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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

Conditional Probability ----------------------- If A and B are DEPENDENT events. The probability of B given A, P(B|A), is given by: P(B|A) = P(A ∩ B)/P(A) ... 1. Note: If the events are INDEPENDENT then: P(A ∩ B) = P(A).P(B) Rearranging 1. we get: P(A ∩ B) = P(B|A).P(A) Bayes Theorem ------------- From before we had: P(A ∩ B) = P(B|A).P(A) We can also write: P(B ∩ A) = P(A|B).P(B) Since P(A ∩ B) = P(B ∩ A), P(B|A).P(A) = P(A|B).P(B) Or, P(A|B) = P(B|A).P(A)/P(B) Where, P(A) is the PRIOR, the initial probability of A. P(A|B) is the POSTERIOR, the probability of A after B. P(B|A)/P(B) is the support B provides for A. From the above Tree diagram we can see that: P(B) = P(A ∩ B) + P(~A ∩ B) = P(A).P(B|A) + P(~A).P(B|~A) Therefore, P(A|B) = P(B|A).P(A)/{P(A).P(B|A) + P(~A).P(B|~A)} We can generalize this to: P(A1|B) = P(B|A1).P(A1)/ΣkP(Bk) = P(B|A1).P(A1)/ΣkP(B|Ak).P(Ak) ≡ P(A1 ∩ B)/ΣkP(Ak ∩ B) This is BAYES' THEOREM. P(B) = ΣkP(Bk) ≡ ΣkP(B|Ak).P(Ak) ≡ ΣkP(Ak ∩ B) is the LAW OF TOTAL PROBABILITY. Bayes' Theorem calculates the probability that an event (A) occurs given knowledge of another event (B). It can also be understood as a way of understanding how the probability that a theory is true is affected by a new piece of evidence. It is used for many purposes, including detecting faults, surveillance, military defence, search-and-rescue operations, medical screening and even email spam filters. Example 1: Consider 5 marbles in a bag: 2 blue and 3 red. What is the probability of: 1. Pulling 2 blue marbles without replacement? 2. At least one blue marble being pulled? 3. Pulling 2 blue marbles with replacement? 4. The first marble being blue given that the second marble is blue with replacement. 1. These are DEPENDENT events. To calculate the probability of pulling a blue marble given that the first marble picked was blue, we use the formula: P(B1 ∩ B2) = P(B2|B1).P(B1) 1 blue given 1 blue ( P(B1 ∩ B2) ): (2/5)(1/4) = 1/10 1 red given 1 blue ( P(R1 ∩ B2) ): (2/5)(3/4) = 3/10 1 blue given 1 red ( P(B1 ∩ R2) ): (3/5)(2/4)* = 3/10 1 red given 1 red ( P(R1 ∩ R2) ): (3/5)(2/4) = 3/10 * Remember a blue marble wasn't picked so there must still be 2. The probabability of 2 blue is 1/10. 2. The probability of at least 1 blue is 1/10 + 3/10 + 3/10 = 7/10. 3. These events are INDEPENDENT and we get: 1 blue given 1 blue ( P(B1 ∩ B2) ): (2/5)(2/5) = 4/25 1 red given 1 blue ( P(R1 ∩ B2) ): (2/5)(3/5) = 6/25 1 blue given 1 red ( P(B1 ∩ R2) ): (3/5)(2/5) = 6/25 1 red given 1 red ( P(R1 ∩ R2) ): (3/5)(3/5) = 9/25 The probabability of 2 blue is 4/25. The probability of at least 1 blue is 4/25 + 6/25 + 6/25 = 14/25. 4. To calculate the probability that the first marble was blue given that the second marble picked was blue, we use Bayes' formula: P(B1|B2) = P(B2|B1).P(B1)/P(B2) Where, P(B1) is the PRIOR, the initial probability of B1. P(B1|B2) is the POSTERIOR, the probability of B1 after B2. P(B2|B1)/P(B2) is the support B provides for B1. But P(B2) = 1/10 + 3/10 from answer 1. Therefore, P(B1|B2) = (2/5)(1/4)/{1/10 + 3/10} ... dotted red lines    = 1/4 So the probability that a blue was picked initially given that a another blue was picked the second time is 'modified' from the prior value of 2/5 to 1/4! Example 2: 1% of women have breast cancer. A woman with breast cancer has a 90% chance of testing positive while a woman without has a 5% chance (false positive). What is the probability a women has breast cancer given that she had a positive test? P(C|+) = P(+|C).P(C)/P(+) + given cancer ( P(B ∩ B) ): (0.01)(0.90) = 0.0090 - given cancer ( P(R ∩ B) ): (0.01)(0.10) = 0.0010 + given no cancer ( P(B ∩ R) ): (0.99)(0.10) = 0.0990 - given no cancer ( P(R ∩ R) ): (0.99)(0.9) = 0.8910 P(C|+) = 0.0090/{0.0090 + 0.0990} = 0.0090/0.1080 = 0.08333 So the probability of having cancer given that the test is positive is actually quite low!.