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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: May 4, 2019

Eigenvectors and Eigenvalues ---------------------------- Eigenvectors are the directions along which a linear transformation acts by stretching/compressing and/or reversing a vector. Eigenvalues give you the factors by which the vectors are stretched/compressed. The basic equation is: A(αv) = αAv = αλv = λ(αv) Where, A is a matrix. α is a scalar. v is a vector The Characteristic Polynomial ----------------------------- The eigenvalues are found from: (A - λI)x = 0 This has a solution only if det(A - λI) = 0 The eigenvectors are found from: A - λI = 0 Example: - - A = | 7 3 | | 3 -1 | - - - - - - Step 1. λI = λ| 1 0 | = | λ 0 | | 0 1 | | 0 λ | - - - - - - - - Step 2 : A - λI = | 7 3 | - | λ 0 | | 3 -1 | | 0 λ | - - - - - - = | 7-λ 3 | | 3 -1-λ | - - - - Step 3: det| 7-λ 3 | = λ2 - 6λ - 16 = (λ - 8)(λ + 2) = 0 | 3 -1-λ | - - ∴ λ = 8 and -2 - - Step 4: | -1 3 | = B | 3 -9 | - - For the eigenvectors (A - λI) = 0 - - - - | -7 3 | - 8| 1 0 | = 0 | 3 -1 | | 0 1 | - - - - Therefore, - - - - - - | -15 3 || x | = | -15x + 3y | = 0 | 3 -9 || y | | 3x - 9y | - - - - - - => x = 3 and y = 1 We can also see this another way: Set Bv = 0 - - - - - - | -1 3 || x1 | = | 0 | | 3 -9 || x2 | | 0 | - - - - - - -x1 + 3x2 = 0 3x1 - 9x2 = 0 ∴ x1 = 3 and x2 = 1 - - - - - - | 7 3 || 3 | = 8| 3 | | 3 -1 || 1 | | 1 | - - - - - - We follow the same procedure for λ = -2 - - | 9 3 | = B | 3 1 | - - Which leads to: x1 = 1 x2 = -3 - - - - - - | 7 3 || 1 | = -2| 1 | | 3 -1 || -3 | | -3 | - - - - - - Notes: For an n x n matrix. Tr = sum of eigenvalues. det = product of eigenvalues. Significance ------------ There are many problems that can be modeled with linear transformations, and the eigenvectors give very simple solutions. For example, consider the system of linear differential equations dx/dt = ax + by dy/dt = cx + dy Solving this system directly is complicated because the 2 equations are coupled to each other. - - - - - - | dy1/dt | = | a b || y1 | | dy2/dt | | c d || y2 | - - - - - - . y = Ay Solutions are of the form: y1 = αexp(λt) and, y2 = βexp(λt) We can write this as: - - y = exp(λt)| α | | β | - - = exp(λt)e d(αexp(λt))/dt = λexp(λt)e λexp(λt)e = λexp(λt)e Now Ae = λe λexp(λt)e = Aexp(λt)e λe = Ae This is a solution for one eigenvalue. We can use the same argument to get a solution for thesecond eigenvalue. The general solution is obtained by taking linear combinations of these two solutions, so the general solution is of the form: y = c1exp(λ1)e1 + c2exp(λ2)e2 - - - - - - | y1 | = c1exp(λ1)| α1 | + c2exp(λ2)| α2 | | y2 |    | β1 |    | β2 | - - - - - - Where c1 and c2 are constants determined using the initial conditions. Example: dx/dt = 4x - y dy/dt = 2x + y We can write this in matrix form as: - - - - - - | dx/dt | = | 4 -1 || x | | dy/dt | | 2 1 || y | - - - - - - Using the above processes the eigenvalues and eigenvectors are: - - λ1 = 3 e1 = | 1 |    | 1 | - - - - λ2 = 2 e1 = | 1/2 |    | 1 | - - - - | x | = Cexp(λ1t)e1 + Dexp(λ2t)e2 | y | - - - - - - - - | x | = Cexp(3t)| 1 | + Dexp(2t)| 1/2 | | y | | 1 | | 1 | - - - - - - x = Cexp(3t) + Dexp(2t)/2 y = 2Cexp(3t) + Dexp(2t) Geometry of Real and Complex Eigenvalues ---------------------------------------- When the eigenvalues are real the vectors along the eigenvectors (bases) are stretched, compressed or reversed. / / λx x // // Aξ = λξ When the eigenvalues are complex the story is a different. Lets go through the analysis to see what happens. If λ = a + ib is complex then the eigenvector, ξ = x + iy, is complex. The eigenvalue equation can now be written as: A(x + iy) = (a + ib)((x + iy) THerefore, Ax + iAy = (ax - by) + i(bx + ay) Which can be broken down as follows: Ax = (ax - by) and, Ay = (bx + ay) Now lets change the basis of the vector space: e1 = x = Re(ξ) e2 = y = Im(ξ) - - - - [A]e = | α12 || e1 |   | α2 α1 || e2 | - - - - - -   = | α1e1 - α2e2 |   | α2e1 + α1e2 | - - Lets look at amore complicated example using the matrix, J1 from so (3,1) that generates rotations in the yz about the x axis. - -   | 0 0 0 0 | iJ1 = | 0 0 0 0 |   | 0 0 0 1 |   | 0 0 -1 0 | - - The eigenvalues, λ, are - 0, -i, i The eigenvectors, ξ, are: - - - - - - - - | 1 | | 0 | | 0 | | 0 | | 0 |, | 1 |, | 0 |, | 0 | | 0 | | 0 | | i | | -i | | 0 | | 0 | | 1 | | 1 | - - - - - - - - - - - - - - - - | 0 0 0 0 || 1 | | 0 | | 1 | | 0 0 0 0 || 0 | = | 0 | = 0| 0 | | 0 0 0 1 || 0 | | 0 | | 0 | | 0 0 -1 0 || 0 | | 0 | | 0 | - - - - - - - - - - - - - - - - | 0 0 0 0 || 0 | | 0 | | 0 | | 0 0 0 0 || 1 | = | 0 | = 0| 1 | | 0 0 0 1 || 0 | | 0 | | 0 | | 0 0 -1 0 || 0 | | 0 | | 0 | - - - - - - - - - - - - - - - - | 0 0 0 0 || 0 | | 0 | | 0 | | 0 0 0 0 || 0 | = | 0 | = -i| 0 | | 0 0 0 1 || i | | 1 | | i | | 0 0 -1 0 || 1 | | -i | | 1 | - - - - - - - - - - - - - - - - | 0 0 0 0 || 0 | | 0 | | 0 | | 0 0 0 0 || 0 | = | 0 | = i| 0 | | 0 0 0 1 || -i | | 1 | | -i | | 0 0 -1 0 || 1 | | i | | 1 | - - - - - - - - Look at the eigenvector corresponding to i. - - - - - - | 0 | | 0 | | 0 | ξ = | 0 | ≡ | 0 | + i| 0 | | -i | | 0 | | -1 | | 1 | | 1 | | 0 | - - - - - - = Re(ξ) + iIm(ξ) = e1 + ie2 Where we have defined e1 and e2 as our basis vectors. Therefore, - - - -   | 0 |  | 0 | e1 = | 0 | and e2 = | 0 |   | 0 |  | -1 |   | 1 |  | 0 | - - - - Thus A represents a rotation in the yz plane. We would also see this if we did a similar calculation for λ = -i. Quantum Mechanics ----------------- The possible states of a quantum mechanical system are represented by unit vectors (called state vectors) residing in a complex Hilbert space (state space). In any measurement on a quantum mechanical system, the value of the observable attained will be one of the eigenvalues of the Hermitian operator that corresponds to the observable. The eigenvalues of each eigenstate correspond to the allowable values of the quantity being measured. Following the measurement, the state of the system will be the corresponding eigenvector. The eigenvectors of a Hermitian operator form a complete set. That is they from an orthonormal basis. The eigenvalues must be real. An eigenvalue which corresponds to two or more different linearly independent eigenvectors is said to be DEGENERATE, - - | 0 1 1 | | 1 0 1 | | 1 1 0 | - - λ1 = -1 λ2 = -1 λ3 = 2 - - - - - - | 1 | | -1 | | 1 | | 1 | | 0 | | 1 | | 0 | | 1 | | 1 | - - - - - -