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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: May 6, 2020

Creation and Annihilation Operators ----------------------------------- Quantum Harmonic Oscillator --------------------------- Consider unit mass hanging on a spring. ---------- / \ ^ / | x \ unit mass . Lagrangian, L = (1/2)x2 + (1/2)ω2x2 where ω = √(K/m) Euler-Lagrange: . d(∂L/∂x)/dt - ∂L/∂x = 0 Leads to .. x = -ω2x with solution x = cosωt Hamiltonian: . . p = ∂L/∂x = x . H = px - L . . = px - [(1/2)x2 + (1/2)ω2x2] . = (1/2)x2 + (1/2)ω2x2 = (1/2)p2 + (1/2)ω2x2 (set m = 1) In CM H can be zero because x and p can be set to zero. Not so in QM because of uncertainty principle. H|ψ> = (1/2)(-ih∂/∂x)(-ih∂/∂x)ψ(x) + (1/2)ω2x2ψ(x) = (-1/2)∂2ψ(x)/∂x2 + (1/2)ω2x2ψ(x) (set h = 1) Schrodinger Equation: i∂ψ/∂t = Hψ = (-1/2)∂2ψ(x)/∂x2 + (1/2)ω2x2ψ(x) Energy Eigenvectors: H|ψ = E|ψ> -(1/2)∂2ψ/∂x2 + ω2x2ψ(x) = Eψ(x) ψ(x) = Aexp(-ω/2)x2) is a solution to this equation where A is the normalization factor. If we plug this into the above equation we get, ω/2 = E - The Ground State H = (1/2)(p2 + ω2x2) where p = -i∂/∂x = (1/2)(ωx + ip)(ωx - ip) = (1/2)(p2 + ω2x2 + iω(px - xp)) - x and p do not commute in QM = (1/2)(p2 + ω2x2 + ω) <- replace (px - xp) by [p,x] = -i Therefore, the corrected Hamiltonian is: H = (1/2)(ωx + ip)(ωx - ip) + ω/2 Define: a = √(ω/2)(x + ip/√(2ω) or (1/√(2ω)(ωx + ip) and a = √(ω/2)(x - ip/√(2ω) or (1/√(2ω)(ωx - ip) Therefore, x = (1/√(2ω))(a + a) and p = -i√(ω/2)(a - a) The -i makes the operator Hermitian. Consider 2 the addition and subtraction of a complex number and its conjugate: x + iy + (x - iy) = 2x = real x + iy - (x - iy) = 2iy = imaginary The resulting commutator is: [a,a] = 1 Returning to the Hamiltonian, H = ω(aa + 1/2) Let us go further with this. You might think: a|n> => |n+1> a|n> => |n-1> So, aa|n> = a|n - 1> = |n> and aa|n> = a|n + 1> = |n> However, this is not correct because [a,a] = 0 and not -1. What we need instead is: a|n> = √(n+1)|n+1> ... A. a|n> = √n|n-1> ... B. Now, aa|n> = a√n|n-1> = √n√n|n> = n|n> ... using A. and aa|n> = a√(n+1)|n+1> = √(n+1)√(n+1)|n> = n+1|n> ... using B. Now [a,a] = aa - aa = (n+1) - n = 1 as before Number Operator --------------- aa|n> = √na|n-1>    = √n√n|n> (after replacing n with n+1)    = n|n> apa= np counts the number of quanta in state p and is referred to as the OCCUPATION NUMBER. These operators change the eigenvalues of the NUMBER OPERATOR, N. Thus, N = Σpnp Knowing that aa = n, we can rewite the Hamiltonian as: H = hω(N + 1/2) (after putting h back in) with eigenvalues of: |0> = (1/2)ω|0> a|0> = (3/2)ω|0> aa|0> = (5/2)ω|0> aaa|0> = (7/2)ω|0> Note regarding interpretation: The energy eigenvales of the harmonic oscillator could be interpreted in 2 ways. The increase in energy of a single quanta or the addition of quanta with energy hω. It is this second idea that we want focus on. Thus, the creation and annihilation operators add and subtract quanta: ap increases quanta (hω) in momentum state p by 1; ap decreases quanta in momentum state p by 1. Commutation Rules for Bosons ---------------------------- From before we have shown that: [a,a]|n> = ((n + 1) - n))|n>   = |n> Thus, for example: a|0> = |1> a|1> = |2> a|2> = |1> a|1> = |0> a|0> = not allowed Therefore, [a,a]|0>(aa - aa)|0> = |0> [a,a]|1>(aa - aa)|1> = |1> Summarizing: [ap,aq] = [ap,aq] = 0 [ap,aq] = δpq (1 when p = q, 0 when p ≠ q) [ap,aq] = -δpq Commuation Rules for Fermions ----------------------------- Bosons can occur in the same state state. However, because of the Pauli Exclusion Principle, Fermions cannot. Fermions can only have 0 or 1 in a state. We define new creation and annihilation operators c and c such that: c|0> = |1> c|1> = |0> c|1> = not allowed c|0> = not allowed cc|0> = not allowed cc|0> = |0> cc|1> = |1> cc|1> = not allowed Now consider combinations of c and c: (cc + cc)|0> = |0> (cc + cc)|1> = |1> Therefore {c,c}|0> = |0> and, {c,c}|1> = |1> Conclusion: c and c follow the ANTICOMMUTATOR relationship. The anticommutator can be written as: {c,c}|n> = ((n + 1) + n))|n>   = (2n + 1)|n> For this to work, n has to equal 0. Therefore, the anticommutation relationship allows a state to be filled only if it is empty. Summarizing: {cp,cq} = {cp,cq} = 0 {cp,cq} = δpq (1 when p = q, 0 when p ≠ q) {cp,cq} = -δpq