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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

Quantum Gravity --------------- Our current understanding of gravity is based on Einstein's theory of General Relativity that describes gravity as a curvature within space-time that is influenced by mass. Conversely, the electromagnetic, weak and strong forces are described by quantum theories that depend on the emdedding of particle fields in the flat space-time of special relativity. Unfortunately, difficulties arise when it comes to constructing a quantum field theory for gravity with the graviton as the force particle. This comes about because the theory obtained is not renormalizable and cannot be used to make meaningful predictions. Let us see why this is true. We start with the Einstein-Hilbert action given by: S = (c4/16πG)∫d4x √(-|g|)R Where G is Newton's constant and R is the curvature scalar. Now G can be written quantum mechanically in terms of the Planck mass, mp as: G = hc/mp2 If we set h = c = 1 we get: G = 1/mp2 and (neglecting the factor of 16π): L = mp2√(-|g|)R The action becomes: S = mp2∫d4 √(-|g|)R = (1/G)∫d4 √(-|g|)R Consider small perturbations around flat Minkowski space: gμν = ημν + Ghμν We can regard ημν as the 'basic' field, hμν as the theorized GRAVITON field and G as the coupling constant. The graviton is expected to be massless (because the gravitational force appears to have unlimited range like the photon) and must be a spin-2 boson. A spin 2 particle is also known as a tensor boson, compared to a spin 0 scalar boson and a spin 1 vector boson. The spin 2 follows from the fact that the source of gravitation is the stress-energy tensor which is of second order (compared to electromagnetism's spin-1 photon, the source of which is a 4 vector, a first order tensor). Also, it can be shown that any massless spin 2 field would give rise to a force indistinguishable from gravitation, because a massless spin 2 field would couple to the stress–energy tensor in the same way that gravitational interactions do. Taylor series review: f(x) = f(a) + f'(a)(x - a) + f''(a)(x - a)2/2 + ... If we set (x - a) = ε so that x = a + ε we get: f(a + ε) = f(a) + f'(a)ε Where, f'(a) = ∂f(x)/∂x|x = a Now we can do the same for the perturbed metric, g = η + Gh. We get: g(η + Gh) = g(η) + Gh∂g(η)/∂g(η) + G2h22g(η)/∂g(η)2)/2 + ... = η + Gh + G2h2/2 + G3h3/6 + ... Now, schematically, R ~ ∂Γ/∂xμ + Γ2 where Γ ~ ∂g/∂xμ ≡ ∂μg (Christoffel symbols) Therefore, R ~ ∂μ2g + (∂μg)2 Now (dropping constants for clarity and simplicity), ∂μg = ∂μh + Gh∂μh + G2h2μh + ... ∂μ2g = ∂μ2h + G2h∂μ2h Therefore, R ~ ∂μ2h + Gh∂μ2h + (∂μh + Gh∂μh + G2h2μh)2 ~ ∂μ2h + Gh∂μ2h + (∂μh + Gh∂μh)2 ~ ∂μ2h + Gh∂μ2h + (∂μh)2 + Gh(∂μh)2 + G2h2(∂μh)2 + Gh(∂μh)2 + G2h2(∂μh)2 + G3h3(∂μh)2 + G2h2(∂μh)2 + G3h3(∂μh)2 + G4h4(∂μh)2 ~ (∂μh)2 + Gh(∂μh)2 + G2h2(∂μh)2 + G3h3(∂μh)2 + G4h4(∂μh)2 + ... Dimensional Comparison ---------------------- Lets now take a look at the dimensions of the Einstein-Hilbert action under unperturbed and perturbed conditions: g and h are dimensionless since ds2 = gμνdxμdxν. However, they are both functions of xμ so: [∂μg] = [∂μh] = [1/L] = [M] [d4x] = [L4] = [1/M4] [Γ] ~ [∂g/∂xμ] ≡ [∂μg] = [1/L] = [M] [R] ~ [∂Γ/∂xμ + Γ2] ≡ [∂μ2g + (∂μg)2] = [1/L2] = [M2] [G] = [1/L2] = [M2] In the unperturbed case: S = (1/G)∫d4x √(-|g|)R = [m2][m-4][m2] We see that S is dimensionless (as it needs to be for the Path Integral formulation in QFT theory). In the perturbed case: S = (1/G)∫d4x √(-|g|){(∂h)2 + Gh(∂h)2 + G2h2(∂h)2 + ...} Graphically, this expansion is an expansion in numbers of loops in Feynman diagrams. At each loop the whole expression should be dimensionless. The first term in {} has [m2] so that is fine. However, the second term has [m4]. In order to make it equal to [m2], we would need to add 2 more powers of momentum in the denominator. Because 1/G is of the order of 1016 Kg, the expression becomes more and more divergent the higher you go in the perturtabive expansion In order to cancel these divergences it would be necessary to introduce an infinite number of counterterms which, in terms of renormalization, makes no sense. The conclusion is that, unlike other quantum field theories s where the coupling constant is dimensionless, the gravitational theory is not renormalizable (at least at very high energies). Effective Field Theory ---------------------- The problem of non-renormalizability of quantum gravity does not necessarily mean that quantum mechanics is incompatible with gravity, only that quantum gravity should be treated as an EFFECTIVE FIELD THEORY for energies well below the Planck scale of 1019 GeV. An effective field theory appromimates an underlying physical theory that describes physical phenomena occurring at a chosen length or energy scale, while ignoring phenomena at shorter distances. Intuitively, it averages over the behavior of the underlying theory at shorter length scales to derive what is hoped to be a simplified model at longer length scales. Effective field theories typically work best when there is a large separation between the length scale of interest and the length scale of the underlying dynamics.