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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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Uniform convergence and pointwise convergence Uniform convergence and pointwise convergence

Description

Target audience: Most of this material should be accessible to anyone who understands what a real-valued function is, and understands the notion of convergence of a sequence of real numbers. This should include most mathematics undergraduates by the end of their first year. An understanding of continuity and of boundedness for real-valued functions defined on various types of domain would help the student to understand the latter part of the material. Target audience: Most of this material should be accessible to anyone who understands what a real-valued function is, and understands the notion of convergence of a sequence of real numbers. This should include most mathematics undergraduates by the end of their first year. An understanding of continuity and of boundedness for real-valued functions defined on various types of domain would help the student to understand the latter part of the material. The aim of this material is to introduce the student to two notions of convergence for sequences of real-valued functions. The notion of pointwise convergence is relatively straightforward, but the notion of uniform convergence is more subtle. Uniform convergence is explained in terms of closed function balls and the new notion of sets absorbing sequences. The differences between the two types of convergence are illustrated with several examples. Some standard facts are also discussed: a uniform limit of continuous functions must be continuous; a uniform limit of bounded functions must be bounded; a uniform limit of unbounded functions must be unbounded. Target audience: Most of this material should be accessible to anyone who understands what a real-valued function is, and understands The aim of this material is to introduce the student to two notions of convergence for sequences of real-valued functions. The notion of pointwise convergence is relatively straightforward, but the notion of uniform convergence is more subtle. Uniform convergence is explained in terms of closed function balls and the new notion of sets absorbing sequences. The differences between the two types of convergence are illustrated with several examples. Some standard facts are also discussed: a uniform limit of continuous functions must be continuous; a uniform limit of bounded functions must be bounded; a uniform limit of unbounded functions must be unbounded. Target audience: Most of this material should be accessible to anyone who understands what a real-valued function is, and understands

Subjects

UNow | UNow | Pointwise convergence | Pointwise convergence | Uniform convergence | Uniform convergence | Pure mathmatics | Pure mathmatics | UKOER | UKOER

License

Except for third party materials (materials owned by someone other than The University of Nottingham) and where otherwise indicated, the copyright in the content provided in this resource is owned by The University of Nottingham and licensed under a Creative Commons Attribution-NonCommercial-ShareAlike UK 2.0 Licence (BY-NC-SA) Except for third party materials (materials owned by someone other than The University of Nottingham) and where otherwise indicated, the copyright in the content provided in this resource is owned by The University of Nottingham and licensed under a Creative Commons Attribution-NonCommercial-ShareAlike UK 2.0 Licence (BY-NC-SA)

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18.100B Analysis I (MIT) 18.100B Analysis I (MIT)

Description

Analysis I covers fundamentals of mathematical analysis: metric spaces, convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. Analysis I covers fundamentals of mathematical analysis: metric spaces, convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations.

Subjects

mathematical analysis | mathematical analysis | convergence of sequences | convergence of sequences | convergence of series | convergence of series | continuity | continuity | differentiability | differentiability | Riemann integral | Riemann integral | sequences and series of functions | sequences and series of functions | uniformity | uniformity | interchange of limit operations | interchange of limit operations | utility of abstract concepts | utility of abstract concepts | construction of proofs | construction of proofs | point-set topology | point-set topology | n-space | n-space

License

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18.100B Analysis I (MIT) 18.100B Analysis I (MIT)

Description

Analysis I covers fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. MIT students may choose to take one of the two versions of 18.100. Option A chooses less abstract definitions and proofs, and gives applications where possible. Option B is more demanding and for students with more mathematical maturity; it places more emphasis on point-set topology and n-space, whereas Option A is concerned primarily with the real line. Analysis I covers fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. MIT students may choose to take one of the two versions of 18.100. Option A chooses less abstract definitions and proofs, and gives applications where possible. Option B is more demanding and for students with more mathematical maturity; it places more emphasis on point-set topology and n-space, whereas Option A is concerned primarily with the real line.

Subjects

mathematical analysis | mathematical analysis | convergence of sequences | convergence of sequences | convergence of series | convergence of series | continuity | continuity | differentiability | differentiability | Reimann integral | Reimann integral | sequences and series of functions | sequences and series of functions | uniformity | uniformity | interchange of limit operations | interchange of limit operations | utility of abstract concepts | utility of abstract concepts | construction of proofs | construction of proofs | point-set topology | point-set topology | n-space | n-space | sequences of functions | sequences of functions | series of functions | series of functions | applications | applications | real variable | real variable | metric space | metric space | sets | sets | theorems | theorems | differentiate | differentiate | differentiable | differentiable | converge | converge | uniform | uniform | 18.100 | 18.100

License

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18.100A Analysis I (MIT) 18.100A Analysis I (MIT)

Description

Analysis I (18.100) in its various versions covers fundamentals of mathematical analysis: continuity, differentiability, some form of the Riemann integral, sequences and series of numbers and functions, uniform convergence with applications to interchange of limit operations, some point-set topology, including some work in Euclidean n-space. MIT students may choose to take one of three versions of 18.100: Option A (18.100A) chooses less abstract definitions and proofs, and gives applications where possible. Option B (18.100B) is more demanding and for students with more mathematical maturity; it places more emphasis from the beginning on point-set topology and n-space, whereas Option A is concerned primarily with analysis on the real line, saving for the last weeks work in 2-space (the pla Analysis I (18.100) in its various versions covers fundamentals of mathematical analysis: continuity, differentiability, some form of the Riemann integral, sequences and series of numbers and functions, uniform convergence with applications to interchange of limit operations, some point-set topology, including some work in Euclidean n-space. MIT students may choose to take one of three versions of 18.100: Option A (18.100A) chooses less abstract definitions and proofs, and gives applications where possible. Option B (18.100B) is more demanding and for students with more mathematical maturity; it places more emphasis from the beginning on point-set topology and n-space, whereas Option A is concerned primarily with analysis on the real line, saving for the last weeks work in 2-space (the pla

Subjects

mathematical analysis | mathematical analysis | convergence of sequences | convergence of sequences | convergence of series | convergence of series | continuity | continuity | differentiability | differentiability | Riemann integral | Riemann integral | sequences and series of functions | sequences and series of functions | uniformity | uniformity | interchange of limit operations | interchange of limit operations | utility of abstract concepts | utility of abstract concepts | construction of proofs | construction of proofs | point-set topology | point-set topology | n-space | n-space

License

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18.100B Analysis I (MIT) 18.100B Analysis I (MIT)

Description

Analysis I covers fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, and interchange of limit operations. Analysis I covers fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, and interchange of limit operations.

Subjects

mathematical analysis | mathematical analysis | convergence of sequences | convergence of sequences | convergence of series | convergence of series | continuity | continuity | differentiability | differentiability | Riemann integral | Riemann integral | sequences and series of functions | sequences and series of functions | uniformity | uniformity | interchange of limit operations | interchange of limit operations | utility of abstract concepts | utility of abstract concepts | construction of proofs | construction of proofs | point-set topology | point-set topology | n-space | n-space

License

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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.

Subjects

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

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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18.327 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.

Subjects

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

License

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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

License

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18.100A Introduction to Analysis (MIT) 18.100A Introduction to Analysis (MIT)

Description

Analysis I (18.100) in its various versions covers fundamentals of mathematical analysis: continuity, differentiability, some form of the Riemann integral, sequences and series of numbers and functions, uniform convergence with applications to interchange of limit operations, some point-set topology, including some work in Euclidean n-space. MIT students may choose to take one of three versions of 18.100: Option A (18.100A) chooses less abstract definitions and proofs, and gives applications where possible. Option B (18.100B) is more demanding and for students with more mathematical maturity; it places more emphasis from the beginning on point-set topology and n-space, whereas Option A is concerned primarily with analysis on the real line, saving for the last weeks work in 2-space (the pla Analysis I (18.100) in its various versions covers fundamentals of mathematical analysis: continuity, differentiability, some form of the Riemann integral, sequences and series of numbers and functions, uniform convergence with applications to interchange of limit operations, some point-set topology, including some work in Euclidean n-space. MIT students may choose to take one of three versions of 18.100: Option A (18.100A) chooses less abstract definitions and proofs, and gives applications where possible. Option B (18.100B) is more demanding and for students with more mathematical maturity; it places more emphasis from the beginning on point-set topology and n-space, whereas Option A is concerned primarily with analysis on the real line, saving for the last weeks work in 2-space (the pla

Subjects

mathematical analysis | mathematical analysis | estimations | estimations | limit of a sequence | limit of a sequence | limit theorems | limit theorems | subsequences | subsequences | cluster points | cluster points | infinite series | infinite series | power series | power series | local and global properties | local and global properties | continuity | continuity | intermediate-value theorem | intermediate-value theorem | convexity | convexity | integrability | integrability | Riemann integral | Riemann integral | calculus | calculus | convergence | convergence | Gamma function | Gamma function | Stirling | Stirling | quantifiers and negation | quantifiers and negation | Leibniz | Leibniz | Fubini | Fubini | improper integrals | improper integrals | Lebesgue integral | Lebesgue integral | mathematical proofs | mathematical proofs | differentiation | differentiation | integration | integration

License

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14.121 Microeconomic Theory I (MIT) 14.121 Microeconomic Theory I (MIT)

Description

This half-semester course provides an introduction to microeconomic theory designed to meet the needs of students in the economics Ph.D. program. Some parts of the course are designed to teach material that all graduate students should know. Others are used to introduce methodologies. Topics include consumer and producer theory, markets and competition, general equilibrium, and tools of comparative statics and their application to price theory. Some topics of recent interest may also be covered. This half-semester course provides an introduction to microeconomic theory designed to meet the needs of students in the economics Ph.D. program. Some parts of the course are designed to teach material that all graduate students should know. Others are used to introduce methodologies. Topics include consumer and producer theory, markets and competition, general equilibrium, and tools of comparative statics and their application to price theory. Some topics of recent interest may also be covered.

Subjects

microeconomic theory | microeconomic theory | demand theory | demand theory | producer theory; partial equilibrium | producer theory; partial equilibrium | competitive markets | competitive markets | general equilibrium | general equilibrium | externalities | externalities | Afriat's theorem | Afriat's theorem | pricing | pricing | robust comparative statics | robust comparative statics | utility theory | utility theory | properties of preferences | properties of preferences | choice as primitive | choice as primitive | revealed preference | revealed preference | classical demand theory | classical demand theory | Kuhn-Tucker necessary conditions | Kuhn-Tucker necessary conditions | implications of Walras?s law | implications of Walras?s law | indirect utility functions | indirect utility functions | theorem of the maximum (Berge?s theorem) | theorem of the maximum (Berge?s theorem) | expenditure minimization problem | expenditure minimization problem | Hicksian demands | Hicksian demands | compensated law of demand | compensated law of demand | Slutsky substitution | Slutsky substitution | price changes and welfare | price changes and welfare | compensating variation | compensating variation | and welfare from new goods | and welfare from new goods | price indexes | price indexes | bias in the U.S. consumer price index | bias in the U.S. consumer price index | integrability | integrability | demand aggregation | demand aggregation | aggregate demand and welfare | aggregate demand and welfare | Frisch demands | Frisch demands | and demand estimation | and demand estimation | increasing differences | increasing differences | producer theory applications | producer theory applications | the LeCh?telier principle | the LeCh?telier principle | Topkis? theorem | Topkis? theorem | Milgrom-Shannon monotonicity theorem | Milgrom-Shannon monotonicity theorem | monopoly pricing | monopoly pricing | monopoly and product quality | monopoly and product quality | nonlinear pricing | nonlinear pricing | and price discrimination | and price discrimination | simple models of externalities | simple models of externalities | government intervention | government intervention | Coase theorem | Coase theorem | Myerson-Sattherthwaite proposition | Myerson-Sattherthwaite proposition | missing markets | missing markets | price vs. quantity regulations | price vs. quantity regulations | Weitzman?s analysis | Weitzman?s analysis | uncertainty | uncertainty | common property externalities | common property externalities | optimization | optimization | equilibrium number of boats | equilibrium number of boats | welfare theorems | welfare theorems | uniqueness and determinacy | uniqueness and determinacy | price-taking assumption | price-taking assumption | Edgeworth box | Edgeworth box | welfare properties | welfare properties | Pareto efficiency | Pareto efficiency | Walrasian equilibrium with transfers | Walrasian equilibrium with transfers | Arrow-Debreu economy | Arrow-Debreu economy | separating hyperplanes | separating hyperplanes | Minkowski?s theorem | Minkowski?s theorem | Existence of Walrasian equilibrium | Existence of Walrasian equilibrium | Kakutani?s fixed point theorem | Kakutani?s fixed point theorem | Debreu-Gale-Kuhn-Nikaido lemma | Debreu-Gale-Kuhn-Nikaido lemma | additional properties of general equilibrium | additional properties of general equilibrium | Microfoundations | Microfoundations | core | core | core convergence | core convergence | general equilibrium with time and uncertainty | general equilibrium with time and uncertainty | Jensen?s inequality | Jensen?s inequality | and security market economy | and security market economy | arbitrage pricing theory | arbitrage pricing theory | and risk-neutral probabilities | and risk-neutral probabilities | Housing markets | Housing markets | competitive equilibrium | competitive equilibrium | one-sided matching house allocation problem | one-sided matching house allocation problem | serial dictatorship | serial dictatorship | two-sided matching | two-sided matching | marriage markets | marriage markets | existence of stable matchings | existence of stable matchings | incentives | incentives | housing markets core mechanism | housing markets core mechanism

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14.381 Statistical Method in Economics (MIT) 14.381 Statistical Method in Economics (MIT)

Description

This course is divided into two sections, Part I and Part II.  Part I, found here, provides an introduction to statistical theory. A brief review of probability will be given mainly as background material, however, it is assumed to be known. Topics include normal distribution, limit theorems, Bayesian concepts, and testing, among others. Part II prepares students for the remainder of the econometrics sequence and and can be found by visiting 14.381 Fall 2006.    This course is divided into two sections, Part I and Part II.  Part I, found here, provides an introduction to statistical theory. A brief review of probability will be given mainly as background material, however, it is assumed to be known. Topics include normal distribution, limit theorems, Bayesian concepts, and testing, among others. Part II prepares students for the remainder of the econometrics sequence and and can be found by visiting 14.381 Fall 2006.   

Subjects

economics | economics | statistics | statistics | sample | sample | population | population | convergence | convergence | limits | limits | method | method | testing | testing | confidence sets | confidence sets | Bayesian | Bayesian

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18.100B Analysis I (MIT)

Description

Analysis I covers fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, and interchange of limit operations.

Subjects

mathematical analysis | convergence of sequences | convergence of series | continuity | differentiability | Riemann integral | sequences and series of functions | uniformity | interchange of limit operations | utility of abstract concepts | construction of proofs | point-set topology | n-space

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18.336 Numerical Methods for Partial Differential Equations (MIT) 18.336 Numerical Methods for Partial Differential Equations (MIT)

Description

This graduate-level course is an advanced introduction to applications and theory of numerical methods for solution of differential equations. In particular, the course focuses on physically-arising partial differential equations, with emphasis on the fundamental ideas underlying various methods. This graduate-level course is an advanced introduction to applications and theory of numerical methods for solution of differential equations. In particular, the course focuses on physically-arising partial differential equations, with emphasis on the fundamental ideas underlying various methods.

Subjects

advection equation | advection equation | heat equation | heat equation | wave equation | wave equation | Airy equation | Airy equation | convection-diffusion problems | convection-diffusion problems | KdV equation | KdV equation | hyperbolic conservation laws | hyperbolic conservation laws | Poisson equation | Poisson equation | Stokes problem | Stokes problem | Navier-Stokes equations | Navier-Stokes equations | interface problems | interface problems | consistency | consistency | stability | stability | convergence | convergence | Lax equivalence theorem | Lax equivalence theorem | error analysis | error analysis | Fourier approaches | Fourier approaches | staggered grids | staggered grids | shocks | shocks | front propagation | front propagation | preconditioning | preconditioning | multigrid | multigrid | Krylov spaces | Krylov spaces | saddle point problems | saddle point problems | finite differences | finite differences | finite volumes | finite volumes | finite elements | finite elements | ENO/WENO | ENO/WENO | spectral methods | spectral methods | projection approaches for incompressible ows | projection approaches for incompressible ows | level set methods | level set methods | particle methods | particle methods | direct and iterative methods | direct and iterative methods

License

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18.100B Analysis I (MIT)

Description

Analysis I covers fundamentals of mathematical analysis: metric spaces, convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations.

Subjects

mathematical analysis | convergence of sequences | convergence of series | continuity | differentiability | Riemann integral | sequences and series of functions | uniformity | interchange of limit operations | utility of abstract concepts | construction of proofs | point-set topology | n-space

License

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18.125 Measure and Integration (MIT) 18.125 Measure and Integration (MIT)

Description

This graduate-level course covers Lebesgue's integration theory with applications to analysis, including an introduction to convolution and the Fourier transform. This graduate-level course covers Lebesgue's integration theory with applications to analysis, including an introduction to convolution and the Fourier transform.

Subjects

Lebesgue integral | Lebesgue integral | convergence theorems | convergence theorems | Lebesgue measure in Rn | Lebesgue measure in Rn | Lpspaces | Lpspaces | Radon-Nikodym Theorem | Radon-Nikodym Theorem | Lebesgue Differentiation Theorem | Lebesgue Differentiation Theorem | Fubini Theorem | Fubini Theorem | Hausdorff measure | Hausdorff measure | Area and Coarea Formulas | Area and Coarea Formulas | measure theory | measure theory | convolution | convolution | Fourier transform | Fourier transform | Lebesque Integration Theory | Lebesque Integration Theory

License

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Regional integration and welfare-state convergence in Europe

Description

Professor Beckfield discusses whether the welfare state convergence is really taking place, or it is just regional integration, especially in the European context. The contemporary institutionalization of a transnational regional political economy in Europe raises questions about the role of regional integration in the convergence of European welfare states. To date, sociological work has emphasized processes of industrialization and globalization as the social changes that may drive increasing similarity among welfare states. Building on neoinstitutionalist theory and the Europeanization literature, we develop the argument that regional integration drives welfare-state convergence by generating, diffusing, and enforcing the adoption of policy scripts concerning "appropriate" European so Wales; http://creativecommons.org/licenses/by-nc-sa/2.0/uk/

Subjects

regional integration | welfare state | social policy | globalisation | the European Union | welfare state convergence | regional integration | welfare state | social policy | globalisation | the European Union | welfare state convergence | 2011-06-06

License

http://creativecommons.org/licenses/by-nc-sa/2.0/uk/

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CMS.100 Introduction to Media Studies (MIT) CMS.100 Introduction to Media Studies (MIT)

Description

This course offers an overview of the social, cultural, political, and economic impact of mediated communication on modern culture. Combines critical discussions with experiments working with different media. Media covered include radio, television, film, the printed word, and digital technologies. Topics include the nature and function of media, core media institutions, and media in transition. This course offers an overview of the social, cultural, political, and economic impact of mediated communication on modern culture. Combines critical discussions with experiments working with different media. Media covered include radio, television, film, the printed word, and digital technologies. Topics include the nature and function of media, core media institutions, and media in transition.

Subjects

mass communication | mass communication | mass film | mass film | television | television | video games | video games | recorded music | recorded music | digital media | digital media | multimedia | multimedia | media literacy | media literacy | social media | social media | media convergence | media convergence

License

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21L.015 Introduction to Media Studies (MIT) 21L.015 Introduction to Media Studies (MIT)

Description

Introduction to Media Studies is designed for students who have grown up in a rapidly changing global multimedia environment and want to become more literate and critical consumers and producers of media. Through an interdisciplinary comparative and historical lens, the course defines "media" broadly as including oral, print, performance, photographic, broadcast, cinematic, and digital cultural forms and practices. The course looks at the nature of mediated communication, the functions of media, the history of transformations in media and the institutions that help define media's place in society. This year’s course will focus on issues of network culture and media convergence, addressing such subjects as Intellectual Property, peer2peer authoring, blogging, and game modification. Introduction to Media Studies is designed for students who have grown up in a rapidly changing global multimedia environment and want to become more literate and critical consumers and producers of media. Through an interdisciplinary comparative and historical lens, the course defines "media" broadly as including oral, print, performance, photographic, broadcast, cinematic, and digital cultural forms and practices. The course looks at the nature of mediated communication, the functions of media, the history of transformations in media and the institutions that help define media's place in society. This year’s course will focus on issues of network culture and media convergence, addressing such subjects as Intellectual Property, peer2peer authoring, blogging, and game modification.

Subjects

Comparative Media Studies | Comparative Media Studies | global multimedia environment | global multimedia environment | literate | literate | critical | critical | consumers | consumers | producers | producers | interdisciplinary | interdisciplinary | comparative | comparative | historical | historical | lens | lens | the course defines oral | the course defines oral | print | print | performance | performance | photographic | photographic | broadcast | broadcast | cinematic | cinematic | digital | digital | cultural | cultural | forms | forms | practices | practices | mediated communication | mediated communication | functions | functions | society | society | network culture | network culture | media convergence | media convergence | Intellectual Property | Intellectual Property | peer2peer authoring | peer2peer authoring | blogging | blogging | game modification | game modification | lens | the course defines oral | lens | the course defines oral

License

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21L.430 Popular Narrative: Masterminds (MIT) 21L.430 Popular Narrative: Masterminds (MIT)

Description

Our purpose is to consider some of the most elaborate and thoughtful efforts to define and delineate "all-mastering," and to consider some of the delineations of "all-mastering the intellect" in various guises - from magicians to master spies to detectives to scientists (mad and otherwise). The major written work of the term will be an ongoing reading journal, which you will circulate to your classmates using an e-mail mailing list. The use of that list is fundamental - it is my intention to generate a sort of ongoing cyberconversation. Our purpose is to consider some of the most elaborate and thoughtful efforts to define and delineate "all-mastering," and to consider some of the delineations of "all-mastering the intellect" in various guises - from magicians to master spies to detectives to scientists (mad and otherwise). The major written work of the term will be an ongoing reading journal, which you will circulate to your classmates using an e-mail mailing list. The use of that list is fundamental - it is my intention to generate a sort of ongoing cyberconversation.

Subjects

Mastering | Mastering | mastery | mastery | narrative | narrative | popular culture | popular culture | media | media | convergence | convergence | film | film | television | television | spies | spies | detectives | detectives | intellect | intellect | magician | magician | scientists | scientists | graduate students | graduate students | journals | journals | SP.492 | SP.492

License

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21G.039J Gender and Japanese Popular Culture (MIT) 21G.039J Gender and Japanese Popular Culture (MIT)

Description

This course examines relationships between identity and participation in Japanese popular culture as a way of understanding the changing character of media, capitalism, fan communities, and culture. It emphasizes contemporary popular culture and theories of gender, sexuality, race, and the workings of power and value in global culture industries. Topics include manga (comic books), hip-hop and other popular music, anime and feature films, video games, contemporary literature, and online communication. Students present analyses and develop a final project based on a particular aspect of gender and popular culture. This course examines relationships between identity and participation in Japanese popular culture as a way of understanding the changing character of media, capitalism, fan communities, and culture. It emphasizes contemporary popular culture and theories of gender, sexuality, race, and the workings of power and value in global culture industries. Topics include manga (comic books), hip-hop and other popular music, anime and feature films, video games, contemporary literature, and online communication. Students present analyses and develop a final project based on a particular aspect of gender and popular culture.

Subjects

gender | gender | Japan | Japan | culture | culture | Pecha Kucha | Pecha Kucha | media theory | media theory | manga | manga | inequality | inequality | economics | economics | robots | robots | technology | technology | anime | anime | anthropology | anthropology | queer | queer | transgender | transgender | hostess club | hostess club | feminist social theory | feminist social theory | gender traits | gender traits | fujoshi | fujoshi | women | women | Princess Jellyfish | Princess Jellyfish | Kuragehime | Kuragehime | convergence culture | convergence culture | participatory culture | participatory culture | capital | capital | debt | debt | power | power | slavery | slavery | sexism | sexism | Takarazuka | Takarazuka | host club | host club | masculinity | masculinity | seduction | seduction | Onnagata | Onnagata | Kabuki theater | Kabuki theater | idols | idols | virtual idol | virtual idol | games | games | Tokyo | Tokyo

License

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6.435 System Identification (MIT) 6.435 System Identification (MIT)

Description

This course is offered to graduates and includes topics such as mathematical models of systems from observations of their behavior; time series, state-space, and input-output models; model structures, parametrization, and identifiability; non-parametric methods; prediction error methods for parameter estimation, convergence, consistency, and asymptotic distribution; relations to maximum likelihood estimation; recursive estimation; relation to Kalman filters; structure determination; order estimation; Akaike criterion; bounded but unknown noise model; and robustness and practical issues. This course is offered to graduates and includes topics such as mathematical models of systems from observations of their behavior; time series, state-space, and input-output models; model structures, parametrization, and identifiability; non-parametric methods; prediction error methods for parameter estimation, convergence, consistency, and asymptotic distribution; relations to maximum likelihood estimation; recursive estimation; relation to Kalman filters; structure determination; order estimation; Akaike criterion; bounded but unknown noise model; and robustness and practical issues.

Subjects

mathematical models | mathematical models | time series | time series | state-space | state-space | input-output models | input-output models | model structures | model structures | parametrization | parametrization | identifiability | identifiability | non-parametric methods | non-parametric methods | prediction error | prediction error | parameter estimation | parameter estimation | convergence | convergence | consistency | consistency | andasymptotic distribution | andasymptotic distribution | maximum likelihood estimation | maximum likelihood estimation | recursive estimation | recursive estimation | Kalman filters | Kalman filters | structure determination | structure determination | order estimation | order estimation | Akaike criterion | Akaike criterion | bounded noise models | bounded noise models | robustness | robustness

License

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15.020 Competition in Telecommunications (MIT) 15.020 Competition in Telecommunications (MIT)

Description

Competition in Telecommunications provides an introduction to the economics, business strategies, and technology of telecommunications markets. This includes markets for wireless communications, local and long-distance services, and customer equipment. The convergence of computers, cable TV and telecommunications and the competitive emergence of the Internet are covered in depth. A number of speakers from leading companies in the industry will give course lectures. Competition in Telecommunications provides an introduction to the economics, business strategies, and technology of telecommunications markets. This includes markets for wireless communications, local and long-distance services, and customer equipment. The convergence of computers, cable TV and telecommunications and the competitive emergence of the Internet are covered in depth. A number of speakers from leading companies in the industry will give course lectures.

Subjects

telephone | telephone | Internet | Internet | communications | communications | economics | economics | business strategy | business strategy | technologies | technologies | wireless | wireless | convergence | convergence | cable television | cable television | governmental regulations | governmental regulations | public policy | public policy | evolution of technology | evolution of technology | computer hardware and software | computer hardware and software | VoIP | VoIP | data and voice traffic | data and voice traffic | network integration | network integration | deregulation | deregulation | cell phones | cell phones | WiFi | WiFi | Internet commerce | Internet commerce | spectrum auctions | spectrum auctions | telecommunications markets | telecommunications markets | competition | competition | wireless communications | wireless communications | long-distance services | long-distance services | computers | computers | satellite TV | satellite TV | telecommunications industry | telecommunications industry | regulation | regulation | technology | technology | market structures | market structures | data traffic | data traffic | voice traffic | voice traffic

License

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6.436J Fundamentals of Probability (MIT) 6.436J Fundamentals of Probability (MIT)

Description

This is a course on the fundamentals of probability geared towards first or second-year graduate students who are interested in a rigorous development of the subject. The course covers most of the topics in MIT course 6.431 but at a faster pace and in more depth. Topics covered include: probability spaces and measures; discrete and continuous random variables; conditioning and independence; multivariate normal distribution; abstract integration, expectation, and related convergence results; moment generating and characteristic functions; Bernoulli and Poisson processes; finite-state Markov chains; convergence notions and their relations; and limit theorems. Familiarity with elementary notions in probability and real analysis is desirable. This is a course on the fundamentals of probability geared towards first or second-year graduate students who are interested in a rigorous development of the subject. The course covers most of the topics in MIT course 6.431 but at a faster pace and in more depth. Topics covered include: probability spaces and measures; discrete and continuous random variables; conditioning and independence; multivariate normal distribution; abstract integration, expectation, and related convergence results; moment generating and characteristic functions; Bernoulli and Poisson processes; finite-state Markov chains; convergence notions and their relations; and limit theorems. Familiarity with elementary notions in probability and real analysis is desirable.

Subjects

Introduction to probability theory | Introduction to probability theory | Probability spaces and measures | Probability spaces and measures | Discrete and continuous random variables | Discrete and continuous random variables | Conditioning and independence | Conditioning and independence | Multivariate normal distribution | Multivariate normal distribution | Abstract integration | Abstract integration | expectation | expectation | and related convergence results | and related convergence results | Moment generating and characteristic functions | Moment generating and characteristic functions | Bernoulli and Poisson process | Bernoulli and Poisson process | Finite-state Markov chains | Finite-state Markov chains | Convergence notions and their relations | Convergence notions and their relations | Limit theorems | Limit theorems | Familiarity with elementary notions in probability and real analysis is desirable | Familiarity with elementary notions in probability and real analysis is desirable

License

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CMS.100 Introduction to Media Studies (MIT) CMS.100 Introduction to Media Studies (MIT)

Description

This course provides a critical analysis of mass media in our culture. Various types of media such as books, films, video games, and online interactions will be discussed and reviewed. This course will also evaluate how information and ideas travel between people on a large scale. This course provides a critical analysis of mass media in our culture. Various types of media such as books, films, video games, and online interactions will be discussed and reviewed. This course will also evaluate how information and ideas travel between people on a large scale.

Subjects

mass communication | mass communication | mass film | mass film | television | television | video games | video games | recorded music | recorded music | digital media | digital media | multimedia | multimedia | media literacy | media literacy | social media | social media | media convergence | media convergence

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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