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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 27, 2018

Time Evolution -------------- Recall from classical mechanics that the Hamiltonian describes how x and p change with time. Thus, it describes flow with time through phase space (the time evolution of phase space). H = p2/2m + U(x) Hamilton's equations are: . ∂H/∂p = x . ∂H/∂x = -p Consider some arbitrary function F(x,p) . . dF(x,p)/dt = (∂F/∂p)p + (∂F/∂x)x = -(∂F/∂p)(∂H/∂x) + (∂F/∂x)(∂H/∂p) = {F,H} where {} are Poisson brackets Therefore, the Hamiltonian is the generator of time evolution. Return to QM: The basic requirement is that the inner product between states does not change with time. This means that whatever the operator is it must be unitary (U). Therefore, |a> -> U(t)|a> and, <b| -> <b|U(t) and, UU = 1 = I (identity operator) and, <b|a> -> <b|U(t)U(t)|a> = <b|a> Therefore, the states are unchanged. Let's apply an operator to move a state forward in time by an infinetisimally small amount, ε. |ψ(t + ε)> = U(ε)|ψ(t)> When ε = 0, U(0) = 1 Consider U(ε) = 1 - (iεH/h) Therefore, U(ε)U(ε) = (1 - iεH/h)(1 + iεH/h) = 1 (iε/h)(H - H) = 0 Thus, H = H |ψ(t + ε)> = (1 - iεH/h)|ψ(t)> Therefore, |ψ(t + ε)> - |ψ(t)> = (-iεH/h)|ψ(t)> Divide both side by ε. The LHS becomes a differential and we get: ih∂|ψ(t)>/∂t = H|ψ(t)> This is the Time dependent Schrodinger equation that describes how a state function changes with time. Solution to the TDSE -------------------- ih∂ψ/∂t = Hψ = (p2/2m)ψ = (1/2m)(-ih∂/∂x)(-ih∂/∂x)ψ ih∂ψ/∂t = -(h2/2m)∂2ψ/∂x2 ∂ψ(x,t)/∂t = (ih/2m)∂2ψ(x,t)/∂x2 Solutions to the equation: The eigenvectors of the momentum operator are: exp(ipx/h) Let ψ(t) = f(t)exp(ipx/h) Plug this into SE: . . ψ = fexp(ipx/h) = (ih/2m)f(ip/h)2exp(ipx/h) . f = (-i/2mh)p2f The solution is, f = exp(-ip2t/2mh) = exp(-iEt/h) ... The QUANTUM PROPAGATOR f is the operator that describes the process of a physical system in time and must be a unitary operator. We can say f*f = ff* = I This leads to: ψ(t) = ψ(x)exp(-iEt/h) = exp(ipx/h)exp(-iEt/h) = exp(ipx/h)exp(-iωt) where E = hω = ih∂/∂t|ψ(t) = H|ψ(t)> = |ψ(t)> = exp(-iHt/h)|ψ(0)> We can also obtain the quantum propagator by evolving a state using n incremental time periods, ε. Therefore, ε = t/n and we can write: f = (1 - itH/hn)n = exp(-itH/h) Therefore, we can write: |ψ(t + ε)> = exp(-itH/h)|ψ(t)> Expectation Value of the Time Derivative of an Operator ------------------------------------------------------- . <O> = <ψ|O|ψ> = <dψ/dt|O|ψ> + <dψ/dt|O|dψ/dt> = -i/h<ψ|HO|ψ> - i<ψ|OH|ψ> = -i/h<ψ|HO - OH|ψ> = -i/h<ψ|[H,O]|ψ> . <O> = -i/h[O,H] This is a very important relationship that we will use frequently. Symmetry Operations ------------------- Let O be a unitary operator corresponding to a rotation or a translation. Let U corresponds to a time translation. OO = (1 - iεO)(1 + iεO) = 1 + iε(O - O) For this to be equal to 1 O - 0 which means that O must be Hermitian. Recall the TDSE: ih∂|ψ>/∂t = H|ψ> Now, operate on ψ with O. ih∂(O|ψ>)/∂t = HO|ψ> LHS: ih∂(O|ψ>)/∂t = Oih∂(|ψ>)/∂t = OH|ψ> RHS: HO|ψ> ∴ OH = HO If O commutes with H (i.e. [O,H] = 0) then there is a symmetry. Therefore, to find symmetries we look for operators that commute with the time evolution operator (the Hamiltonian). This is the quantum mechanical equivalent of NOTHER'S THEOREM in classical mechanics which states that if there is a symmetry there is an associated conserved quantity. In this case the conserved quantity is the energy. A symmetry is a unitary operation that commutes with the Hamiltonian and preserves the relationship between states. If the states have a particular relationship, that relationhip will be maintained. For example, if the states are orthogonal they will remain orthogonal. We can replace U with I - iεH and O with I - iεO to get the commutator: (I - iεH/h)(I - iεO/h) - (I - iεO/h)(I - iεH/h) = I - iεO/h - iεH/h - ε2HO/h2 - (1 - iεH/h - iεO/h - ε2OH/h2) = 0 H and G are referred to as GROUP GENERATORS and both are hermitian. Consider the following examples. Translation Symmetry -------------------- O|ψ(x)> = |ψ(x - ε)> = ψ(x) - ε(∂ψ/∂x) + O(ε2) ... Taylor series Therefore, O = I - ε(∂/∂x) = I - iεp/h since p = -ih∂/∂x Therefore, the generator of translation is p/h. Now H = p2/2m so the commutator is: [p2/2m, p/h] This is equal to 0 and so momentum is the conserved quantity. Rotation Symmetry ----------------- O|ψ(θ)> = |ψ(θ - ε)> = ψ(θ) - ε(∂ψ/∂θ) Therefore, O = I - ε∂ψ/∂θ By analogy with the momentum generator, we can define the angular momentum generator, L, as: L = -ih(∂/∂θ) Therefore, O = I - iεL/h The eigenvalue equation is: L|ψ> = m|ψ> -ih∂ψ(θ)/∂θ = mψ(θ) Therefore, for angular momentum the eigenvectors and eigenvalues are: Eigenvector: ψ(θ) = exp(-imθ/h) Eigenvalue: m Likewise, for the opposite direction we get: Eigenvector: ψ(θ) = exp(imθ/h) Eigenvalue: -m The energies corresponding to m and -m are the same but the states are different therefore the energy levels are at first glance appear to be DEGENERATE. However, this degeneracy can be easily broken by the presence of a magnetic field. Therefore, rotation symmetry by itself is not enough to tell you that the levels are truly degenerate. In order to prove this we need to add another symmetry - reflection symmetry. Reflection Symmetry ------------------- Reflection symmetry is a discrete symmetry. ^ | |---> | ------------------------ | |---> | v Consider the angular momentun again. Consider the reflection operator, M, which relects about the x axis. M|ψ(θ)> = |ψ(-θ)> If there is a reflection symmetry, the energy associated with clockwise angular momentum states has to equal the energy associated with anti-clockwise angular momentum states. It can also be shown that if any 2 symmetries don't commute with each other it implies degeneracy. We will show this for [M,L]. Consider, [M,L]exp(imθ/h) = (ML - LM)exp(imθ/h) = MLexp(imθ/h) - LMmxp(imθ/h) = Mmexp(imθ/h) - Lexp(-imθ/h) = mexp(-imθ/h) + mmexp(-imθ/h) ≠ 0 Closed Symmetry Groups --------------------- Consider 2 hermitian operators A, B. Assume: [A,H] = 0 [B,H] = 0 [A,B] = iC If C is indeed a new operator. It is easy to show that this new operator also commutes with H. [C,H] = [[A,B],H] - [H,[A,B]] = (AB - BA)H - H(AB - BA) = ABH - BAH - HAB + HBA = 0 In this case, the process of commutation is 'closed' in that any 2 generators produce the third generator in the group. As an example, consider the following 2D rotation: δx = -εy δy = εx δψ = (∂ψ/∂x)δx + (∂ψ/∂y)δy = -ε(∂ψ/∂x)y + ε(∂ψ/∂y)x = iεLzψ = -iεypx + iεxpy Therefore, Lz = xpy - ypx This is the z component of L = r x p. Likewise. Lx = ypy - ypx Ly = zpy - ypx Now calculate [Lx,Ly]: [Lx,Ly] = [ypz - zpy, zpx - xpz]    = -ihypx + ihxpy    = ih(xpy - ypx)    = ihLz The fact that [Lx,y,z,H] = 0 implies that angular momentum is conserved.