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Astronomy

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Celestial Coordinates
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Celestial Navigation
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Distance Units
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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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Penrose Diagrams
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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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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

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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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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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 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 Qubit
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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

test

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test

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 Vacuum Catastrophe ---------------------- Vacuum energy is an underlying background energy that exists in space throughout the entire Universe. In Quantum Field Theory, the fields are comprised of harmonic oscillators with energy E = (n + 1/2)hω. Each oscillator has a zero point energy of E = (1/2)hω. That is to say that at 0 displacement, the oscillator still has some energy. The 0 point energy is consequence of the Uncertainty Principle. Lets look at the analysis for the photon field. Consider 2 dimensional momentum space. k1 | | ---+ 2,2 | | | | +--- | 1,1 --------------- k2 k = 2π/λ = 2πn/L Let n = 1 so the box has sides of length, k = 2π/L. Consider a volume in momemtum space, Vk. The number of modes, N, in a cube with sides, L is given by: N = Vk/k3 = L3Vk/(2π)3 The volume density of modes, n, is given by: n = N/Total volume of space = N/L3 = Vk/(2π)3 This is effectively the density of n-space The number of modes between k and k + dk is the volume of k space in a thin spherical shell. Thus, dn = (1/2π)34πk2dk Now setting, h = c = 1 we get E = k = ω. Each mode has an energy ω/2 associated with it. Therefore, dρ = (1/2π)34πω3/2dω or, ρ = (1/2π)3∫dω 2πω3 = ω4/8π2 = hω4/8π2c3 after restoring h and c. There is an extra factor of 2 here to account for the fact that the photon has 2 polarization states. Evidence for the existence of vacuum energy can be found in the CASIMIR EFFECT. The Casimir Effect ------------------ The Casimir effect is a small attractive force that acts between two close parallel uncharged, grounded, conducting plates. On the interior surface of the plates the electric field must be zero since a non-zero value would dissipate energy and violate the condition of equilibrium. This imposes a constraint on the nodes of the field between the plates. The distances between the plates can only fit 1/2 period sine waves. The vacuum energy contains contributions from all wavelengths, except those excluded by the spacing between plates. As the plates draw together, more wavelengths are excluded and the vacuum energy decreases. The decrease in energy means there must be a force doing work on the plates as they move. .. .. || || ->|| ||<- P || || .. .. <- x -> The lack of low frequency modes between the plates lowers the vacuum energy between them. The introduces a small correction term: Δρ = -π2A/720x4 Where A is the area of the plates. Thus, the modified ρ is: ρ = ω4/8π2 - π2A/720x4 The difference in energy results in a pressure, P, acting on the plates. If the spacing between the plates is increased by dx, the work done is: W = PdV = PAdx P = -(1/A)dU/dx = -π2240x4 The force acting on the plates is the CASIMIR FORCE. One may view the force as arising due to the radiation pressure of the vacuum. The Casimir Effect has been observed in the laboratory and is considered to be evidence of the existance of vacuum energy. The Casimir does not measure the value of the vacuum energy, but only identifies that it exists. Other Fields ------------ All fields have a zero-point energy. Therefore, the vacuum energy can be viewed as the combination of all zero-point fields. The above equation is also a fair approximation for these other quantum fields. Therefore: ρ = Σω4/4π3 over all Bosonic and Fermionic fields. This is clearly divergent as ω -> ∞. This divergence can also be interpreted in terms of Feynman loops diagrams with virtual particle pairs that blink into existence and then annihilate in a timespan too short to observe. Now, Fermionic fields contribute a negative amount -ω/2 for each degree of freedom to the vacuum energy. While Supersymmetry might provide some cancellations at higher energies it does not produce work for lower energies. This is due to the fact that Supersymmetry must be spontaneously broken and the masses of the superpartners are not equal (if the masses were equal physicists would have discovered them by now). Thus, even if Supersymmetry turns out to be real, there remains a considerable vacuum energy governed by equations of the above form. This zero-point energy density of the vacuum, due to all quantum fields, is enormously large, even when we cut off the largest allowable frequencies based on plausible physical arguments. QFT is at the heart of modern day physics and has been spectacularly successful. Such an enormous vacuum energy is generally not important in these field theories because we are only interested in energy differences and can use normal ordering procedures to eliminate the contributions from the zero-point field energies. This has the effect of subtracting out the vacuum energy regardless of its size. However, a large vacuum energy presents a huge problem for gravity. To see this, consider the following classical argument involving the Sun and Pluto. Using Newton's and Gauss's Law of Universal Gravity we can write: FS = GMm/r2 and, FV = M(r)m/r2 Now, M(r) = 4πr3ρ/3 Therefore, FV = 4πρGmr/3 Newtonian mechanics does an excellent job of describing planetary motion therefore we must conclude FV << FS 4πρGmr/3 << GMm/r2 ∴ 4πρr/3 << M/r2 ρ < 3M/4πr3 For M = 1.99 x 1030kg (Sun's mass) and r = 5.9 x 1012m (orbital radius of Pluto), we get ρ < 2.3 x 10-9 kg/m3. Cosmological Constant Problem ----------------------------- The vacuum energy is one possible explanation for the Cosmological constant and the source of dark energy. The estimated mass density of the Universe is (including Dark matter) thought to be roughly equal to 10-26 kg/m3. If the energy density of the vacuum is not close to this value, the behavior of the Universe would be seriously affected and the curvature and expansion of space would not agree with astronomical observation. This can be seen from the equation: .. (a/a) = -4πG(ρ + 3p)/3 + Λ/3 Which describes the acceleration of the expansion of the Universe (see the note on "Geometries of the Universe"). So, a large vacuum energy presents a huge problem for General Relativity because the absolute amount of vacuum energy has a real physical meaning. In fact, the Cosmological constant and the vacuum energy differ by about an astonishing 120 orders of magnitude! This is the infamous "Cosmological constant problem" which remains one of the greatest unsolved mysteries of physics in the modern era. See also the note on the "Fine Tuning Problem".