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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 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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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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Penrose Diagrams
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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

Game Theory

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The Truel .

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 Computing

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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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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 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 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

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

Maxwell's Equations ------------------- Integral form: Ε = ∫E.dl = -∂φB/∂t where φB = BA (Faraday) ∫B.dl = μ0I + μ0ε0∂φE/∂t where φE = EA (Ampere - Maxwell) The displacement current correction was added to Ampere's law by Maxwell to explain the presence of a magnetic field between capacitor plates even though no current is threading through the path. ∫E.dA = Q/ε0 (Gauss for E) ∫B.dA = 0 (Gauss for B, no monopoles) Differential form: ∇ x E = -∂B/∂t ... 1. ∇ x B = μ0J + μ0ε0∂E/∂t ... 2. ∇.E = ρ/ε0 ∇.B = 0 The differential and integral formulations of the equations are mathematically equivalent, by the divergence theorem in the case of Gauss's law and Gauss's law for magnetism, and by Stokes theorem in the case of Faraday's law and Ampere's law. Proof: Gauss's Law ----------- ∫∫E.dS = q/ε0 s q = ∫∫∫ρdV v ∫∫E.dS = (1/ε0)∫∫∫ρdV s v Divergence Theorem: ∫∫∫divEdV = ∫∫E.dS v S = (1/ε0)∫∫∫ρdV v ∫∫∫(divE - ρ/ε)dV = 0 v divE - ρ/ε = 0 Faraday's Law ------------- ∫∫B.dS = ρ/ε0 s If we replace E with B and set ρ = 0 (since there is no such thing as magenetic charge (monopole)), we get: divB = 0 φ = ∫∫B.dS s V = -dφ/dt and V = ∮E.dr (E = V/r) c ∮E.dr = -dφ/dt = -d/dt{∫∫B.dS} c s = -∫∫(∂B/∂t).dS) s Now apply Stokes Theorem, ∮E.dr = ∫∫curlE.dS, to get: c s ∫∫{curlE + ∂B/∂t).dS = 0 s So that: curlE + ∂B/∂t = 0 Ampere's Law ------------ ∮B.dr = μ0I c Now, I = ∫∫j.dS s Therefore, ∮B.dr = ∫∫μ0j.dS c s Now apply Stokes Theorem to get: ∫∫(curlB - μ0j).dS = 0 s So that: curlB = μ0j with Maxwell's correction we get: curlB = μ0(j + ε0∂E/∂t) Both the differential and integral formulations are useful. The integral formulation can often be used to simply and directly calculate fields from symmetric distributions of charges and currents. On the other hand, the differential formulation is a more natural starting point for calculating the fields in more complicated (less symmetric) situations. Wave Equation: From 1. ∇ x (∇ x E) = - ∂(∇ x B)/∂t Sub 2. ∇ x (∇ x E) = -μ0ε02E/∂t2 Use identity ∇ x (∇xE) = -∇2E + ∇(∇.E) ρ = J = 0 (free space vacuum ) so, ∇.E = 0 => ∇2E = μ0ε02E/∂t2 One solution is E = E0sin{2π(x - vt)/λ} Substitute into above => v2 = 1/μ0ε0 = c22E = (1/c2)∂2E/∂t2 Similarly, we can also do From 2. ∇ x (∇ x B) = ∇ x {μ0ε0∂E/∂t} ∇(∇.B) - ∇2B = μ0ε0∂(∇ x E)/∂t but ∇.B = 0 so ∇2B = μ0ε0∂(-∂B/∂t)/∂t ∇2B = μ0ε02B/∂t2 This has a solution similar to E: B = B0sin{2π(x - vt)/λ} Light is an self sustaining e-m wave! Changing E field => changing B field => changing E field. The B field is perpendicular to the E field and both have the same phase (i.e. zero phase difference). From 1 again. ∇ x E = -∂B/∂t Therefore, ∂E0sin{2π(x - vt)/λ}/∂x = -∂B0sin{2π(x - vt)/λ}/∂t E0(2π/λ)cos{2π(x - vt)/λ} = B0(2πv/λ)cos{2π(x - vt)/λ} E0 = B0v v = c = E0/B0 Thus, the ratio of the electric to magnetic fields in an electromagnetic wave in free space is always equal to the speed of light. Energy density of E field: UE = ε0E2/2 J/m3 This is derived from the energy stored in a capacitor. Energy density of B field: UB = B2/2μo J/m3 This is derived from the energy stored in a solenoid. UTotal = ε0E2/2 + B2/2μo Now B = E/c so UB = E2/2μ0c2 Now, c2 = 1/ε0μ0 so UB = E2ε0μ0/2μ0 = E2/2ε0 Thus, we can write UTotal in the following equivalent forms: UTotal = ε0E2 ≡ B2o ... substituting E = cB in the UE equation. ≡ cε0EB The intensity can be found by taking the energy density (energy per unit volume) at a point in space and multiplying it by the velocity at which the energy is moving. The resulting vector has the units of power divided by area. I0 = cU0 = cε0E02/2 + cB02/2μo = cε0E02 ≡ cB02o ≡ c2ε0E0B0 Now, c2 = 1/(μ0ε0) Therefore, I0 = {1/(μ0ε0)}{ε0E0B0} = (1/μ0)EB This is the POYNTING VECTOR. This more formally defined as: - S = (1/μ0)ExB = (1/μ0)EBsinθ = (1/μ0)EB since E and B are orthogonal The Poynting vector represents the rate of energy transport per unit area (energy flux) in W per meter2). The modulus of S, |S| is equal to the intensity, I. All electromagnetic waves (radio, light, X-rays, etc.) obey the inverse-square law thus the intensity of an electromagnetic wave is proportional to the inverse of the square of the distance from a point source. ERMS = E0/√2 and BRMS = B0/√2 IAverage = cε0ERMS2 = cBRMS2o Photons ------- In the quantum description, the electromagnetic field is an observable property of photons. Photons can be thought of as mini E/M wave segments that consists of an oscillating electric field component, E, and an oscillating magnetic field component, B. The electric and magnetic fields are orthogonal (perpendicular) to each other, and they are orthogonal to the direction of propogation of the photon. The E and B fields flip direction as the photon travels. The number of oscillations that occur in one second is the frequency, f. The superposition of a sufficiently large number of photons has the characteristics of a continuous electromagnetic wave. The energy of the photon is given by E = hf and a wavelength is equal to c/f. There is a correspondence between the energy of a photon stream and the Poynting vector in the classical approach.