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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: March 6, 2020

Symmetry and Conservation Laws - Noether's Theorem -------------------------------------------------- Calculus of Variations ---------------------- Consider a critical point (minima). The definition is that ∂f(x)/∂x = 0. A small displacement from x0 does not change the value of the function to a first order in the displacement from that point. We can see this from Taylor's theorem: f(x) = f(x0 + δx) = f(x0) + f'(x0)δx + 1/2f''(xo)δx2 + ... where δx is referred to as the variation of x. In general: δf(x) = f'(x)δx (i.e. the change in f(x) = the rate of change of f(x) wrt x * the change in x). There is a multi-variable version of the Taylor series: f(x,y) = f(x0,y0) + (∂f/∂x)δx + (∂f/∂y)δy For the Lagrangian we can write: . . . L(qi,q) = L + Σ{(∂L/∂qi)δqi + (∂L/∂qi)δqi} i So the change in L is just:
. . δL = Σ{(∂L/∂qi)δqi + (∂L/∂qi)δqi} i
and the change in the Action is given by: . . δS = ∫Σ{(∂L/∂qi)δqi + (∂L/∂qi)δqi}dt i This is referred to as the first variation in the action. FROM THIS POINT ON WE WILL DROP THE Σ FROM THE EQUATIONS TO MINIMIZE COMPLEXITY ON THE UNDERSTANDING THAT SUMMATION OVER i IS IMPLIED. Integration by parts of the second term in the integral yields the following result: . . . (∂L/∂qi)δqi = (∂L/∂qi)δqi - d/dt(∂L/∂qi)δqi Substituting back into the equation and rearranging gives:
. . t2 δS = ∫dt{(∂L/∂qi) - d/dt(∂L/∂qi)}δqi + [(∂L/∂qi)δqi] t1
In the case where the endpoint of the trajectories are the same, the last term vanishes and we are left with the E-L equations. Now, if the original trajectory is a solution to the equations of motion, the variation in the E-L equation is equal to zero and there is no variation in the action. Linear Translational Symmetry ----------------------------- But what happens if we shift the entire trajectory so the endpoints change as in the following diagram? Now we need to figure out what we need to do with the last term. If we assume that any symmetry operation doesn't change the action, then the last term must also vanish when the endpoints change. This implies that the quantity at t2 is equal to the quantity at t1 or in other words, the quantity is conserved. Thus, . (∂L/∂qi)δqi = 0 and so, . δS = ∫dt{(∂L/∂qi) - d/dt(∂L/∂qi)}δqi Now the term inside the {} is the Euler-Lagrange equation: . d(∂L/∂qi)/dt - ∂L/∂qi = 0 Consequently δS = 0 and there is a conserved quantity. To find the conserved quantity we need to look at δL. . . δL = (∂L/∂qi)δqi + (∂L/∂qi)δqi . . Now, (∂L/∂qi) = pi and ∂L/∂qi = pi. pi (ofter written as Πi) is called the "canonical momentum conjugate to the coordinate qi". For example, . . . L = mx2/2 ∴ ∂L/∂x = mx = p Using the Euler-Lagrange equation: . d(∂L/∂qi)/dt - ∂L/∂qi = 0 It follows that: dpi/dt = ∂L/∂qi Therefore, δL becomes: . . δL = piδqi + piδqi Now d(FG)/dt = FG + FG. Therefore, we can write δL as: δL = d(piδqi)/dt For symmetry we require that δL = 0 Therefore, d(piδqi)/dt = 0 In general, we can replace δqi with εfi(q) where fi(q) is a function that defines the particular symmetry operation (in this case a linear translation). Thus, we can write: d(piεfi(q))/dt = 0 But ε is just a small number (i.e. a constant) so, d(pifi(q))/dt = dQ/dt = 0 Where Q = pifi(q) is the NOETHER CHARGE equal to the conserved quantity. Example: Consider a linear translation along the x axis. . L = (1/2)mx2 f(x) = 1 . . Q = Πxf(x) => ∂L/∂x = mx = the conserved quantity Rotational Symmetry ------------------- Rotate axes by small angle ε. The 2D rotation matrix is: - - - - - - | x' | = | cosε -sinε || x | | y' | | sinε cosε || y | - - - - - - Thus, x' = xcosε - ysinε y' = xsinε + ycosε If ε = 0 => (x,0), if ε = 90° => (0,-x) Basically, the x direction becomes the y direction and the y direction becomes the = -x direction. Now if ε is very small and is in radians then sinε ~ ε and cosε ~ 1. Therefore we can write: fx = δx = -εy fy = δy = εx Q = pxfx + pyfy   . . = -ypx + xpy => xmy - ymx = z component of the angular momentum, L = r ^ p. The conserved quantity is the angular momentum, L. Time Translation Symmetry ------------------------- Time translation t -> t + δt where δt = εT (T is the generator of time evolution, ε is infinitesimal) A positive time shift in the trajectory cause q to move backwards. The change in q can be viewed as being in the vertical, t, or the horizontal, q. Let's look at the horizontal shift. The translation is q(t) -> q(t - ε) The trajectories can be viewed as being the same as a coordinate transform with overhanging pieces A and B. Note: A shift in q represents a negative coordinate shift. . . tB δS = ∫dt{(∂L/∂q) - d/dt(∂L/∂q)δq} + [(∂L/∂q)δq]| tA . tA = 0 + δq∂L/∂q| tB Now we need to include the contributions from A and B, . tA δS = δq∂L/∂q| + A - B tB Using Taylor series we can write, . . q(t - ε) = q(t) - εq therefore δq = -εq If ε is sufficiently small, the action for A and B is ∫Lε. We can write, SA = εL(tA) and SB = εL(tB). Therefore, we end up with: . . tA δS = -ε(∂L/∂q)q}| + εL(tA) - εL(tB) tB . . . . = ε[L - (∂L/∂q)q] - ε[L - (∂L/∂q)q] tA tB As in the case of linear translation, this term must equal 0 on the assumption that the symmetry operation does not change the action. Its value at tA must equal its value at tB. Again, ε is just a small constant that is irrelevant. Therefore, we can conclude from the above that the conserved quantity is the energy: . Q = L - qipi = -H Or, . The HAMILTONIAN, H = -L + qipi Example: . . L = 1/2mx2 - U(x) therefore Π = mx . . . H = -1/2mx2 + U(x) + xmx . = 1/2mx2 + U(x) QED Inclusion of Explicit Time Dependence ------------------------------------- . The Lagrangian has an implicit time dependence (qi). We now want to consider an explicit time dependence i.e., . L(q,q,t). Example: L = (1/2){mv2 - k(t)x2} ... k changes with time. To see if L changes with time we look at dL/dt. . . .. dL/dt = (∂L/∂qi)qi + (∂L/∂qi)qi + ∂L/∂t \ Now, explicit time term . . . (∂L/∂qi)(qi) = piqi and, . .. .. (∂L/∂qi)(qi) = piqi Therefore, . . .. dL/dt = piqi + piqi + ∂L/∂t Which can be written as: . . . dL/dt = d(piqi)/dt + ∂L/∂t using d(FG)/dt = FG + FG . From before piqi = H + L dL/dt = dH/dt + dL/dt + ∂L/∂t 0 = dH/dt + ∂L/∂t H varies with time only if the Lagrangian has an explicit time dependence. If a system is time translation independent then ∂L/∂t = 0 and H is conserved. Summary ------- In general Noether's theorem can be written as: t -> t + δt where δt = ΣεT Where T is the generator of time evolution. q -> q + δq where δq = ΣεQ Where Q is the generator of coordinates. . . . {(∂L/∂q)q - L}T - (∂L/∂q)Q For translational and rotational invariance, T = 0, Q = 1: . p = ∂L/∂q   . . L = -ypx + xpy => xmy - ymx = z For time invariance, T = 1 and Q = 0: . . H = (∂L/∂q)q - L In summary, Noether's theorem states that each symmetry of a system leads to a physically conserved quantity. Symmetry under translation corresponds to the conservation of momentum, symmetry under rotation corresponds to the conservation of angular momentum, symmetry in time corresponds to the conservation of energy.