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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: November 18, 2021 ✓

Band Theory of Solids --------------------- Linear Combination of Atomic Orbitals ------------------------------------- The linear combinations of atomic orbitals (LCAO) approach can be used to estimate the molecular orbitals that are formed upon bonding between a molecule’s constituent atoms. Similar to an atomic orbital, a Schrödinger equation can be constructed for a molecular orbital as well. Linear combinations of atomic orbitals, or the sums and differences of the atomic wavefunctions, provide approximate solutions to these molecular Schrodinger equations. These sums and differences represent constuctive and destuctive interference between the individual atomic wavefunctions. For simple diatomic molecules, the obtained wavefunctions are represented mathematically by the equations: ΨConstructive = ψ1 + ψ2 ΨDestructive = ψ1 - ψ2 Where the atomic orbitals ψi are the combined spatial and spin wavefunctions. If we compute the probabilities for each situation we see that |ψConstructive|2 ≠ 0 whereas |ψDestructive|2 = 0. This indicates that there is high electron density between the nuclei in the constructive case that results in a net attractive force between the atoms (see more on this below). The net result is the splitting of orbital energy levels into SINGLET states as shown below. antibonding MO (destructive) σ+ - spend most of their time away from regions between 2 nuclei. | E v ^ -------- | / \ | AO atom 1 ----- ------- AO atom 2 \ / -------- ^ | bonding MO (constructive) σ - spend most of their time between 2 nuclei. The orbitals get filled starting at the lowest energy. The bonding orbitals get filled first followed by the anti-bonding orbitals. The molecular stability is determined by the BOND ORDER which is defined as: Bond Order = (1/2){No. of electrons in the BO - No. of electrons in the ABO} The higher the BO, the greater the stability. A BO of 0 indicates the molecule is completely unstable. Consider 2 H atoms. The individual AOs combine to form 2 MOs. The Pauli Exclusion Principle dictates that a state can only contains 2 electrons with opposite spins. Thus, the lowest energy configuration for the system is when an H2 molecule is formed. The Bond Order is 1. The physical interpretation of this is as follows: When the hydrogen atoms are brought together, two new forces of attraction appear because of the attraction between the electron on one atom and the proton on the other. But two forces of repulsion are also created because the two negatively charged electrons repel each other, as do the two positively charged protons. The force of repulsion between the protons can be minimized by placing the pair of electrons between the two nuclei. The distance between the electron on one atom and the nucleus of the other is now smaller than the distance between the two nuclei. As a result, the force of attraction between each electron and the nucleus of the other atom is larger than the force of repulsion between the two nuclei, as long as the nuclei are not brought too close together. If the nuclei are close enough together to share the pair of electrons, but not so close that repulsion between the nuclei becomes too large then the net result of pairing the electrons and placing them between the two nuclei is a system that is more stable than a pair of isolated atoms (). The hydrogen atoms in an H2 molecule are therefore held together (or bonded) by the sharing of a pair of electrons. In the case of 3 H atoms, the atomic orbitals combine to form 3 molecular orbitals. Again, each orbital can hold 2 electrons. However, there are only 3 electrons in the system, so the lowest energy situation is for 2 electrons to be in the lowest energy state, 1 electron in the next highest state and 0 in the highest energy state. However, the overall energy is still higher than the H2 system. That is why we don't get H3 molecules in nature. In this case the Bond Order is (1/2){2 - 1} = 1/2 which indicates that H3 is not that stable. Likewise, we can extend this idea to a 2 He atom system. There are 2 resulting MOs each containing 2 electrons. At a simplifed level, there is no energy advantage of forming a molecule so the He atoms don't combine. The Bond Order is (1/2){2 - 2} = 0 which confirms this. In general, we can summarize as follows: When N atoms are brought together, each individual energy state splits into a band of N energy levels. There will be bands corresponding to the 1s, 2s etc. atomic orbitals. The width of the band depends on the strength of the interaction and the overlap between neighboring atoms. Two or more energy bands may coincide in energy at special positions or there may be gaps between bands. This leads to the distinction between metals, insulators and semiconductors. Degenerate Orbitals - Hund's Rule --------------------------------- In the case of degenerate orbitals (i.e. orbitals with the same energy) Hund's rule is applied. Hund's rule basically says that the electrons don't pair up until they have to. They space out with one in each orbital. Only when they run out of orbitals with equal energy do they begin to pair up. A good analogy is an empty bus. Assume that everyone boarding the bus is travelling alone. Because they would rather sit by themselves than next to a stranger, passengers will individually fill the rows even if there is space next other passengers. It is not until all the empty rows are filled that passengers will begin sitting next to strangers and fill the row. Quantitative Band Theory --------------------------- Density of States Function: It is impractical to solve the Schrodinger equation for energy for every molecular orbital. Fortunately, there is an easier way which provides a good approximation in most situations. If we consider the electrons being confined inside a potential well where the dimensions of the well are equal to the dimensions of the metal we can use the FREE ELECTRON THEORY to derive a density of states function g(E). The density of states equals the density per unit volume and energy of the number of solutions to Schrodinger's equation. Free Electron Theory: In the FE theory a metal is described in terms of a potential well. ------------ --------- V(x) | | | | | | --------- 0 L Inside the well, V(x) = 0 The time independent Schrodinger equation is: ∂2ψ/∂x2 + 2mEψ/h2 = 0 or, ∂2ψ/∂x2 = -2mEψ/h2 = -k2ψ Where k = √(2mE)/h This has the solution: ψ(x) = Asinkx + Bcoskx ≡ Cexp(ikx) + Dexp(-ikx) The boundary conditions are: ψ(0) = ψ(L) = 0 ψ(0) = Asink0 + Bcosk0 = 0 => B = 0 and, ψ(L) = AsinkL = 0 => A = 0 or sinkL = 0 In the latter case, this implies that k = nπ/L Now, E = p2/2m = h2k2/2m = n2π2h2/2mL2 Thus, only certain values of energy are allowed which correspond to n = 0, 1, 2, 3 ... This result can be generalized to 3D as follows: E = π2h2|n|2/2mL2 where |n|2 = nx2 + ny2 + nz2 Density of States: E = π2h2|n|2/2mL2 |n| = √(2mL2E/π2h2) Number of states in n-space with energy less than E is given by the volume of a sphere of radius, R = √(2L2mE/π2h2) Therefore, Total number of states = (4π/3)(2mL2E/π2h2)3/2 Real space occupies 1/8 of sphere. Thus, Number of real states = (4π/3)(2mL2E/π2h2)3/2 which must equal ∫g(E)dE. E ∫g(E)dE = (1/6π2h3)(2m)3/2E3/2 0 Differentiate this to obtain: g(E) = (1/4π2h3)(2m)3/2E1/2 Accounting for the Pauli Exclusion Principle we get: g(E) = (1/2π2h3)(2m)3/2E1/2 What happens if the PE inside the well is not 0 but is periodic as would be the case in a normal metal? Bloch Function -------------- The Bloch function is a wavefunction for a particle in a periodically-repeating environment, most commonly an electron in a crystal. Bloch's theorem states that the energy eigenfunction for such a system may be written as the product of a plane wave envelope function and a periodic function (periodic Bloch function), uk(x), that has the same periodicity as the potential (uk(x) = uk(x + a)). The Bloch function is the solution to Schrodinger Equation with a potential V(x). Thus: (p2/2m + V(x))ψ = Eψ ψ = uk(x)e-ikx Kronig-Penney Model ------------------- Model crystal as a series of potential wells. <---- a -----> ----- ----------- ----------- --------- | | | | | | V| | | | | | -- -- -- b The solution to the time independent Schrodinger Equation is: cos(ka) = Psin(αa)/αa + cos(αa) where P = mbaV/h2 and α = √(2mE/h2) Note: Plotting the RHS against αa reveals that the electron in a periodic potential can only have energies that lie in certain bands. If P = 0, α = k and we get the free electron situation: E = p2/2m = h2k2/2m. When P is finite there are discontinuities at k = nπ/a. E-k curve --------- E \ | / . | . . | . | ^ | | Forbidden energy . | . v <-- dE/dk = 0 . | . \ | /<-- point of inflection d2/dk2 =0 \ | / . | . --------+--------. .---------+---------------- k -π/a +π/a <-- 1st Brillouin --> zone The resulting curve is no longer parabolic. However, if the forbidden energy gap is relatively large, the parabolic free-electron (E = h2k2/2m + V) approximation can be used at the extrema of the bands (the .. regions). The slope of the curve at the discontinuity is 0. However, if the gap is small this proves to be ineffective because the interaction between bands cannot be ignored. In this case it is necessary to use k.p perturbation theory to determine the electron energy dispersion. The periodic function uk satisfies the following Schrodinger-type equation: Hk(x) = Ekuk(x) We can write Hk(x) as the sum of an unperturbed Hamiltonian and a perturbation term: Hk = H0 + Hk' where H0 = (p2/2m + V) and Hk' = hk.p/m + h2k2/2m This combines to: Hk = p2/2m + V + hk.p/m + h2k2/2m were k.p = kx(-ih∂/∂x) + ky(-ih∂/∂y) + kz(-ih∂/∂z) Which can be solved to derive and expression for the dispersion of E: E = En0 + h2k2/2m + (h2/m2m≠n|<ψn0|k.p|ψm0|2/(En0 - Em0) Effective Mass ------------- vg = ∂ω/∂k = (1/h)∂E/∂k since E = hω The work done by an electric field, ε, is dE: dE = eεvgdt = eε((1/h)dE/dk)dt ∴ dE.dk/dt = eε(1/h)dE ∴ dk/dt = eε(1/h) dvg/dt = (1/h)(d2E/dkdt) = (1/h)(d2E/dk2)(dk/dt) = (1/h)(d2E/dk2)eε(1/h) = (1/h2)(d2E/dk2)eε Now F = ma = mdvg/dt = eε. Therefore, dvg/dt = (1/h2)(d2E/dk2)mdvg/dt 1 = (1/h2)(d2E/dk2)m Define: m* = h2/{d2E/dk2} For free electron E = h2k2/2m so d2E/dk2 = h2/m which yields m* = m as expected. The electron velocity increases from 0 at k = 0 (accelerating so d2E/dk2 is positive). The maximum velocity is at the point of inflection at k = π/2a (d2E/dk2 = 0). The electron velocity then decreases to 0 again at k = π/a (decelerating so d2E/dk2 is negative). Thus, we have the situation where m* changes from positive to negative as k increases. The negative mass corresponds to a HOLE.