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Astronomy

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Astronomical Distance Units .
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Celestial Coordinates .
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Celestial Navigation .
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Location of North and South Celestial Poles .

Chemistry

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Avogadro's Number .
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Balancing Chemical Equations
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Stochiometry
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The Periodic Table .

Classical Mechanics

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Blackbody Radiation .

Classical Physics

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Archimedes Principle
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Bernoulli Principle
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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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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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Poiseuille's Law
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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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The Gas Laws
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The Laws of Thermodynamics
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The Zeeman Effect .
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Young's Modulus

Climate Change

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Keeling Curve .

Cosmology

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Baryogenesis
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Cosmic Background Radiation and Decoupling .
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CPT Symmetries
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Dark Matter .
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Friedmann-Robertson-Walker Equations .
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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
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Vacuum Energy .

Finance and Accounting

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Amortization
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Annuities
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Brownian Model of Financial Markets .
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Capital Structure
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Dividend Discount Formula
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Lecture Notes on International Financial Management
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NPV and IRR
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Periodically and Continuously Compounded Interest
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Repurchase versus Dividend Analysis

Game Theory

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The Truel .

General Relativity

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Basis One-forms .
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Catalog of Spacetimes .
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Curvature and Parallel Transport
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Einstein's Field Equations
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Geodesics
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Gravitational Waves
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Hyperbolic Motion and Rindler Coordinates .
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Quantum Gravity
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Ricci Decomposition .
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Ricci Flow .
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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 Dirac Equation in Curved Spacetime .
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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 Light Cone .
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The Metric Tensor .
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The Principle of Least Action in Relativity .
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Vierbein (Frame) Fields

Group Theory

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Basic Group Theory .
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Basic Representation Theory .
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Building Groups From Other Groups .
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Sets, Groups, Modules, Rings and Vector Spaces
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Symmetric Groups .
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The Integers Modulo n Under + and x .

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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Binomial Theorem (Pascal's Triangle)
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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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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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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 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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Volume Integrals

Microeconomics

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Marginal Revenue and Cost

Nuclear Physics

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Radioactive Decay

Particle Physics

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Feynman Diagrams and Loops
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Field Dimensions
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Helicity, Chirality and Weyl Spinors .
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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 .

Probability and Statistics

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Box and Whisker Plots
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Buffon's Needle .
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Categorical Data - Crosstabs
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Chebyshev's Theorem
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Chi Squared Goodness of Fit
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Conditional Probability
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Confidence Intervals
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Data Types
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Expected Value
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Factor Analysis
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Hypothesis Testing
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Linear Regression
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Monte Carlo Methods
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Non Parametric Tests
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One-Way ANOVA
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Pearson Correlation
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Permutations and Combinations
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Pooled Variance and Standard Error
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Probability Distributions
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Probability Rules
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Sample Size Determination
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Sampling Distributions
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Set Theory - Venn Diagrams
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Stacked and Unstacked Data
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Stem Plots, Histograms and Ogives
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Survey Data - Likert Item and Scale
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Tukey's Test
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Two-Way ANOVA

Programming and Computer Science

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Hashing
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How this site works ...
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More Programming Topics
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MVC Architecture
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Open Systems Interconnection (OSI) Standard - TCP/IP Protocol
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Public Key Encryption

Quantum Computing

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Density Operators and Mixed States .
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Entangled States .
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The Qubit .

Quantum Field Theory

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Creation and Annihilation Operators
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Field Operators for Bosons and Fermions
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Lagrangians in Quantum Field Theory
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Path Integral Formulation
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Relativistic Quantum Field Theory

Quantum Mechanics

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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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Heisenberg Uncertainty Principle
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Ladder Operators .
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Multi Electron Wavefunctions .
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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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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 Quantum Harmonic Oscillator .
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The Schrodinger, Heisenberg and Dirac Pictures
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The WKB Approximation
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Time Dependent Perturbation Theory
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Time Evolution and Symmetry Operations
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Time Independent Perturbation Theory
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Wavepackets

Semiconductor Reliability

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The Weibull Distribution

Solid State Electronics

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Band Theory of Solids .
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Fermi-Dirac Statistics .
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Intrinsic and Extrinsic Semiconductors .
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The MOSFET
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The P-N Junction

Special Relativity

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4-vectors .
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Electromagnetic (Faraday) Tensor .
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Energy and Momentum in Special Relativity, E = mc2 .
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Invariance of the Velocity of Light .
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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 Continuity Equation .
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The Lorentz Group .

Statistical Mechanics

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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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Bardeen–Cooper–Schrieffer Theory
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BCS Theory
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Cooper Pairs
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Introduction to Superconductivity .
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Superconductivity (Lectures 1 - 10)
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Superconductivity (Lectures 11 - 20)

Supersymmetry (SUSY) and Grand Unified Theory (GUT)

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Chiral Superfields
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Generators of a Supergroup
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Grassmann Numbers
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Introduction to Supersymmetry
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The Gauge Hierarchy Problem

The Standard Model

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Electroweak Unification (Glashow-Weinberg-Salam)
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Gauge Theories (Yang-Mills)
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Gravitational Force and the Planck Scale
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Introduction to the Standard Model
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Isospin, Hypercharge, Weak Isospin and Weak Hypercharge
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Quantum Flavordynamics and Quantum Chromodynamics
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Special Unitary Groups and the Standard Model - Part 1 .
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Special Unitary Groups and the Standard Model - Part 2
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Special Unitary Groups and the Standard Model - Part 3 .
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Standard Model Lagrangian
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The Higgs Mechanism
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The Nature of the Weak Interaction

Topology

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Units, Constants and Useful Formulas

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Constants
Last modified: January 26, 2018

Sampling Distributions ---------------------- If we draw a sample size, n, from a given population and compute a statistic (mean, standard deviation, proportion) for each sample. The probability distribution of the statisitic is called a sampling distibution. Central Limit Theorem --------------------- For large enough sample sizes* the sample MEANS follow N(μ,σ/√n) where n is the sample size. In other words, the mean of all the sample means is equal to the population mean. This applies regardless of the distribution of the parent population. As the sample size is increased the standard deviation, skew and kurtosis of the sampling distribution decreases. The SD of the sampling distribution of the mean is called the STANDARD ERROR OF THE MEAN. SE = σ/√n = σsample means We can generalize this to: SE = (σ/√n)((N - n)/(N - 1)) Where N is the population size. For infinite N this reduces to: SE = σ/√n as before For small N it reduces to: SE = σ * A sample size ≥ 30 is generally considered to be the minimum for the CLT to apply. Distribution of Means --------------------- _ If x is the mean of a random sample of size n taken from a normal population having mean μ and variance σ2, then we can state the following: Large Sample Size (n > 30) -------------------------- _ z = (x - μ)/σ/√n Generally, the population standard deviation is not given. Under these circumstances it is necessary to compute it from the sample using:    _ s2 = Σi(x - xi)2/(n - 1) Why use n - 1? To answer that we need to look at biassed versus unbiassed estimators. Biassed versus Unbiassed Estimators ----------------------------------- Consider a small distribution: 5 7 12 N = 3 n = 2 μ = 8 σ2 = Σ(x - μ)2/N = 8.67 σ = 2.94 _   _ s2 = Σ(x - x)2/(n - 1) = Σ(x - x)2/1 _ Sample x s2 s s22 ------ --- ---- ---- ---- 5 5 5.0 0.0 0.00 0.00 5 7 6.0 2.0 1.41 0.23 5 12 8.5 24.5 4.95 2.83 7 5 6.0 2.0 1.41 0.23 7 7 7.0 0.0 0.00 0.00 7 12 9.5 12.5 3.54 1.44 12 5 8.5 24.5 4.95 2.83 12 7 9.5 12.5 3.54 1.44 12 12 12.0 0.0 0.00 0.00 Average 8.0 8.67 2.20 Notes: _ - x follows a t distribution (normal dostribution if the sample size is large). - s is a biassed estimator of σ - (n - 1)s22 follows the χ2 distribution. - A denominator of n gives the biassed estimator for σ2. - A denominator n - 1 gives the unbiassed estimator for σ2. Small Sample Size (n ≤ 30) -------------------------- If n ≤ 30 then we must use the t-statistic instead:   _ tv = (x - μ)/s/√n with v = n - 1 degrees of freedom. The t-distribution has the following properties: mean: μ = 0 variance: σ2 = v/(v - 2), where v is the degrees of freedom and v > 2 The variance is always greater than 1, although it is close to 1 when there are many degrees of freedom. With infinite degrees of freedom, the t distribution is the same as the standard normal distribution. Distribution of Proportions --------------------------- If p is the proportion (probability) of successes in the population, then: σp = √(p(1 - p)/n) The sampling distribution of p is a discrete rather than a continuous distribution. It is approximately normally distributed if n is fairly large and p is not close to 0 or 1. A general rule of thumb is that the approximation is good when: np and n(1 - p) are both ≥ 10 _ z = (p - μ)/σ/√n As before, generally, p and σ are not given for the original population. Under these circumstances it is necessary to compute them from the sample. Thus, p = psample Distribution of Variances ------------------------- If s2 is the variance of a random sample of size n taken from a normal poulation having the variance σ2, then χ2 = (n - 1)s22 with v = n - 1 degrees of freedom. The χ2 distribution has the following properties: mean: μ = v variance: σ2 = 2v Example: An optical firm purchases glass to be ground into lenses and past experience has shown that the variance of the refractive index = 1.26 x 10-4. A shipment is received and a sample of 20 pieces is pulled. The measured variance of the sample is 2.10 x 10-4. Should the sample be rejected? H0: σ2 = s2 H1: σ2 ≠ s2 χ2 = (20 - 1)2.10 x 10-4/1.26 x 10-4 = 31.66 From tables, χ20.05 for v = n - 1 = 19 is equal to 30.144. Since 31.66 > 30.144 the result is significant at the 0.05 level and there is sufficient reason to reject H0. F Distribution -------------- If s12 and s22 are the variances of independent random samples of size n1 and n2 taken from two normal populations having the same variance, then F = s12/s22 with v1 = n1 - 1 and v2 = n2 - 1 degrees of freedom. The F distribution has the following properties: mean: μ = v2/(v2 - 2) for v2 > 2. variance: σ2 = [2v22(v1 + v2 - 2)]/[v1(v2 - 2)2(v2 - 4)] for v2 > 4. Example: Suppose you randomly select 7 marbles from company 1's production line and 12 marbles from company 2's production line and measure their diameters. Assume you are given: s1 = 1.0 and s2 = 1.1 F = s12/s22 = 1/1.21 = 0.83 H0: σ1 = σ2 H1: σ1 ≠ σ2 From tables, F0.05 for v1 = n1 - 1 = 6 and v2 = n2 - 1 = 11 is equal to 3.0946. Since 0.83 < 3.0946 the result is not significant at the 0.05 and there is insufficient reason to reject H0.