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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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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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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: January 27, 2018

The Schrodinger, Heisenberg and Dirac Pictures ---------------------------------------------- In the Schrodinger picture the operators do not change with time but the states do. In the Heisenberg picture the operators change with time but the states do not. <O> = <ψ(t)|O(0)|ψ(t)> ... Schrodinger. <O(t)> = <ψ(0)|O(t)|ψ(0)> ... Heisenberg. Using the completeness relationship (see Essential Mathematics of QM) we can expand the wavefunction, ψ, as: |ψ> = Σαj(t)|ej> |dψ/dt> = Σdαj/dt|ej> Now using the TDSE we can also write: |dψ/dt> = Σαj(t)(-iH/h)|ej> Therefore, by comparison: dαj/dt = -iαjH/h The solution to this equation is: αj(t) = αj(0)exp(-iHt/h) We cab use this information to compute the expectation value as of O(t) follows: <O(t)> = <ψ(t)|O|ψ(t)> = <Σαk*exp(iEkt/h)ek|0|Σαjexp(-iEjt/h)ej> = ΣΣαk*exp(iEkt/hjexp(-iEjt/h)<ek|0|ej> = ΣΣαkjexp(i(Ek - Ej)t/h)<ek|0|ej> = ΣΣαkj[exp(i(Ek - Ej)t/h)Okj] So the term in [] is the time evolution of the operator which can be written as: O(t) = exp(iEkt/h)Okjexp(-iEjt/h) Heisenberg Equation ------------------- Using the product rule. dO(t)/dt = (i/h)Ekexp(iEkt/h)Oexp(-iEjt/h) + exp(iEkt/h)(∂O/∂t)exp(iEjt/h) + (i/h)exp(iEkt/h)O(-Ej)exp(-iEjt/h) = (i/h)exp(iEkt/h)(EkO - OEj)exp(-iEjt/h) + exp(iEkt/h)(∂O/∂t)exp(-iEjt/h) = (i/h)(EkO(t) - O(t)Ej) + exp(iEkt/h)(∂O/∂t)exp(-iEjt/h) If we let Ej = Ek = E we get: dO(t)/dt = (i/h)[E,O(t)] + exp(iEt/h)(∂O/∂t)exp(-iEt/h) This is the Heisenberg equation. Dirac (Interaction) Picture --------------------------- In the Dirac picture, both the wavefunctions and the operators carry time-dependence. The interaction picture allows for operators to act on the state vector at different times and forms the basis for quantum field theory and many other newer methods. <O> = <ψ(t)|O(t)|ψ(t)> ... Dirac. HS = H0,S + H1,S (1 not I) = Free + Interaction A state vector in the interaction picture is defined as: ψI(t) = exp(iH0,St/h)|ψS(t)> or, ψS(t) = exp(-iH0,St/h)|ψI(t)> An operator in the interaction picture is defined as: OI(t) = exp(iH0,St/h)OSexp(-iH0St/h) For the free Hamiltonian, the interaction picture and Schrodinger pictures coincide: H0,I(t) = exp(iH0,St/h)H0,Sexp(-iH0,St/h) = H0,S(t) For the interacting Hamiltonian, the interaction picture is: H1,I = exp(iH0,St/h)H1,Sexp(-iH0,St/h) Time Evolution of States ------------------------ The Schrodinger equation for states in the interaction picture can be derived starting from the Schrodinger picture as follows: ih∂|ψS(t)>/∂t = HSS> Rewrite in the Dirac picture: ih∂exp(-iH0,St/h)|ψI(t)>/∂t = (H0,S + H1,S)exp(-iH0,St/h)|ψI(t)> Calculate, ih(-iH0,S/h)exp(-iH0,St/h)|ψI(t)> + ihexp(-iH0,St/h)∂|ψI>/∂t = H0,Sexp(-iH0,St/h)|ψI(t)> + H1,Sexp(-iH0,St/h)|ψI(t)> Simplify, H0,Sexp(-iH0,St/h)|ψI(t)> + ihexp(-iH0,St/h)∂|ψI>/∂t = H0,Sexp(-iH0,St/h)|ψI(t)> + H1,Sexp(-iH0,St/h)|ψI(t)> ihexp(-iH0,St/h)∂|ψI>/∂t = H1,Sexp(-iH0,St/h)|ψI(t)> ih∂|ψI>/∂t = exp(iH0,St/h)H1,Sexp(-iH0,St/h)|ψI(t)> Finally, ih∂|ψI>/∂t = H1,II(t)> This the SCHWINGER-TOMONAGA equation. Time Evolution of Operators --------------------------- If the operator OS itself does not have an explicit time dependence (is not a function of time) then the corresponding time evolution for OI(t) is given by: ∂OI(t)/∂t = -i/h[H1,I,H0,S] This is proven in the notes on "Time Evolution and Symmetry Operations".