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Astronomy

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Celestial Coordinates
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Celestial Navigation
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Distance Units
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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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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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Penrose Diagrams
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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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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

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 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 Qubit
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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

test

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test

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

Feynman Path Integrals ---------------------- Feynman Path Integrals are an alternative formulation of quantum field theory. Unlike the canonical approach which is based on creation and annihilation operators, the path integral approach generalizes the action principle of classical mechanics. The use of path integrals replaces the classical notion of a single, unique history for a system with a sum over an infinity of possible set of paths (histories) to compute a quantum amplitude. Feynman showed that an amplitude computed in this way will also obey the Schrodinger equation for the Hamiltonian corresponding to the given action. Dirac pointed out that the propagator was analogous to the quantity exp(iS/h) where S is the time integral of the Lagrangian in classical mechanics. Using this, Feynman postulated that the probability amplitude for a particular space-time path is given by: exp(iSΓ/h) Where SΓ = ∫Ldt, the classical action calculated along a path, Γ. The propagator is given by: K(x,y) = ∫exp(iSΓ/h)Dx = ∫exp((i/h)∫Ldt)Dx Where Dx represents all possible paths in space-time. In order to find the overall probability amplitude for a given process it is necessary to sum over the space of all possible space-time paths of the system in between the initial and final states, including paths that are absurd by classical standards. This different to the classical case where the goal is to minimize the action along a single trajectory. The initial and final states are then related to each other by: φ(x) = ∫dy K(x,y)φ(y) Green's Functions ----------------- Consider the differential equation: Du(x) = f(x) Where D is a linear differential operator. The Green's function, G(x,u) of D is defined by the equation: DG(x,u) = δ(x - u) Where u is a dummy variable used to describe the position of the δ function. The solution for u(x) is given by: u(x) = ∫duG(x,u)f(u) 0 This is the same form as the Feynman propagator, K. Time Slicing ------------ The Lagrangian describes interactions at a point or, because of the derivative terms, neighboring points that are very close by. Therefore, to obtain the propagator for a finite time interval it is necessary to use the method of TIME SLICING. The idea is to break the time interval up into N equal slices. In the limit N -> ∞, the integral over positions at each time slice can be said to be an integral over all possible paths. The exponent becomes a time-integral of the Lagrangian, namely the action for each path. Using the property that <b|a> ≡ <a|n><n|b> and <b|a> = ∫b*a dx we can write: <xf,tf|xi,ti> = ∫ ... ∫<xi,ti|x1,t1>dx1<x1,t1|x2,t2>dx2<x2,t2|x3,t3> ... dxN<xN,tN|xf,tf> = ∫ ... ∫exp(iS(xi,ti;x1,t1)/h)dx1exp(iS(x1,t1;x2,t2)/h)dx2 ... dxNexp(iS(xN,tN;xf,tf)/h) = ∫ ... ∫exp(iS(xi,ti;x1,t1)/h)exp(iS(x1,t1;x2,t2)/h) ... exp(iS(xN,tN;xf,tf)/h)dx1dx2 ... dxN Therefore, to create a more general process involving motion it is necessary to compound (multiply) individual actions, S as follows: t xf, tf | / | / S | o | / S | o | / S | o | / S | xi, ti | ----------------- x Non Relativistic Free Particle ----------------------------- The Lagrangian for a free particle is: . L = mx2/2 y t . K(x,y) = ∫exp(i∫(x2/2)dt)Dx with h = m = 1 x 0 Splitting this into time slices yields: K(x,y) = ∫Πtexp{(i/2)([x(t + ε) - x(t))/ε]2ε}Dx If we replace the i with -1 the problem becomes easier to solve. We will re-insert the i at the end. Doing this is analagous to a Wick rotation (see below). Therefore, K(x,y) = ∫Πtexp{(1/2)([x(t + ε) - x(t))/ε]2ε}Dx Each factor in the product is a Gaussian integral centered at x(t) with variance ε. The expression is a convolution of Gaussians. K(x,y) can be represented by the K(x,y) = Gε * Gε ... Gε The number of G's, n, is given by t/ε If we take the Fourier transform the convolutions become multiplications. Thus, of this we can write: K(p) = Gε(p)n The FT of Gε(p) is another Gaussian: Gε = exp(-εp2/2) Therefore, K(p) = exp(-tp2/2) The FT of this gives: K(x,y) ∝ exp(-(x - y)2/2t) K(x,y) is a solution to the equation: ∂K/∂t = (1/2)∇2K Proof: ∂K/∂t = (x - y)2exp(-(x - y)2/2t)/2t2 ∂K/∂x = (x - y)exp(-(x - y)2/2t)/t ∂2K/∂x2 = (x - y)2exp(-(x - y)2/2t)/t2 If we now insert the i back into the equation we get: K(x,y) ∝ exp(i(x - y)2/2t) and, ∂K/∂t = (i/2)∇2K This is the Schrodinger equation with ψ replaced with K. Extension to Fields ------------------ The FPI approach can be extend to fields by using the Lagrangian density: S[φ] = ∫d4xL(φ,∂μφ) Where the the 'paths' or histories being considered are not the motions of a single particle, but the possible time evolutions of a field over all space. The [φ] implies that the probability amplitude is computed over all possible combinations of values that the field could have anywhere in space–time. Thus, the field representation is a sum over all field configurations, whereas the particle representation is a sum over different particle paths. S[φ] is a functional of φ meaning that the domain is no longer a region of space, but a space of functions that are in turn functions of space and time. The Lagrangian for the Klein-Gordon equation (real scalar field) is: L = ∂μφ∂μφ - m2φ2 Therefore, S[φ] = ∫d4x (∂μφ∂μφ - m2φ2) and, K(x,y) = ∫Dφ exp(iS[φ]) Where Dφ represents all possible field configurations in space-time. Therefore, K(x,y) = ∫Dφ exp(i∫d4x (∂μφ∂μφ - m2φ2)) This is a functional integral. Functional integration is a special branch of mathematics that we will not concern ourselves with. Instead, we will just state the result: K(x,y) = ∫d4p/(2π)4 exp(-ip(x-y))/(p2 - m2 + iε) Which is the same result that we get from the canonical approach. For a Fermionic field the situation is more complex and requires the use of GRASSMANN ALGEBRA to derive the propagator. Grassmann algebra is discussed in the section on supersymmetry. Much of the current study surrounding QFT is devoted to the properties of functional integrals, and how such integrals can be made to be mathematically precise. Lattice Gauge Theory -------------------- Lattice Gauge Theory is an alternate technique that overcomes the challenges associated with functional integration and lends itself to modern day computational methods. In lattice gauge theory, the spacetime is Wick rotated into Euclidean space and discretized into a lattice. A Wick rotation involves replacing t by (it) in the Minskowki metric. When this is done the result is the Euclidean metric. Minkowski metric: (ds2) = -(dt)2 + (dx)2 + (dy)2 + (dz)2 Euclidean metric: (ds2) = (dτ)2 + (dx)2 + (dy)2 + (dz)2 Problems are easier are generally easier to solve in Euclidean space than they are in Minkowski space. The resulting solution may then, under reverse substitution, yield a solution to the original problem. Based on the concept of wave-particle duality, we can represent the field in terms of the number of quanta at each value of (x,t) and the number of quanta at each (x',t'). This is referred to as the FOCK/ CANONICAL representation of a quantum field. Therefore, φ now consists of creation and annihilation operators. We can the calculate the probability amplitude that the particle content went from (x,t) to (x',t'). To do this, we construct a lattice where each cell is associated with an action, Si. t +-+-+-+-+-+-+-+-+-+ φf | | | | | | | | | | +-+-+-+-+-+-+-+-+-+ | | | | | | | | | | ^ +-+-+-+-+-+-+-+-+-+ | | | | | | | | | | | | +-+-+-+-+-+-+-+-+-+ history (analagous to trajectory) | | | | | | | | | | +-+-+-+-+-+-+-+-+-+ | | | | | | | | | | +-+-+-+-+-+-+-+-+-+ | | | | | | | | | | +-+-+-+-+-+-+-+-+-+ φi <final state| S1S2S3 ... |initial state> To get the probability amplitude, PΓ for the path replace ΠiSi with Πi(1 + iSi). PΓ can now be written as: PΓ = (1 + iS1)(1 + iS2)(1 + iS3) ... Now 1 + iSi can be expanded as exp(iSi) Thus, we can write after including h to match Feynman: PΓ = exp(i/h)S1) exp(i/h)S2) exp(i/h)S3) = exp((i/hiSi) Now, S = ∫Ld4x. For a very small volume, ε, we can approximate S as: S = εL Therefore, we can write: PΓ = exp((iε/hiLi) What kinds of things in the Lagrangian correspond to a particle moving from one cell to another? The Lagrangian contains the following derivatives: (1/2)(∂φ/∂t)2 = (1/2){φ(x,t) - φ(x,t')/Δt}2 (1/2)(∂φ/∂x)2 = (1/2){φ(x,t) - φ(x',t)/Δx}2 As a result, the action (the ∫ of the Lagrangian over each cell) will contain terms in φ(x,t) and φ(x',t'). As we stated before, the ones of interest for motion are the cross-terms φ(x,t)φ(x',t') since they annihilated and create particles at different points. If we just focus on these terms and ignore the rest for now we can conceptually write: PΓ = exp(iΣφ(x,t)φ(x',t')) Note that this is a greatly simplified treatment designed to illustrate the basic concepts and is not rigorous. Now, the exponential can be expanded as: 1 + iφ(x,t)φ(x',t') + (1/2!)Σφ(x,t)φ(x',t')Σφ(x',t')φ(x'',t'') ... Consider: +---+---+---+ 2 | | f | | +---+---+---+ 1 | | | | +---+---+---+ 0 | i | | | +---+---+---+ 0 1 2 Such a process could be represented by: 1 + iφ(0,0)φ(1,0) + (1/2!)φ(1,0)φ(1,1)φ(1,1)φ(1,2) or 1 + iφ(0,0)φ(0,1) + (1/2!)φ(0,1)φ(0,2)φ(0,2)φ(1,2) The Σ's in the original expansion would act to sum all of the possible paths to get from (0,0) to (1,2) Of course, there are other terms in the Lagrangian that we haven't included. For example there is the term m2φ2/2 that creates and annihilates a particle at the same point and adds mass to the otherwise massless processes describe above. For particles with mass, the mass is included for weighting the probability amplitude for the path. We could also have a term like gφ3 which represents a 3 particle interaction at a vertex. Similar to the above g, the coupling constant, is included for weighting the probabilty amplitude for the path. In general, the Lagrangians can get extremely complicated and difficult to solve.