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18.366 Random Walks and Diffusion (MIT) 18.366 Random Walks and Diffusion (MIT)

Description

This graduate-level subject explores various mathematical aspects of (discrete) random walks and (continuum) diffusion. Applications include polymers, disordered media, turbulence, diffusion-limited aggregation, granular flow, and derivative securities. This graduate-level subject explores various mathematical aspects of (discrete) random walks and (continuum) diffusion. Applications include polymers, disordered media, turbulence, diffusion-limited aggregation, granular flow, and derivative securities.

Subjects

Discrete and continuum modeling of diffusion processes in physics | chemistry | and economics | Discrete and continuum modeling of diffusion processes in physics | chemistry | and economics | central limit theorems | central limit theorems | continuous-time random walks | continuous-time random walks | Levy flights | Levy flights | correlations | correlations | extreme events | extreme events | mixing | mixing | renormalization | renormalization | and percolation | and percolation | percolation | percolation

License

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18.366 Random Walks and Diffusion (MIT) 18.366 Random Walks and Diffusion (MIT)

Description

Mathematical modeling of diffusion phenomena: Central limit theorems, the continuum limit, first passage, persistence, continuous-time random walks, Levy flights, fractional calculus, random environments, advection-diffusion, nonlinear diffusion, free-boundary problems. Applications may include polymers, disordered media, turbulence, diffusion-limited aggregation, granular flow, and derivative securities. Mathematical modeling of diffusion phenomena: Central limit theorems, the continuum limit, first passage, persistence, continuous-time random walks, Levy flights, fractional calculus, random environments, advection-diffusion, nonlinear diffusion, free-boundary problems. Applications may include polymers, disordered media, turbulence, diffusion-limited aggregation, granular flow, and derivative securities.

Subjects

Discrete and continuum modeling of diffusion processes in physics | Discrete and continuum modeling of diffusion processes in physics | chemistry | chemistry | and economics | and economics | central limit theorems | central limit theorems | ontinuous-time random walks | ontinuous-time random walks | Levy flights | Levy flights | correlations | correlations | extreme events | extreme events | mixing | mixing | renormalization | renormalization | and percolation | and percolation

License

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3.016 Mathematics for Materials Scientists and Engineers (MIT) 3.016 Mathematics for Materials Scientists and Engineers (MIT)

Description

The class will cover mathematical techniques necessary for understanding of materials science and engineering topics such as energetics, materials structure and symmetry, materials response to applied fields, mechanics and physics of solids and soft materials. The class uses examples from 3.012 to introduce mathematical concepts and materials-related problem solving skills. Topics include linear algebra and orthonormal basis, eigenvalues and eigenvectors, quadratic forms, tensor operations, symmetry operations, calculus of several variables, introduction to complex analysis, ordinary and partial differential equations, theory of distributions, fourier analysis and random walks.Technical RequirementsMathematica® software is required to run the .nb files found on this course site. The class will cover mathematical techniques necessary for understanding of materials science and engineering topics such as energetics, materials structure and symmetry, materials response to applied fields, mechanics and physics of solids and soft materials. The class uses examples from 3.012 to introduce mathematical concepts and materials-related problem solving skills. Topics include linear algebra and orthonormal basis, eigenvalues and eigenvectors, quadratic forms, tensor operations, symmetry operations, calculus of several variables, introduction to complex analysis, ordinary and partial differential equations, theory of distributions, fourier analysis and random walks.Technical RequirementsMathematica® software is required to run the .nb files found on this course site.

Subjects

energetics | energetics | materials structure and symmetry: applied fields | materials structure and symmetry: applied fields | mechanics and physics of solids and soft materials | mechanics and physics of solids and soft materials | linear algebra | linear algebra | orthonormal basis | orthonormal basis | eigenvalues | eigenvalues | eigenvectors | eigenvectors | quadratic forms | quadratic forms | tensor operations | tensor operations | symmetry operations | symmetry operations | calculus | calculus | complex analysis | complex analysis | differential equations | differential equations | theory of distributions | theory of distributions | fourier analysis | fourier analysis | random walks | random walks | mathematical technicques | mathematical technicques | materials science | materials science | materials engineering | materials engineering | materials structure | materials structure | symmetry | symmetry | applied fields | applied fields | materials response | materials response | solids mechanics | solids mechanics | solids physics | solids physics | soft materials | soft materials | multi-variable calculus | multi-variable calculus | ordinary differential equations | ordinary differential equations | partial differential equations | partial differential equations | applied mathematics | applied mathematics | mathematical techniques | mathematical techniques

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6.262 Discrete Stochastic Processes (MIT) 6.262 Discrete Stochastic Processes (MIT)

Description

Includes audio/video content: AV lectures. Discrete stochastic processes are essentially probabilistic systems that evolve in time via random changes occurring at discrete fixed or random intervals. This course aims to help students acquire both the mathematical principles and the intuition necessary to create, analyze, and understand insightful models for a broad range of these processes. The range of areas for which discrete stochastic-process models are useful is constantly expanding, and includes many applications in engineering, physics, biology, operations research and finance. Includes audio/video content: AV lectures. Discrete stochastic processes are essentially probabilistic systems that evolve in time via random changes occurring at discrete fixed or random intervals. This course aims to help students acquire both the mathematical principles and the intuition necessary to create, analyze, and understand insightful models for a broad range of these processes. The range of areas for which discrete stochastic-process models are useful is constantly expanding, and includes many applications in engineering, physics, biology, operations research and finance.

Subjects

probability | probability | Poisson processes | Poisson processes | finite-state Markov chains | finite-state Markov chains | renewal processes | renewal processes | countable-state Markov chains | countable-state Markov chains | Markov processes | Markov processes | countable state spaces | countable state spaces | random walks | random walks | large deviations | large deviations | martingales | martingales

License

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3.016 Mathematics for Materials Scientists and Engineers (MIT) 3.016 Mathematics for Materials Scientists and Engineers (MIT)

Description

This course covers the mathematical techniques necessary for understanding of materials science and engineering topics such as energetics, materials structure and symmetry, materials response to applied fields, mechanics and physics of solids and soft materials. The class uses examples from the materials science and engineering core courses (3.012 and 3.014) to introduce mathematical concepts and materials-related problem solving skills. Topics include linear algebra and orthonormal basis, eigenvalues and eigenvectors, quadratic forms, tensor operations, symmetry operations, calculus of several variables, introduction to complex analysis, ordinary and partial differential equations, theory of distributions, and fourier analysis. Users may find additional or updated materials at Professor C This course covers the mathematical techniques necessary for understanding of materials science and engineering topics such as energetics, materials structure and symmetry, materials response to applied fields, mechanics and physics of solids and soft materials. The class uses examples from the materials science and engineering core courses (3.012 and 3.014) to introduce mathematical concepts and materials-related problem solving skills. Topics include linear algebra and orthonormal basis, eigenvalues and eigenvectors, quadratic forms, tensor operations, symmetry operations, calculus of several variables, introduction to complex analysis, ordinary and partial differential equations, theory of distributions, and fourier analysis. Users may find additional or updated materials at Professor C

Subjects

energetics | energetics | visualization | visualization | graph | graph | plot | plot | chart | chart | materials science | materials science | DMSE | DMSE | structure | structure | symmetry | symmetry | mechanics | mechanics | physicss | physicss | solids and soft materials | solids and soft materials | linear algebra | linear algebra | orthonormal basis | orthonormal basis | eigenvalue | eigenvalue | eigenvector | eigenvector | quadratic form | quadratic form | tensor operation | tensor operation | symmetry operation | symmetry operation | calculus | calculus | complex analysis | complex analysis | differential equations | differential equations | ODE | ODE | solution | solution | vector | vector | matrix | matrix | determinant | determinant | theory of distributions | theory of distributions | fourier analysis | fourier analysis | random walk | random walk | Mathematica | Mathematica | simulation | simulation

License

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5.72 Statistical Mechanics (MIT) 5.72 Statistical Mechanics (MIT)

Description

This course discusses the principles and methods of statistical mechanics. Topics covered include classical and quantum statistics, grand ensembles, fluctuations, molecular distribution functions, other concepts in equilibrium statistical mechanics, and topics in thermodynamics and statistical mechanics of irreversible processes. This course discusses the principles and methods of statistical mechanics. Topics covered include classical and quantum statistics, grand ensembles, fluctuations, molecular distribution functions, other concepts in equilibrium statistical mechanics, and topics in thermodynamics and statistical mechanics of irreversible processes.

Subjects

statistical mechanics | statistical mechanics | quantum | quantum | statistics | statistics | atoms | atoms | materials | materials | master equations | master equations | random walk | random walk | langevin | langevin | fokker | fokker | planck | planck | probability theory | probability theory | bloch-redfield | bloch-redfield | navier-stokes | navier-stokes | hydrodynamic | hydrodynamic | scattering | scattering | projection operator | projection operator | thermodynamics | thermodynamics

License

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6.055J The Art of Approximation in Science and Engineering (MIT) 6.055J The Art of Approximation in Science and Engineering (MIT)

Description

This course teaches simple reasoning techniques for complex phenomena: divide and conquer, dimensional analysis, extreme cases, continuity, scaling, successive approximation, balancing, cheap calculus, and symmetry. Applications are drawn from the physical and biological sciences, mathematics, and engineering. Examples include bird and machine flight, neuron biophysics, weather, prime numbers, and animal locomotion. Emphasis is on low-cost experiments to test ideas and on fostering curiosity about phenomena in the world. This course teaches simple reasoning techniques for complex phenomena: divide and conquer, dimensional analysis, extreme cases, continuity, scaling, successive approximation, balancing, cheap calculus, and symmetry. Applications are drawn from the physical and biological sciences, mathematics, and engineering. Examples include bird and machine flight, neuron biophysics, weather, prime numbers, and animal locomotion. Emphasis is on low-cost experiments to test ideas and on fostering curiosity about phenomena in the world.

Subjects

approximation | approximation | science | science | engineering | engineering | managing complexity | managing complexity | divide and conquer | divide and conquer | heterogeneous hierarchies | heterogeneous hierarchies | homogeneous hierarchies | homogeneous hierarchies | proportional reasoning | proportional reasoning | conservation/box models | conservation/box models | dimensional analysis | dimensional analysis | special cases | special cases | extreme cases | extreme cases | discretization | discretization | spring models | spring models | symmetry | symmetry | invariance | invariance | discarding information | discarding information | oil imports | oil imports | tree representations | tree representations | gold | gold | random walks | random walks | UNIX | UNIX | triangle bisection | triangle bisection | pentagonal heat flow | pentagonal heat flow | jump heights | jump heights | simple calculus | simple calculus | drag | drag | cycling | cycling | swimming | swimming | flying | flying | flight | flight | algebraic symmetry | algebraic symmetry | densities | densities | hydrogen size | hydrogen size | bending of light | bending of light | Buckingham Pi Theorem | Buckingham Pi Theorem | pulley acceleration | pulley acceleration | waves | waves

License

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6.856J Randomized Algorithms (MIT) 6.856J Randomized Algorithms (MIT)

Description

This course examines how randomization can be used to make algorithms simpler and more efficient via random sampling, random selection of witnesses, symmetry breaking, and Markov chains. Topics covered include: randomized computation; data structures (hash tables, skip lists); graph algorithms (minimum spanning trees, shortest paths, minimum cuts); geometric algorithms (convex hulls, linear programming in fixed or arbitrary dimension); approximate counting; parallel algorithms; online algorithms; derandomization techniques; and tools for probabilistic analysis of algorithms. This course examines how randomization can be used to make algorithms simpler and more efficient via random sampling, random selection of witnesses, symmetry breaking, and Markov chains. Topics covered include: randomized computation; data structures (hash tables, skip lists); graph algorithms (minimum spanning trees, shortest paths, minimum cuts); geometric algorithms (convex hulls, linear programming in fixed or arbitrary dimension); approximate counting; parallel algorithms; online algorithms; derandomization techniques; and tools for probabilistic analysis of algorithms.

Subjects

Randomized Algorithms | Randomized Algorithms | algorithms | algorithms | efficient in time and space | efficient in time and space | randomization | randomization | computational problems | computational problems | data structures | data structures | graph algorithms | graph algorithms | optimization | optimization | geometry | geometry | Markov chains | Markov chains | sampling | sampling | estimation | estimation | geometric algorithms | geometric algorithms | parallel and distributed algorithms | parallel and distributed algorithms | parallel and ditributed algorithm | parallel and ditributed algorithm | parallel and distributed algorithm | parallel and distributed algorithm | random sampling | random sampling | random selection of witnesses | random selection of witnesses | symmetry breaking | symmetry breaking | randomized computational models | randomized computational models | hash tables | hash tables | skip lists | skip lists | minimum spanning trees | minimum spanning trees | shortest paths | shortest paths | minimum cuts | minimum cuts | convex hulls | convex hulls | linear programming | linear programming | fixed dimension | fixed dimension | arbitrary dimension | arbitrary dimension | approximate counting | approximate counting | parallel algorithms | parallel algorithms | online algorithms | online algorithms | derandomization techniques | derandomization techniques | probabilistic analysis | probabilistic analysis | computational number theory | computational number theory | simplicity | simplicity | speed | speed | design | design | basic probability theory | basic probability theory | application | application | randomized complexity classes | randomized complexity classes | game-theoretic techniques | game-theoretic techniques | Chebyshev | Chebyshev | moment inequalities | moment inequalities | limited independence | limited independence | coupon collection | coupon collection | occupancy problems | occupancy problems | tail inequalities | tail inequalities | Chernoff bound | Chernoff bound | conditional expectation | conditional expectation | probabilistic method | probabilistic method | random walks | random walks | algebraic techniques | algebraic techniques | probability amplification | probability amplification | sorting | sorting | searching | searching | combinatorial optimization | combinatorial optimization | approximation | approximation | counting problems | counting problems | distributed algorithms | distributed algorithms | 6.856 | 6.856 | 18.416 | 18.416

License

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15.997 Practice of Finance: Advanced Corporate Risk Management (MIT) 15.997 Practice of Finance: Advanced Corporate Risk Management (MIT)

Description

This is a course in how corporations make use of the insights and tools of risk management. Most courses on derivatives, futures and options, and financial engineering are taught from the viewpoint of investment bankers and traders in the securities. This course is taught from the point of view of the manufacturing corporation, the utility, the software firm—any potential end-user of derivatives, but not the dealer. Most related courses focus on the extensive taxonomy of instruments and the complex models developed to price them, and on ways to exploit mispricing. While this course will make use of some of these pricing models, the focus is on how corporations use the insights and models to improve their operations, to increase the value of their real assets, or to create the financi This is a course in how corporations make use of the insights and tools of risk management. Most courses on derivatives, futures and options, and financial engineering are taught from the viewpoint of investment bankers and traders in the securities. This course is taught from the point of view of the manufacturing corporation, the utility, the software firm—any potential end-user of derivatives, but not the dealer. Most related courses focus on the extensive taxonomy of instruments and the complex models developed to price them, and on ways to exploit mispricing. While this course will make use of some of these pricing models, the focus is on how corporations use the insights and models to improve their operations, to increase the value of their real assets, or to create the financi

Subjects

risk | risk | corporate finance | corporate finance | risk management | risk management | hedging | hedging | derivatives | derivatives | trading operations | trading operations | pricing risk | pricing risk | liability management | liability management | financial policy | financial policy | valuation | valuation | discounted cash flow | discounted cash flow | asset management | asset management | transaction hedging | transaction hedging | market volatility | market volatility | foreign currency derivatives | foreign currency derivatives | interest rate risk | interest rate risk | liability structure | liability structure | strategic management | strategic management | Modigliani-Miller Theory of hedging | Modigliani-Miller Theory of hedging | dynamic models | dynamic models | monte carlo simulation | monte carlo simulation | random walk model | random walk model | binomial tree | binomial tree | mispricing | mispricing | risk neutral pricing | risk neutral pricing

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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18.366 Random Walks and Diffusion (MIT) 18.366 Random Walks and Diffusion (MIT)

Description

This graduate-level subject explores various mathematical aspects of (discrete) random walks and (continuum) diffusion. Applications include polymers, disordered media, turbulence, diffusion-limited aggregation, granular flow, and derivative securities. This graduate-level subject explores various mathematical aspects of (discrete) random walks and (continuum) diffusion. Applications include polymers, disordered media, turbulence, diffusion-limited aggregation, granular flow, and derivative securities.

Subjects

Discrete and continuum modeling of diffusion processes in physics | chemistry | and economics | Discrete and continuum modeling of diffusion processes in physics | chemistry | and economics | central limit theorems | central limit theorems | continuous-time random walks | continuous-time random walks | Levy flights | Levy flights | correlations | correlations | extreme events | extreme events | mixing | mixing | renormalization | renormalization | and percolation | and percolation | percolation | percolation

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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5.72 Statistical Mechanics (MIT) 5.72 Statistical Mechanics (MIT)

Description

This course discusses the principles and methods of statistical mechanics. Topics covered include classical and quantum statistics, grand ensembles, fluctuations, molecular distribution functions, other concepts in equilibrium statistical mechanics, and topics in thermodynamics and statistical mechanics of irreversible processes. This course discusses the principles and methods of statistical mechanics. Topics covered include classical and quantum statistics, grand ensembles, fluctuations, molecular distribution functions, other concepts in equilibrium statistical mechanics, and topics in thermodynamics and statistical mechanics of irreversible processes.

Subjects

statistical mechanics | statistical mechanics | quantum | quantum | statistics | statistics | atoms | atoms | materials | materials | master equations | master equations | random walk | random walk | langevin | langevin | fokker | fokker | planck | planck | probability theory | probability theory | bloch-redfield | bloch-redfield | navier-stokes | navier-stokes | hydrodynamic | hydrodynamic | scattering | scattering | projection operator | projection operator | thermodynamics | thermodynamics

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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18.445 Introduction to Stochastic Processes (MIT) 18.445 Introduction to Stochastic Processes (MIT)

Description

This course is an introduction to Markov chains, random walks, martingales, and Galton-Watsom tree. The course requires basic knowledge in probability theory and linear algebra including conditional expectation and matrix. This course is an introduction to Markov chains, random walks, martingales, and Galton-Watsom tree. The course requires basic knowledge in probability theory and linear algebra including conditional expectation and matrix.

Subjects

probability | probability | Stochastic Processes | Stochastic Processes | Markov chains | Markov chains | random walks | random walks | martingales | martingales | Galton-Watsom tree | Galton-Watsom tree | linear algebra | linear algebra

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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18.103 Fourier Analysis (MIT) 18.103 Fourier Analysis (MIT)

Description

This course continues the content covered in 18.100 Analysis I. Roughly half of the subject is devoted to the theory of the Lebesgue integral with applications to probability, and the other half to Fourier series and Fourier integrals. This course continues the content covered in 18.100 Analysis I. Roughly half of the subject is devoted to the theory of the Lebesgue integral with applications to probability, and the other half to Fourier series and Fourier integrals.

Subjects

Fourier series | Fourier series | Fourier analysis | Fourier analysis | partial sums | partial sums | waves | waves | Boolean rings | Boolean rings | Hilbert Space | Hilbert Space | Orthonormal bases | Orthonormal bases | Lp theory | Lp theory | Fourier integrals | Fourier integrals | measure | measure | central limit theorem | central limit theorem | brownian motion | brownian motion | Lebesgue integral | Lebesgue integral | periodic functions | periodic functions | Fourier coefficients | Fourier coefficients | Parseval's formula | Parseval's formula | Bernoulli sequence | Bernoulli sequence | random walks | random walks | probability theory | probability theory | Lebesgue measure | Lebesgue measure

License

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18.177 Universal Random Structures in 2D (MIT) 18.177 Universal Random Structures in 2D (MIT)

Description

This graduate-level course introduces students to some fundamental 2D random objects, explains how they are related to each other, and explores some open problems in the field. This graduate-level course introduces students to some fundamental 2D random objects, explains how they are related to each other, and explores some open problems in the field.

Subjects

continuum random tree | continuum random tree | stable Levy tree | stable Levy tree | stable looptree | stable looptree | Gaussian free field | Gaussian free field | Schramm-Loewner evolution | Schramm-Loewner evolution | percolation | percolation | uniform spanning tree | uniform spanning tree | loop-erased random walk | loop-erased random walk | Ising model | Ising model | FK cluster model | FK cluster model | conformal loop ensemble | conformal loop ensemble | Brownian loop soup | Brownian loop soup | random planar map | random planar map | Liouville | Liouville | quantum gravity | quantum gravity | Brownian map | Brownian map | Brownian snake | Brownian snake | diffusion limited aggregation | diffusion limited aggregation | first passage percolation | first passage percolation | and dielectric breakdown model | and dielectric breakdown model | imaginary geometry | imaginary geometry | quantum zipper | quantum zipper | peanosphere | peanosphere | quantum Loewner evolution | quantum Loewner evolution

License

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20.430J Fields, Forces, and Flows in Biological Systems (MIT) 20.430J Fields, Forces, and Flows in Biological Systems (MIT)

Description

This course covers the fundamental driving forces for transport—chemical gradients, electrical interactions, and fluid flow—as applied to the biology and biophysics of molecules, cells, and tissues. This course covers the fundamental driving forces for transport—chemical gradients, electrical interactions, and fluid flow—as applied to the biology and biophysics of molecules, cells, and tissues.

Subjects

diffusion | diffusion | molecular diffusion | molecular diffusion | diffusion-reaction | diffusion-reaction | conduction | conduction | convection | convection | biological systems | biological systems | fields | fields | electrical double layers | electrical double layers | Maxwell stress tensor | Maxwell stress tensor | physiological systems | physiological systems | fluid | fluid | solid | solid | equations of motion | equations of motion | case study | case study | electrode interfaces | electrode interfaces | transduction | transduction | random walk | random walk | Stokes-Einstein | Stokes-Einstein | Fick's laws | Fick's laws | reaction | reaction | Damköhler number | Damköhler number

License

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Fundamentals of Materials Science: Diffusion

Description

This set of animations provides an introduction to the mechanisms and driving forces of diffusion. It demonstartes some of the processes in which it is observed. From TLP: Diffusion

Subjects

diffusionsubstitutional | interstitial | random walk | Fick | law | DoITPoMS | University of Cambridge | animation | corematerials | ukoer

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Enhancing Physics Knowledge for Teaching – Transport properties

Description

In this session we’ll look at transport properties in fluids. By transport properties we mean the flows that restore a system to equilibrium, so for example, the flow of heat to eliminate a temperature gradient.

Subjects

sfsoer | ukoer | brownian motion | random walks | diffusion | fluctuations | dissipation | Physical sciences | F000

License

Attribution-Noncommercial-Share Alike 2.0 UK: England & Wales Attribution-Noncommercial-Share Alike 2.0 UK: England & Wales http://creativecommons.org/licenses/by-nc-sa/2.0/uk/ http://creativecommons.org/licenses/by-nc-sa/2.0/uk/

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Fundamentals of Materials Science: Diffusion

Description

This set of animations provides an introduction to the mechanisms and driving forces of diffusion. It demonstartes some of the processes in which it is observed. From TLP: Diffusion

Subjects

diffusionsubstitutional | interstitial | random walk | fick | law | doitpoms | university of cambridge | animation | corematerials | ukoer | Engineering | H000

License

Attribution-Noncommercial-Share Alike 2.0 UK: England & Wales Attribution-Noncommercial-Share Alike 2.0 UK: England & Wales http://creativecommons.org/licenses/by-nc-sa/2.0/uk/ http://creativecommons.org/licenses/by-nc-sa/2.0/uk/

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5.72 Statistical Mechanics (MIT)

Description

This course discusses the principles and methods of statistical mechanics. Topics covered include classical and quantum statistics, grand ensembles, fluctuations, molecular distribution functions, other concepts in equilibrium statistical mechanics, and topics in thermodynamics and statistical mechanics of irreversible processes.

Subjects

statistical mechanics | quantum | statistics | atoms | materials | master equations | random walk | langevin | fokker | planck | probability theory | bloch-redfield | navier-stokes | hydrodynamic | scattering | projection operator | thermodynamics

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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18.366 Random Walks and Diffusion (MIT)

Description

This graduate-level subject explores various mathematical aspects of (discrete) random walks and (continuum) diffusion. Applications include polymers, disordered media, turbulence, diffusion-limited aggregation, granular flow, and derivative securities.

Subjects

Discrete and continuum modeling of diffusion processes in physics | chemistry | and economics | central limit theorems | continuous-time random walks | Levy flights | correlations | extreme events | mixing | renormalization | and percolation | percolation

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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18.366 Random Walks and Diffusion (MIT)

Description

Mathematical modeling of diffusion phenomena: Central limit theorems, the continuum limit, first passage, persistence, continuous-time random walks, Levy flights, fractional calculus, random environments, advection-diffusion, nonlinear diffusion, free-boundary problems. Applications may include polymers, disordered media, turbulence, diffusion-limited aggregation, granular flow, and derivative securities.

Subjects

Discrete and continuum modeling of diffusion processes in physics | chemistry | and economics | central limit theorems | ontinuous-time random walks | Levy flights | correlations | extreme events | mixing | renormalization | and percolation

License

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3.016 Mathematics for Materials Scientists and Engineers (MIT)

Description

The class will cover mathematical techniques necessary for understanding of materials science and engineering topics such as energetics, materials structure and symmetry, materials response to applied fields, mechanics and physics of solids and soft materials. The class uses examples from 3.012 to introduce mathematical concepts and materials-related problem solving skills. Topics include linear algebra and orthonormal basis, eigenvalues and eigenvectors, quadratic forms, tensor operations, symmetry operations, calculus of several variables, introduction to complex analysis, ordinary and partial differential equations, theory of distributions, fourier analysis and random walks.Technical RequirementsMathematica® software is required to run the .nb files found on this course site.

Subjects

energetics | materials structure and symmetry: applied fields | mechanics and physics of solids and soft materials | linear algebra | orthonormal basis | eigenvalues | eigenvectors | quadratic forms | tensor operations | symmetry operations | calculus | complex analysis | differential equations | theory of distributions | fourier analysis | random walks | mathematical technicques | materials science | materials engineering | materials structure | symmetry | applied fields | materials response | solids mechanics | solids physics | soft materials | multi-variable calculus | ordinary differential equations | partial differential equations | applied mathematics | mathematical techniques

License

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15.997 Practice of Finance: Advanced Corporate Risk Management (MIT)

Description

This is a course in how corporations make use of the insights and tools of risk management. Most courses on derivatives, futures and options, and financial engineering are taught from the viewpoint of investment bankers and traders in the securities. This course is taught from the point of view of the manufacturing corporation, the utility, the software firm—any potential end-user of derivatives, but not the dealer. Most related courses focus on the extensive taxonomy of instruments and the complex models developed to price them, and on ways to exploit mispricing. While this course will make use of some of these pricing models, the focus is on how corporations use the insights and models to improve their operations, to increase the value of their real assets, or to create the financi

Subjects

risk | corporate finance | risk management | hedging | derivatives | trading operations | pricing risk | liability management | financial policy | valuation | discounted cash flow | asset management | transaction hedging | market volatility | foreign currency derivatives | interest rate risk | liability structure | strategic management | Modigliani-Miller Theory of hedging | dynamic models | monte carlo simulation | random walk model | binomial tree | mispricing | risk neutral pricing

License

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18.177 Universal Random Structures in 2D (MIT)

Description

This graduate-level course introduces students to some fundamental 2D random objects, explains how they are related to each other, and explores some open problems in the field.

Subjects

continuum random tree | stable Levy tree | stable looptree | Gaussian free field | Schramm-Loewner evolution | percolation | uniform spanning tree | loop-erased random walk | Ising model | FK cluster model | conformal loop ensemble | Brownian loop soup | random planar map | Liouville | quantum gravity | Brownian map | Brownian snake | diffusion limited aggregation | first passage percolation | and dielectric breakdown model | imaginary geometry | quantum zipper | peanosphere | quantum Loewner evolution

License

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18.103 Fourier Analysis (MIT)

Description

This course continues the content covered in 18.100 Analysis I. Roughly half of the subject is devoted to the theory of the Lebesgue integral with applications to probability, and the other half to Fourier series and Fourier integrals.

Subjects

Fourier series | Fourier analysis | partial sums | waves | Boolean rings | Hilbert Space | Orthonormal bases | Lp theory | Fourier integrals | measure | central limit theorem | brownian motion | Lebesgue integral | periodic functions | Fourier coefficients | Parseval's formula | Bernoulli sequence | random walks | probability theory | Lebesgue measure

License

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