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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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Kinematics
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Kinetic Theory of Gases
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One Dimensional Wave Equation
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Phase and Group Velocity
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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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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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Planck Units
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Stephen Hawking's PhD Thesis
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The Big Bang Model

Finance and Accounting

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Annuities
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Capital Structure
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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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Ricci Decomposition
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Ricci Flow
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Tensors
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The Area Metric
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The Equivalence Principal
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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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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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Contravariant and Covariant Components of a Vector
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Grassmann and Clifford Algebras
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Heron's Formula
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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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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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Data Types
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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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Stacked and Unstacked Data
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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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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 (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

Strings in Curved Spacetime --------------------------- First consider how a point particle, x, in the diagram below, moves on (in) the surface of a sphere. The center of mass (COM) of the particle is always coincident with the particle's center and will travel along a geodesic in the absence of external forces. The energy of the particle is: E = (mv2/2) = p2/2m The angular momentum is L = r x p L is quantized so L also equals nh, The energy equation becomes: E = p2/2m = L2/2mr2 = L2/2I = n2h2/2I Where I is the moment of inertia (MOM) of the particle. Now an excited string is no longer a point particle. Exciting oscillator modes (increasing the energy) causes the string to stretch. Classically, this corresponds to: E = kx2/2 = T2x2/2 Where, T = √k is the string tension with units of energy per unit length. To get some idea about how much an open string stretches we make use of the formula: x(σ,τ) = Σxn(τ)cos(nσ) n=0 = xCOM + Σxn(τ)cos(nσ) n=1 = xCOM + Σn((an- + an+)/√n)cos(nσ) The length of the string can be considerd to be the average deviation of x(σ) from the COM. For convenience we calculate the quantum mechanical expectation value of (x(σ) - xCOM)2 as, <0|(x(σ) - xCOM)2|0> = <0|Σnm((an- + an+)/√n)((am- + am+)/√m)cos(nσ)cos(mσ)|0> = Σnm((an-an+)/n)cos2(nσ) for n = m = (1/2)Σn(1/n) since cos2(nσ) ~ 1/2 = (1/2)ln(n) So the distance from the COM increases logarithmically with n. Although the increase is slow (ln(1000) = 6.9) the string grows in length and the stretching spreads out its mass so that its COM is no longer necessariy coincident with its center. This creates a problem for strings in curved spacetime. To see why consider the following diagrams showing a string on the surface of a sphere. We see that ss the string grows in length its COM gets closer and closer to e COM of the sphere. r decreases and the MOM shrinks, ultimately going to 0. At this point the energy becomes infinite. This is very similar to the problem of ultraviolet divergences in Feynman loop diagrams. It turns out that this divergence occurs for all curved geometries - not just the sphere. To avoid the situation where we would have impose a cutoff to the oscillator modes to prevent divergences we instead look for a background spacetime geometry that doesn't change as more modes are added to the string. Since strings have energy (mass) they act as sources in the spacetime background in which the string propagates. Therefore, the addition/subtraction of oscillator modes can change the effective metric tensor of the background. In addition, we require that the effective geometry created also has to satisfy Einstein's field equations. In order meet the requirement for mode independence, we need to apply the concept of RICCI FLOW. The Ricci flow equation is written as: δgμν = -2Rμν For a Riemannian manifold the Ricci flow is a PDE that evolves the metric tensor over 'time'. It is analogous to the diffusion of heat. Informally, it can be viewed as a process that produces an equilibrium geometry for a manifold for which the Ricci curvature is constant. Clearly, if we don't want the metric to change as we add more modes, the RHS must be set to 0. Therefore, Rμν = 0 This geometry is referred to as being RICCI-FLAT. Ricci-Flat ---------- Consider Einstein's field equations with no source. Rμν - (1/2)gμνRαα = 0 Where Rαα is the trace. Rμν - (1/2)gμνRαα = 0 Rμν - (1/2)δμνRαα = 0 Rαα - 2Rαα = 0 since δμν = 4 (4 D Kronecker) Thus, the curvature scalar disappears and we are left with: Rμν = 0 Rμν represents the part of curvature that derives from the presence of matter (i.e. that part due to the stress energy tensor Tμν). The remaining components of the Riemann tensor (called the WEYL tensor) represents the intrinsic gravitational field in the absence of matter or non gravitational fields. The intrinsic spacetime containing only gravitational radiation will satisfy Rμν = 0 but will also have Rμνσλ ≠ 0. With Rμν = 0 we are left with the Weyl tensor, also called the CONFORMAL TENSOR. If the Weyl tensor vanishes the manifold is said to be CONFORMALLY FLAT. Any 2-dimensional (smooth) Riemannian manifold is conformally flat. What this means is that each point on the manifold has a neighborhood that can be mapped to flat space by a conformal transformation. A conformal transformation can be defined as a special case of general coordinate transformation, x -> x', such that x' = f(x) has the following effect on the metric: g'μν = (∂xρ/∂x'μ)(∂xσ/∂x'ν)gρσ It acts as a rescaling of the metric, x -> x. This rescaling of the metric is often referred to as a WEYL transformation and the two terms are often used interchangeably, although they are in fact different things. A Weyl transformation takes us to a coordinate system where the metric has the same form as the one we started with, but the the proper distance between points have been changed by a scale factor while preserving the angles between all lines. Therefore, under a Weyl transformation: gμν = Ω2gμν It is usual to write Ω2 = exp(2f). In 2 dimensions f is realized by analytic functions, f(z), where z is complex. Thus, f(z) = u(x,y) + iv(x,y) satisfy the Cauchy-Riemann equations. ∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x Ricci flatness is a requirement for all string theories but fortunately it turns out there are many many topologies that are Ricci flat. Of particular interest in our simple analyses is the Ricci flat torus. However, more advanced analyses model strings on Ricci flat CALABI-YAU MANIFOLDS.