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: December 17, 2021 ✓

Basic Group Theory ------------------ Symbols ------- ∈ = is in ∉ = not in ∀ = for all ∅ = empty ∃ = exists ⊂ = subset of ℤ = set of all integers ℂ = set of all complex numbers ℝ = set of real numbers ℚ = set of rational numbers Definition ---------- A group, (G,*) is a set of elements gi = {a,b,c ...} that satisfy the following axioms: Note: * is the operation + or x (only 1 operation is allowed). 1. Closure: a and b are elements then their composition must also be a member of the group. Thus, a, b ∈ G -> a*b ∈ G 2. Identity: There is some element, e, such that e*a = a. Thus, e ∈ G -> e*a ∈ G for any a ε G 3. Inverse: For any element, a, there is an inverse element a-1 such that a*a-1 = e. Thus, a, a-1 ∈ G then aa-1 ∈ G 4. Associativity: a*(b*c) = (a*b)*c Example 1. (ℤ,+) = {... -3,-2,-1,0,1,2,3,4 ...} Closure: a,b ∈ ℤ -> a + b ∈ ℤ Identity: a + 0 = 0 + a = a Inverse: a-1 = -a since a-1 + a = e Associativity: (a + b) + c = a + (b + c) Example 2. (ℤ,x) = {... -3,-2,-1,0,1,2,3,4 ...} Closure: a,b ∈ ℤ -> a x b ∈ ℤ Identity: a x 1 = 1 + a = a x Inverse: 2 x integer = 1 has no solution in ℤ Associativity: (a x b) x c = a x (b x c) Example 3. GL(n,ℝ) is the group of all n x n matrices with non zero determinants under matrix multiplication. - - - - - - Closure: | a b || e f | = | ae + bg af + bh | | c d || g h | = | ce + dg cf + dh | - - - - - - - - Identity: I = | 1 0 | ∴ A x I = I x A | 0 1 | - - - - - - Inverse: | a b |-1 = 1/(ad- bc)| d -b | | c d |   | -c a | - - - - Associativity: (A x B) x C = A x (B x C) Commutativity: A x B ≠ B x A x Commutativity ------------- ABELIAN: [gi,gj] = 0 NON-ABELIAN: [gi,gj] ≠ 0 Examples: Integer multiplication is Abelian. Matrix multiplication is generally non-Abelian. Cayley Tables ------------- Cayley tables allow us to construct verify the group closure property. They have the following properties. 1. Square 2. No duplicate elements allwed in rows or columns. This dictates how the rows and columns are formed regardless of the group operation. 3. Symmetric about diagonal indicates the group is abelian. * = group operation. 2nd order group: * | e | a --+---+---- e | e | a --+---+---- a | a | e This is the same as (isomorphic to) ℤ (mod 2) (or ℤ2). 3rd order group: * | e | a | b --+---+---+--- e | e | a | b --+---+---+--- a | a | b | e --+---+---+--- b | b | e | a This is the same as (isomorphic to) ℤ (mod 3) (or ℤ3). 4th order groups are a little more complicated because there is more than one way to organize the Cayley, but the same principles apply. Klein 4 group: + | e | a | b | c --+---+---+---+--- e | e | a | b | c --+---+---+---+--- a | a | e | c | b --+---+---+---+--- b | b | c | e | a --+---+---+---+--- c | c | b | a | e ℤ (mod 4) (or ℤ4) + | e | a | b | c --+---+---+---+--- e | e | a | b | c --+---+---+---+--- a | a | b | c | e --+---+---+---+--- b | b | c | e | a --+---+---+---+--- c | c | e | a | b 4th roots of unity {1,-1,i,-i}: x | e | a | b | c --+---+---+---+--- e | e | a | b | c --+---+---+---+--- a | a | e | c | b --+---+---+---+--- b | b | c | a | e --+---+---+---+--- c | c | b | e | a Modular Additive and Multiplicative Inverses -------------------------------------------- In modular arithmetic, the modular additive and multiplicative inverses are also defined. It is the number a such that a + x = 0 (mod n) or ax = 0 (mod n). These inverse always exist. For example, consider the group ℤ3 + | 0 1 2 --+------ 0 | 0 1 2 1 | 1 2 0 2 | 2 0 1 1 + 2 = 0 (mod 3) and, x | 0 1 2 --+------ 0 | 0 0 0 1 | 0 1 2 2 | 0 2 1 2 x 2 = 1 (mod 3) The modular additive and multiplicative inverses should not be confused with the ordinary inverses i.e. 2 + (-2) = 0 and 2(1/2) = 1. Subgroups --------- (G,*) = {a,b,c,d, ...} then (H,*) = {a,b} form a subgroup. Note: e and the inverse must be in the subgroup. Example 1. (ℤ,+) Consider the even integers (2ℤ,+) Closure: a,b ∈ 2ℤ -> a + b &eisin; 2ℤ Identity: a + 0 = 0 + a = a Inverse: a-1 = -a Associativity: (a + b) + c = a + (b + c) It turns out that all we need to show is closure and the existence of inverses to show that we have found a subgroup. What about the odd integers (2ℤ + 1,+) x Closure: a + b = even number Inverse: a-1 = -a Example 2. Consider rotations by 90° of a square in 2D with composition equal to the addition of rotation angles. Therefore, G = {I, R90, R180, R270}. This is the symmetry group of a square. A B ------ | | {I, R90, R180, R270} | | ------ D C We can represent all of the rotation operations by:     I R90 R180 R270 --------------------- I    | I    R90  R180 R270 R90  | R90  R180 R270 I R180 | R180 R270 I    R90 R270 | R270 I    R90  R180 Notes: * | e --+--- is the trivial subgroup. e | e The group itself is a subgroup. A subgroup that is not the group itself is called a proper subgroup. Group Extensions ---------------- Consider the symmetry group of the square again. Reflection about vertical center line: B A ------- y | | ^ {mv} | O | | | | -->x ------- C D   - - mv = | -1 0 |   | 0 1 |   - - - - - - - - | -1 0 || x | = | -x | | 0 1 || y | | y | - - - - - - Reflection about horizontal center line: D C ------- | | {R180mv} | O | | | ------- A B - - - - - - - - - - | cosθ -sinθ || x | = | -1 0 || x | = | -x | | sinθ cosθ || y | | 0 -1 || y | | -y | - - - - - - - - - - - - - - - - | -1 0 || -x | = | x | | 0 1 || -y | | -y | - - - - - - Reflection about upper-left to lower-right diagonal: A D ------- | | {R90mv} | O | | | ------- B C Remembering that a clockwise rotation of 90° corresponds to a terminal angle of 270°, we get: - - - - - - - - - - | cosθ -sinθ || x | = | 0 1 || x | = | y | | sinθ cosθ || y | | -1 0 || y | | -x | - - - - - - - - - - - - - - - - | -1 0 || y | = | -y | | 0 1 || -x | | -x | - - - - - - Therefore: A B (-1,1) (1,1) ------- | | | O | | | ------- (-1,-1) (1,-1) D C x -> -y y -> -x A: (-1,1) -> (1,-1) = C B: (1,1) -> (-1,-1) = B C: (1,-1) -> (-1,1) = A D: (-1,-1) -> (1,1) = D A D (-1,1) (1,1) ------- | | | O | | | ------- (-1,-1) (1,-1) B C Reflection about lower-left to upper-right diagonal: C B ------- | | {R270mv} | O | | | ------- D A Remembering that a clockwise roation of 270° corresponds to a terminal angle of 90°, we get: - - - - - - - - - - | cosθ -sinθ || x | = | 0 -1 || x | = | -y | | sinθ cosθ || y | | 1 0 || y | | x | - - - - - - - - - - - - - - - - | -1 0 || -y | = | y | | 0 1 || x | | x | - - - - - - Therefore: A B (-1,1) (1,1) ------- | | | O | | | ------- (-1,-1) (1,-1) D C x -> y y -> x A: (-1,1) -> (1,-1) = C B: (1,1) -> (1,1) = B C: (1,-1) -> (-1,1) = A D: (-1,-1) -> (-1,-1) = D C B (-1,1) (1,1) ------= | | | O | | | ------- (-1,-1) (1,-1) D A