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

Hubble's Law ------------ Hubble compared recession velocities of galaxies by measuring their redshift and comparing this with their distance. The distances were obtained using Cepheid variable stars found in the galaxies. Cepheid variable stars were discovered by Henrietta Leavitt and are objects whose period of variability is related to their average luminosity. By comparing stars at assumed comparable distances, she found that the ones with longer periods had higher luminosities. Subsequent to Leavitt's discovery, Harlow Shapley used parallax methods to measure the distance to one of these cepheid variables. The allowed him to determine the relationship between the period of the variable and its absolute luminosity. Now the distance to any cepheid variable could be found by measuring its period, determining its absolute luminosity, comparing this to its apparent luminosity and applying the inverse square law. Hubble found that there was a positive relationship between the recession velocity, v, and the proper distance, D, and associated this with the expansion of the universe. Thus: v = H0D Where H0 is the HUBBLE CONSTANT. This is HUBBLE'S LAW. Hubble's law tells us that velocity is dependent on distance. Gravitationally interacting galaxies move relative to each other independent of the expansion of the Universe. These relative velocities need to be accounted for separately. Hubble's Law can be derived in terms of the expansion of space in the following manner: O a a a a a o----o----o----o----o----> x x y Point O is fixed. a is the scale factor and is a function of t. It represents the relative expansion of the universe. It is easy to see that as the system is stretched in the x direction, the further away the points are from O, the faster they must be moving. D = an where n is an integer that represents the number of a's. So x->y = 2. D is the distance between the objects. . . D = v = an . = a(a/a)n . = (a/a)D = H0D where H0 is the HUBBLE CONSTANT (3 x 10-18 s-1) In 3D D = a√{(Δx)2 + (Δy)2 + (Δz)2} . . D = (a/a)D as before Since a is function of time, the coordinate mesh expands with time. --------------- / / / / /----/----/----/ / / / / --------------- ^ ------------ | / / / / | time /---/---/---/ | / / / / ------------ a(t) --------- / / / / a(t) /--/--/--/ / / / / --------- H0 corresponds to the LHS of the FRW equations at the time of observation. The current estimate of H0 is about 72 (km/s)/Mpc (1 Mpc ~ 3.26 x 106 light years). So at 3.26 mly the velocity is 72 km/s, at 7.52 mly it is 144 km/s and so on. Cosmological Redshift --------------------- Consider a light wave and an observer moving away from the source with velocity, v. Therefore, λS + vt = ct or, t = λS/(c - v) = c/(c - v)fS = 1/(1 - β)fS where β = v/c The observer will measure this time to be: tO = t/γ where γ = 1/√(1 - β2) Now, fO = 1/tO = γ(1 - β)fS Therefore, fS/fO = √[(1 + β)/(1 - β)] The resulting redshift, z, is: z = (λO - λS)/λS ≡ (fS - fO)/fO = λOS - 1 = fS/fO - 1 = √[(1 + v/c)/(1 - v/c)] - 1 = (1 + v/c + v2/2c2 + ...) - 1 For v<<c we can write: z ~ v/c Therefore, in terms of the Hubble constant we get: z = H0D/c . = (a/a)D/c Now z = (fS - fS)/fS ~ df/f = v/c = Hdt since D = cdt = da/a Equating, and solving we get: a0/aS = z + 1 It is important to note that cosmological redshift is different to Doppler shift. In Doppler Shift, the wavelength of the emitted radiation depends on the motion of the object at the instant the photons are emitted. In cosmological redshift, the wavelength at which the radiation is originally emitted is stretched as it travels through expanding space. Cosmological redshift results from the expansion of space itself and not from the motion of an individual body. Distance Measurement -------------------- We can use the above relationship to estimate the distance to objects. D = cz/H0 The modern discovery of TYPE 1A supernovae has enabled further refinement to these techniques by allowing measurements at much greater distances. Type Ia supernovae occur in a binary system where 2 stars orbit one another. One of the stars is a white dwarf. The other can be a giant star or even a smaller white dwarf. The white dwarf pulls material off its companion star, adding that matter to itself. Eventually, when the white dwarf reaches a certain mass a nuclear reaction occurs, causing the white dwarf to explode and produce a burst of light 5 billion times brighter than the Sun. Because the chain reaction always happens in the same way, and at the same mass, the brightness of these Type Ia supernovae are also always the same. Using this knowledge, the distance to the galaxy containing the supernova can be computed by comparing the observed and absolute intensities and again applying the inverse square law. Comoving and Proper Distance ---------------------------- Comoving and proper distance are two closely related distance measures used by cosmologists to define distances between objects. Proper distance corresponds to where a distant object would be at a specific moment of cosmological time, which can change over time due to the expansion of the universe. Comoving distance factors out the expansion of the universe, giving a distance that does not change in time due to the expansion of space. Comoving distance and proper distance are defined to be equal at the present time. Age of Universe --------------- Hubble time: 1/H0 ~ 14 billion years.