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 Mechanics

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Blackbody Radiation .

Classical Physics

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Archimedes Principle
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Bernoulli Principle
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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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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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Poiseuille's Law
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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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The Gas Laws
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The Laws of Thermodynamics
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The Zeeman Effect .
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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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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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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
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Vacuum Energy .

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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Basis One-forms .
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Catalog of Spacetimes .
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Curvature and Parallel Transport
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Einstein's Field Equations
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Geodesics
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Gravitational Waves
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Hyperbolic Motion and Rindler Coordinates .
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Quantum Gravity
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Ricci Decomposition .
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Ricci Flow .
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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 Dirac Equation in Curved Spacetime .
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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 Light Cone .
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The Metric Tensor .
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The Principle of Least Action in Relativity .
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Vierbein (Frame) Fields

Group Theory

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Basic Group Theory .
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Basic Representation Theory .
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Building Groups From Other Groups .
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Sets, Groups, Modules, Rings and Vector Spaces
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Symmetric Groups .
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The Integers Modulo n Under + and x .

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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Binomial Theorem (Pascal's Triangle)
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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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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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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 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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Volume Integrals

Microeconomics

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Marginal Revenue and Cost

Nuclear Physics

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Radioactive Decay

Particle Physics

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Feynman Diagrams and Loops
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Field Dimensions
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Helicity, Chirality and Weyl Spinors .
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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 .

Probability and Statistics

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Box and Whisker Plots
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Buffon's Needle .
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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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Density Operators and Mixed States .
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Entangled States .
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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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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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Heisenberg Uncertainty Principle
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Ladder Operators .
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Multi Electron Wavefunctions .
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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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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 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 (Faraday) Tensor .
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Energy and Momentum in Special Relativity, E = mc2 .
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Invariance of the Velocity of Light .
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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 Continuity Equation .
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The Lorentz Group .

Statistical Mechanics

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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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Bardeen–Cooper–Schrieffer Theory
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BCS Theory
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Cooper Pairs
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Introduction to Superconductivity .
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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
Last modified: May 3, 2022 ✓

The Light Cone -------------- In special and general relativity, a light cone is the path that a flash of light, emanating from a single event (a single point in space and a single moment in time) and traveling in all directions, would take through spacetime. Light cones are always oriented in the time direction and light rays always travel at 45° angles. Consider the (1,-1,-1,-1) form of the metric. For the (1,-1,-1,-1) form of the metric: Lightlike (LL): c2dt2 - dx2 = 0 v = c Timelike (TL): dτ2 = c2dt2 - dx2 > 0 v < c Spacelike (SL): ds2 = -c2dt2 + dx2 < 0 v > c For the (-1,1,1,1) form of the metric: Lightlike (LL): -c2dt2 + dx2 = 0 v = c Timelike (TL): dτ2 = -c2dt2 + dx2 < 0 v < c Spacelike (SL): ds2 = c2dt2 - dx2 > 0 v > c Therefore, ds2 = -dτ2 or, ds = idτ In the timelike case, time is the bigger factor. The distance light travels is greater than the spatial separation. 2 events can take place at the same position but not at the same time. Therefore, a causal relation between the two events is possible, i.e. one event can cause or influence the other. Timelike events are inside the light cone. In the spacelike case, space is the bigger factor. The spatial separation is greater than the distance light travels. 2 events can take place at the same time but not at the same position. Therefore, any causal relation between the two events is impossible, i.e. one cannot affect the other. Spacelike events are outside the light cone. Simply put, they show how far something is apart compared to ct. In the timelike case, if you are fast enough, you can be at event A and at event B, it is only a matter of time until you see the second event. In the spacelike case, the events C and D are too far apart in space. You cannot see both of them together, no matter how fast you travel. As soon as event C happened and you go as fast as possible, event D will have happened before you arrive there. Rindler Wedge ------------- In region I the worldlines are timelike. At first sight this would seem to be at odds with the definition of the light cone where region I is spacelike and region II is timelike. However, this is misleading. The light cone requires that the worldlines pass through the origin in the Minkowski frame (X = 0,T = 0), which they clearly do not. If an observer did pass through the origin, then indeed region I would be spacelike and not reachable. What is really happening is that every point (event) along the hyperbolae will have their own past and future light cones. All events from the perspective of the observer at a particular point are then either outside (spacelike), inside (timelike), or on (light like) the associated light cone. Since observers traveling along the hyperbolae can never exceed the speed of light, these worldlines stay within the light cone and are, therefore, always timelike. In region II, space and time trade places. The spatial coordinate in region I becomes timelike, and the time coordinate becomes spacelike, i.e. Schwarzchild metric: dτ2 = (1 - rS/r)dt2 - (1/(1 - rS/r))dr2 - (1/c2)r2Ω22 For r < rS the metric becomes: ds2 = -(1 - rS/r)dt2 + (1/(1 - rS/r))dr2 + (1/c2)r2Ω22 Rindler metric: dτ2 = (Rα/c)22 - dR2 ds2 = -(Rα/c)22 + dR2 Consequently, the hyperbolas in region II are spacelike. In these diagrams, light rays are constrained to travel at 45° and are shown in pink. Understanding this fact allows any number of different scenarios to be investigated. For example, consider Alice and Bob in a rocket. Alice is ejected at point, P, and follows the blue trajectory. At Q Bob shines a light ray at Alice which she receives. At R Bob sends out another light ray which Alice receives just as she hits the singularity. At point S Bob shines a third light ray at Alice but this doesn't reach her because she has already hit the singularity and no longer exists! From Bob's perspective, he see's Alice falling towards the event horizon, E, at a slower and slower rate, seemingly taking forever (ω = ∞) to get to the point where she gets 'stuck' on the horizon. Once Alice is behind the event horizon she cannot escape. In order to do so she would have to exceed the speed of light which is not possible. Her fate is sealed and she is torn apart by the extreme gravitational tidal forces as she heads towards the singularity.