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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

Lorentz Transform ----------------- The Lorentz transformation converts between two different observers' measurements of space and time, where one observer is in constant motion with respect to the other. In classical physics (Galilean relativity), the only conversion believed necessary was x' = x - vt, describing how the origin of one observer's coordinate system slides through space with respect to the other's, at speed v and along the x-axis of each frame. According to special relativity, this is only a good approximation at speeds small compared to the speed of light, and in general the result is not just an offsetting of the x coordinates; lengths and times are distorted as well. Consider 2 cartesian coordinate systems (x,y,z) and (x',y',z') such that the x and x' axes are coincident and the other 2 axes are parallel (i.e. a boost along the x-axis). The Lorentz transformation for time is: t - vx/c2 t' = -------- √1-v2/c2 Consider a time interval t1' - t2' = γ{t1 - vx1/c2 - t2 + vx2/c2} The time measurements in the moving frame are made at the same location so the expression reduces to: t12' = γt12 = γτ A clock in motion slows down as observed by a stationary observer in an inertial reference frame. t12' = time interval we observe in the other reference frame τ = time observed in observers own frame of reference ('Proper Time') γ = 1/√(1 - v2/c2) The Lorentz Transformation for spacial coordinates is: x - vt x' = ------ √1-v2/c2 y' = y z' = z Consider a length measurement x1' - x2' = γ{x1 - vt1 - x2 + vt2} The length measurements in the moving frame are made at the same time so the expression reduces to: x12' = γx12 which can also be written x = γx0 where x0 is the 'Proper Length' The Lorentz Transform can also be written as a matrix (see the note on the Lorentz Group for more details). For a boost of a contravariant vector in the x-direction we get: - - - - | γ -βγ 0 0 || ct | | -βγ γ 0 0 || x | where β = v/c | 0 0 1 0 || y | | 0 0 0 1 || z | - - - - - - | γct - βγx | = | -βγct + γx | | y | | z | - - For a boost of a covariant vector in the x-direction we get: - - - - | γ βγ 0 0 ||-ct | | βγ γ 0 0 || x | | 0 0 1 0 || y | | 0 0 0 1 || z | - - - - - - | γβx - γct | = | γx - βγct | | y | | z | - - Note: The boost matrix for a covariant vector is the inverse of the boost for a contravariant vector. (see notes on the Lorentz Group). Therefore, the product of the 2 resulting vectors is equal to: - - - - | (γct - βγx) (-βγct + γx) y z|| γβx - γct | - - | γx - βγct | | y | | z | - - = (γct - βγx)(γβx - γct) + (-βγct + γx)(γx - βγct) + y2 + z2 = γ2(ct - βx)(βx - ct) + γ2(-βct + x)(x - βct) + y2 + z2 = γ2[(ct - βx)(βx - ct) + (-βct + x)(x - βct)] + y2 + z2 = γ2[(-c2t2 - β2x2 + 2ctβx) + (-2βctx + x2 + β2c2t2)] + y2 + z2 = γ2[-c2t2 - β2x2 + x2 + β2c2t2] + y2 + z2 = γ2[-c2t2(1 - β2) + x2(1 - β2] + y2 + z2 = γ2(1 - β2)[-c2t2 + x2] + y2 + z2 = -c2t2 + x2 + y2 + z2 Note: the above matrix will be different for a boost in either the y or z direction. Note: Proper time can be a little confusing. It is the time measured by an observer in his own frame of reference regardless of whether that frame is moving or not. Thus a clock moving along a world line will show the proper time to the observer carrying the clock. Likewise for the proper length. Rapidity and Boosts ------------------- The Lorentz transformations for boosts can also be derived in a way that resembles circular rotations in 3d space using the hyperbolic functions. For the moment consider rotations in circular space: A rotation of x and y by θ around the z axis is given by: x' = xcosθ + ysinθ y' = -xsinθ + ycosθ z' = z t' = t The length of the vector, R, has to be the same in both frames. Therefore, x2 + y2 = x'2 + y'2 The rotation can also be written as the matrix. _ _ | cosθ sinθ 0 0| R = |-sinθ cosθ 0 0| | 0 0 1 0| | 0 0 0 1| - - Boosts ------ Consider a boost along the x coordinate. t t' | | / x = vt | | / | |/ o A | +----- x' | / | / | / | / ------------------- x The origin of the prime frame is moving on the line x = vt. The point A in the prime frame is at (x',t'). Therefore: x' = x - vt t = t' But this doesn't work for a light ray x = ±ct because x' = ct - vt = (c - v)t and x' = -ct - vt = -(c + v)t violates the idea that the laws of physics are the same in all inertial reference frames. To compensate for the fact that the speed of light in free space has to be the same value in all inertial reference frames. We start with, x2 - c2t2 = x'2 - c2t'2 Note: Although not discussed in this particular note, each side of the equation is the 4-vector formalism xμxμ for 2 dimensions where xμ = x - ct and xμ = x + ct This is the equation of a hyperbola. Instead, of using circular geometry using sines and cosines we now need to switch to the hyperbolic functions sinh and cosh. The tranformations that satisfy x2 - c2t2 = x'2 - c2t'2 are: x' = xcoshζ - ctsinhζ t' = -(x/c)sinhζ + tcoshζ y' = y z' = z In the prime frame x' = x - vt We can write: x - vt = xcoshζ - ctsinhζ ∴ x - xcoshζ = -ctsinhζ + vt ∴ x(1 - coshζ) = t(v - csinhζ) ∴ v(1 - coshζ) = (v - csinhζ) ∴ vcoshζ = csinhζ) ∴ v = ctanhζ or tanhζ = v/c tanh2ζ = v2/c2 Now, tanh2ζ = 1 - sech2ζ ∴ sech2 = 1 - v2/c2 ∴ 1/cosh2ζ = 1 - v2/c2 ∴ coshζ = 1/√(1 - v2/c2) Now, sinhζ = tanhζcoshζ = (v/c)/√(1 - v2/c2) Substituting into the original coordinate transformation formulas give. x' = x/√(1 - v2/c2) - (v/c)ct/√(1 - v2/c2) = (x - vt)/√(1 - v2/c2) ζ is the RAPIDITY defined as tanh-1β where β = v/c. ζ is the hyperbolic angle. It is analagous to but does not have the dimensions of velocity. In summary, the connections between γ, β and ζ are: β = v/c = tanhζ γ = 1/√(1 - v2/c2/) = coshζ βγ = (v/c)/√(1 - v2/c2) = sinhζ Rotational Transform -------------------- In addition to boosts we can also have rotations about the 3 different spatial axes. In this case the Lorentz transformation matrix is: - - | 1 0 0 0 | | 0 x x x | where the x's represent a 3D rotation matrix. | 0 x x x | | 0 x x x | - - In general there can be combination of boosts and rotations. Active versus Passive Transformations ------------------------------------- An active transformation is a transformation which actually changes the physical position of a point. A passive transformation is a change in the coordinate system in which the object is described. Lorentz Transformation of Fields -------------------------------- For a basic scalar field: φ(x) -> φ'(x) = φ(Λ-1x) The inverse appears in the argument because we want to express φ'(x) in terms of the 'untransformed' field, φ. In other words, the transformed field evaluated at the transformed point gives the same value as the original field evaluated at the point before it was transformed. For example consider a rotation, R, of a field about the z axis. z | | | | (1,0,0) --------o---- x / A / B o (0,1,0) / y The point A goes to B. So from B's perspective, A looks like R-1(0,1,0) = (1,0,0). The definition of a Lorentz invariant theory is that if φ(x) solves the equations of motion then φ(Λ-1x) also solves the equations of motion. The derivative of φ(Λ-1x) can be found as follows: Let yν = (Λ-1)νμxμ We can use the Chain rule to write: (∂'φ(xμ)/∂xμ) = (∂yν)/∂xμ)(∂φ(yν)/∂yν) Now, ∂yν/∂xμ = ∂(Λ-1x)νμ/∂xμ = (Λ-1)νμ So we end up with: (∂φ'(xμ)/∂xμ) = (Λ-1)νμ(∂φ(yν)/∂yν)    ≡ (Λ-1)νμνφ(y) Therefore, the derivative transforms like a vector which is not surprising since derivative of a scalar field is a vector. We can find the second derivative as follows: (∂φ(x))2 = ∂μφ(x)∂νφ(x)ημν   = (Λ-1)ρμρφ(y)(Λ-1)σνσφ(y)ημν   = ∂ρφ(y)∂σφ(y)ηρσ since ΛρμΛσνημν = ηρσ   = (∂φ(y))2 We now show that the Klein-Gordon, Dirac, and Weyl equations are Lorentz invariance. Klein-Gordon: (∂2 + m2)φ = 0 (∂2 + m2)φ'(x) -> ((Λ-1)νμν-1)νμν + m2)φ(Λ-1x) = (gνσνσ + m2)φ(Λ-1x) = (∂2 + m2)φ(Λ-1x) = 0 Dirac: (iγμμ - m)ψ = 0 In the case of the Dirac field the spinor also carries an orientation that can be rotated or boosted in spacetime. Therefore, ψα = S[Λ]αβψβ-1x) (iγμμ - m)ψ(x) -> (iγμ-1)νμν - m)S[Λ]ψ(Λ-1x) Multiply from the right by S[Λ]S[Λ]-1 = I to get: = S[Λ]S[Λ]-1(iγμ-1)νμν - m)S[Λ]ψ(Λ-1x) = S[Λ](iS[Λ]-1γμS[Λ](Λ-1)νμν - m)S[Λ]ψ(Λ-1x) = S[Λ](i(Λ)μσγσ-1)νμν - m)S[Λ]ψ(Λ-1x) = S[Λ]{iγνν - m)ψ(Λ-1x) = 0 Weyl: iσμμψL = 0 iσμμψL -> (iσμ-1)νμν)S[Λ]ψL-1x) Using the same process as above we get: = S[Λ]{iσννL-1x) = 0