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Astronomy

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Astronomical Distance Units .
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Celestial Coordinates .
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Celestial Navigation .
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Location of North and South Celestial Poles .

Chemistry

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Avogadro's Number
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Balancing Chemical Equations
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Stochiometry
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The Periodic Table .

Classical Physics

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Archimedes Principle
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Bernoulli Principle
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Blackbody (Cavity) Radiation and Planck's Hypothesis
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Center of Mass Frame
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Comparison Between Gravitation and Electrostatics
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Compton Effect .
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Coriolis Effect
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Cyclotron Resonance
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Dispersion
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Doppler Effect
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Double Slit Experiment
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Elastic and Inelastic Collisions .
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Electric Fields
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Error Analysis
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Fick's Law
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Fluid Pressure
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Gauss's Law of Universal Gravity .
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Gravity - Force and Acceleration
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Hooke's law
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Ideal and Non-Ideal Gas Laws (van der Waal)
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Impulse Force
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Inclined Plane
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Inertia
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Kepler's Laws
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Kinematics
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Kinetic Theory of Gases .
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Kirchoff's Laws
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Laplace's and Poisson's Equations
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Lorentz Force Law
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Maxwell's Equations
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Moments and Torque
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Nuclear Spin
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One Dimensional Wave Equation .
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Pascal's Principle
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Phase and Group Velocity
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Planck Radiation Law .
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Poiseuille's Law
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Radioactive Decay
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Refractive Index
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Rotational Dynamics
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Simple Harmonic Motion
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Specific Heat, Latent Heat and Calorimetry
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Stefan-Boltzmann Law
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The Gas Laws
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The Laws of Thermodynamics
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The Zeeman Effect .
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Wien's Displacement Law
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Young's Modulus

Climate Change

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Keeling Curve .

Cosmology

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Baryogenesis
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Cosmic Background Radiation and Decoupling
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CPT Symmetries
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Dark Matter
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Friedmann-Robertson-Walker Equations
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Geometries of the Universe
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Hubble's Law
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Inflation Theory
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Introduction to Black Holes .
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Olbers' Paradox
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Penrose Diagrams
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Planck Units
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Stephen Hawking's Last Paper .
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Stephen Hawking's PhD Thesis .
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The Big Bang Model

Finance and Accounting

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Amortization
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Annuities
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Brownian Model of Financial Markets
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Capital Structure
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Dividend Discount Formula
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Lecture Notes on International Financial Management
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NPV and IRR
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Periodically and Continuously Compounded Interest
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Repurchase versus Dividend Analysis

Game Theory

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The Truel .

General Relativity

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Accelerated Reference Frames - Rindler Coordinates
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Catalog of Spacetimes .
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Curvature and Parallel Transport
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Dirac Equation in Curved Spacetime
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Einstein's Field Equations
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Geodesics
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Gravitational Time Dilation
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Gravitational Waves
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One-forms
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Quantum Gravity
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Relativistic, Cosmological and Gravitational Redshift
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Ricci Decomposition
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Ricci Flow
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Stress-Energy Tensor
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Stress-Energy-Momentum Tensor
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Tensors
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The Area Metric
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The Equivalence Principal
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The Essential Mathematics of General Relativity
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The Induced Metric
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The Metric Tensor
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Vierbein (Frame) Fields
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World Lines Refresher

Lagrangian and Hamiltonian Mechanics

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Classical Field Theory .
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Euler-Lagrange Equation
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Ex: Newtonian, Lagrangian and Hamiltonian Mechanics
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Hamiltonian Formulation .
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Liouville's Theorem
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Symmetry and Conservation Laws - Noether's Theorem

Macroeconomics

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Lecture Notes on International Economics
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Lecture Notes on Macroeconomics
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Macroeconomic Policy

Mathematics

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Amplitude, Period and Phase
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Arithmetic and Geometric Sequences and Series
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Asymptotes
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Augmented Matrices and Cramer's Rule
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Basic Group Theory
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Basic Representation Theory
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Binomial Theorem (Pascal's Triangle)
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Building Groups From Other Groups
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Completing the Square
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Complex Numbers
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Composite Functions
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Conformal Transformations .
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Conjugate Pair Theorem
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Contravariant and Covariant Components of a Vector
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Derivatives of Inverse Functions
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Double Angle Formulas
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Eigenvectors and Eigenvalues
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Euler Formula for Polyhedrons
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Factoring of a3 +/- b3
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Fourier Series and Transforms .
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Fractals
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Gauss's Divergence Theorem
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Grassmann and Clifford Algebras .
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Heron's Formula
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Index Notation (Tensors and Matrices)
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Inequalities
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Integration By Parts
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Introduction to Conformal Field Theory .
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Inverse of a Function
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Law of Sines and Cosines
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Line Integrals, ∮
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Logarithms and Logarithmic Equations
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Matrices and Determinants
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Matrix Exponential
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Mean Value and Rolle's Theorem
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Modulus Equations
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Orthogonal Curvilinear Coordinates .
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Parabolas, Ellipses and Hyperbolas
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Piecewise Functions
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Polar Coordinates
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Polynomial Division
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Quaternions 1
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Quaternions 2
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Regular Polygons
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Related Rates
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Sets, Groups, Modules, Rings and Vector Spaces
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Similar Matrices and Diagonalization .
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Spherical Trigonometry
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Stirling's Approximation
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Sum and Differences of Squares and Cubes
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Symbolic Logic
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Symmetric Groups
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Tangent and Normal Line
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Taylor and Maclaurin Series .
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The Essential Mathematics of Lie Groups
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The Integers Modulo n Under + and x
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The Limit Definition of the Exponential Function
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Tic-Tac-Toe Factoring
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Trapezoidal Rule
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Unit Vectors
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Vector Calculus
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Volume Integrals

Microeconomics

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Marginal Revenue and Cost

Particle Physics

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Feynman Diagrams and Loops
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Field Dimensions
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Helicity and Chirality
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Klein-Gordon and Dirac Equations
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Regularization and Renormalization
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Scattering - Mandelstam Variables
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Spin 1 Eigenvectors .
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The Vacuum Catastrophe

Probability and Statistics

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Box and Whisker Plots
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Categorical Data - Crosstabs
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Chebyshev's Theorem
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Chi Squared Goodness of Fit
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Conditional Probability
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Confidence Intervals
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Data Types
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Expected Value
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Factor Analysis
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Hypothesis Testing
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Linear Regression
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Monte Carlo Methods
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Non Parametric Tests
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One-Way ANOVA
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Pearson Correlation
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Permutations and Combinations
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Pooled Variance and Standard Error
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Probability Distributions
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Probability Rules
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Sample Size Determination
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Sampling Distributions
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Set Theory - Venn Diagrams
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Stacked and Unstacked Data
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Stem Plots, Histograms and Ogives
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Survey Data - Likert Item and Scale
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Tukey's Test
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Two-Way ANOVA

Programming and Computer Science

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Hashing
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How this site works ...
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More Programming Topics
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MVC Architecture
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Open Systems Interconnection (OSI) Standard - TCP/IP Protocol
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Public Key Encryption

Quantum Computing

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The Qubit .

Quantum Field Theory

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Creation and Annihilation Operators
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Field Operators for Bosons and Fermions
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Lagrangians in Quantum Field Theory
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Path Integral Formulation
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Relativistic Quantum Field Theory

Quantum Mechanics

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Basic Relationships
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Bell's Theorem
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Bohr Atom
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Clebsch-Gordan Coefficients .
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Commutators
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Dyson Series
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Electron Orbital Angular Momentum and Spin
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Entangled States
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Heisenberg Uncertainty Principle
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Ladder Operators .
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Multi Electron Wavefunctions
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Pauli Exclusion Principle
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Pauli Spin Matrices
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Photoelectric Effect
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Position and Momentum States
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Probability Current
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Schrodinger Equation for Hydrogen Atom
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Schrodinger Wave Equation
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Schrodinger Wave Equation (continued)
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Spin 1/2 Eigenvectors
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The Differential Operator
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The Essential Mathematics of Quantum Mechanics
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The Observer Effect
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The Quantum Harmonic Oscillator .
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The Schrodinger, Heisenberg and Dirac Pictures
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The WKB Approximation
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Time Dependent Perturbation Theory
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Time Evolution and Symmetry Operations
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Time Independent Perturbation Theory
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Wavepackets

Semiconductor Reliability

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The Weibull Distribution

Solid State Electronics

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Band Theory of Solids .
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Fermi-Dirac Statistics .
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Intrinsic and Extrinsic Semiconductors
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The MOSFET
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The P-N Junction

Special Relativity

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4-vectors .
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Electromagnetic 4 - Potential
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Energy and Momentum, E = mc2
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Lorentz Invariance
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Lorentz Transform
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Lorentz Transformation of the EM Field
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Newton versus Einstein
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Spinors - Part 1 .
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Spinors - Part 2 .
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The Lorentz Group
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Velocity Addition

Statistical Mechanics

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Black Body Radiation
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Entropy and the Partition Function
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The Harmonic Oscillator
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The Ideal Gas

String Theory

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Bosonic Strings
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Extra Dimensions
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Introduction to String Theory
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Kaluza-Klein Compactification of Closed Strings
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Strings in Curved Spacetime
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Toroidal Compactification

Superconductivity

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BCS Theory
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Introduction to Superconductors
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Superconductivity (Lectures 1 - 10)
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Superconductivity (Lectures 11 - 20)

Supersymmetry (SUSY) and Grand Unified Theory (GUT)

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Chiral Superfields
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Generators of a Supergroup
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Grassmann Numbers
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Introduction to Supersymmetry
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The Gauge Hierarchy Problem

The Standard Model

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Electroweak Unification (Glashow-Weinberg-Salam)
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Gauge Theories (Yang-Mills)
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Gravitational Force and the Planck Scale
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Introduction to the Standard Model
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Isospin, Hypercharge, Weak Isospin and Weak Hypercharge
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Quantum Flavordynamics and Quantum Chromodynamics
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Special Unitary Groups and the Standard Model - Part 1 .
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Special Unitary Groups and the Standard Model - Part 2
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Special Unitary Groups and the Standard Model - Part 3 .
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Standard Model Lagrangian
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The Higgs Mechanism
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The Nature of the Weak Interaction

Topology

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Units, Constants and Useful Formulas

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Constants
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Formulas
Last modified: January 26, 2018

Kaluza-Klein Compactification of Closed Strings ----------------------------------------------- For a closed string at H, the angular momentum in the compactified dimensions, y, around the cylinder is: L = p x r = nh where r is the radius of compactification. Therefore, setting m = 1 we get: p = nh/r The energy is given by: E = pc = mc2 = cnh/r Therefore, m2 = n2h2/cr2 This referred to as a KALUZA-KLEIN particle. For the closed string wound around the cylinder in the compactified dimensions, the energy is: E = mc2 = 2πwrT Where T is the tension (the energy per unit length) and w is the WINDING NUMBER equal to the number of times the string is wound around the cylinder in the compactified dimensions. Therefore, m2 = (2πwrT)2/c4   = w2r2/α'2c4 Where α' = 1/2πT is the string length. The total mass squared (setting h = c = 1) is given by: m2 = n2/r2 + w2r2/α'2 + energy (mass) from oscillator modes. Ignoring the last term for now, it is possible to identify 2 different mass scales for a particle. A very crucial feature of this is that the quantized momentum modes of a closed string are not distinguishable from the the winding modes of the string in the compact dimension. This creates a symmetry between small and large distances called T-DUALITY. T-duality is a particular example of the idea of DUALITY in physics. Duality refers to a situation where two seemingly different physical systems turn out to be equivalent to each other. If two theories are related by a duality, it means that one theory can be transformed in some way so that it ends up looking just like the other theory. In other words, the two theories are mathematically different descriptions of the same phenomena. Here we have one theory strings propagating in spacetime shaped like a cylinder of radius, r, while in the other theory strings are propagating on a spacetime shaped like a cylinder of radius 1/r. The two theories are equivalent in the sense that all observable quantities in one description are identified with quantities in the dual description. In this case, the momentum in one description takes discrete values that are equal to the number of times the string winds around the circle in the dual description. Therefore, T-Duality for CLOSED strings can be summarized as follows: n <-> w and 1/r <-> r General Relativity in 5 Dimensions ---------------------------------- Kaluza–Klein theory is a unified field theory of gravitation and electromagnetism built around the idea of a 5th dimension beyond the 4 of spacetime. The theory hypothesises a 5D metric of the form:    - - g'ab ≡ | gμν + φ2 φ2Aμ |    | φ2Aν  φ2  |    - -    - -    ≡ | g'μν g' |    | g'ν5 g'55 |    - - Where Aμ is the vector potential and φ is the scalar potential. a and b span 5 dimensions and so we get a 4D gμν with an extra dimension defined in terms of Aμ and φ. We can write: g'μν = gμν + φ2, g' = gν5 = φ2Aμ and g'55 = φ2 The associated inverse metric is,    - - g'ab ≡ | gμν -Aμ    |    | -Aν gαβAαAβ + 1/φ2 |    - - This metric implies: ds2 = g'abdxadxb = gμνdxμdxν + φ2(Aνdxν + dx5)2 The Einstein field equations are obtained by using the 5D metric to construct 5D Christoffel symbols and then using these to construct the 5D Ricci tensor. The solutions to the field equations are most easily found for the the vacuum case i.e. where the presence of matter is not considered and the space is Ricci flat, meaning R'ab = 0. Also, to simplify the analysis, Kaluza also introduced the hypothesis known as the "cylinder condition", ∂g'ab/∂x5 = 0, that no component of the 5D metric depends on the fifth dimension. Without this assumption, the field equations of 5D relativity are enormously more complex. Later, Klein explained the cylinder condition by hypothesizing that the 5th dimension was both 'curled up' and microscopic. Kaluza also set φ equal to a constant. Under these conditions the solutions can be shown to be: R'55 = 0 leads to: gμνμνφ = (1/4)φ3FαβFαβ It shows that the electromagnetic field is a source for φ. R' = 0 leads to: (1/2)gβμμ3Fαβ) = 0 It has the form of the vacuum Maxwell equations if the scalar field is constant. R'μν - (1/2)g'μνR' = 0 leads to: Rμν - (1/2)gμνR = (1/2)φ2(gαβFμαFνβ - (1/4)gμνFαβFαβ) + (1/φ)(∇μνφ - gμνgμνμνφ) Where ∇μ is the covariant derivative from GR and Fμα is the electromagnetic tensor. This last equation shows that the electromagnetic tensor that emerges from the 5D vacuum solutions is a source for the 4D equations of gravity. This is a remarkable result. The right hand side of Einstein's field equations in 4D is the stress- energy-momentum tensor, so it seems there is an equivalence between the electromagnetic field (charge) and the flow of momentum of the Kaluza-Klein particle (H in the above diagram) in the 5th dimension. In other words, the electromagnetic field with charge as the source corresponds to the gravitation forces of the 5 dimensional theory. But what about the winding number? Is there a corresponding field to the electromagnetic field whose source are winding numbers instead of momenta. It turns out there is. The field is called the KALB-RAMOND field, Bμν. The Kalb-Ramond field is another form of the vector potential, Aμ that has two indices instead of one. This difference is related to the fact that while Aμ is integrated over the world lines of particles to get its contributions to the action, the Kalb–Ramond field must be integrated over the 2D worldsheet of the string. Thus, while the action for a charged particle moving in Aμ has the form: S = -q∫dxμAμ A string coupled to the Kalb–Ramond field has the form: S = -∫dxμdxνBμν So we can summarize by saying that T duality also exists for charge. On the one hand there is an electromagnetic field associated whose sources are the momenta. On the other hand there is an analagous field, the Kalb-Ramond field, whose sources are the winding numbers. In both cases, the fields correspond to the gravitation forces of the 5th dimensional theory. The attraction and repulsion between strings originates with the sign of the momentum quantum number, n, and the sign of the winding number, w. Strings with opposite momenta and winding number will attract each other while strings with the same signs will repel each other.