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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 26, 2018

Generators of a Supergroup -------------------------- Creation and Annihilation Operators ----------------------------------- Supersymmetry is a symmetry that interchanges bosons with fermions and vice versa. The creation and annihilation operators of bosons and fermions obey the following commutation rules: Bosons: [ai-,aj+] = δij [ai+,aj+] = [ai-,aj-] = 0 Fermions: {ci-,cj+} = δij {ci+,cj+} = {ci-,cj-} = 0 for i = j This is reflective of the Pauli Exclusiom Principle that disallows 2 or more fermions to bein thesame state. Both together: [a+,c+] = [a-,c-] = [a+,c-] = [a-,c+] = 0 Interchange of 2 bosons: [ai+,aj+] = ai+aj+ - aj+ai+ = 0 => ai+aj+ = aj+ai+ Therefore, ai+aj+|0> = aj+ai+|0> Interchange of 2 fermions: {ci+,cj+} = ci+cj+ + cj+ci+ = 0 => ci+cj+ = -cj+ci+ Therefore, ci+cj+|0> = -cj+ci+|0> We can see this diagramatically as follows: Boson: φ(xA,xB) = φ(xB,xA) - symmetric [φ(xA),φ(xB)] = [φ(xB),φ(xA)] Fermion: ψ(xA,xB) = -ψ(xB,xA) - antisymmetric {ψ(xA),ψ(xB)} = -{φ(xB),φ(xA)} {θAB} = -{θBA} Grassmann Algebra ----------------- There is no limit to number of bosons in a state. Therefore, the field operators can grow large. On the other hand, the Pauli Exclusion Principle prohibits 2 or more fermions from occupying the same state because (c+)2 = (c-)2 = 0. Therefore, the behavior of these fields is encoded in the commutator and anti-commutator relationships respectively. There is a special algebra that describes the anti-commuting properties of fermionic fields called GRASSMANN ALGEBRA. In Grassmann algebra, ordinary numbers that commute are replaced by numbers that anti-commute. These anti-commuting numbers are extremely useful in keeping track of fermionic fields. In essence, Fermionic fields are really Grassmann numbers. COMMUTING NUMBERS ARE REFERRED TO AS EVEN ELEMENTS OF THE GRASSMANN ALGEBRA AND CORRESPOND TO DIRECTLY MEASURABLE QUANTITIES. ANTI-COMMUTING NUMBERS ARE REFERRED TO AS ODD ELEMENTS OF THE GRASSMANN ALGEBRA AND CORRESPOND TO QUANTUM MECHANICAL PROBABILITIES. BOSONIC FIELDS ARE EVEN ELEMENTS AND FERMIONIC FIELDS ARE ODD ELEMENTS. Generators of a Supergroup, Q ----------------------------- Lie Algebra ----------- The Lie algebra of the infinitessimal generators for even elements, G, is the familiar: [Gi,Gj] = iεijkGk In Supersymmetry, we for look the infinitessimal group generators that transform a bosonic field to a fermionic field and vice versa. The generators are defined as: Q = (√m)a+c- ... removes a fermion and adds a boson Q = (√m)a-c+ ... adds a fermion and remove a boson Because they are composed of the products a+c- or a-c+, which are even and odd elements respectively, these generators represent the odd elements of the Grassmann algebra. Therefore, they follow the same anti- commutation rules as the creation and annihilation operators for fermions. Therefore, {Q,Q} = m(a+c-a+c-) = 0 since (c-)2 = 0. Likewise, {Q,Q} = 0 It is important to note that the Q and Q operators act on a SINGLE particle. Because QQ = 0, the operations cannot be compounded in the same way that other generators can (i.e. generators of rotation). In this sense they are different from other generators that we have encountered. The other anti-commutations are: {Q,Q} = m(a-c+a-c+) = 0 since (c+)2 = 0 {Q,Q} = m(a+c-a-c+ + a-c+a+c-)   = m(a+a-c-c+ + a-a+c+c- + a+a-c+c- - a+a-c+c-)   = m(a+a- + c+c-)   = m(nB + nF)   = 2H ... assumes boson mass = fermion mass   = 2i∂/i∂t   = 2p0 p0 is the energy component of pμ which we will encounter when we extend the discussion to spacetime. From the rules for Grassmann algebra, a mix of Grassmann (odd elements) and ordinary numbers (even elements) are always COMMUTED. H is an even element. The commutator of the Q's with H are: [Q,H] = [Q,H] = 0 Proof: H|ψ> = E|ψ> Q = Symmetry operation (rotation/translation) HQ|ψ> = EQ|ψ> = QE|ψ> = QH|ψ> (HQ - QH)|ψ> = 0 [Q,H] = -[H,Q] = 0 This means that Q and Q are both conserved quantities. Therefore, internal symmetries that take one particle to another involve time translations. In the Standard Model, for example, isospin symmetries take protons to neutrons, and color symmetries take red quarks to blue quarks but neither of these symmetries involves spacetime. This is something quite new! There is another way to think about this. Instead of time translation, we can regard the supersymmetry generators as shifts in the coordinates which are Grassmann numbers (Grassmann coordinates). Think about a superfield, Φ, defined in terms of Grassmann _ _ numbers, θ, θ, and time, t, Φ(θ,θ,t). A simple field like this may look like: _ _ _ Φ(t) = φ(t) + θψ(t) + ψ(t)θ + D(t)θθ E = B O = F O = F E = B Consider the following symmetry transformations: θ -> θ + ξ ... spatial translation _ _ _ θ -> θ + ξ ... spatial translation _ _ t -> t + iξθ + iξθ ... time translation invariance. Where ξ is a infinitesimally small Grassmann number. (Note: ξ2 = 0 so it behave like ε in Group Theory). The change in Φ is given by: _ _ _ _ _ δΦ(θ,θ,t) = ξ∂Φ/∂θ + ξ∂Φ/∂θ + iξθ∂Φ/∂t + iξθ∂Φ/∂t _ _ = ξ(∂/∂θ + iθ∂/∂t)Φ + ξ[...]Φ _ = (ξQ + ξQ)Φ Where, _ Q = ∂/∂θ + iθ∂/∂t and _ Q = ∂/∂θ + iθ∂/∂t Therefore, _ _ {Q,Q} = {∂/∂θ + iθ∂/∂t,∂/∂θ + iθ∂/∂t} Making use of the following properties of Grassmann numbers: _ (∂/∂θ)(∂/∂θ) = θθ = 0 _ _ (∂/∂θ)θ = (∂/∂θ)θ = 1 Leaves us with, _ _ {Q,Q} = i(∂/∂θ)θ(∂/∂t) + i(∂/∂θ)θ(∂/∂t) {Q,Q} = -2i∂/∂t as we calculated before. So there is alternative to the creation and annihilation formulation of the Q's that involves differential operators that involves a new type of spacetime involving anti-commuting coordinates. They generate energy in the ordinary coordinates _ via the iθ∂/∂t term, but also perform spatial transformations in the Grassmann coordinates via the ∂/∂θ term.