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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

The one-way ANOVA is a method for extending the two-sample t-test for independent samples to three or more samples. ANOVA is an acronym for ANalysis Of VAriance. The acronym is a little misleading since we are actually analyzing means not variances. In a one-way ANOVA there is one dependent variable and one independent variable. Means Model ----------- yij = μi + eij Where, yij = measured value on jth subject in ith group μi = mean value for group i eij = random error about μi In analysis of variance (ANOVA), the sum of squares helps express the total variation that can be attributed to various factors. Sum of Squares -------------- Let: k = number of treatments (columns) r = number of groups in each treatment (rows). n = total number of groups (rows x columns) yij = ith observation in jth column. _ y = grand mean = Σyij/n _ yj = column means = Σiyij/r    _ Total SS = Σ(yij - y)2   _  _ SST = Σjr(yj - y)2 _ SSE = Σij(yij - yj)2 Mean Total SS = Total SS/(n - 1) MSST = SST/(k - 1) MSSE = SSE/(n - k) Assumptions: 1. Samples are randomly selected from the k treatment populations. 2. All k treatment populations are normal. 3. All k treatment variances are equal. Hypotheses: H0: μ1 = μ2 ... = μn H1: μi ≠ μj for at least one i and j Test statistic: F test = = MSST/MSSE If H0 is rejected how do we determine which samples are different? Two methods are used: LSD and Bonferroni. Both methods use the t-test. SPSS: Analyze>Compare Means>One-Way ANOVA Post Hoc => check LSD and Bonferroni (more accurate) df SS MS F ----- --- ------------------ --------- Treatment k - 1 SST MSST = SST/(k - 1) MSST/MSSE Error n - k SSE MSSE = SSE/(n - k) Total n - 1 Tot SS *** *** Total ≠ column totals. Why ANOVA and not t-test? 1. Comparing three groups using t-tests would require that 3 t-tests be conducted. This increases the chances of making a type I error. 2. The t-test does not make use of all of the available information from which the samples were drawn. For example, in a comparison of group 1 vs. group 2, the information from group 3 is neglected. An ANOVA makes use of the entire data set. Pairwise Comparisons -------------------- The number of ways that the means can be compared is given by: c = k(k - 1)/2 Thus, for example, if there are 3 treatments c = 3. These combinations are: μ1 with μ2 μ1 with μ3 μ2 with μ3 A simple completely randomized design example: SAT Score means Boys Girls Sample 1 540 530 Sample 2 530 540 Sample 3 540 520 _ yj 536.66 530 _ y = (540 + 530 + 540 + 530 + 540 + 520)/6 = 533.33 k = 2, r = 3, c = 2, n = 6    _ Total SS = Σ(yij - y)2 = (540 - 533.33)2 + (530 - 533.33)2 + (540 - 533.33)2 + (530 - 533.33)2 + (540 - 533.33)2 + (520 - 533.33)2 = 333.334   _  _ SST = Σjr(yj - y)2 = 3(536.66 - 533.33)2 + 3(530 - 533.33)2 = 33.267 + 33.267 = 66.533 SSE = Σij(yij - yj)2 = (540 - 536.66)2 + (530 - 536.66)2 + (540 - 536.66)2 + (530 - 530)2 + (540 - 530)2 + (520 - 530)2 = 266.667 Note that: Total SS = SST + SSE = 66.533 + 266.667 = 333.2000 Mean Total SS = Total SS/(n - 1) = 333.334/5 = 66.667 MSST = SST/(k - 1) = 66.533 MSSE = SSE/(n - k) = 266.667/4 = 66.667 F = MSST/MSSE = 66.533/66.667 = 0.9980 MSST df = k - 1 = 1 MSSE df = n - k = 4 F1,4 = 7.7086 for α = 0.05 Summarizing: df SS MS F -- ------ ------ ------ Treatment 1 66.533 66.533 0.9980 Error 4 266.667 66.667 When 2 samples are being compared, the t and F tests are equivalent. F = t2 _ _ x1 - x2 t = --------------- √s2(1/n1 + 1/n2) s2 = MSE = 66.667 536.66 - 530 = ------------------ √66.667(1/3 + 1/3) = 6.66/6.66 = 1 A simple randomized block design example: With a randomized block design, the experimenter divides subjects into subgroups called blocks, such that the variability within blocks is less than the variability between blocks. Then, subjects within each block are randomly assigned to treatment conditions. Compared to a completely randomized design, this design reduces variability within treatment conditions and potential confounding, producing a better estimate of treatment effects. SPSS Output: df SS MS F ------------- --- ------------------------- --------- Treatment k - 1 SST MSST = SST/(k - 1) MSST/MSSE Block b - 1 SSB MSSB = SSB/(b - 1) MSSB/MSSE Error n - b - k + 1 SSE MSSE = SSE/(n - b - k + 1) Total n - 1 Tot SS *** *** Total ≠ column totals. SAT Score means Boys Girls Block Means School 1 540 530 535 School 2 530 540 535 School 3 540 520 530 _ yj 536.66 530 _ y = (540 + 530 + 540 + 530 + 540 + 520)/6 = 533.33 k = 2, r = 3, c = 2, n = 6, b = 3    _ Total SS = Σ(yij - y)2 = (540 - 533.33)2 + (530 - 533.33)2 + (540 - 533.33)2 + (530 - 533.33)2 + (540 - 533.33)2 + (520 - 533.33)2 = 333.334   _  _ SST = Σjr(yj - y)2 = 3(536.66 - 533.33)2 + 3(530 - 533.33)2 = 33.267 + 33.267 = 66.533   _  _ SSB = Σik(yi - y)2 = 2(535 - 533.33)2 + 2(535 - 533.33)2 + 2(530 - 533.33)2 = 5.578 + 5.578 + 22.218 = 33.374 SSE = Total SS - SST - SSB = 333.334 - 66.533 - 33.374 = 233.427 MSST = SST/(k - 1) = 66.533 MSSE = SSE/(n - b - k + 1) = 233.427/(6 - 3 - 2 + 1) = 116.714 MSSB = SSB/(b - 1) = 33.374/2 = 16.687 F = MSST/MSSE = 66.533/116.714 = 0.5701 MSST df = k - 1 = 1 MSSE df = n - b - k + 1 = 2 MSSB df = b - 1 = 2 F1,2 = 18.5128 for α = 0.05 This randomized block design removes school as a potential source of variability and as a potential confounding variable. Summarizing: df SS MS F -- ------- ------- ------ Treatment 1 66.533 66.533 0.5701 Block 2 33.374 16.687 0.1429 Error 2 266.667 116.714