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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

The Nature of the Weak Interaction ---------------------------------- The Feynman path integral representation gives the probability amplitude of going from point A to point B (distance s) in terms of the propagator. The propagator is a function of space-time distance and is also crucially dependent on the mass of the particle. Now, in experiments we typically measure momentum. The laws of conservation of momentum hold and therefore the difference in momentum between the particles on the LHS of the diagram must equal the momentum of the emitted W boson. From QM we know that distance and momentum space are related through the Fourier Transform. From this, it is possible to specify the propagator in terms of momentum space. The probability amplitude, δ(p), that a particle with mass, m will travel s in momentum space is of the form: δ(p) = 1/(p2 - m2 + iε) The small imaginary term, iε, is put there to ensure the correct boundary conditions are met and handle the singularity at p2 = m2. The limit ε -> 0 is taken at the end of the calculation. Therefore, for low momenta, δ = -1/m2 Thus, the probability of traveling distance s falls off very rapidly with mass. The combined probability amplitude of the process shown in the diagram can thus be written as: gW2/mW2 = GF ... the FERMI G The net result is the weak interactions are weak because of the large mass of the W boson. The probability of the above process occurring is very roughly: gW4/mW4 where gW is of the order of the fine structure constant equal to (1/4πε0)(e2/hc) or about 1/137 and mW is the mass of the W boson. This is a very small number! Therefore, most of the time this process doesn't happen - the W boson is more likely to be continuously emitted and absorbed as shown in the diagram below. Note: m = 0 for a virtual photon. For low k processes which are very typical in QED, the probability amplitude of the process is very much larger than it is for the W boson. The mass associated with the virtual W boson, mW, is not the same as the mass of the real W particle detected in the lab (~80 GeV). The latter case involves the Higgs mechanism - virtual particles do not. The mass of the virtual boson is of the order of 100 times the proton mass or about 10GeV. However, it depends on probabilities and comes from the fact the W carries some 4-momentum from the quarks to the electron/neutrino. The question that arises is 'how does the W boson, which is much heavier than either n or p, get its energy?'. The answer to this is related to the difference between virtual and real particles (see below). In a nutshell, the virtual particle gets its mass by APPARENTLY violating conservation laws within the bounds of the energy-time uncertainty principle. A somewhat poor analogy is that the W boson borrows energy from the field but has to repay it before time runs out! Charge Conservation ------------------- The relationship between charge, weak isospin and weak hypercharge is given by: Q = I3 + YW/2 Consider the LHS: Qd = (-1/2) + (1/6) = (-1/3) Qu = (1/2) + (1/6) = (2/3) Therefore, Q changes by -1 Consider the RHS: Qν = (-1/2) + (1/2) = (0) Qe = (-1/2) + (-1/2) = (-1) Therefore, Q changes by -1 Virtual versus Real Particles ----------------------------- In quantum field theory real particles are viewed as being observable excitations of underlying quantum fields. Virtual particles are also viewed as excitations of the underlying fields, but are not observable. They are 'temporary' in the sense that they appear in calculations, but are not detected as single particles. Virtual particles are essentially mathematical constructs that only exist for extremely short periods of time and correspond to the propagators appearing in the Feynman diagrams. Where δ(s) is the PROPAGATOR and g is a coupling constant. A virtual particle does not necessarily carry the same mass as the corresponding real particle and, because it is 'short-lived', the energy-time uncertainty principle allows it to APPEAR not to conserve energy and momentum. In other words, a virtual particle does not precisely obey the formula m2c4 = E2 - p2c2 for a period of time corresponding to its very short lifetime. Another way of saying this is that there can be fluctuations in energy, ΔE within some time Δt, that obeys ΔEΔt ~ h. This is expressed by the phrase 'off mass shell'. Actual particles, of course, never violate conservation laws. The longer a virtual particle appears to 'live', the more closely it adheres to the 'mass-shell' relationship associated with a real particle. Virtual particles are the force carriers. Higher energy (mass) particles cannot travel as far as lower-energy virtual particles. This explains why the electric, strong, and weak forces diminish with distance. If we assume the particles in the above diagram are electrons, the electron on the right could shoot out a photon. Having given up some of its energy and momentum, that electron would recoil. Next the emitted photon would run into the other electron. Upon absorbing the photon’s energy and momentum, this electron would careen off toward the upper left. This is the basis of the repulsive force. * This same phenomenon can also take place with tunnelling. An alpha particle tunnelling out of a nucleus SEEMINGLY has to violate conservations laws for a very short period of time to surmount the potential barrier that keeps it in the nucleus. Because virtual particles only exist for short periods of time, observing them requires a large amount of energy. For example, the W particle emission and absorbtion process illustrated above take place at enormously high frequencies. In order to see it in the lab requires the generation of extremely high energy photons (E = hf). Unfortunately, as a consequence of QM where observation disturbs the system, the amount of energy that needs to supplied actually ends up creating the particle that you are trying to observe. Thus, in a process like the above, it is possible to observe real W bosons but you have to hit a proton or neutron with an energy that ultimately results in the ejection of the W particle. This is a much different process than the weak interaction. Resonances ---------- When the incoming real particles have enough energy a real instead of a virtual particle can be produced. The process becomes 'on shell' and is greatly enhanced - a RESONANCE. Resonances are described by the BREIT-WIGNER probability distribution: f(E) ∝ 1/(E2 - m2 + imΓ) Where Γ is the RESONANCE WIDTH (or decay width) = h/τ and τ is the lifetime of the particle and m is the mass f the particle. The distribution arises from the propagator of an unstable particle which has a denominator of the form p2 - m2 + iε. Production of on-shell particles in colliders can be accomplished in different ways. One way is to collide electrons and positrons (LEP). Consider W and Z bosons generated in this way. Now, these bosons only live for a short period of time before they decay into different products with probabilities defined by their BRANCHING RATIOS. At 70%, the most likely decay process for the W and Z bosons is that they will decay into quark antiquark pairs (hadrons). Thus, we can write. _ _ ee -> Z or W -> qq (JETS) Even if the bosons decay before they can reach the detectors, their presence can be detected by keeping track of the total rate of these final states. The resonances for these both look like: The Z particle is strongly peaked and comes in at ~ 91 GeV. In this particular experiment the W bosons were produced in pairs. The combined mass of the W+W- combination is not strongly peaked and is between 160 GeV and 200 GeV (80 - 100 GeV per W). Typically all decay processes (called CHANNELS) are considered in experiments. Not all channels will exhibit resonances. More accurate measurements of the W± masses have been made by colliding protons with antiprotons (Tevatron) and observing the Lepton channels. These experiments have shown the W± mass to be very close to 80.4 GeV.