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18.104 Seminar in Analysis: Applications to Number Theory (MIT) 18.104 Seminar in Analysis: Applications to Number Theory (MIT)

Description

18.104 is an undergraduate level seminar for mathematics majors. Students present and discuss subject matter taken from current journals or books. Instruction and practice in written and oral communication is provided. The topics vary from year to year. The topic for this term is Applications to Number Theory. 18.104 is an undergraduate level seminar for mathematics majors. Students present and discuss subject matter taken from current journals or books. Instruction and practice in written and oral communication is provided. The topics vary from year to year. The topic for this term is Applications to Number Theory.

Subjects

Infinitude of the primes | Infinitude of the primes | Summing powers of integers | Summing powers of integers | Bernoulli polynomials | Bernoulli polynomials | sine product formula | sine product formula | $\zeta(2n)$ | $\zeta(2n)$ | Fermat's Little Theorem | Fermat's Little Theorem | Fermat's Great Theorem | Fermat's Great Theorem | Averages of arithmetic functions | Averages of arithmetic functions | arithmetic-geometric mean | arithmetic-geometric mean | Gauss' theorem | Gauss' theorem | Wallis's formula | Wallis's formula | Stirling's formula | Stirling's formula | prime number theorem | prime number theorem | Riemann's hypothesis | Riemann's hypothesis | Euler's proof of infinitude of primes | Euler's proof of infinitude of primes | Density of prime numbers | Density of prime numbers | Euclidean algorithm | Euclidean algorithm | Golden Ratio | Golden Ratio

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12.620J Classical Mechanics: A Computational Approach (MIT) 12.620J Classical Mechanics: A Computational Approach (MIT)

Description

12.620J covers the fundamental principles of classical mechanics, with a modern emphasis on the qualitative structure of phase space. The course uses computational ideas to formulate the principles of mechanics precisely. Expression in a computational framework encourages clear thinking and active exploration.The following topics are covered: the Lagrangian formulation, action, variational principles, and equations of motion, Hamilton's principle, conserved quantities, rigid bodies and tops, Hamiltonian formulation and canonical equations, surfaces of section, chaos, canonical transformations and generating functions, Liouville's theorem and Poincaré integral invariants, Poincaré-Birkhoff and KAM theorems, invariant curves and cantori, nonlinear resonances, resonance ov 12.620J covers the fundamental principles of classical mechanics, with a modern emphasis on the qualitative structure of phase space. The course uses computational ideas to formulate the principles of mechanics precisely. Expression in a computational framework encourages clear thinking and active exploration.The following topics are covered: the Lagrangian formulation, action, variational principles, and equations of motion, Hamilton's principle, conserved quantities, rigid bodies and tops, Hamiltonian formulation and canonical equations, surfaces of section, chaos, canonical transformations and generating functions, Liouville's theorem and Poincaré integral invariants, Poincaré-Birkhoff and KAM theorems, invariant curves and cantori, nonlinear resonances, resonance ov

Subjects

classical mechanics | classical mechanics | phase space | phase space | computation | computation | Lagrangian formulation | Lagrangian formulation | action | action | variational principles | variational principles | equations of motion | equations of motion | Hamilton's principle | Hamilton's principle | conserved quantities | conserved quantities | rigid bodies and tops | rigid bodies and tops | Hamiltonian formulation | Hamiltonian formulation | canonical equations | canonical equations | surfaces of section | surfaces of section | chaos | chaos | canonical transformations | canonical transformations | generating functions | generating functions | Liouville's theorem | Liouville's theorem | Poincar? integral invariants | Poincar? integral invariants | Poincar?-Birkhoff | Poincar?-Birkhoff | KAM theorem | KAM theorem | invariant curves | invariant curves | cantori | cantori | nonlinear resonances | nonlinear resonances | resonance overlap | resonance overlap | transition to chaos | transition to chaos | chaotic motion | chaotic motion | 12.620 | 12.620 | 6.946 | 6.946 | 8.351 | 8.351

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12.010 Computational Methods of Scientific Programming (MIT) 12.010 Computational Methods of Scientific Programming (MIT)

Description

This course introduces programming languages and techniques used by physical scientists: FORTRAN, C, C++, MATLAB®, and Mathematica®. Emphasis is placed on program design, algorithm development and verification, and comparative advantages and disadvantages of different languages. Students first learn the basic usage of each language, common types of problems encountered, and techniques for solving a variety of problems encountered in contemporary research: examination of data with visualization techniques, numerical analysis, and methods of dissemination and verification. No prior programming experience is required.Technical RequirementsAny number of development tools can be used to compile and run the .c and .f files found on this course site. C++ compiler is required to This course introduces programming languages and techniques used by physical scientists: FORTRAN, C, C++, MATLAB®, and Mathematica®. Emphasis is placed on program design, algorithm development and verification, and comparative advantages and disadvantages of different languages. Students first learn the basic usage of each language, common types of problems encountered, and techniques for solving a variety of problems encountered in contemporary research: examination of data with visualization techniques, numerical analysis, and methods of dissemination and verification. No prior programming experience is required.Technical RequirementsAny number of development tools can be used to compile and run the .c and .f files found on this course site. C++ compiler is required to

Subjects

programming languages | techniques used by physical scientists | programming languages | techniques used by physical scientists | FORTRAN | FORTRAN | C | C | C++ | C++ | Matlab | Matlab | Mathematica | Mathematica | program design | program design | algorithm development and verification | algorithm development and verification | comparative advantages and disadvantages of different languages | comparative advantages and disadvantages of different languages | examination of data with visualization techniques | examination of data with visualization techniques | numerical analysis | numerical analysis | methods of dissemination and verification | methods of dissemination and verification | algorithms | algorithms | formula | formula | formulae | formulae | computer programs | computer programs | graphics | graphics | computing languages | computing languages | structure | structure | documentation | documentation | program interface | program interface | syntax | syntax | advanced modeling | advanced modeling | simulation systems | simulation systems

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13.013J Dynamics and Vibration (MIT) 13.013J Dynamics and Vibration (MIT)

Description

Introduction to dynamics and vibration of lumped-parameter models of mechanical systems. Three-dimensional particle kinematics. Force-momentum formulation for systems of particles and for rigid bodies (direct method). Newton-Euler equations. Work-energy (variational) formulation for systems particles and for rigid bodies (indirect method). Virtual displacements and work. Lagrange's equations for systems of particles and for rigid bodies. Linearization of equations of motion. Linear stability analysis of mechanical systems. Free and forced vibration of linear damped lumped parameter multi-degree of freedom models of mechanical systems. Application to the design of ocean and civil engineering structures such as tension leg platforms. Introduction to dynamics and vibration of lumped-parameter models of mechanical systems. Three-dimensional particle kinematics. Force-momentum formulation for systems of particles and for rigid bodies (direct method). Newton-Euler equations. Work-energy (variational) formulation for systems particles and for rigid bodies (indirect method). Virtual displacements and work. Lagrange's equations for systems of particles and for rigid bodies. Linearization of equations of motion. Linear stability analysis of mechanical systems. Free and forced vibration of linear damped lumped parameter multi-degree of freedom models of mechanical systems. Application to the design of ocean and civil engineering structures such as tension leg platforms.

Subjects

umped-parameter models | umped-parameter models | lumped-parameter models | lumped-parameter models | mechanical systems | mechanical systems | Three-dimensional particle kinematics | Three-dimensional particle kinematics | Force-momentum formulation | Force-momentum formulation | Newton-Euler equations | Newton-Euler equations | Work-enery (variational) formulation | Work-enery (variational) formulation | systems particles | systems particles | rigid bodies (indirect method) | rigid bodies (indirect method) | Virtual displacements | Virtual displacements | Lagrange's equations | Lagrange's equations | Linear stability analysis | Linear stability analysis | 1.053J | 1.053J | 13.013 | 13.013 | 1.053 | 1.053

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2.003J Dynamics and Vibration (13.013J) (MIT) 2.003J Dynamics and Vibration (13.013J) (MIT)

Description

Introduction to dynamics and vibration of lumped-parameter models of mechanical systems. Three-dimensional particle kinematics. Force-momentum formulation for systems of particles and for rigid bodies (direct method). Newton-Euler equations. Work-energy (variational) formulation for systems particles and for rigid bodies (indirect method). Virtual displacements and work. Lagrange's equations for systems of particles and for rigid bodies. Linearization of equations of motion. Linear stability analysis of mechanical systems. Free and forced vibration of linear damped lumped parameter multi-degree of freedom models of mechanical systems. Application to the design of ocean and civil engineering structures such as tension leg platforms. This subject was originally offered in Course 13 (Departm Introduction to dynamics and vibration of lumped-parameter models of mechanical systems. Three-dimensional particle kinematics. Force-momentum formulation for systems of particles and for rigid bodies (direct method). Newton-Euler equations. Work-energy (variational) formulation for systems particles and for rigid bodies (indirect method). Virtual displacements and work. Lagrange's equations for systems of particles and for rigid bodies. Linearization of equations of motion. Linear stability analysis of mechanical systems. Free and forced vibration of linear damped lumped parameter multi-degree of freedom models of mechanical systems. Application to the design of ocean and civil engineering structures such as tension leg platforms. This subject was originally offered in Course 13 (Departm

Subjects

umped-parameter models | umped-parameter models | lumped-parameter models | lumped-parameter models | mechanical systems | mechanical systems | Three-dimensional particle kinematics | Three-dimensional particle kinematics | Force-momentum formulation | Force-momentum formulation | Newton-Euler equations | Newton-Euler equations | Work-enery (variational) formulation | Work-enery (variational) formulation | systems particles | systems particles | rigid bodies (indirect method) | rigid bodies (indirect method) | Virtual displacements | Virtual displacements | Lagrange's equations | Lagrange's equations | Linear stability analysis | Linear stability analysis | 13.013J | 13.013J | 13.013 | 13.013 | 1.053 | 1.053

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16.225 Computational Mechanics of Materials (MIT) 16.225 Computational Mechanics of Materials (MIT)

Description

16.225 is a graduate level course on Computational Mechanics of Materials. The primary focus of this course is on the teaching of state-of-the-art numerical methods for the analysis of the nonlinear continuum response of materials. The range of material behavior considered in this course includes: linear and finite deformation elasticity, inelasticity and dynamics. Numerical formulation and algorithms include: variational formulation and variational constitutive updates, finite element discretization, error estimation, constrained problems, time integration algorithms and convergence analysis. There is a strong emphasis on the (parallel) computer implementation of algorithms in programming assignments. The application to real engineering applications and problems in engineering science is 16.225 is a graduate level course on Computational Mechanics of Materials. The primary focus of this course is on the teaching of state-of-the-art numerical methods for the analysis of the nonlinear continuum response of materials. The range of material behavior considered in this course includes: linear and finite deformation elasticity, inelasticity and dynamics. Numerical formulation and algorithms include: variational formulation and variational constitutive updates, finite element discretization, error estimation, constrained problems, time integration algorithms and convergence analysis. There is a strong emphasis on the (parallel) computer implementation of algorithms in programming assignments. The application to real engineering applications and problems in engineering science is

Subjects

Computational Mechanics | Computational Mechanics | Computation | Computation | Mechanics | Mechanics | Materials | Materials | Numerical Methods | Numerical Methods | Numerical | Numerical | Nonlinear Continuum Response | Nonlinear Continuum Response | Continuum | Continuum | Deformation | Deformation | Elasticity | Elasticity | Inelasticity | Inelasticity | Dynamics | Dynamics | Variational Formulation | Variational Formulation | Variational Constitutive Updates | Variational Constitutive Updates | Finite Element | Finite Element | Discretization | Discretization | Error Estimation | Error Estimation | Constrained Problems | Constrained Problems | Time Integration | Time Integration | Convergence Analysis | Convergence Analysis | Programming | Programming | Continuum Response | Continuum Response | Computational | Computational | state-of-the-art | state-of-the-art | methods | methods | modeling | modeling | simulation | simulation | mechanical | mechanical | response | response | engineering | engineering | aerospace | aerospace | civil | civil | material | material | science | science | biomechanics | biomechanics | behavior | behavior | finite | finite | deformation | deformation | elasticity | elasticity | inelasticity | inelasticity | contact | contact | friction | friction | coupled | coupled | numerical | numerical | formulation | formulation | algorithms | algorithms | Variational | Variational | constitutive | constitutive | updates | updates | element | element | discretization | discretization | mesh | mesh | generation | generation | error | error | estimation | estimation | constrained | constrained | problems | problems | time | time | convergence | convergence | analysis | analysis | parallel | parallel | computer | computer | implementation | implementation | programming | programming | assembly | assembly | equation-solving | equation-solving | formulating | formulating | implementing | implementing | complex | complex | approximations | approximations | equations | equations | motion | motion | dynamic | dynamic | deformations | deformations | continua | continua | plasticity | plasticity | rate-dependency | rate-dependency | integration | integration

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18.327 Wavelets, Filter Banks and Applications (MIT) 18.327 Wavelets, Filter Banks and Applications (MIT)

Description

Wavelets are localized basis functions, good for representing short-time events. The coefficients at each scale are filtered and subsampled to give coefficients at the next scale. This is Mallat's pyramid algorithm for multiresolution, connecting wavelets to filter banks. Wavelets and multiscale algorithms for compression and signal/image processing are developed. Subject is project-based for engineering and scientific applications. Wavelets are localized basis functions, good for representing short-time events. The coefficients at each scale are filtered and subsampled to give coefficients at the next scale. This is Mallat's pyramid algorithm for multiresolution, connecting wavelets to filter banks. Wavelets and multiscale algorithms for compression and signal/image processing are developed. Subject is project-based for engineering and scientific applications.

Subjects

Discrete-time filters | Discrete-time filters | convolution | convolution | Fourier transform | Fourier transform | owpass and highpass filters | owpass and highpass filters | Sampling rate change operations | Sampling rate change operations | upsampling and downsampling | upsampling and downsampling | ractional sampling | ractional sampling | interpolation | interpolation | Filter Banks | Filter Banks | time domain (Haar example) and frequency domain | time domain (Haar example) and frequency domain | conditions for alias cancellation and no distortion | conditions for alias cancellation and no distortion | perfect reconstruction | perfect reconstruction | halfband filters and possible factorizations | halfband filters and possible factorizations | Modulation and polyphase representations | Modulation and polyphase representations | Noble identities | Noble identities | block Toeplitz matrices and block z-transforms | block Toeplitz matrices and block z-transforms | polyphase examples | polyphase examples | Matlab wavelet toolbox | Matlab wavelet toolbox | Orthogonal filter banks | Orthogonal filter banks | paraunitary matrices | paraunitary matrices | orthogonality condition (Condition O) in the time domain | orthogonality condition (Condition O) in the time domain | modulation domain and polyphase domain | modulation domain and polyphase domain | Maxflat filters | Maxflat filters | Daubechies and Meyer formulas | Daubechies and Meyer formulas | Spectral factorization | Spectral factorization | Multiresolution Analysis (MRA) | Multiresolution Analysis (MRA) | requirements for MRA | requirements for MRA | nested spaces and complementary spaces; scaling functions and wavelets | nested spaces and complementary spaces; scaling functions and wavelets | Refinement equation | Refinement equation | iterative and recursive solution techniques | iterative and recursive solution techniques | infinite product formula | infinite product formula | filter bank approach for computing scaling functions and wavelets | filter bank approach for computing scaling functions and wavelets | Orthogonal wavelet bases | Orthogonal wavelet bases | connection to orthogonal filters | connection to orthogonal filters | orthogonality in the frequency domain | orthogonality in the frequency domain | Biorthogonal wavelet bases | Biorthogonal wavelet bases | Mallat pyramid algorithm | Mallat pyramid algorithm | Accuracy of wavelet approximations (Condition A) | Accuracy of wavelet approximations (Condition A) | vanishing moments | vanishing moments | polynomial cancellation in filter banks | polynomial cancellation in filter banks | Smoothness of wavelet bases | Smoothness of wavelet bases | convergence of the cascade algorithm (Condition E) | convergence of the cascade algorithm (Condition E) | splines | splines | Bases vs. frames | Bases vs. frames | Signal and image processing | Signal and image processing | finite length signals | finite length signals | boundary filters and boundary wavelets | boundary filters and boundary wavelets | wavelet compression algorithms | wavelet compression algorithms | Lifting | Lifting | ladder structure for filter banks | ladder structure for filter banks | factorization of polyphase matrix into lifting steps | factorization of polyphase matrix into lifting steps | lifting form of refinement equationSec | lifting form of refinement equationSec | Wavelets and subdivision | Wavelets and subdivision | nonuniform grids | nonuniform grids | multiresolution for triangular meshes | multiresolution for triangular meshes | representation and compression of surfaces | representation and compression of surfaces | Numerical solution of PDEs | Numerical solution of PDEs | Galerkin approximation | Galerkin approximation | wavelet integrals (projection coefficients | moments and connection coefficients) | wavelet integrals (projection coefficients | moments and connection coefficients) | convergence | convergence | Subdivision wavelets for integral equations | Subdivision wavelets for integral equations | Compression and convergence estimates | Compression and convergence estimates | M-band wavelets | M-band wavelets | DFT filter banks and cosine modulated filter banks | DFT filter banks and cosine modulated filter banks | Multiwavelets | Multiwavelets

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18.104 Seminar in Analysis: Applications to Number Theory (MIT)

Description

18.104 is an undergraduate level seminar for mathematics majors. Students present and discuss subject matter taken from current journals or books. Instruction and practice in written and oral communication is provided. The topics vary from year to year. The topic for this term is Applications to Number Theory.

Subjects

Infinitude of the primes | Summing powers of integers | Bernoulli polynomials | sine product formula | $\zeta(2n)$ | Fermat's Little Theorem | Fermat's Great Theorem | Averages of arithmetic functions | arithmetic-geometric mean | Gauss' theorem | Wallis's formula | Stirling's formula | prime number theorem | Riemann's hypothesis | Euler's proof of infinitude of primes | Density of prime numbers | Euclidean algorithm | Golden Ratio

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16.888 Multidisciplinary System Design Optimization (MIT) 16.888 Multidisciplinary System Design Optimization (MIT)

Description

This course is mainly focused on the quantitative aspects of design and presents a unifying framework called "Multidisciplinary System Design Optimization" (MSDO). The objective of the course is to present tools and methodologies for performing system optimization in a multidisciplinary design context, focusing on three aspects of the problem: (i) The multidisciplinary character of engineering systems, (ii) design of these complex systems, and (iii) tools for optimization. There is a version of this course (16.60s) offered through the MIT Professional Institute, targeted at professional engineers. This course is mainly focused on the quantitative aspects of design and presents a unifying framework called "Multidisciplinary System Design Optimization" (MSDO). The objective of the course is to present tools and methodologies for performing system optimization in a multidisciplinary design context, focusing on three aspects of the problem: (i) The multidisciplinary character of engineering systems, (ii) design of these complex systems, and (iii) tools for optimization. There is a version of this course (16.60s) offered through the MIT Professional Institute, targeted at professional engineers.

Subjects

optimization | optimization | multidisciplinary design optimization | multidisciplinary design optimization | MDO | MDO | subsystem identification | subsystem identification | interface design | interface design | linear constrained optimization fomulation | linear constrained optimization fomulation | non-linear constrained optimization formulation | non-linear constrained optimization formulation | scalar optimization | scalar optimization | vector optimization | vector optimization | systems engineering | systems engineering | complex systems | complex systems | heuristic search methods | heuristic search methods | tabu search | tabu search | simulated annealing | simulated annealing | genertic algorithms | genertic algorithms | sensitivity | sensitivity | tradeoff analysis | tradeoff analysis | goal programming | goal programming | isoperformance | isoperformance | pareto optimality | pareto optimality | flowchart | flowchart | design vector | design vector | simulation model | simulation model | objective vector | objective vector | input | input | discipline | discipline | output | output | coupling | coupling | multiobjective optimization | multiobjective optimization | optimization algorithms | optimization algorithms | tradespace exploration | tradespace exploration | numerical techniques | numerical techniques | direct methods | direct methods | penalty methods | penalty methods | heuristic techniques | heuristic techniques | SA | SA | GA | GA | approximation methods | approximation methods | sensitivity analysis | sensitivity analysis | isoperformace | isoperformace | output evaluation | output evaluation | MSDO framework | MSDO framework

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15.083J Integer Programming and Combinatorial Optimization (MIT) 15.083J Integer Programming and Combinatorial Optimization (MIT)

Description

The course is a comprehensive introduction to the theory, algorithms and applications of integer optimization and is organized in four parts: formulations and relaxations, algebra and geometry of integer optimization, algorithms for integer optimization, and extensions of integer optimization. The course is a comprehensive introduction to the theory, algorithms and applications of integer optimization and is organized in four parts: formulations and relaxations, algebra and geometry of integer optimization, algorithms for integer optimization, and extensions of integer optimization.

Subjects

theory | theory | algorithms | algorithms | integer optimization | integer optimization | formulations and relaxations | formulations and relaxations | algebra and geometry of integer optimization | algebra and geometry of integer optimization | algorithms for integer optimization | algorithms for integer optimization | extensions of integer optimization | extensions of integer optimization | 15.083 | 15.083

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8.512 Theory of Solids II (MIT) 8.512 Theory of Solids II (MIT)

Description

This is the second term of a theoretical treatment of the physics of solids. Topics covered include linear response theory; the physics of disorder; superconductivity; the local moment and itinerant magnetism; the Kondo problem and Fermi liquid theory. This is the second term of a theoretical treatment of the physics of solids. Topics covered include linear response theory; the physics of disorder; superconductivity; the local moment and itinerant magnetism; the Kondo problem and Fermi liquid theory.

Subjects

Linear response theory | Linear response theory | Fluctuation dissipation theorem | Fluctuation dissipation theorem | Scattering experiment | Scattering experiment | f-sum rule | f-sum rule | Physics of disorder | Physics of disorder | Kubo formula for conductivity | Kubo formula for conductivity | Conductance and sensitivity to boundary conditions | Conductance and sensitivity to boundary conditions | Scaling theory of localization | Scaling theory of localization | Mott variable range hopping | Mott variable range hopping | Superconductor | Superconductor | Transverse response | Transverse response | Landau diamagnetism | Landau diamagnetism | Microscopic derivation of London equation | Microscopic derivation of London equation | Effect of disorder | Effect of disorder | Quasiparticles and coherence factors | Quasiparticles and coherence factors | Tunneling and Josephson effect | Tunneling and Josephson effect | Magnetism | Magnetism | Local moment magnetism | Local moment magnetism | exchange interaction | exchange interaction | Ferro- and anti-ferro magnet and spin wave theory | Ferro- and anti-ferro magnet and spin wave theory | Band magnetism | Band magnetism | Stoner theory | Stoner theory | spin density wave | spin density wave | Local moment in metals | Local moment in metals | Friedel sum rule | Friedel sum rule | Friedel-Anderson model | Friedel-Anderson model | Kondo problem | Kondo problem | Fermi liquid theory | Fermi liquid theory | Electron Green?s function | Electron Green?s function

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13.853 Computational Ocean Acoustics (MIT) 13.853 Computational Ocean Acoustics (MIT)

Description

This course examines wave equations for fluid and visco-elastic media, wave-theory formulations of acoustic source radiation and seismo-acoustic propagation in stratified ocean waveguides, and Wavenumber Integration and Normal Mode methods for propagation in plane-stratified media. Also covered are Seismo-Acoustic modeling of seabeds and ice covers, seismic interface and surface waves in a stratified seabed, Parabolic Equation and Coupled Mode approaches to propagation in range-dependent ocean waveguides, numerical modeling of target scattering and reverberation clutter in ocean waveguides, and ocean ambient noise modeling. Students develop propagation models using all the numerical approaches relevant to state-of-the-art acoustic research. This course examines wave equations for fluid and visco-elastic media, wave-theory formulations of acoustic source radiation and seismo-acoustic propagation in stratified ocean waveguides, and Wavenumber Integration and Normal Mode methods for propagation in plane-stratified media. Also covered are Seismo-Acoustic modeling of seabeds and ice covers, seismic interface and surface waves in a stratified seabed, Parabolic Equation and Coupled Mode approaches to propagation in range-dependent ocean waveguides, numerical modeling of target scattering and reverberation clutter in ocean waveguides, and ocean ambient noise modeling. Students develop propagation models using all the numerical approaches relevant to state-of-the-art acoustic research.

Subjects

Wave equations | Wave equations | fluid and visco-elastic media | fluid and visco-elastic media | Wave-theory formulations | Wave-theory formulations | acoustic source radiation | acoustic source radiation | seismo-acoustic propagation | seismo-acoustic propagation | stratified ocean waveguides | stratified ocean waveguides | Wavenumber Integration | Wavenumber Integration | Normal Mode | Normal Mode | propagation in plane-stratified media | propagation in plane-stratified media | Seismo-Acoustic modeling | Seismo-Acoustic modeling | Seismic interface | Seismic interface | surface waves | surface waves | stratified seabed | stratified seabed | Parabolic Equation | Parabolic Equation | Coupled Mode | Coupled Mode | range-dependent ocean waveguides | range-dependent ocean waveguides | Numerical modeling | Numerical modeling | target scattering | target scattering | reverberation clutter | reverberation clutter | Ocean ambient noise modeling | Ocean ambient noise modeling | Fluid media | Fluid media | visco-elastic media | visco-elastic media | plane-stratified media | plane-stratified media | ice covers | ice covers | 2.068 | 2.068

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8.08 Statistical Physics II (MIT) 8.08 Statistical Physics II (MIT)

Description

Probability distributions for classical and quantum systems. Microcanonical, canonical, and grand canonical partition-functions and associated thermodynamic potentials. Conditions of thermodynamic equilibrium for homogenous and heterogenous systems. Applications: non-interacting Bose and Fermi gases; mean field theories for real gases, binary mixtures, magnetic systems, polymer solutions; phase and reaction equilibria, critical phenomena. Fluctuations, correlation functions and susceptibilities, and Kubo formulae. Evolution of distribution functions: Boltzmann and Smoluchowski equations. Probability distributions for classical and quantum systems. Microcanonical, canonical, and grand canonical partition-functions and associated thermodynamic potentials. Conditions of thermodynamic equilibrium for homogenous and heterogenous systems. Applications: non-interacting Bose and Fermi gases; mean field theories for real gases, binary mixtures, magnetic systems, polymer solutions; phase and reaction equilibria, critical phenomena. Fluctuations, correlation functions and susceptibilities, and Kubo formulae. Evolution of distribution functions: Boltzmann and Smoluchowski equations.

Subjects

Probability distributions | Probability distributions | quantum systems | quantum systems | Microcanonical | Microcanonical | canonical | canonical | grand canonical partition-functions | grand canonical partition-functions | thermodynamic potentials | thermodynamic potentials | Conditions of thermodynamic equilibrium for homogenous and heterogenous systems | Conditions of thermodynamic equilibrium for homogenous and heterogenous systems | non-interacting Bose and Fermi gases | non-interacting Bose and Fermi gases | mean field theories for real gases | mean field theories for real gases | binary mixtures | binary mixtures | magnetic systems | magnetic systems | polymer solutions | polymer solutions | phase and reaction equilibria | phase and reaction equilibria | critical phenomena | critical phenomena | Fluctuations | Fluctuations | correlation functions and susceptibilities | correlation functions and susceptibilities | Kubo formulae | Kubo formulae | Evolution of distribution functions | Evolution of distribution functions | Boltzmann and Smoluchowski equations | Boltzmann and Smoluchowski equations | correlation functions | correlation functions | susceptibilities | susceptibilities

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RES.2-002 Finite Element Procedures for Solids and Structures (MIT) RES.2-002 Finite Element Procedures for Solids and Structures (MIT)

Description

Includes audio/video content: AV lectures. Finite element analysis is now widely used for solving complex static and dynamic problems encountered in engineering and the sciences. In these two video courses, Professor K. J. Bathe, a researcher of world renown in the field of finite element analysis, teaches the basic principles used for effective finite element analysis, describes the general assumptions, and discusses the implementation of finite element procedures for linear and nonlinear analyses. These videos were produced in 1982 and 1986 by the MIT Center for Advanced Engineering Study. Includes audio/video content: AV lectures. Finite element analysis is now widely used for solving complex static and dynamic problems encountered in engineering and the sciences. In these two video courses, Professor K. J. Bathe, a researcher of world renown in the field of finite element analysis, teaches the basic principles used for effective finite element analysis, describes the general assumptions, and discusses the implementation of finite element procedures for linear and nonlinear analyses. These videos were produced in 1982 and 1986 by the MIT Center for Advanced Engineering Study.

Subjects

finite element method | finite element method | statics | statics | dynamics | dynamics | linear analysis | linear analysis | nonlinear analysis | nonlinear analysis | computer modeling | computer modeling | engineering design | engineering design | solids | solids | structures | structures | wave propagation | wave propagation | vibration | vibration | collapse | collapse | buckling | buckling | Lagrangian formulation | Lagrangian formulation | truss | truss | beam | beam | plate | plate | shell | shell | elastic materials | elastic materials | plastic materials | plastic materials | creep | creep | ADINA | ADINA | numerical integration methods | numerical integration methods | mode superposition | mode superposition

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16.346 Astrodynamics (MIT) 16.346 Astrodynamics (MIT)

Description

Includes audio/video content: AV selected lectures. This course covers the fundamentals of astrodynamics, focusing on the two-body orbital initial-value and boundary-value problems with applications to space vehicle navigation and guidance for lunar and planetary missions, including both powered flight and midcourse maneuvers. Other topics include celestial mechanics, Kepler's problem, Lambert's problem, orbit determination, multi-body methods, mission planning, and recursive algorithms for space navigation. Selected applications from the Apollo, Space Shuttle, and Mars exploration programs are also discussed. Includes audio/video content: AV selected lectures. This course covers the fundamentals of astrodynamics, focusing on the two-body orbital initial-value and boundary-value problems with applications to space vehicle navigation and guidance for lunar and planetary missions, including both powered flight and midcourse maneuvers. Other topics include celestial mechanics, Kepler's problem, Lambert's problem, orbit determination, multi-body methods, mission planning, and recursive algorithms for space navigation. Selected applications from the Apollo, Space Shuttle, and Mars exploration programs are also discussed.

Subjects

space navigation | space navigation | two body problem | two body problem | boundary value problem | boundary value problem | Kepler | Kepler | astrodynamics | astrodynamics | orbital transfer | orbital transfer | satellite | satellite | hyperbolic orbits | hyperbolic orbits | planetary flybys | planetary flybys | hypergeometric functions | hypergeometric functions | flight guidance | flight guidance | three body problem | three body problem | Clohessy-Wiltshire equation | Clohessy-Wiltshire equation | Hodograph plane | Hodograph plane | Battin-vaughan formulation | Battin-vaughan formulation | atmospheric drag | atmospheric drag | disturbing function | disturbing function

License

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2.003J Dynamics and Control I (MIT) 2.003J Dynamics and Control I (MIT)

Description

Introduction to the dynamics and vibrations of lumped-parameter models of mechanical systems. Kinematics. Force-momentum formulation for systems of particles and rigid bodies in planar motion. Work-energy concepts. Virtual displacements and virtual work. Lagrange's equations for systems of particles and rigid bodies in planar motion. Linearization of equations of motion. Linear stability analysis of mechanical systems. Free and forced vibration of linear multi-degree of freedom models of mechanical systems; matrix eigenvalue problems. Introduction to numerical methods and MATLAB® to solve dynamics and vibrations problems. Introduction to the dynamics and vibrations of lumped-parameter models of mechanical systems. Kinematics. Force-momentum formulation for systems of particles and rigid bodies in planar motion. Work-energy concepts. Virtual displacements and virtual work. Lagrange's equations for systems of particles and rigid bodies in planar motion. Linearization of equations of motion. Linear stability analysis of mechanical systems. Free and forced vibration of linear multi-degree of freedom models of mechanical systems; matrix eigenvalue problems. Introduction to numerical methods and MATLAB® to solve dynamics and vibrations problems.

Subjects

dynamics and vibrations of lumped-parameter models | dynamics and vibrations of lumped-parameter models | mechanical systems | mechanical systems | Kinematics | Kinematics | Force-momentum formulation | Force-momentum formulation | systems of particles | systems of particles | rigid bodies in planar motion | rigid bodies in planar motion | Work-energy concepts | Work-energy concepts | Virtual displacements | Virtual displacements | virtual work | virtual work | Lagrange's equations | Lagrange's equations | Linearization of equations of motion | Linearization of equations of motion | Linear stability analysis | Linear stability analysis | Free vibration | Free vibration | forced vibration | forced vibration | linear multi-degree of freedom models | linear multi-degree of freedom models | matrix eigenvalue problems | matrix eigenvalue problems | numerical methods | numerical methods | MATLAB | MATLAB

License

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2.068 Computational Ocean Acoustics (13.853) (MIT) 2.068 Computational Ocean Acoustics (13.853) (MIT)

Description

This course examines wave equations for fluid and visco-elastic media, wave-theory formulations of acoustic source radiation and seismo-acoustic propagation in stratified ocean waveguides, and Wavenumber Integration and Normal Mode methods for propagation in plane-stratified media. Also covered are Seismo-Acoustic modeling of seabeds and ice covers, seismic interface and surface waves in a stratified seabed, Parabolic Equation and Coupled Mode approaches to propagation in range-dependent ocean waveguides, numerical modeling of target scattering and reverberation clutter in ocean waveguides, and ocean ambient noise modeling. Students develop propagation models using all the numerical approaches relevant to state-of-the-art acoustic research. This course was originally offered in Course 13 ( This course examines wave equations for fluid and visco-elastic media, wave-theory formulations of acoustic source radiation and seismo-acoustic propagation in stratified ocean waveguides, and Wavenumber Integration and Normal Mode methods for propagation in plane-stratified media. Also covered are Seismo-Acoustic modeling of seabeds and ice covers, seismic interface and surface waves in a stratified seabed, Parabolic Equation and Coupled Mode approaches to propagation in range-dependent ocean waveguides, numerical modeling of target scattering and reverberation clutter in ocean waveguides, and ocean ambient noise modeling. Students develop propagation models using all the numerical approaches relevant to state-of-the-art acoustic research. This course was originally offered in Course 13 (

Subjects

Wave equations | Wave equations | fluid and visco-elastic media | fluid and visco-elastic media | Wave-theory formulations | Wave-theory formulations | acoustic source radiation | acoustic source radiation | seismo-acoustic propagation | seismo-acoustic propagation | stratified ocean waveguides | stratified ocean waveguides | Wavenumber Integration | Wavenumber Integration | Normal Mode | Normal Mode | propagation in plane-stratified media | propagation in plane-stratified media | Seismo-Acoustic modeling | Seismo-Acoustic modeling | Seismic interface | Seismic interface | surface waves | surface waves | stratified seabed | stratified seabed | Parabolic Equation | Parabolic Equation | Coupled Mode | Coupled Mode | range-dependent ocean waveguides | range-dependent ocean waveguides | Numerical modeling | Numerical modeling | target scattering | target scattering | reverberation clutter | reverberation clutter | Ocean ambient noise modeling | Ocean ambient noise modeling | Fluid media | Fluid media | visco-elastic media | visco-elastic media | plane-stratified media | plane-stratified media | ice covers | ice covers

License

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8.512 Theory of Solids II (MIT) 8.512 Theory of Solids II (MIT)

Description

This is the second term of a theoretical treatment of the physics of solids. Topics covered include linear response theory; the physics of disorder; superconductivity; the local moment and itinerant magnetism; the Kondo problem and Fermi liquid theory. This is the second term of a theoretical treatment of the physics of solids. Topics covered include linear response theory; the physics of disorder; superconductivity; the local moment and itinerant magnetism; the Kondo problem and Fermi liquid theory.

Subjects

Linear response theory | Linear response theory | Fluctuation dissipation theorem | Fluctuation dissipation theorem | Scattering experiment | Scattering experiment | f-sum rule | f-sum rule | Physics of disorder | Physics of disorder | Kubo formula for conductivity | Kubo formula for conductivity | Conductance and sensitivity to boundary conditions | Conductance and sensitivity to boundary conditions | Scaling theory of localization | Scaling theory of localization | Mott variable range hopping | Mott variable range hopping | Superconductor | Superconductor | Transverse response | Transverse response | Landau diamagnetism | Landau diamagnetism | Microscopic derivation of London equation | Microscopic derivation of London equation | Effect of disorder | Effect of disorder | Quasiparticles and coherence factors | Quasiparticles and coherence factors | Tunneling and Josephson effect | Tunneling and Josephson effect | Magnetism | Magnetism | Local moment magnetism | Local moment magnetism | exchange interaction | exchange interaction | Ferro- and anti-ferro magnet and spin wave theory | Ferro- and anti-ferro magnet and spin wave theory | Band magnetism | Band magnetism | Stoner theory | Stoner theory | spin density wave | spin density wave | Local moment in metals | Local moment in metals | Friedel sum rule | Friedel sum rule | Friedel-Anderson model | Friedel-Anderson model | Kondo problem | Kondo problem | Fermi liquid theory | Fermi liquid theory | Electron Green?s function | Electron Green?s function

License

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8.325 Relativistic Quantum Field Theory III (MIT) 8.325 Relativistic Quantum Field Theory III (MIT)

Description

This course is the third and last term of the quantum field theory sequence. Its aim is the proper theoretical discussion of the physics of the standard model. Topics include: quantum chromodynamics; the Higgs phenomenon and a description of the standard model; deep-inelastic scattering and structure functions; basics of lattice gauge theory; operator products and effective theories; detailed structure of the standard model; spontaneously broken gauge theory and its quantization; instantons and theta-vacua; topological defects; introduction to supersymmetry. This course is the third and last term of the quantum field theory sequence. Its aim is the proper theoretical discussion of the physics of the standard model. Topics include: quantum chromodynamics; the Higgs phenomenon and a description of the standard model; deep-inelastic scattering and structure functions; basics of lattice gauge theory; operator products and effective theories; detailed structure of the standard model; spontaneously broken gauge theory and its quantization; instantons and theta-vacua; topological defects; introduction to supersymmetry.

Subjects

gauge symmetry | gauge symmetry | confinement | confinement | renormalization | renormalization | asymptotic freedom | asymptotic freedom | anomalies | anomalies | instantons | instantons | zero modes | zero modes | gauge boson and Higgs spectrum | gauge boson and Higgs spectrum | fermion multiplets | fermion multiplets | CKM matrix | CKM matrix | unification in SU(5) and SO(10) | unification in SU(5) and SO(10) | phenomenology of Higgs sector | phenomenology of Higgs sector | lepton and baryon number violation | lepton and baryon number violation | nonperturbative (lattice) formulation | nonperturbative (lattice) formulation

License

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8.08 Statistical Physics II (MIT) 8.08 Statistical Physics II (MIT)

Description

This course covers probability distributions for classical and quantum systems. Topics include: Microcanonical, canonical, and grand canonical partition-functions and associated thermodynamic potentials. Also discussed are conditions of thermodynamic equilibrium for homogenous and heterogenous systems. The course follows 8.044, Statistical Physics I, and is second in this series of undergraduate Statistical Physics courses. This course covers probability distributions for classical and quantum systems. Topics include: Microcanonical, canonical, and grand canonical partition-functions and associated thermodynamic potentials. Also discussed are conditions of thermodynamic equilibrium for homogenous and heterogenous systems. The course follows 8.044, Statistical Physics I, and is second in this series of undergraduate Statistical Physics courses.

Subjects

Probability distributions | Probability distributions | quantum systems | quantum systems | Microcanonical | canonical | and grand canonical partition-functions | Microcanonical | canonical | and grand canonical partition-functions | thermodynamic potentials | thermodynamic potentials | Conditions of thermodynamic equilibrium for homogenous and heterogenous systems | Conditions of thermodynamic equilibrium for homogenous and heterogenous systems | non-interacting Bose and Fermi gases | non-interacting Bose and Fermi gases | mean field theories for real gases | mean field theories for real gases | binary mixtures | binary mixtures | magnetic systems | magnetic systems | polymer solutions | polymer solutions | phase and reaction equilibria | phase and reaction equilibria | critical phenomena | critical phenomena | Fluctuations | Fluctuations | correlation functions and susceptibilities | and Kubo formulae | correlation functions and susceptibilities | and Kubo formulae | Evolution of distribution functions: Boltzmann and Smoluchowski equations | Evolution of distribution functions: Boltzmann and Smoluchowski equations

License

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8.325 Relativistic Quantum Field Theory III (MIT) 8.325 Relativistic Quantum Field Theory III (MIT)

Description

This is the third and last term of the quantum field theory sequence. The course is devoted to the standard model of particle physics, including both its conceptual foundations and its specific structure, and to some current research frontiers that grow immediately out of it. This is the third and last term of the quantum field theory sequence. The course is devoted to the standard model of particle physics, including both its conceptual foundations and its specific structure, and to some current research frontiers that grow immediately out of it.

Subjects

gauge symmetry | gauge symmetry | confinement | confinement | renormalization | renormalization | asymptotic freedom | asymptotic freedom | anomalies | anomalies | instantons | instantons | zeromodes | zeromodes | gauge boson and Higgs spectrum | gauge boson and Higgs spectrum | fermion multiplets | fermion multiplets | CKM matrix | CKM matrix | unification in SU(5) andSO(10) | unification in SU(5) andSO(10) | phenomenology of Higgs sector | phenomenology of Higgs sector | lepton andbaryon number violation | lepton andbaryon number violation | nonperturbative (lattice)formulation | nonperturbative (lattice)formulation

License

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11.967 Special Studies in Urban Studies and Planning: Economic Development Planning Skills (MIT) 11.967 Special Studies in Urban Studies and Planning: Economic Development Planning Skills (MIT)

Description

This intensive and brief 4-day seminar, taught during MIT's Independent Activities Period in January, uses a case set in Hartford, Vermont to introduce economic development planning skills to students in the Master in City Planning (MCP) Degree Program. It introduces analytical tools that are used to assess local economic development conditions, issues, and opportunities as part of formulating economic development plans. The course is designed to provide MCP students with skills needed for applied economic development planning work in other courses, particularly Economic Development Planning (11.438) and Revitalizing Urban Main Streets (11.439). This intensive and brief 4-day seminar, taught during MIT's Independent Activities Period in January, uses a case set in Hartford, Vermont to introduce economic development planning skills to students in the Master in City Planning (MCP) Degree Program. It introduces analytical tools that are used to assess local economic development conditions, issues, and opportunities as part of formulating economic development plans. The course is designed to provide MCP students with skills needed for applied economic development planning work in other courses, particularly Economic Development Planning (11.438) and Revitalizing Urban Main Streets (11.439).

Subjects

local development | local development | economic development | economic development | conditions | conditions | issues | issues | opportunites | opportunites | formulating economic development plans | formulating economic development plans | Hartford | VT | Hartford | VT | economic development plans | economic development plans | urban main streets | urban main streets | development planning | development planning

License

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12.950 Atmospheric and Oceanic Modeling (MIT) 12.950 Atmospheric and Oceanic Modeling (MIT)

Description

The numerical methods, formulation and parameterizations used in models of the circulation of the atmosphere and ocean will be described in detail. Widely used numerical methods will be the focus but we will also review emerging concepts and new methods. The numerics underlying a hierarchy of models will be discussed, ranging from simple GFD models to the high-end GCMs. In the context of ocean GCMs, we will describe parameterization of geostrophic eddies, mixing and the surface and bottom boundary layers. In the atmosphere, we will review parameterizations of convection and large scale condensation, the planetary boundary layer and radiative transfer. The numerical methods, formulation and parameterizations used in models of the circulation of the atmosphere and ocean will be described in detail. Widely used numerical methods will be the focus but we will also review emerging concepts and new methods. The numerics underlying a hierarchy of models will be discussed, ranging from simple GFD models to the high-end GCMs. In the context of ocean GCMs, we will describe parameterization of geostrophic eddies, mixing and the surface and bottom boundary layers. In the atmosphere, we will review parameterizations of convection and large scale condensation, the planetary boundary layer and radiative transfer.

Subjects

numerical methods | numerical methods | formulation | formulation | parameterizations | parameterizations | models of the circulation of the atmosphere and ocean | models of the circulation of the atmosphere and ocean | numerics underlying a hierarchy of models | numerics underlying a hierarchy of models | simple GFD models | simple GFD models | high-end GCMs | high-end GCMs | ocean GCMs | ocean GCMs | parameterization of geostrophic eddies | parameterization of geostrophic eddies | mixing | mixing | surface and bottom boundary layers | surface and bottom boundary layers | atmosphere | atmosphere | parameterizations of convection | parameterizations of convection | large scale condensation | large scale condensation | planetary boundary layer | planetary boundary layer | radiative transfer | radiative transfer | finite difference method | finite difference method | Spatial discretization | Spatial discretization | numerical dispersion | numerical dispersion | Series expansion | Series expansion | Time-stepping | Time-stepping | Space-time discretization | Space-time discretization | Shallow water dynamics | Shallow water dynamics | Barotropic models | Barotropic models | Quasi-geostrophic equations | Quasi-geostrophic equations | Quasi-geostrophic models | Quasi-geostrophic models | Eddy parameterization | Eddy parameterization | Vertical coordinates | Vertical coordinates | primitive equations | primitive equations | Boundary layer parameterizations | Boundary layer parameterizations

License

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14.731 Economic History (MIT) 14.731 Economic History (MIT)

Description

This course is a survey of world economic history, and it introduces economics students to the subject matter and methodology of economic history. It is designed to expand the range of empirical settings in students' research by drawing upon historical material and long-run data. Topics are chosen to show a wide variety of historical experience and illuminate the process of industrialization. The emphasis will be on questions related to labor markets and economic growth. This course is a survey of world economic history, and it introduces economics students to the subject matter and methodology of economic history. It is designed to expand the range of empirical settings in students' research by drawing upon historical material and long-run data. Topics are chosen to show a wide variety of historical experience and illuminate the process of industrialization. The emphasis will be on questions related to labor markets and economic growth.

Subjects

Economic History | Economic History | industrialization | industrialization | demographic change | demographic change | policies | policies | Applied Economics | Applied Economics | formulate and test hypotheses | formulate and test hypotheses | labor history | labor history | discrimination | discrimination | technology | technology | institutions | institutions | financial crises | financial crises | migration | migration | recovery after shocks | recovery after shocks | wages | wages | inequality | inequality | health | health | stock market regulation | stock market regulation

License

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14.731 Economic History (MIT) 14.731 Economic History (MIT)

Description

This course offers a comprehensive survey of world economic history, designed to introduce economics graduate students to the subject matter and methodology of economic history. Topics are chosen to show a wide variety of historical experience and illuminate the process of industrialization. A final term paper is due at the end of the course. This course offers a comprehensive survey of world economic history, designed to introduce economics graduate students to the subject matter and methodology of economic history. Topics are chosen to show a wide variety of historical experience and illuminate the process of industrialization. A final term paper is due at the end of the course.

Subjects

Economic History | Economic History | industrialization | industrialization | demographic change | demographic change | policies | policies | Applied Economics | Applied Economics | formulate and test hypotheses | formulate and test hypotheses

License

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