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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: April 9, 2020

Gauge Theories (Yang-Mills) --------------------------- Abelian Gauge Theory -------------------- A gauge theory is a field theory in which some GLOBAL continuous symmetry of the theory is replaced by a stricter LOCAL continuous symmetry requirement. The imposition of a local symmetry requires the introduction of new fields that make the Lagrangian invariant under the local transformation. For each group generator there necessarily arises a corresponding field (usually a vector field) called the gauge field. Consider the following Lagrangian for a simple complex field: L = ∂μφ∂μφ* - m2φφ* It is easy to see that the global transformation φ' = φexp(iθ) leaves the Lagrangian invariant. Now consider if one wished to not only make global changes of phase but also local transformations of the form φ' = φexp(iθ(x)). Now the situation is not so straightforward since θ is now a function of x and the kinetic term picks up a derivative of θ(x). As a result the action is no longer invariant under this type of change. In order to make it invariant and enforce such a symmetry, one must rewrite the transformation law so that there is a new type of derivative Dμφ, which under the change of phase on φ transforms in the same fashion, Dμφ -> exp(iθ(x))Dμφ. Dμ is called the GAUGE COVARIANT DERIVATIVE and is defined as: Dμ = ∂μ + igAμ Where Aμ is a new quantity called the GAUGE FIELD and g is a coupling constant. Therefore, gauge fields are included in the Lagrangian to ensure its invariance under the local transformations. Proof: φ' = φexp(iθ(x)) φ*' = φ*exp(-iθ(x)) Dμ'φDμ'φ* = (∂μφ' + igAμ'φ')(∂μφ*' - igAμ'φ*')    = (∂μφ + i∂μθφ + igAμ'φ)exp(iθφ) (∂μφ* - i∂μθφ* - igAμ'φ*)exp(-iθφ)    = (∂μφ + i∂μθφ + igAμ'φ) (∂μφ* - i∂μθφ* - igAμ'φ*) To get back to the original form we need Aμ' to transform as follows: Aμ' = Aμ - (1/g)∂μθ and Aμ' = Aμ - (1/g)∂μθ Therefore, Dμ'φDμ'φ* = (∂μφ + i∂μθφ + ig(Aμ - (1/g)∂μθ)φ) (∂μφ* - i∂μθφ* - ig(Aμ - (1/g)∂μθ)φ*)    = (∂μφ + igAμφ)(∂μφ* - igAμφ*)    = DμφDμφ* Also, m2φφ* -> m2exp(iθ)φexp(-iθ)ψ* = m2φφ* Aside: The above also applies to the Dirac Lagrangian. _ L = ψ(iγμμ - m)ψ ψ -> exp(iθ)ψ _ _ ψ -> exp(-iθ)ψ Now, ∂μ(ψexp(1θ)) = exp(iθ)∂μψ + ψi∂μθexp(iθ) The covariant derivative is Dμ = ∂μ + igAμ Where, Aμ -> Aμ - (1/g)∂μθ Therefore, Dμ = ∂μ + ig(Aμ - (1/g)∂μθ) Dμ(ψexp(iθ)) = ∂μ(ψexp(iθ)) + iψexp(iθ)gAμ - iψexp(iθ)∂μθ   = exp(1θ)∂μψ + iψexp(iθ)∂μθ + iψexp(iθ)gAμ - iψexp(iθ)∂μθ   = exp(1θ)∂μψ + iψexp(iθ)gAμ   = exp(1θ)[∂μψ + iψgAμ]   = exp(1θ)[∂μ + igAμ  = exp(1θ)Dμψ Therefore, _ ψiγμDμψ _ => ψexp(-iθ)iγμDμ(ψexp(iθ)) _ => ψexp(-iθ)exp(iθ)iγμDμψ _ => ψiγμDμψ Also, _ _ mψψ => mexp(iθ)ψexp(-iθ)ψ _ => mψψ Invariance of Aμ by Itself ------------------------- We have looked at how the introduction of the gauge field makes φ invariant under a local transformation. However, the gauge field itself must also have its own gauge invariant dynamic term in the Lagrangian. Consider the construction, Fμν, defined as: Fμν = ∂μAν - ∂νAμ Intuitively, this is a good choice since it contains the first derivatives, ∂μ, of the field that is characterisitic of all Lagrangians. If we plug in Aμ' = Aμ - (1/q)∂μθ into the above we get: Fμν = ∂μAν + ∂μνθ - (∂νAμ + ∂νμθ)   = ∂μAν - ∂νAμ Thus, under the gauge transformation, Fμν, remains unchanged. To make this both gauge and Lorentz invariant and also match the quadratic nature of the other terms in the Lagrangian, it is necessary to form the product: FμνFμν We can now rewite our a Lagrangian for the gauge field as: L = -FμνFμν The is the Lagrangian that describes that electromagnetic field in the absence of any charges (complex fields are charged fields so in the absence of charge φ = 0). Fμν is referred to as the FIELD STRENGTH TENSOR. In the Abelian case this is equivalent to the ELECTROMAGNETIC TENSOR. Note: By convention L is normally wriiten as L = -(1/4)FμνFμν In the abelian U(1) theory there is 1 gauge fields whose quanta is the photon. Non-Abelian Gauge Theory (Yang-Mills) ------------------------------------- Yang and Mills extended the above Abelian theory to Non-Abelian groups. Therefore, it extends the U(1) gauge theory to a gauge theory based on the SU(N) group. Yang–Mills theory seeks to describe the behavior of elementary particles using these non- Abelian Lie groups and forms the basis of our understanding of the Standard Model of particle physics. The Lagrangian for the gauge fields is: L = -(1/4)FaμνFaμν Aside: This is the kinetic term of the PROCA ACTION that describes a massive spin-1 field of mass, m, in Minkowski spacetime (i.e. the W and Z bosons). The Proca action is given by: L = -(1/4)FμνFμν + m2AμAμ The mass term of the Proca Lagrangian is not invariant under a gauge transformation since (Aμ + ∂μθ)(Aμ + ∂μθ) ≠ m2AμAμ. The consequence is that m = 0 unless the symmetry is spontaneously broken. This discussed in detail in the section on the Higgs mechanism. Continuing, the gauge covariant derivative becomes: Dμ = ∂μ - igTaAaμ Where Ta are the group generators that satisfy the Lie algebra: [Ta,Tb] = ifabcTc [Dμ,Dν] = -igTaFaμν Proof: [Dμ,Dν] = igT(∂μAν - ∂νAμ + gT[Aμ,Aν]    ≡ igTa(∂μAaν - ∂νAaμ + gfabcAbμAcν)    = igTaFaμν Note that in the abelian case Faμν reduces to the electromanetic tensor and [Dμ,Dν] = igFμν. SU(2) ----- φ' = φexp(iθiσi/2) φ*' = φ*exp(-iθiσi/2) [σi/2,σj/2] = iεijkσk/2 Dμ = ∂μ - igWμiσi/2 Gμνi = ∂μWνi - ∂νWμi + fijkWμjWνk In the non-abelian SU(2) theory there are 3 gauge fields whose quanta are the W+, W- and Z bosons. SU(3) ----- φ' = φexp(iθiλi/2) φ*' = φ*exp(-iθiλi/2) [λi/2,λj/2] = ifijkλk/2 Dμ = ∂μ - ig''Gμiλi Gμν = ∂μGνi - ∂νGμi + fijkGμjGνk In the non-abelian SU(3) theory there are 8 gauge fields whose quanta are the gluons. U(1) ⊗ SU(2) ⊗ SU(3) --------------------- Dμ = ∂μ - ig''Gμiλi/2 - igWμiσi/2 - ig'YBμ Footnote: Gauge symmetry is a powerfull symmetry particularly in the context of General Relativity where coordinates vary from place to place based on the curvature of spacetime. QED is described by a U(1) group that represents electric charge. The unified electroweak interaction is described by SU(2) ⊗ U(1) group where the U(1) group represents the weak hypercharge, YW, rather than the electric charge. QCD is an SU(3) Yang–Mills theory. The massless bosons from the SU(2) ⊗ U(1) theory mix after spontaneous symmetry breaking (Higgs mechanism) to produce the 3 massive weak bosons, and the photon field. The Standard Model combines the strong interaction with the unified electroweak interaction (unifying the weak and electromagnetic interaction) through the symmetry group U(1) ⊗ SU(2) ⊗ SU(3). At the current time, however, the strong interaction has not been unified with the electroweak interaction, but from the observed running of the coupling constants it is believed they all converge to a single value at very high energies.