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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 Mechanics

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Blackbody Radiation .

Classical Physics

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Archimedes Principle
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Bernoulli Principle
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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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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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Poiseuille's Law
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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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The Gas Laws
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The Laws of Thermodynamics
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The Zeeman Effect .
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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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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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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
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Vacuum Energy .

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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Basis One-forms .
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Catalog of Spacetimes .
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Curvature and Parallel Transport
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Einstein's Field Equations
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Geodesics
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Gravitational Waves
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Hyperbolic Motion and Rindler Coordinates .
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Quantum Gravity
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Ricci Decomposition .
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Ricci Flow .
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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 Dirac Equation in Curved Spacetime .
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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 Light Cone .
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The Metric Tensor .
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The Principle of Least Action in Relativity .
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Vierbein (Frame) Fields

Group Theory

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Basic Group Theory .
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Basic Representation Theory .
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Building Groups From Other Groups .
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Sets, Groups, Modules, Rings and Vector Spaces
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Symmetric Groups .
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The Integers Modulo n Under + and x .

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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Binomial Theorem (Pascal's Triangle)
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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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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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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 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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Volume Integrals

Microeconomics

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Marginal Revenue and Cost

Nuclear Physics

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Radioactive Decay

Particle Physics

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Feynman Diagrams and Loops
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Field Dimensions
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Helicity, Chirality and Weyl Spinors .
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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 .

Probability and Statistics

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Box and Whisker Plots
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Buffon's Needle .
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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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Density Operators and Mixed States .
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Entangled States .
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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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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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Heisenberg Uncertainty Principle
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Ladder Operators .
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Multi Electron Wavefunctions .
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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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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 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 (Faraday) Tensor .
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Energy and Momentum in Special Relativity, E = mc2 .
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Invariance of the Velocity of Light .
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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 Continuity Equation .
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The Lorentz Group .

Statistical Mechanics

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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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Bardeen–Cooper–Schrieffer Theory
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BCS Theory
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Cooper Pairs
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Introduction to Superconductivity .
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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
Last modified: February 20, 2022 ✓

Dark Matter ----------- The Virial Theorem ------------------ The Virial theorem relates the total kinetic energy of a self-gravitating body due to the motions of its constituent parts, K, to the gravitational potential energy, U, of the body. Consider N point paricles in a system. We define the VIRIAL, G, as: G := Σipi.ri Where pi and ri are the momentum and position of particle i. Then: dG/dt = Σi(dpi/dt).ri + Σipi.(dri/dt) Now, the force on the particle, Fi, is: Fi = miai = dpi/dt and pi = mi(dri/dt) Therefore we can write: dG/dt = ΣiFi.ri + Σimi(dri/dt)2 = ΣiFi.ri + Σimivi2 = ΣiFi.ri + 2K (K = mv2/2) The time average is given by: τ <dG/dt> = (1/τ)∫(ΣiFi.ri + 2K)dt 0 = <Fi.ri> + 2<K> If the system is cyclical such that it returns to its initial state after an interval, τ, then τ can be chosen to be equal to the cycle period and, therefore, dG/dt reduces to 0. In this case: <K> = (-1/2)Σi<Fi.ri> Now Fi = -∂U/∂ri (U is PE) <K> = (-1/2)<U> or, 2<K> + <U> = 0 ... 1. Astronomy has made great use of the Virial Theorem as a way of measuring gravitational mass. The gravitational potential, U, for a cluster of N galaxies each separated by a distance, r, is: U = -GNm2/r = -GM2/r Likewise, the KE is: K = Mv2/2 Using 1. we get: Mv2/2 = (1/2)GM2/r ∴ v2 = GM/r Therefore, the total mass of the cluster is: M = 2v2/G The left-hand side of the equation can be estimated if we can measure the motions of bodies in the system. Thanks to the Doppler shift, we can determine the motion in the radial direction very easily. Mass-Luminosity Relationship ---------------------------- The mass–luminosity equation gives the relationship between a star's mass and its luminosity. A rough approximation can be obtained by modelling stars as ideal gases (Eddington). L/LO = (M/MO)α Where LO and MO are the luminosity and mass of the Sun respectively and, α, is number dependent on the mass range of the star. L/LO ~ 0.23(M/MO)2.3  M < 0.43MO L/LO = (M/MO)4.0 0.43MO < M < 2MO L/LO ~ 1.5(M/MO)3.5 2MO < M < 20MO L/LO ~ 3200(M/MO) M > 20MO Dark Matter ----------- The existence of Dark Matter was first proposed by the Swiss astronomer, Fritz Zwicky in 1933. Zwicky focussed on the COMA CLUSTER of galaxies and measured the Doppler shift of their spectra to find their velocities. He then used the Virial theorem to determine their masses. In parallel he also determined their masses using the mass-luminosity relationship described above. He discovered that the 2 methods yielded masses that differed greatly. The dynamical mass from the Virial theorem turned out to be about 400 times the luminosity mass. Because of this Zwicky believed that there must be a large amount of invisible matter within the cluster that he termed "dark matter". Galactic Rotation Curves ------------------------ Four decades after Zwicky's initial observations new techniques were used to analyze the rotation speeds of galaxies. The new data provided the first pieces of evidence that large quantities of non-luminous mass might exist outside the visible region of most galaxies. The velocity of a mass, m, orbiting at radius r around a mass M is given by: v = √(MG/r) (F = mv2/r = GMm/r2) Stars in tentacles of spinning galaxies don't show this - v doesn't change with √(1/r) but is roughly constant. The implication of this is that the mass is not concentrated in the center of the galaxy but is distributed out to where the star is located. In other words the M(r), is a constant * r. Furthermore, whatever this mass is it does not radiate energy. Note: If we know r and v then we can compute M(r). v = √(M(r)G/r ) ∴ M(r) ~ r Cosmologists speculate that this dark matter may be made of particles produced shortly after the Big Bang. These particles would be very different from ordinary "baryonic matter" and hypothetically referred to as WIMPs (Weakly Interacting Massive Particles) or "non- baryonic matter". A WIMP is thought to be a new elementary particle which interacts via gravity and potentially other weak forces that are not part of the current Standard Model. The DENSITY PARAMETER, Ω, is comprised of the following: Ω = Ωmass,0 + Ωrad,0 + ΩΛ,0 Data from the CMB indicates that Ωmass,0 = 0.27 ± 0.04. But the amount of ordinary or baryonic matter is only calculated to be 0.044 ± 0.004. Therefore, observable baryonic matter constitutes only about 17% of the matter of the Universe, with balance being dark matter.