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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

Helicity and Chirality ---------------------- Helicity corresponds to 'handedness' but is dependent on the reference frame. For a massive particle it is possible to get ahead of the particle and look back. If one does, the handedness of the particle will be reversed. Therefore, the notion of helicity is not intrinsic to the particle. Chirality, on the other hand, is intrinsic to the particle and is independent of reference frames. Particles with different chirality are truly different. In the case of a massless particle that is moving at the speed of light, the chirality and helicity are the same - it is not possible to ever get ahead of such a particle and look back. While chirality and helicity are not the same, they become approximately equivalent at high energies. Chirality is very important in physics, indeed the Standard Model is a chiral theory since it differentiates between left and right-handed particles. The spin angular momentum of a particle can be oriented in any direction. However, what if we want to to find the component of spin in the direction of motion. This leads us to the definition of helicity as the projection of the spin onto the direction of momentum. Therefore, h = S.p where ----> right-handed helicity <---- left-handed helicity --------> p This quantity is conserved. Helicity can be visualized as follows: The gray arrow represents the direction of motion. The red and blue arrows represent the handedness and NOT the actual spin of the particle which is an intrinsic property unrelated to classical momentum. Right handed particles follow the right hand rule. If your thumb points in the direction of the gray arrow, then your fingers wrap in the direction of the red arrow. Conversely, a particle is 'left-handed' because it follows the left hand rule. If your thumb points in the direction of the gray arrow, then your fingers wrap in the direction of the blue arrow. Weyl Spinors ------------ We can write the 4-component Dirac spinor in terms of two 2-component WEYL SPINORS. - - - - ψ = | φ | ≡ | ψL | | χ | | ψR | - - - - The reason it makes sense to decompose a Dirac spinor this way is that, ψL and ψR transform independently under Lorentz transformations (they do not mix). The technical way to express this is to say that a Dirac spinor forms a reducible representation of the Lorentz group, whereas Weyl spinors form irreducible representations. This implies that from a purely mathematical point of view, Weyl spinors may be considered more 'fundamental' than Dirac spinors. Before we go any further we need to introduce the 5th GAMMA MATRIX. This is defined as:   - -   | 0 0 1 0 | γ5 = iγ0γ1γ2γ3 = | 0 0 0 1 |   | 1 0 0 0 |   | 0 1 0 0 |   - - Where, - - - - γi = | 0 σi | and γ0 = | I2 0 |  | -σi 0 |   | 0 -I2 | - - - - A Dirac field, ψ, can be projected onto its left-chiral and right-chiral components by: ψL = (1/2)(I4 - γ5)ψ and ψR = (1/2)(I4 + γ5)ψ There is another representation of the Gamma matrices called the WEYL REPRESENTATION. In the Weyl representation the γi matrices are unchanged but the γ0 matrix is different. - - - - γi = | 0 σi | and γ0 = | 0 I2 |  | -σi 0 |   | I2 0 | - - - - γ5 becomes:   - -   | -1 0 0 0 | γ5 = iγ0γ1γ2γ3 = | 0 -1 0 0 |   | 0 0 1 0 |   | 0 0 0 1 |   - - and the chiral projections take a simple form:   - -   -   - ψL = | I2 0 |ψ and ψR = | 0 0   | 0  0 |  | 0 I2 |   - -   -   - Or, - - -  - - - | I2 0 || ψL | = | ψL | | 0  0 || ψR | | 0  | - - -  - - - and, - - -  - - - | 0 0  || ψL | = | 0  | | 0 I2 || ψR | | ψR | - - -  - - - In the chiral representation, the eigenstates of γ5 are: - - - -   - - | -I2 0 || ψL | = ±1| ψL | | 0 I2 || ψR | | ψR | - - - -   - - By a slight abuse of language, we will say ψL has a chirality of -1 (left-chiral), whereas ψR has a chirality of +1 (right-chiral). This is an abuse of language because it is actually the 4-component spinor with two components set to 0 that has a definite chirality, not the 2-component spinors themselves. The spinors transform in the same way under rotations, but oppositely under boosts. The Weyl Equations ------------------ From the discussion of the The Dirac equation, the Dirac spinor, ψ, can be written as a linear combination of left-handed and right-handed spinors. Therefore, ψ = ψL + ψR and ψ = ψL + ψR Note: In Group Theory language this means that the Dirac spinor representation of the Lorentz group is reducible. It decomposes into two irreducible representations, acting only on 2-component Weyl spinors. ψR is in the (1/2,0) representation of the Lorentz group while ψL is in the (0,1/2) representation. The Dirac spinor lies in the (1/2,0) ⊕ (0,1/2) representation. _ We can use the relationship ψ = ψγ0, we get: _ ψψ = ψγ0ψ - - - - - - = | ψL ψR || 0 1 || ψL | - - | 1 0 || ψR | - - - - = ψLψR + ψRψL In the Weyl representaion we can use the more compact form: - -   | 0  σμ | γμ = | _    |   | σμ 0  | - -   _ Where by analogy with γ0 and γμ, σμ and σμ are defined as: σμ = (σ0xyz) σ0 is sometimes considered as the Identity matrix. Thus, σμ = (I,σxyz)   = (1,σi) and, _ σμ = (1,-σi) _ Note: The here is nothing to do with the Dirac adjoint. It is purely convention. We can now write the Dirac Lagrangian as:   _ L = iψLσμμψL + iψRσμμψR - m(ψLψR + ψRψL) For a massless particle the equations of motion derived from the Euler-Lagrange equations are:  _ iσμμψL + iσμμψR = 0 Therefore, iσμμψL = 0 and, _ iσμμψR = 0 These are the WEYL EQUATIONS for a massless spin 1/2 particle where ψL and ψR are again the Weyl Spinors. It is important to note that massive fermions like electrons are not Weyl spinors, they are Dirac spinors. The subtlety is that the Dirac spinors consist of 2 massless Weyl spinors that are mixed by the mass term. For example, an 'electron' propagating in space and interacting with a mass due to the Higgs field connects a left-chiral electron with a right-chiral antipositron. This means that the two types of particles are exhibiting quantum mixing. The 'physical electron' which is propagating through space is a combination of eL and eR particles that swap back and forth. At a particular point in time the particle may be eL, but if you observe it a moment later, the very same particle might manifest itself as eR. For a more details about this, please see the discussion regarding the Higgs Field.