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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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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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Kinematics
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One Dimensional Wave Equation
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Planck Radiation 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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Quantum Gravity
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Relativistic, Cosmological and Gravitational Redshift
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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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Basic Group Theory
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Building Groups From Other Groups
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Complex Numbers
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Contravariant and Covariant Components of a Vector
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Grassmann and Clifford Algebras
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Integration By Parts
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Introduction to Conformal Field Theory
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Symmetric 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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Confidence Intervals
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Factor Analysis
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Hypothesis Testing
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Linear Regression
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Permutations and Combinations
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Set Theory - Venn Diagrams
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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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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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Photoelectric Effect
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Position and Momentum States
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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

Non Parametric Tests -------------------- Up until now it has been assumed that the underlying distributions from which the samples have been drawn is normally distributed OR the sample sizes are big enough (typically greater than 30) so that the central limit theorem applies (i.e. the means of the samples follow a normal distribution). But what if these assertions are not true - we are dealing with small sample sizes and we know nothing about the parent population? Enter non parametric testing! There are 6 non-parametric tests of importance. The Sign Test ------------- This is used for testing hypotheses about the central tendency of a non-parametric distribution. It provides inferences about the population median, η, rather than the population mean, μ. H0: η = η0 H1: η < η0 or η > η0   η ≠ η0 Test statistic: S = # of measurements < (or >) η0 (1 tailed) = Larger of # of measurements < η0 and > η0 (2 tailed) p value: P(x ≥ S) (1 tailed) 2P(x ≥ S) (2 tailed) Where x has a binomial distribution with n = p = 0.5. P(x ≥ S) = Probability of S successes in n trials. Reject H0 if p value ≤ α Large Samples ------------ If n ≥ 30 we can use the normal approximation to the binomial. Z = [(S - 0.5) - 0.5n]/0.5√n Where 0.5 is the 'continuity correction', the mean, np = 0.5n and the standard deviation is √(npq) = √(n(0.5)(0.5)) = 0.5√n Example: Determine whether η is less than 1.00. H0: η = 1.00 H1: η < 1.00 Test Results ------------ 0.78 0.51 3.79 0.23 0.77 0.98 0.96 0.89 S = # of measurements < η = 7 P(x ≥ 7) = Probability of 7 successes in 8 trials = 1 - P(x ≤ 6) = 1 - 0.965 from Binomial tables = 0.035 Therefore, reject H0 at the α = 0.05 level. The Wilcoxon Rank Sum (a.k.a The Mann-Whitney U Test) ----------------------------------------------------- This is the non-parametric analog of the independent samples t-test. H0: The 2 distributions D1 and D2 are identical. H1: D1 is shifted to the right of D2.   D1 is shifted to the left of D2.   D1 is shifted either to the left or the right of D2. Test statistic: Rank every observation as though they were all drawn from the same population with Rank 1 being the lowest observation. Assign ascending ranks to observation with the same value and divide by the number of observations with the same value. For example Rank: 1 3 2 4 8 5 7 6 9 10 Observation: 1.1 2.3 2.1 3.3 4.2 3.3 4.0 3.3 4.4 5.2 => 1 3 2 5* 8 5* 7 5* 9 10 * 3.3 => (4 + 5 + 6)/3 = 5 To find the test statstic we take the sum of the ranks for the smaller sample size (or either if n1 = n2). Use the Wilcoxon Rank Sum Tables to obtain the upper and lower rejection regions, TU and TL. n1 corresponds to columns and n2 corresponds to rows. Rejection region: If n1 < n2: T1 ≥ TU : T1 ≤ TL (1 tailed). T1 ≥ TU or T1 ≤ TL (2 tailed). If n2 < n1: T2 ≥ TU : T2 ≤ TL (1 tailed). T2 ≥ TU or T2 ≤ TL (2 tailed). Large Samples ------------ If n1 ≥ 10 and n2 ≥ 10 we can use the Z statistic for the hypthesis test. T1 - (n1(n1 + n2 + 1))/2 Z = ----------------------- √(n1n2(n1 + n2 + 1)/12) Example: Comparison of drug reaction times. Drug A Drug B Reaction Time Rank Reaction Time Rank ------------- ---- ------------- ---- 1.96 4 2.11 6 2.24 7 2.43 9 1.71 2 2.07 5 2.41 8 2.71 11 1.62 1 2.50 10 1.93 3 2.84 12 2.88 13 -- -- Rank Sum 25 66 Sample Size, n 6 7 H0: The reaction times for A and B have the same   probability distribution. H1: The reaction times for A are shifted to the left   or right of the reaction times for B. T1 = 25 T2 = 66 From the Wilcoxon Paired Difference Rank Sum Tables: n1 = 6, n2 = 7, α = 0.05 => TL = 28 and TU = 56 (2 tailed) Therefore, reject H0 at the α = 0.05 level because T1 (the smaller sample size) falls in the rejection region. t-test comparison: The t-value is -2.9927. The 2 tailed p-value is 0.0122. The result is significant at p < 0.05. Note: If we had a 1 tailed test with H1: A is shifted to the left of B then TL = 30 and TU = 54 (1 tailed) for α = 0.05. In this case T1 is not in the rejection region so we would accept H0. Wilcoxon Paired Difference Signed Rank Test ------------------------------------------- This is the non-parametric analog of the paired t-test. H0: The 2 distributions D1 and D2 are identical. H1: D1 is shifted to the right of D2.   D1 is shifted to the left of D2.   D1 is shifted either to the left or the right of D2. Test statistic: Calculate the ranks of the absolute values of the differences. Rank every observation as though they were all drawn from the same population with Rank 1 being the lowest observation. Determine the sum of the ranks of the positive and negative differences of the original measurements. 0 differences are eliminated and n reduced accordingly. Find T0 from the Wilcoxon Paired Difference Signed Rank Tables with n = number of pairs. Rejection region: Consider the difference between 2 distributions D1 - D2. 1 tailed: If D1 is to the right of D2, more positive differences should occur. Therefore, T+ > T- and the rejection criterion is T- ≤ T0. If D1 is to the left of D2, more negative differences should occur. Therefore, T- > T+ and the rejection criterion is T+ ≤ T0 2 tailed: If D1 is shifted to the left or right of D2 then the rejection criterion is the smaller of T- or T+ ≤ T0. Large Samples ------------ If n1 ≥ 10 and n2 ≥ 10 we can use the Z statistic for the hypthesis test. T1 - n(n + 1)/4 Z = --------------------- √(n(n + 1)(2n + 1)/24) Example: Before and after test. A B (A - B) |A - B| Rank Modified Rank -- -- ------- ------- ---- ------------ 12 8 4 4 4 4.5 16 10 6 6 7 7 8 9 -1 1 1 1 10 8 2 2 2 2 19 12 7 7 8 8 14 17 -3 3 3 3 12 4 8 8 9 9 10 6 4 4 5 4.5 12 17 -5 5 6 6 16 4 12 12 10 10 H0: A and B have the same probability distribution. H1: A and B have different probability distributions. T- = 10 T+ = 45 From the Wilcoxon Paired Difference Signed Rank Tables: n = 10, α = 0.05 => T0 = 8 (2 tailed) Therefore, accept H0 at the α = 0.05 level because T- is not in the rejection region. t-test comparison: The t-value is 2.0466. The 2 tailed p-value is 0.0710. The result is not significant at p < 0.05. Note: If we had a 1 tailed test with H1: A is shifted to the left of B then T0 = 11 for α = 0.05. In this case T+ is not in the rejection region so we would again accept H0. Kruskal-Wallis H Test --------------------- This is the non-parametric analog of the ANOVA test for a completely randomized design. H0: The distributions Dk are identical. H1: At least 2 of the distributions differ in location. Test statistic:   _  _ H = (12/n(n + 1))Σnj(Rj - R)2 or, equivalently: H = (12/n(n + 1))ΣRj2/nj - 3(n + 1) Where, Rj = Rank sum for sample j. nj = number of measurements in jth sample. _ Rj = Rj/nj = mean rank sum for sample j. _ R = (n + 1)/2 = mean of all ranks. n = n1 + n2 + ... + nk = total sample size. = number of ranks. Rejection region: H > χα2 with k - 1 df Example: Comparison of unoccupied bed space for 3 different hospitals on 10 successive days. H1 H2 H3 Beds Rank Beds Rank Beds Rank ---- ---- ---- ---- ---- ---- 6 5 34 25 13 5 38 27 28 19 35 26 3 2 42 30 19 15 17 13 13 9.5 4 3 11 8 40 29 29 20 30 21 31 22 0 1 15 11 9 7 7 6 16 12 32 23 33 24 25 17 39 28 18 14 5 4 27 18 24 16 --- ----- ----- Rank suns 120 210.5 134.5 H0: Distributions for all 3 hospitals are the same. H1: At least 2 of the hospitals have distributions   that differ in location. n1 = n2 = n3 = 10 _ R1 = 120/10 = 12.0 _ R2 = 210.5/10 = 21.05 _ R3 = 134.5/10 = 13.45 _ R = (30 + 1)/2 = 15.5 => H = 6.097 From the χ2 tables with α = 0.05 and (k - 1) = 2 df we get χ0.052 = 5.991 Since H > 5.991 we can reject H0. Friedman F Test --------------- This is the non-parametric analog of the ANOVA test for a randomized block design. Test statistic:   _  _ F = (12b/k(k + 1))Σ(Rj - R)2 or, equivalently: F = (12/bk(k + 1))ΣRj2 - 3b(k + 1) Where, b = number of blocks. k = number of treatments. nj = number of measurements in jth sample. Rj = Rank sum for jth treatment where the rank of each measurement is found relative to its position in its own block (the measurements can are ranked in blocks). _ Rj = Rj/nj = mean rank sum for sample j. _ R = (n + 1)/2 = mean of all ranks where n = number of ranks. Rejection region: F > χα2 with k - 1 df Example: Drug reaction times. Block Drug A Rank Drug B Rank Drug C Rank ----- ------ ---- ------ ---- ------ ---- 1 1.21 1 1.48 2 1.56 3 2 1.63 1 1.85 2 2.01 3 3 1.42 1 2.06 3 1.70 2 4 2.43 2 1.98 1 2.64 3 5 1.16 1 1.27 2 1.48 3 6 1.94 1 2.44 2 2.81 3 -- -- -- Rank sums 7 12 17 H0: The reaction times for A, B and C have the same   probability distributions. H1: At least 2 of the drugs have reaction times whose   distributions differ by location. k = 3 b = 9 _ R1 = 7/6 = 1.167 _ R2 = 12/6 = 2.0 _ R3 = 17/6 = 2.833 _ R = (3 + 1)/2 = 2.0 => F = 8.33 From the χ2 tables with α = 0.05 and (k - 1) = 2 df we get χ0.052 = 5.991 Spearman's Rank Correlation --------------------------- This is the non-parametric analog of Pearson correlation. SSuv rS = ---------- √(SSuuSSvv) Where: _ _ SSuv = Σ(ui - u)(vi - v) ≡ ΣuiΣvi/n _ SSuu = Σ(ui - u)2 ≡ Σ(ui)2 - (Σui)2/n _ SSvv = Σ(vi - v)2 ≡ Σ(vi)2 - (Σvi)2/n n is the number of data points. ui is the rank of the ith observation in sample 1. vi is the rank of the jth observation in sample 2. In this case the independent variables are ranked separately. Example: Cigarettes smoked per day vesus birth weight. Cigs/day Rank Weight Rank -------- ---- ------ ---- 12 1 7.7 2 15 2 8.1 3 35 5 6.9 1 21 4 8.2 4.5 20 3 8.2 4.5 SSuu = 10 SSvv = 9.5 SSuv = -0.5 rS = -0.5/√(10*9.5) = -0.051