cases as follows: dτ2 = dt2 - a(t)2[dr2 + ξ(r)222] ... 1. Where, ξ(r) = r k = 0 ξ(r) = sinr k = +1 ξ(r) = sinhr k = -1 The above FRW metrics fullfill the COSMOLOGICAL PRINCIPLE that the distribution of matter in the universe is homogeneous and isotropic when viewed on a large enough scale. All three geometries are classes of Riemannian geometry that are based on three possible states for parallel lines - Never meeting (flat or Euclidean) - Must cross (spherical) - Always divergent (hyperbolic) Alternatively, one can think of triangles where for a flat universe the angles of a triangle sum to 180 degrees, in a closed universe the sum must be greater than 180, in an open universe the sum must be less than 180. Note: These geometries should not in any way be construed as objects embedded in a higher dimension like a sphere in space. Rather, they are they geometries that are intrinsic to space itself. Note, this curvature is similar to spacetime curvature due to stellar masses except that in this case it is the entire mass of the universe that determines the curvature. Measuring Curvature ------------------- One way to visualize positive and negative curvature is via the use of STEREGRAPHIC PROJECTIONS. In flat space angles subtended by objects at a distance, r, with a diameter, d, vary as θ = d/r. In positively curved space, angles subtended by objects vary as d/sinr. In negatively curved space angles subtended by objects vary as d/sinhr. This is equivalent to saying that, compared to flat space, objects in positively curved space appear to be larger while objects in negatively curved space appear to be smaller. Similarly, if the universe is flat it would be expected that that the number of galaxies out to a distance, r, increases proportionally to r2. If the universe has positive curvature it would be expected that the number of galaxies increases more slowly than r2 (N ∝ sin2r). If the universe has negative curvature it would be expected that the number of galaxies increases faster than N ∝ r2 (N ∝ sinh2r). The curvature of the universe is revealed by whether the number of galaxies per volume increases more slowly or more quickly than the flat space case. The number of galaxies at different distances (or redshifts) has been as has the size of prominent ripples in the CMB. In both cases the indications are that space seems to be flat. The Critical Density -------------------- The general case of the FRW equation can be written as: . (a/a)2 + k/a2 = 8πGρR/3 + 8πGρM/3 + Λ/3 . (a/a)2 + k/a2 = 8πGρ/3 Define the critical density, ρc, such that Ω = ρ/ρc The critical density corresponds to the case where k = 0 and so: ρc = 3H2/8πG This is calculated today to be 9.47 x 10-27kg/m3 A value of Ω < 1 corresponds to a universe that expands forever (open with k < 0). A value of Ω > 1 corresponds to a universe that will eventually stop expanding and collapse (closed with k > 0). Setting Ω = 1 (i.e. k = 0) we get: 1 = ΩR + ΩM + ΩΛ Where R stands for radiation and M stands for mass. Determination of the Ω's is an ongoing process in cosmology. Direct observations indicate that the radiation component can be ignored. To get values for the remaining terms, physicists construct complex mathematical models to describe experimental results. One such model is the Lambda-CDM* model. Basically, the model describes the the existence and structure of the CMB, the accelerating expansion of the universe observed in the redshifted light from distant galaxies and supernovae, the abundances of the light elements hydrogen, helium and lithium and the large scale distribution of galaxies. It is based on the FRW metric and the FRW equations and their solutions. The inputs consists of a number of different parameters that include the baryonic density, the dark matter density and the dark energy density. An initial 'guess' is made for the values of each of these parameters, and the model output is compared against physical observations. The process is iterative as parameters are adjusted to improve the fit to the observed data. Values of the Ω's at the present time are indicated to be: * Cold Dark Matter ΩR = 0 ΩM ~ 0.3 ΩΛ ~ 0.7 k ~ 0 General Relativity and the FRW Equation --------------------------------------- The FRW equations can be derived from Einstein's field equations using the metric shown in equation 1. From these metrics it is possible to calculate the Einstein tensor and set it equal to the pressure and energy of the matter in the universe. The Einstein field equations are: Rμν - (1/2)gμνR + gμνΛ= 4πGTμν From the 00 component we get: R00 - (1/2)g00R + gμνΛ = 4πGρ ρ = T00 This leads to (assuming c = 1): . (a/a)2 + k/a2 - Λ/3 = 8πGρ/3 and, using this equation and the trace of Einstein's field equations, we get: .. (a/a) - Λ/3 = -4πG(ρ + 3p)/3 The second equation states that both the energy density and the . pressure, p, cause the expansion rate of the universe, a, to decrease. This is a consequence of gravitation, with pressure playing a similar role to that of mass density. The cosmological constant, on the other hand, causes an acceleration in the expansion of the universe.